6: Fourier Transform • Fourier Series as T → ∞ • Fourier Transform • Fourier Transform Examples • Dirac Delta Function • Dirac Delta Function: Scaling and Translation • Dirac Delta Function: Products and Integrals • Periodic Signals • Duality • Time Shifting and Scaling • Gaussian Pulse 6: Fourier Transform • Summary

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 1 / 12 Fourier Series as T → ∞

6: Fourier Transform i2πnF t • Fourier Series as Fourier Series: u(t) = n∞= Une T → ∞ −∞ • Fourier Transform • Fourier Transform P Examples • Dirac Delta Function • Dirac Delta Function: Scaling and Translation • Dirac Delta Function: Products and Integrals • Periodic Signals • Duality • Time Shifting and Scaling • Gaussian Pulse • Summary

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 2 / 12 Fourier Series as T → ∞

6: Fourier Transform i2πnF t • Fourier Series as Fourier Series: u(t) = n∞= Une T → ∞ −∞ • Fourier Transform 1 • Fourier Transform The harmonic frequenciesP are nF ∀n and are spaced F = apart. Examples T • Dirac Delta Function • Dirac Delta Function: Scaling and Translation • Dirac Delta Function: Products and Integrals • Periodic Signals • Duality • Time Shifting and Scaling • Gaussian Pulse • Summary

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 2 / 12 Fourier Series as T → ∞

6: Fourier Transform i2πnF t • Fourier Series as Fourier Series: u(t) = n∞= Une T → ∞ −∞ • Fourier Transform 1 • Fourier Transform The harmonic frequenciesP are nF ∀n and are spaced F = apart. Examples T • Dirac Delta Function • Dirac Delta Function: As T gets larger, the harmonic spacing becomes smaller. Scaling and Translation • Dirac Delta Function: e.g. T = 1 s ⇒ F = 1Hz Products and Integrals • Periodic Signals • Duality • Time Shifting and Scaling • Gaussian Pulse • Summary

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 2 / 12 Fourier Series as T → ∞

6: Fourier Transform i2πnF t • Fourier Series as Fourier Series: u(t) = n∞= Une T → ∞ −∞ • Fourier Transform 1 • Fourier Transform The harmonic frequenciesP are nF ∀n and are spaced F = apart. Examples T • Dirac Delta Function • Dirac Delta Function: As T gets larger, the harmonic spacing becomes smaller. Scaling and Translation • Dirac Delta Function: e.g. T = 1 s ⇒ F = 1Hz Products and Integrals • Periodic Signals T = 1 day ⇒ F = 11.57 µHz • Duality • Time Shifting and Scaling • Gaussian Pulse • Summary

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 2 / 12 Fourier Series as T → ∞

6: Fourier Transform i2πnF t • Fourier Series as Fourier Series: u(t) = n∞= Une T → ∞ −∞ • Fourier Transform 1 • Fourier Transform The harmonic frequenciesP are nF ∀n and are spaced F = apart. Examples T • Dirac Delta Function • Dirac Delta Function: As T gets larger, the harmonic spacing becomes smaller. Scaling and Translation • Dirac Delta Function: e.g. T = 1 s ⇒ F = 1Hz Products and Integrals • Periodic Signals T = 1 day ⇒ F = 11.57 µHz • Duality • Time Shifting and Scaling If T → ∞ then the harmonic spacing becomes zero, the sum becomes an • Gaussian Pulse • Summary integral and we get the Fourier Transform: + ∞ i2πft u(t) = f= U(f)e df −∞ R

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 2 / 12 Fourier Series as T → ∞

6: Fourier Transform i2πnF t • Fourier Series as Fourier Series: u(t) = n∞= Une T → ∞ −∞ • Fourier Transform 1 • Fourier Transform The harmonic frequenciesP are nF ∀n and are spaced F = apart. Examples T • Dirac Delta Function • Dirac Delta Function: As T gets larger, the harmonic spacing becomes smaller. Scaling and Translation • Dirac Delta Function: e.g. T = 1 s ⇒ F = 1Hz Products and Integrals • Periodic Signals T = 1 day ⇒ F = 11.57 µHz • Duality • Time Shifting and Scaling If T → ∞ then the harmonic spacing becomes zero, the sum becomes an • Gaussian Pulse • Summary integral and we get the Fourier Transform: + ∞ i2πft u(t) = f= U(f)e df −∞ Here, U(f), is the spectralR density of u(t).

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 2 / 12 Fourier Series as T → ∞

6: Fourier Transform i2πnF t • Fourier Series as Fourier Series: u(t) = n∞= Une T → ∞ −∞ • Fourier Transform 1 • Fourier Transform The harmonic frequenciesP are nF ∀n and are spaced F = apart. Examples T • Dirac Delta Function • Dirac Delta Function: As T gets larger, the harmonic spacing becomes smaller. Scaling and Translation • Dirac Delta Function: e.g. T = 1 s ⇒ F = 1Hz Products and Integrals • Periodic Signals T = 1 day ⇒ F = 11.57 µHz • Duality • Time Shifting and Scaling If T → ∞ then the harmonic spacing becomes zero, the sum becomes an • Gaussian Pulse • Summary integral and we get the Fourier Transform: + ∞ i2πft u(t) = f= U(f)e df −∞ Here, U(f), is the spectralR density of u(t). • U(f) is a continuous function of f .

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 2 / 12 Fourier Series as T → ∞

6: Fourier Transform i2πnF t • Fourier Series as Fourier Series: u(t) = n∞= Une T → ∞ −∞ • Fourier Transform 1 • Fourier Transform The harmonic frequenciesP are nF ∀n and are spaced F = apart. Examples T • Dirac Delta Function • Dirac Delta Function: As T gets larger, the harmonic spacing becomes smaller. Scaling and Translation • Dirac Delta Function: e.g. T = 1 s ⇒ F = 1Hz Products and Integrals • Periodic Signals T = 1 day ⇒ F = 11.57 µHz • Duality • Time Shifting and Scaling If T → ∞ then the harmonic spacing becomes zero, the sum becomes an • Gaussian Pulse • Summary integral and we get the Fourier Transform: + ∞ i2πft u(t) = f= U(f)e df −∞ Here, U(f), is the spectralR density of u(t). • U(f) is a continuous function of f . • U(f) is complex-valued.

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 2 / 12 Fourier Series as T → ∞

6: Fourier Transform i2πnF t • Fourier Series as Fourier Series: u(t) = n∞= Une T → ∞ −∞ • Fourier Transform 1 • Fourier Transform The harmonic frequenciesP are nF ∀n and are spaced F = apart. Examples T • Dirac Delta Function • Dirac Delta Function: As T gets larger, the harmonic spacing becomes smaller. Scaling and Translation • Dirac Delta Function: e.g. T = 1 s ⇒ F = 1Hz Products and Integrals • Periodic Signals T = 1 day ⇒ F = 11.57 µHz • Duality • Time Shifting and Scaling If T → ∞ then the harmonic spacing becomes zero, the sum becomes an • Gaussian Pulse • Summary integral and we get the Fourier Transform: + ∞ i2πft u(t) = f= U(f)e df −∞ Here, U(f), is the spectralR density of u(t). • U(f) is a continuous function of f . • U(f) is complex-valued. • u(t) real ⇒ U(f) is conjugate symmetric ⇔ U(−f) = U(f)∗.

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 2 / 12 Fourier Series as T → ∞

6: Fourier Transform i2πnF t • Fourier Series as Fourier Series: u(t) = n∞= Une T → ∞ −∞ • Fourier Transform 1 • Fourier Transform The harmonic frequenciesP are nF ∀n and are spaced F = apart. Examples T • Dirac Delta Function • Dirac Delta Function: As T gets larger, the harmonic spacing becomes smaller. Scaling and Translation • Dirac Delta Function: e.g. T = 1 s ⇒ F = 1Hz Products and Integrals • Periodic Signals T = 1 day ⇒ F = 11.57 µHz • Duality • Time Shifting and Scaling If T → ∞ then the harmonic spacing becomes zero, the sum becomes an • Gaussian Pulse • Summary integral and we get the Fourier Transform: + ∞ i2πft u(t) = f= U(f)e df −∞ Here, U(f), is the spectralR density of u(t). • U(f) is a continuous function of f . • U(f) is complex-valued. • u(t) real ⇒ U(f) is conjugate symmetric ⇔ U(−f) = U(f)∗. • Units: if u(t) is in volts, then U(f)df must also be in volts ⇒ U(f) is in volts/Hz (hence “spectral density”).

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 2 / 12 Fourier Transform

6: Fourier Transform i2πnF t • Fourier Series as Fourier Series: u(t) = n∞= Une T → ∞ −∞ • Fourier Transform • Fourier Transform P Examples • Dirac Delta Function • Dirac Delta Function: Scaling and Translation • Dirac Delta Function: Products and Integrals • Periodic Signals • Duality • Time Shifting and Scaling • Gaussian Pulse • Summary

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 3 / 12 Fourier Transform

6: Fourier Transform i2πnF t • Fourier Series as Fourier Series: u(t) = n∞= Une T → ∞ −∞ • Fourier Transform The summation is over a set of equally spaced frequencies • Fourier Transform P 1 Examples fn = nF where the spacing between them is ∆f = F = . • Dirac Delta Function T • Dirac Delta Function: Scaling and Translation • Dirac Delta Function: Products and Integrals • Periodic Signals • Duality • Time Shifting and Scaling • Gaussian Pulse • Summary

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 3 / 12 Fourier Transform

6: Fourier Transform i2πnF t • Fourier Series as Fourier Series: u(t) = n∞= Une T → ∞ −∞ • Fourier Transform The summation is over a set of equally spaced frequencies • Fourier Transform P 1 Examples fn = nF where the spacing between them is ∆f = F = . • Dirac Delta Function T • Dirac Delta Function: i2πnF t Scaling and Translation Un = u(t)e− • Dirac Delta Function: Products and Integrals • Periodic Signals • Duality • Time Shifting and Scaling • Gaussian Pulse • Summary

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 3 / 12 Fourier Transform

6: Fourier Transform i2πnF t • Fourier Series as Fourier Series: u(t) = n∞= Une T → ∞ −∞ • Fourier Transform The summation is over a set of equally spaced frequencies • Fourier Transform P 1 Examples fn = nF where the spacing between them is ∆f = F = . • Dirac Delta Function T • Dirac Delta Function: i2πnF t 0.5T i2πnF t Scaling and Translation Un = u(t)e− = ∆f u(t)e− dt • t= 0.5T Dirac Delta Function: − Products and Integrals • Periodic Signals R • Duality • Time Shifting and Scaling • Gaussian Pulse • Summary

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 3 / 12 Fourier Transform

6: Fourier Transform i2πnF t • Fourier Series as Fourier Series: u(t) = n∞= Une T → ∞ −∞ • Fourier Transform The summation is over a set of equally spaced frequencies • Fourier Transform P 1 Examples fn = nF where the spacing between them is ∆f = F = . • Dirac Delta Function T • Dirac Delta Function: i2πnF t 0.5T i2πnF t Scaling and Translation Un = u(t)e− = ∆f u(t)e− dt • t= 0.5T Dirac Delta Function: − Products and Integrals • Periodic Signals Spectral Density: If u(t) has finite energy,R Un → 0 as ∆f → 0. • Duality • Time Shifting and Scaling • Gaussian Pulse • Summary

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 3 / 12 Fourier Transform

6: Fourier Transform i2πnF t • Fourier Series as Fourier Series: u(t) = n∞= Une T → ∞ −∞ • Fourier Transform The summation is over a set of equally spaced frequencies • Fourier Transform P 1 Examples fn = nF where the spacing between them is ∆f = F = . • Dirac Delta Function T • Dirac Delta Function: i2πnF t 0.5T i2πnF t Scaling and Translation Un = u(t)e− = ∆f u(t)e− dt • t= 0.5T Dirac Delta Function: − Products and Integrals • Periodic Signals Spectral Density: If u(t) has finite energy,R Un → 0 as ∆f → 0. So we • Duality define a spectral density, U(f ) = Un , on the set of frequencies {f }: • Time Shifting and Scaling n ∆f n • Gaussian Pulse Un 0.5T i πf t • Summary 2 n U(fn) = ∆f = t= 0.5T u(t)e− dt − R

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 3 / 12 Fourier Transform

6: Fourier Transform i2πnF t • Fourier Series as Fourier Series: u(t) = n∞= Une T → ∞ −∞ • Fourier Transform The summation is over a set of equally spaced frequencies • Fourier Transform P 1 Examples fn = nF where the spacing between them is ∆f = F = . • Dirac Delta Function T • Dirac Delta Function: i2πnF t 0.5T i2πnF t Scaling and Translation Un = u(t)e− = ∆f u(t)e− dt • t= 0.5T Dirac Delta Function: − Products and Integrals • Periodic Signals Spectral Density: If u(t) has finite energy,R Un → 0 as ∆f → 0. So we • Duality define a spectral density, U(f ) = Un , on the set of frequencies {f }: • Time Shifting and Scaling n ∆f n • Gaussian Pulse Un 0.5T i πf t • Summary 2 n U(fn) = ∆f = t= 0.5T u(t)e− dt − we can write [Substitute Un = U(fn)∆f] R i2πfnt u(t) = n∞= U(fn)e ∆f −∞ P

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 3 / 12 Fourier Transform

6: Fourier Transform i2πnF t • Fourier Series as Fourier Series: u(t) = n∞= Une T → ∞ −∞ • Fourier Transform The summation is over a set of equally spaced frequencies • Fourier Transform P 1 Examples fn = nF where the spacing between them is ∆f = F = . • Dirac Delta Function T • Dirac Delta Function: i2πnF t 0.5T i2πnF t Scaling and Translation Un = u(t)e− = ∆f u(t)e− dt • t= 0.5T Dirac Delta Function: − Products and Integrals • Periodic Signals Spectral Density: If u(t) has finite energy,R Un → 0 as ∆f → 0. So we • Duality define a spectral density, U(f ) = Un , on the set of frequencies {f }: • Time Shifting and Scaling n ∆f n • Gaussian Pulse Un 0.5T i πf t • Summary 2 n U(fn) = ∆f = t= 0.5T u(t)e− dt − we can write [Substitute Un = U(fn)∆f] R i2πfnt u(t) = n∞= U(fn)e ∆f −∞ Fourier Transform:PNow if we take the limit as ∆f → 0, we get u(t) = ∞ U(f)ei2πftdf −∞ R

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 3 / 12 Fourier Transform

6: Fourier Transform i2πnF t • Fourier Series as Fourier Series: u(t) = n∞= Une T → ∞ −∞ • Fourier Transform The summation is over a set of equally spaced frequencies • Fourier Transform P 1 Examples fn = nF where the spacing between them is ∆f = F = . • Dirac Delta Function T • Dirac Delta Function: i2πnF t 0.5T i2πnF t Scaling and Translation Un = u(t)e− = ∆f u(t)e− dt • t= 0.5T Dirac Delta Function: − Products and Integrals • Periodic Signals Spectral Density: If u(t) has finite energy,R Un → 0 as ∆f → 0. So we • Duality define a spectral density, U(f ) = Un , on the set of frequencies {f }: • Time Shifting and Scaling n ∆f n • Gaussian Pulse Un 0.5T i πf t • Summary 2 n U(fn) = ∆f = t= 0.5T u(t)e− dt − we can write [Substitute Un = U(fn)∆f] R i2πfnt u(t) = n∞= U(fn)e ∆f −∞ Fourier Transform:PNow if we take the limit as ∆f → 0, we get u(t) = ∞ U(f)ei2πftdf −∞ ∞ i2πft U(f) =R t= u(t)e− dt −∞ R

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 3 / 12 Fourier Transform

6: Fourier Transform i2πnF t • Fourier Series as Fourier Series: u(t) = n∞= Une T → ∞ −∞ • Fourier Transform The summation is over a set of equally spaced frequencies • Fourier Transform P 1 Examples fn = nF where the spacing between them is ∆f = F = . • Dirac Delta Function T • Dirac Delta Function: i2πnF t 0.5T i2πnF t Scaling and Translation Un = u(t)e− = ∆f u(t)e− dt • t= 0.5T Dirac Delta Function: − Products and Integrals • Periodic Signals Spectral Density: If u(t) has finite energy,R Un → 0 as ∆f → 0. So we • Duality define a spectral density, U(f ) = Un , on the set of frequencies {f }: • Time Shifting and Scaling n ∆f n • Gaussian Pulse Un 0.5T i πf t • Summary 2 n U(fn) = ∆f = t= 0.5T u(t)e− dt − we can write [Substitute Un = U(fn)∆f] R i2πfnt u(t) = n∞= U(fn)e ∆f −∞ Fourier Transform:PNow if we take the limit as ∆f → 0, we get u(t) = ∞ U(f)ei2πftdf [Fourier Synthesis] −∞ ∞ i2πft [Fourier Analysis] U(f) =R t= u(t)e− dt −∞ R

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 3 / 12 Fourier Transform

6: Fourier Transform i2πnF t • Fourier Series as Fourier Series: u(t) = n∞= Une T → ∞ −∞ • Fourier Transform The summation is over a set of equally spaced frequencies • Fourier Transform P 1 Examples fn = nF where the spacing between them is ∆f = F = . • Dirac Delta Function T • Dirac Delta Function: i2πnF t 0.5T i2πnF t Scaling and Translation Un = u(t)e− = ∆f u(t)e− dt • t= 0.5T Dirac Delta Function: − Products and Integrals • Periodic Signals Spectral Density: If u(t) has finite energy,R Un → 0 as ∆f → 0. So we • Duality define a spectral density, U(f ) = Un , on the set of frequencies {f }: • Time Shifting and Scaling n ∆f n • Gaussian Pulse Un 0.5T i πf t • Summary 2 n U(fn) = ∆f = t= 0.5T u(t)e− dt − we can write [Substitute Un = U(fn)∆f] R i2πfnt u(t) = n∞= U(fn)e ∆f −∞ Fourier Transform:PNow if we take the limit as ∆f → 0, we get u(t) = ∞ U(f)ei2πftdf [Fourier Synthesis] −∞ ∞ i2πft [Fourier Analysis] U(f) =R t= u(t)e− dt −∞ R Un For non-periodic signals Un → 0 as ∆f → 0 and U(fn) = ∆f remains finite.

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 3 / 12 Fourier Transform

6: Fourier Transform i2πnF t • Fourier Series as Fourier Series: u(t) = n∞= Une T → ∞ −∞ • Fourier Transform The summation is over a set of equally spaced frequencies • Fourier Transform P 1 Examples fn = nF where the spacing between them is ∆f = F = . • Dirac Delta Function T • Dirac Delta Function: i2πnF t 0.5T i2πnF t Scaling and Translation Un = u(t)e− = ∆f u(t)e− dt • t= 0.5T Dirac Delta Function: − Products and Integrals • Periodic Signals Spectral Density: If u(t) has finite energy,R Un → 0 as ∆f → 0. So we • Duality define a spectral density, U(f ) = Un , on the set of frequencies {f }: • Time Shifting and Scaling n ∆f n • Gaussian Pulse Un 0.5T i πf t • Summary 2 n U(fn) = ∆f = t= 0.5T u(t)e− dt − we can write [Substitute Un = U(fn)∆f] R i2πfnt u(t) = n∞= U(fn)e ∆f −∞ Fourier Transform:PNow if we take the limit as ∆f → 0, we get u(t) = ∞ U(f)ei2πftdf [Fourier Synthesis] −∞ ∞ i2πft [Fourier Analysis] U(f) =R t= u(t)e− dt −∞ R Un For non-periodic signals Un → 0 as ∆f → 0 and U(fn) = ∆f remains finite. However, if u(t) contains an exactly periodic component, then the corresponding U(fn) will become infinite as ∆f → 0. E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 3 / 12 Fourier Transform

6: Fourier Transform i2πnF t • Fourier Series as Fourier Series: u(t) = n∞= Une T → ∞ −∞ • Fourier Transform The summation is over a set of equally spaced frequencies • Fourier Transform P 1 Examples fn = nF where the spacing between them is ∆f = F = . • Dirac Delta Function T • Dirac Delta Function: i2πnF t 0.5T i2πnF t Scaling and Translation Un = u(t)e− = ∆f u(t)e− dt • t= 0.5T Dirac Delta Function: − Products and Integrals • Periodic Signals Spectral Density: If u(t) has finite energy,R Un → 0 as ∆f → 0. So we • Duality define a spectral density, U(f ) = Un , on the set of frequencies {f }: • Time Shifting and Scaling n ∆f n • Gaussian Pulse Un 0.5T i πf t • Summary 2 n U(fn) = ∆f = t= 0.5T u(t)e− dt − we can write [Substitute Un = U(fn)∆f] R i2πfnt u(t) = n∞= U(fn)e ∆f −∞ Fourier Transform:PNow if we take the limit as ∆f → 0, we get u(t) = ∞ U(f)ei2πftdf [Fourier Synthesis] −∞ ∞ i2πft [Fourier Analysis] U(f) =R t= u(t)e− dt −∞ R Un For non-periodic signals Un → 0 as ∆f → 0 and U(fn) = ∆f remains finite. However, if u(t) contains an exactly periodic component, then the corresponding U(fn) will become infinite as ∆f → 0. We will deal with it. E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 3 / 12 <

Fourier Transform Examples

6: Fourier Transform • Fourier Series as Example 1: T → ∞ at • Fourier Transform e− t ≥ 0 • Fourier Transform u(t) = Examples (0 t < 0 • Dirac Delta Function • Dirac Delta Function: Scaling and Translation • Dirac Delta Function: Products and Integrals • Periodic Signals • Duality • Time Shifting and Scaling • Gaussian Pulse • Summary

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 4 / 12 <

Fourier Transform Examples

6: Fourier Transform 1 a=2 • Fourier Series as Example 1: 0.5 T → ∞ at u(t) 0 • Fourier Transform e− t ≥ 0 -5 0 5 • Fourier Transform u(t) = Time (s) Examples (0 t < 0 • Dirac Delta Function • Dirac Delta Function: Scaling and Translation • Dirac Delta Function: Products and Integrals • Periodic Signals • Duality • Time Shifting and Scaling • Gaussian Pulse • Summary

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 4 / 12 <

Fourier Transform Examples

6: Fourier Transform 1 a=2 • Fourier Series as Example 1: 0.5 T → ∞ at u(t) 0 • Fourier Transform e− t ≥ 0 -5 0 5 • Fourier Transform u(t) = Time (s) Examples (0 t < 0 • Dirac Delta Function • Dirac Delta Function: i2πft Scaling and Translation U(f) = ∞ u(t)e− dt • Dirac Delta Function: Products and Integrals −∞ • Periodic Signals R • Duality • Time Shifting and Scaling • Gaussian Pulse • Summary

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 4 / 12 <

Fourier Transform Examples

6: Fourier Transform 1 a=2 • Fourier Series as Example 1: 0.5 T → ∞ at u(t) 0 • Fourier Transform e− t ≥ 0 -5 0 5 • Fourier Transform u(t) = Time (s) Examples (0 t < 0 • Dirac Delta Function • Dirac Delta Function: i2πft Scaling and Translation U(f) = ∞ u(t)e− dt • Dirac Delta Function: −∞ Products and Integrals at i2πft • ∞ Periodic Signals = R0 e− e− dt • Duality • Time Shifting and Scaling • Gaussian Pulse R • Summary

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 4 / 12 <

Fourier Transform Examples

6: Fourier Transform 1 a=2 • Fourier Series as Example 1: 0.5 T → ∞ at u(t) 0 • Fourier Transform e− t ≥ 0 -5 0 5 • Fourier Transform u(t) = Time (s) Examples (0 t < 0 • Dirac Delta Function • Dirac Delta Function: i2πft Scaling and Translation U(f) = ∞ u(t)e− dt • Dirac Delta Function: −∞ Products and Integrals at i2πft • ∞ Periodic Signals = R0 e− e− dt • Duality ( a i2πf)t • Time Shifting and Scaling = ∞ e − − dt • Gaussian Pulse R0 • Summary R

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 4 / 12 <

Fourier Transform Examples

6: Fourier Transform 1 a=2 • Fourier Series as Example 1: 0.5 T → ∞ at u(t) 0 • Fourier Transform e− t ≥ 0 -5 0 5 • Fourier Transform u(t) = Time (s) Examples (0 t < 0 • Dirac Delta Function • Dirac Delta Function: i2πft Scaling and Translation U(f) = ∞ u(t)e− dt • Dirac Delta Function: −∞ Products and Integrals at i2πft • ∞ Periodic Signals = R0 e− e− dt • Duality ( a i2πf)t • Time Shifting and Scaling = ∞ e − − dt • Gaussian Pulse R0 1 ( a i2πf)t 1 • Summary e ∞ = Ra+−i2πf − − 0 = a+i2πf  

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 4 / 12 <

Fourier Transform Examples

6: Fourier Transform 1 a=2 • Fourier Series as Example 1: 0.5 T → ∞ at u(t) 0 • Fourier Transform e− t ≥ 0 -5 0 5 • Fourier Transform u(t) = Time (s) Examples 0.5 0 t < 0 0.4 • ( 0.3 Dirac Delta Function 0.2 • Dirac Delta Function: |U(f)| 0.1 Scaling and Translation i2πft -5 0 5 U(f) = ∞ u(t)e− dt Frequency (Hz) • Dirac Delta Function: −∞ Products and Integrals at i2πft • ∞ Periodic Signals = R0 e− e− dt • Duality ( a i2πf)t • Time Shifting and Scaling = ∞ e − − dt • Gaussian Pulse R0 1 ( a i2πf)t 1 • Summary e ∞ = Ra+−i2πf − − 0 = a+i2πf  

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 4 / 12 Fourier Transform Examples

6: Fourier Transform 1 a=2 • Fourier Series as Example 1: 0.5 T → ∞ at u(t) 0 • Fourier Transform e− t ≥ 0 -5 0 5 • Fourier Transform u(t) = Time (s) Examples 0.5 0 t < 0 0.4 • ( 0.3 Dirac Delta Function 0.2 • Dirac Delta Function: |U(f)| 0.1 Scaling and Translation i2πft -5 0 5 U(f) = ∞ u(t)e− dt Frequency (Hz) • Dirac Delta Function: −∞ 0.5 Products and Integrals at i2πft • ∞ 0 Periodic Signals = R0 e− e− dt • Duality -0.5 ( a i2πf)t (rad/pi)

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 4 / 12 Fourier Transform Examples

6: Fourier Transform 1 a=2 • Fourier Series as Example 1: 0.5 T → ∞ at u(t) 0 • Fourier Transform e− t ≥ 0 -5 0 5 • Fourier Transform u(t) = Time (s) Examples 0.5 0 t < 0 0.4 • ( 0.3 Dirac Delta Function 0.2 • Dirac Delta Function: |U(f)| 0.1 Scaling and Translation i2πft -5 0 5 U(f) = ∞ u(t)e− dt Frequency (Hz) • Dirac Delta Function: −∞ 0.5 Products and Integrals at i2πft • ∞ 0 Periodic Signals = R0 e− e− dt • Duality -0.5 ( a i2πf)t (rad/pi)

Example 2:   a t v(t) = e− | |

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 4 / 12 Fourier Transform Examples

6: Fourier Transform 1 a=2 • Fourier Series as Example 1: 0.5 T → ∞ at u(t) 0 • Fourier Transform e− t ≥ 0 -5 0 5 • Fourier Transform u(t) = Time (s) Examples 0.5 0 t < 0 0.4 • ( 0.3 Dirac Delta Function 0.2 • Dirac Delta Function: |U(f)| 0.1 Scaling and Translation i2πft -5 0 5 U(f) = ∞ u(t)e− dt Frequency (Hz) • Dirac Delta Function: −∞ 0.5 Products and Integrals at i2πft • ∞ 0 Periodic Signals = R0 e− e− dt • Duality -0.5 ( a i2πf)t (rad/pi)

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 4 / 12 Fourier Transform Examples

6: Fourier Transform 1 a=2 • Fourier Series as Example 1: 0.5 T → ∞ at u(t) 0 • Fourier Transform e− t ≥ 0 -5 0 5 • Fourier Transform u(t) = Time (s) Examples 0.5 0 t < 0 0.4 • ( 0.3 Dirac Delta Function 0.2 • Dirac Delta Function: |U(f)| 0.1 Scaling and Translation i2πft -5 0 5 U(f) = ∞ u(t)e− dt Frequency (Hz) • Dirac Delta Function: −∞ 0.5 Products and Integrals at i2πft • ∞ 0 Periodic Signals = R0 e− e− dt • Duality -0.5 ( a i2πf)t (rad/pi)

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 4 / 12 Fourier Transform Examples

6: Fourier Transform 1 a=2 • Fourier Series as Example 1: 0.5 T → ∞ at u(t) 0 • Fourier Transform e− t ≥ 0 -5 0 5 • Fourier Transform u(t) = Time (s) Examples 0.5 0 t < 0 0.4 • ( 0.3 Dirac Delta Function 0.2 • Dirac Delta Function: |U(f)| 0.1 Scaling and Translation i2πft -5 0 5 U(f) = ∞ u(t)e− dt Frequency (Hz) • Dirac Delta Function: −∞ 0.5 Products and Integrals at i2πft • ∞ 0 Periodic Signals = R0 e− e− dt • Duality -0.5 ( a i2πf)t (rad/pi)

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 4 / 12 Fourier Transform Examples

6: Fourier Transform 1 a=2 • Fourier Series as Example 1: 0.5 T → ∞ at u(t) 0 • Fourier Transform e− t ≥ 0 -5 0 5 • Fourier Transform u(t) = Time (s) Examples 0.5 0 t < 0 0.4 • ( 0.3 Dirac Delta Function 0.2 • Dirac Delta Function: |U(f)| 0.1 Scaling and Translation i2πft -5 0 5 U(f) = ∞ u(t)e− dt Frequency (Hz) • Dirac Delta Function: −∞ 0.5 Products and Integrals at i2πft • ∞ 0 Periodic Signals = R0 e− e− dt • Duality -0.5 ( a i2πf)t (rad/pi)

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 4 / 12 Fourier Transform Examples

6: Fourier Transform 1 a=2 • Fourier Series as Example 1: 0.5 T → ∞ at u(t) 0 • Fourier Transform e− t ≥ 0 -5 0 5 • Fourier Transform u(t) = Time (s) Examples 0.5 0 t < 0 0.4 • ( 0.3 Dirac Delta Function 0.2 • Dirac Delta Function: |U(f)| 0.1 Scaling and Translation i2πft -5 0 5 U(f) = ∞ u(t)e− dt Frequency (Hz) • Dirac Delta Function: −∞ 0.5 Products and Integrals at i2πft • ∞ 0 Periodic Signals = R0 e− e− dt • Duality -0.5 ( a i2πf)t (rad/pi)

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 4 / 12 Fourier Transform Examples

6: Fourier Transform 1 a=2 • Fourier Series as Example 1: 0.5 T → ∞ at u(t) 0 • Fourier Transform e− t ≥ 0 -5 0 5 • Fourier Transform u(t) = Time (s) Examples 0.5 0 t < 0 0.4 • ( 0.3 Dirac Delta Function 0.2 • Dirac Delta Function: |U(f)| 0.1 Scaling and Translation i2πft -5 0 5 U(f) = ∞ u(t)e− dt Frequency (Hz) • Dirac Delta Function: −∞ 0.5 Products and Integrals at i2πft • ∞ 0 Periodic Signals = R0 e− e− dt • Duality -0.5 ( a i2πf)t (rad/pi)

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 4 / 12 Fourier Transform Examples

6: Fourier Transform 1 a=2 • Fourier Series as Example 1: 0.5 T → ∞ at u(t) 0 • Fourier Transform e− t ≥ 0 -5 0 5 • Fourier Transform u(t) = Time (s) Examples 0.5 0 t < 0 0.4 • ( 0.3 Dirac Delta Function 0.2 • Dirac Delta Function: |U(f)| 0.1 Scaling and Translation i2πft -5 0 5 U(f) = ∞ u(t)e− dt Frequency (Hz) • Dirac Delta Function: −∞ 0.5 Products and Integrals at i2πft • ∞ 0 Periodic Signals = R0 e− e− dt • Duality -0.5 ( a i2πf)t (rad/pi)

i2πft |V(f)| V (f) = ∞ v(t)e dt 0 − -5 0 5 −∞ Frequency (Hz) 0 R at i2πft ∞ at i2πft = e e− dt + 0 e− e− dt −∞ 0 1 e(a i2πf)t 1 e( a i2πf)t ∞ = Ra i2πf − R + a+−i2πf − − 0 −1 1 −∞ 2a = a i2πf + a+i2πf = a2+4π2f 2 −    

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 4 / 12 Fourier Transform Examples

6: Fourier Transform 1 a=2 • Fourier Series as Example 1: 0.5 T → ∞ at u(t) 0 • Fourier Transform e− t ≥ 0 -5 0 5 • Fourier Transform u(t) = Time (s) Examples 0.5 0 t < 0 0.4 • ( 0.3 Dirac Delta Function 0.2 • Dirac Delta Function: |U(f)| 0.1 Scaling and Translation i2πft -5 0 5 U(f) = ∞ u(t)e− dt Frequency (Hz) • Dirac Delta Function: −∞ 0.5 Products and Integrals at i2πft • ∞ 0 Periodic Signals = R0 e− e− dt • Duality -0.5 ( a i2πf)t (rad/pi)

i2πft |V(f)| V (f) = ∞ v(t)e dt 0 − -5 0 5 −∞ Frequency (Hz) 0 R at i2πft ∞ at i2πft = e e− dt + 0 e− e− dt −∞ 0 1 e(a i2πf)t 1 e( a i2πf)t ∞ = Ra i2πf − R + a+−i2πf − − 0 −1 1 −∞ 2a = + = 2 2 2 [v(t) real+symmetric a i2πf  a+i2πf  a +4π f   − ⇒ V (f) real+symmetric] E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 4 / 12 Dirac Delta Function

6: Fourier Transform • We define a unit area pulse of width w as Fourier Series as 4 T → ∞ 1 • Fourier Transform −0.5w ≤ x ≤ 0.5w 2 • w δ (x) Fourier Transform d (x) = 3 Examples w 0 0 otherwise -3 -2 -1 0 1 2 3 • Dirac Delta Function ( x • Dirac Delta Function: Scaling and Translation • Dirac Delta Function: Products and Integrals • Periodic Signals • Duality • Time Shifting and Scaling • Gaussian Pulse • Summary

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 5 / 12 Dirac Delta Function

6: Fourier Transform • We define a unit area pulse of width w as Fourier Series as 4 T → ∞ 1 • Fourier Transform −0.5w ≤ x ≤ 0.5w 2 • w δ (x) Fourier Transform d (x) = 3 Examples w 0 0 otherwise -3 -2 -1 0 1 2 3 • Dirac Delta Function ( x • Dirac Delta Function: Scaling and Translation This pulse has the property that its integral equals • Dirac Delta Function: Products and Integrals 1 for all values of w: • Periodic Signals • Duality ∞ d (x)dx = 1 • x= w Time Shifting and Scaling −∞ • Gaussian Pulse • Summary R

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 5 / 12 Dirac Delta Function

6: Fourier Transform • We define a unit area pulse of width w as Fourier Series as 4 T → ∞ 1 δ (x) • Fourier Transform −0.5w ≤ x ≤ 0.5w 2 0.5 • w δ (x) Fourier Transform d (x) = 3 Examples w 0 0 otherwise -3 -2 -1 0 1 2 3 • Dirac Delta Function ( x • Dirac Delta Function: Scaling and Translation This pulse has the property that its integral equals • Dirac Delta Function: Products and Integrals 1 for all values of w: • Periodic Signals • Duality ∞ d (x)dx = 1 • x= w Time Shifting and Scaling −∞ • Gaussian Pulse If we make w smaller, the pulse height increases to preserve unit area. • Summary R

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 5 / 12 Dirac Delta Function

6: Fourier Transform • We define a unit area pulse of width w as δ (x) Fourier Series as 4 0.2 T → ∞ 1 δ (x) • Fourier Transform −0.5w ≤ x ≤ 0.5w 2 0.5 • w δ (x) Fourier Transform d (x) = 3 Examples w 0 0 otherwise -3 -2 -1 0 1 2 3 • Dirac Delta Function ( x • Dirac Delta Function: Scaling and Translation This pulse has the property that its integral equals • Dirac Delta Function: Products and Integrals 1 for all values of w: • Periodic Signals • Duality ∞ d (x)dx = 1 • x= w Time Shifting and Scaling −∞ • Gaussian Pulse If we make w smaller, the pulse height increases to preserve unit area. • Summary R

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 5 / 12 Dirac Delta Function

6: Fourier Transform • We define a unit area pulse of width w as δ (x) Fourier Series as 4 0.2 T → ∞ 1 δ (x) • Fourier Transform −0.5w ≤ x ≤ 0.5w 2 0.5 • w δ (x) Fourier Transform d (x) = 3 Examples w 0 0 otherwise -3 -2 -1 0 1 2 3 • Dirac Delta Function ( x • Dirac Delta Function: Scaling and Translation This pulse has the property that its integral equals • Dirac Delta Function: Products and Integrals 1 for all values of w: • Periodic Signals • Duality ∞ d (x)dx = 1 • x= w Time Shifting and Scaling −∞ • Gaussian Pulse If we make w smaller, the pulse height increases to preserve unit area. • Summary R We define the Dirac delta function as δ(x) = limw 0 dw(x) →

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 5 / 12 Dirac Delta Function

6: Fourier Transform • We define a unit area pulse of width w as δ (x) Fourier Series as 4 0.2 T → ∞ 1 δ (x) • Fourier Transform −0.5w ≤ x ≤ 0.5w 2 0.5 • w δ (x) Fourier Transform d (x) = 3 Examples w 0 0 otherwise -3 -2 -1 0 1 2 3 • Dirac Delta Function ( x • Dirac Delta Function: Scaling and Translation This pulse has the property that its integral equals • Dirac Delta Function: Products and Integrals 1 for all values of w: • Periodic Signals • Duality ∞ d (x)dx = 1 • x= w Time Shifting and Scaling −∞ • Gaussian Pulse If we make w smaller, the pulse height increases to preserve unit area. • Summary R We define the Dirac delta function as δ(x) = limw 0 dw(x) → • δ(x) equals zero everywhere except at x = 0 where it is infinite.

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 5 / 12 Dirac Delta Function

6: Fourier Transform • We define a unit area pulse of width w as δ (x) Fourier Series as 4 0.2 T → ∞ 1 δ (x) • Fourier Transform −0.5w ≤ x ≤ 0.5w 2 0.5 • w δ (x) Fourier Transform d (x) = 3 Examples w 0 0 otherwise -3 -2 -1 0 1 2 3 • Dirac Delta Function ( x • Dirac Delta Function: Scaling and Translation This pulse has the property that its integral equals • Dirac Delta Function: Products and Integrals 1 for all values of w: • Periodic Signals • Duality ∞ d (x)dx = 1 • x= w Time Shifting and Scaling −∞ • Gaussian Pulse If we make w smaller, the pulse height increases to preserve unit area. • Summary R We define the Dirac delta function as δ(x) = limw 0 dw(x) → • δ(x) equals zero everywhere except at x = 0 where it is infinite. • However its area still equals 1 ⇒ ∞ δ(x)dx = 1 −∞ R

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 5 / 12 Dirac Delta Function

6: Fourier Transform • We define a unit area pulse of width w as δ (x) Fourier Series as 4 0.2 T → ∞ 1 δ (x) • Fourier Transform −0.5w ≤ x ≤ 0.5w 2 0.5 • w δ (x) Fourier Transform d (x) = 3 Examples w 0 0 otherwise -3 -2 -1 0 1 2 3 • Dirac Delta Function ( x • Dirac Delta Function: Scaling and Translation This pulse has the property that its integral equals 1 • Dirac Delta Function: δ(x) Products and Integrals 1 for all values of w: 0.5 • Periodic Signals 0 • Duality ∞ d (x)dx = 1 -3 -2 -1 0 1 2 3 • x= w x Time Shifting and Scaling −∞ • Gaussian Pulse If we make w smaller, the pulse height increases to preserve unit area. • Summary R We define the Dirac delta function as δ(x) = limw 0 dw(x) → • δ(x) equals zero everywhere except at x = 0 where it is infinite. • However its area still equals 1 ⇒ ∞ δ(x)dx = 1 −∞ • We plot the height of δ(x) as its areaR rather than its true height of ∞.

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 5 / 12 Dirac Delta Function

6: Fourier Transform • We define a unit area pulse of width w as δ (x) Fourier Series as 4 0.2 T → ∞ 1 δ (x) • Fourier Transform −0.5w ≤ x ≤ 0.5w 2 0.5 • w δ (x) Fourier Transform d (x) = 3 Examples w 0 0 otherwise -3 -2 -1 0 1 2 3 • Dirac Delta Function ( x • Dirac Delta Function: Scaling and Translation This pulse has the property that its integral equals 1 • Dirac Delta Function: δ(x) Products and Integrals 1 for all values of w: 0.5 • Periodic Signals 0 • Duality ∞ d (x)dx = 1 -3 -2 -1 0 1 2 3 • x= w x Time Shifting and Scaling −∞ • Gaussian Pulse If we make w smaller, the pulse height increases to preserve unit area. • Summary R We define the Dirac delta function as δ(x) = limw 0 dw(x) → • δ(x) equals zero everywhere except at x = 0 where it is infinite. • However its area still equals 1 ⇒ ∞ δ(x)dx = 1 −∞ • We plot the height of δ(x) as its areaR rather than its true height of ∞. δ(x) is not quite a proper function: it is called a generalized function.

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 5 / 12 Dirac Delta Function: Scaling and Translation

6: Fourier Transform • Fourier Series as Translation: δ(x − a) δ T → ∞ 1 (x) • Fourier Transform δ(x) is a pulse at x = 0 • Fourier Transform 0 Examples • Dirac Delta Function -1 • -3 -2 -1 0 1 2 3 Dirac Delta Function: x Scaling and Translation • Dirac Delta Function: Products and Integrals • Periodic Signals • Duality • Time Shifting and Scaling • Gaussian Pulse • Summary

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 6 / 12 Dirac Delta Function: Scaling and Translation

6: Fourier Transform • Fourier Series as Translation: δ(x − a) δ δ T → ∞ 1 (x) (x-2) • Fourier Transform δ(x) is a pulse at x = 0 • Fourier Transform 0 Examples δ(x − a) is a pulse at x = a • Dirac Delta Function -1 • -3 -2 -1 0 1 2 3 Dirac Delta Function: x Scaling and Translation • Dirac Delta Function: Products and Integrals • Periodic Signals • Duality • Time Shifting and Scaling • Gaussian Pulse • Summary

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 6 / 12 Dirac Delta Function: Scaling and Translation

6: Fourier Transform • Fourier Series as Translation: δ(x − a) δ δ T → ∞ 1 (x) (x-2) • Fourier Transform δ(x) is a pulse at x = 0 • Fourier Transform 0 Examples δ(x − a) is a pulse at x = a • Dirac Delta Function -1 • -3 -2 -1 0 1 2 3 Dirac Delta Function: x Scaling and Translation Amplitude Scaling: bδ(x) • Dirac Delta Function: Products and Integrals δ(x) has an area of 1 • Periodic Signals • Duality • Time Shifting and Scaling • Gaussian Pulse • Summary

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 6 / 12 Dirac Delta Function: Scaling and Translation

6: Fourier Transform • Fourier Series as Translation: δ(x − a) δ δ T → ∞ 1 (x) (x-2) • Fourier Transform δ(x) is a pulse at x = 0 • Fourier Transform 0 Examples δ(x − a) is a pulse at x = a • Dirac Delta Function -1 • -3 -2 -1 0 1 2 3 Dirac Delta Function: x Scaling and Translation Amplitude Scaling: bδ(x) • Dirac Delta Function: Products and Integrals δ x has an area of ∞ δ x dx • ( ) 1 ⇔ ( ) = 1 Periodic Signals −∞ • Duality • Time Shifting and Scaling R • Gaussian Pulse • Summary

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 6 / 12 Dirac Delta Function: Scaling and Translation

6: Fourier Transform • Fourier Series as Translation: δ(x − a) δ δ T → ∞ 1 (x) (x-2) • Fourier Transform δ(x) is a pulse at x = 0 • Fourier Transform 0 Examples δ(x − a) is a pulse at x = a -0.5δ(x+2) • Dirac Delta Function -1 • -3 -2 -1 0 1 2 3 Dirac Delta Function: x Scaling and Translation Amplitude Scaling: bδ(x) • Dirac Delta Function: Products and Integrals δ x has an area of ∞ δ x dx • ( ) 1 ⇔ ( ) = 1 Periodic Signals −∞ • Duality • Time Shifting and Scaling bδ(x) has an area of b sinceR • Gaussian Pulse ∞ (bδ(x)) dx • Summary −∞ R

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 6 / 12 Dirac Delta Function: Scaling and Translation

6: Fourier Transform • Fourier Series as Translation: δ(x − a) δ δ T → ∞ 1 (x) (x-2) • Fourier Transform δ(x) is a pulse at x = 0 • Fourier Transform 0 Examples δ(x − a) is a pulse at x = a -0.5δ(x+2) • Dirac Delta Function -1 • -3 -2 -1 0 1 2 3 Dirac Delta Function: x Scaling and Translation Amplitude Scaling: bδ(x) • Dirac Delta Function: Products and Integrals δ x has an area of ∞ δ x dx • ( ) 1 ⇔ ( ) = 1 Periodic Signals −∞ • Duality • Time Shifting and Scaling bδ(x) has an area of b sinceR • Gaussian Pulse ∞ (bδ(x)) dx = b ∞ δ(x)dx • Summary −∞ −∞ R R

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 6 / 12 Dirac Delta Function: Scaling and Translation

6: Fourier Transform • Fourier Series as Translation: δ(x − a) δ δ T → ∞ 1 (x) (x-2) • Fourier Transform δ(x) is a pulse at x = 0 • Fourier Transform 0 Examples δ(x − a) is a pulse at x = a -0.5δ(x+2) • Dirac Delta Function -1 • -3 -2 -1 0 1 2 3 Dirac Delta Function: x Scaling and Translation Amplitude Scaling: bδ(x) • Dirac Delta Function: Products and Integrals δ x has an area of ∞ δ x dx • ( ) 1 ⇔ ( ) = 1 Periodic Signals −∞ • Duality • Time Shifting and Scaling bδ(x) has an area of b sinceR • Gaussian Pulse ∞ (bδ(x)) dx = b ∞ δ(x)dx = b • Summary −∞ −∞ R R

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 6 / 12 Dirac Delta Function: Scaling and Translation

6: Fourier Transform • Fourier Series as Translation: δ(x − a) δ δ T → ∞ 1 (x) (x-2) • Fourier Transform δ(x) is a pulse at x = 0 • Fourier Transform 0 Examples δ(x − a) is a pulse at x = a -0.5δ(x+2) • Dirac Delta Function -1 • -3 -2 -1 0 1 2 3 Dirac Delta Function: x Scaling and Translation Amplitude Scaling: bδ(x) • Dirac Delta Function: Products and Integrals δ x has an area of ∞ δ x dx • ( ) 1 ⇔ ( ) = 1 Periodic Signals −∞ • Duality • Time Shifting and Scaling bδ(x) has an area of b sinceR • Gaussian Pulse ∞ (bδ(x)) dx = b ∞ δ(x)dx = b • Summary −∞ −∞ b canR be a complex numberR (on a graph, we then plot only its magnitude)

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 6 / 12 Dirac Delta Function: Scaling and Translation

6: Fourier Transform • Fourier Series as Translation: δ(x − a) δ δ T → ∞ 1 (x) (x-2) • Fourier Transform δ(x) is a pulse at x = 0 • Fourier Transform 0 Examples δ(x − a) is a pulse at x = a -0.5δ(x+2) • Dirac Delta Function -1 • -3 -2 -1 0 1 2 3 Dirac Delta Function: x Scaling and Translation Amplitude Scaling: bδ(x) • Dirac Delta Function: Products and Integrals δ x has an area of ∞ δ x dx • ( ) 1 ⇔ ( ) = 1 Periodic Signals −∞ • Duality • Time Shifting and Scaling bδ(x) has an area of b sinceR • Gaussian Pulse ∞ (bδ(x)) dx = b ∞ δ(x)dx = b • Summary −∞ −∞ b canR be a complex numberR (on a graph, we then plot only its magnitude) Time Scaling: δ(cx)

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 6 / 12 Dirac Delta Function: Scaling and Translation

6: Fourier Transform • Fourier Series as Translation: δ(x − a) δ δ T → ∞ 1 (x) (x-2) • Fourier Transform δ(x) is a pulse at x = 0 • Fourier Transform 0 Examples δ(x − a) is a pulse at x = a -0.5δ(x+2) • Dirac Delta Function -1 • -3 -2 -1 0 1 2 3 Dirac Delta Function: x Scaling and Translation Amplitude Scaling: bδ(x) • Dirac Delta Function: Products and Integrals δ x has an area of ∞ δ x dx • ( ) 1 ⇔ ( ) = 1 Periodic Signals −∞ • Duality • Time Shifting and Scaling bδ(x) has an area of b sinceR • Gaussian Pulse ∞ (bδ(x)) dx = b ∞ δ(x)dx = b • Summary −∞ −∞ b canR be a complex numberR (on a graph, we then plot only its magnitude) Time Scaling: δ(cx) : ∞ c > 0 x= δ(cx)dx −∞ R

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 6 / 12 Dirac Delta Function: Scaling and Translation

6: Fourier Transform • Fourier Series as Translation: δ(x − a) δ δ T → ∞ 1 (x) (x-2) • Fourier Transform δ(x) is a pulse at x = 0 • Fourier Transform 0 Examples δ(x − a) is a pulse at x = a -0.5δ(x+2) • Dirac Delta Function -1 • -3 -2 -1 0 1 2 3 Dirac Delta Function: x Scaling and Translation Amplitude Scaling: bδ(x) • Dirac Delta Function: Products and Integrals δ x has an area of ∞ δ x dx • ( ) 1 ⇔ ( ) = 1 Periodic Signals −∞ • Duality • Time Shifting and Scaling bδ(x) has an area of b sinceR • Gaussian Pulse ∞ (bδ(x)) dx = b ∞ δ(x)dx = b • Summary −∞ −∞ b canR be a complex numberR (on a graph, we then plot only its magnitude) Time Scaling: δ(cx) dy : ∞ ∞ [sub ] c > 0 x= δ(cx)dx = y= δ(y) c y = cx −∞ −∞ R R

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 6 / 12 Dirac Delta Function: Scaling and Translation

6: Fourier Transform • Fourier Series as Translation: δ(x − a) δ δ T → ∞ 1 (x) (x-2) • Fourier Transform δ(x) is a pulse at x = 0 • Fourier Transform 0 Examples δ(x − a) is a pulse at x = a -0.5δ(x+2) • Dirac Delta Function -1 • -3 -2 -1 0 1 2 3 Dirac Delta Function: x Scaling and Translation Amplitude Scaling: bδ(x) • Dirac Delta Function: Products and Integrals δ x has an area of ∞ δ x dx • ( ) 1 ⇔ ( ) = 1 Periodic Signals −∞ • Duality • Time Shifting and Scaling bδ(x) has an area of b sinceR • Gaussian Pulse ∞ (bδ(x)) dx = b ∞ δ(x)dx = b • Summary −∞ −∞ b canR be a complex numberR (on a graph, we then plot only its magnitude) Time Scaling: δ(cx) dy : ∞ ∞ [sub ] c > 0 x= δ(cx)dx = y= δ(y) c y = cx −∞ −∞ 1 ∞ = c y= δ(y)dy R R −∞ R

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 6 / 12 Dirac Delta Function: Scaling and Translation

6: Fourier Transform • Fourier Series as Translation: δ(x − a) δ δ T → ∞ 1 (x) (x-2) • Fourier Transform δ(x) is a pulse at x = 0 • Fourier Transform 0 Examples δ(x − a) is a pulse at x = a -0.5δ(x+2) • Dirac Delta Function -1 • -3 -2 -1 0 1 2 3 Dirac Delta Function: x Scaling and Translation Amplitude Scaling: bδ(x) • Dirac Delta Function: Products and Integrals δ x has an area of ∞ δ x dx • ( ) 1 ⇔ ( ) = 1 Periodic Signals −∞ • Duality • Time Shifting and Scaling bδ(x) has an area of b sinceR • Gaussian Pulse ∞ (bδ(x)) dx = b ∞ δ(x)dx = b • Summary −∞ −∞ b canR be a complex numberR (on a graph, we then plot only its magnitude) Time Scaling: δ(cx) dy : ∞ ∞ [sub ] c > 0 x= δ(cx)dx = y= δ(y) c y = cx −∞ −∞ 1 ∞ 1 = c y= δ(y)dy = c R R −∞ R

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 6 / 12 Dirac Delta Function: Scaling and Translation

6: Fourier Transform • Fourier Series as Translation: δ(x − a) δ δ T → ∞ 1 (x) (x-2) • Fourier Transform δ(x) is a pulse at x = 0 • Fourier Transform 0 Examples δ(x − a) is a pulse at x = a -0.5δ(x+2) • Dirac Delta Function -1 • -3 -2 -1 0 1 2 3 Dirac Delta Function: x Scaling and Translation Amplitude Scaling: bδ(x) • Dirac Delta Function: Products and Integrals δ x has an area of ∞ δ x dx • ( ) 1 ⇔ ( ) = 1 Periodic Signals −∞ • Duality • Time Shifting and Scaling bδ(x) has an area of b sinceR • Gaussian Pulse ∞ (bδ(x)) dx = b ∞ δ(x)dx = b • Summary −∞ −∞ b canR be a complex numberR (on a graph, we then plot only its magnitude) Time Scaling: δ(cx) dy : ∞ ∞ [sub ] c > 0 x= δ(cx)dx = y= δ(y) c y = cx −∞ −∞ 1 ∞ 1 = c y= δ(y)dy = c R R −∞ : ∞ c < 0 x= δ(cx)dx R −∞ R

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 6 / 12 Dirac Delta Function: Scaling and Translation

6: Fourier Transform • Fourier Series as Translation: δ(x − a) δ δ T → ∞ 1 (x) (x-2) • Fourier Transform δ(x) is a pulse at x = 0 • Fourier Transform 0 Examples δ(x − a) is a pulse at x = a -0.5δ(x+2) • Dirac Delta Function -1 • -3 -2 -1 0 1 2 3 Dirac Delta Function: x Scaling and Translation Amplitude Scaling: bδ(x) • Dirac Delta Function: Products and Integrals δ x has an area of ∞ δ x dx • ( ) 1 ⇔ ( ) = 1 Periodic Signals −∞ • Duality • Time Shifting and Scaling bδ(x) has an area of b sinceR • Gaussian Pulse ∞ (bδ(x)) dx = b ∞ δ(x)dx = b • Summary −∞ −∞ b canR be a complex numberR (on a graph, we then plot only its magnitude) Time Scaling: δ(cx) dy : ∞ ∞ [sub ] c > 0 x= δ(cx)dx = y= δ(y) c y = cx −∞ −∞ 1 ∞ 1 = c y= δ(y)dy = c R R −∞ dy : ∞ −∞ [sub ] c < 0 x= δ(cx)dx = yR=+ δ(y) c y = cx −∞ ∞ R R

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 6 / 12 Dirac Delta Function: Scaling and Translation

6: Fourier Transform • Fourier Series as Translation: δ(x − a) δ δ T → ∞ 1 (x) (x-2) • Fourier Transform δ(x) is a pulse at x = 0 • Fourier Transform 0 Examples δ(x − a) is a pulse at x = a -0.5δ(x+2) • Dirac Delta Function -1 • -3 -2 -1 0 1 2 3 Dirac Delta Function: x Scaling and Translation Amplitude Scaling: bδ(x) • Dirac Delta Function: Products and Integrals δ x has an area of ∞ δ x dx • ( ) 1 ⇔ ( ) = 1 Periodic Signals −∞ • Duality • Time Shifting and Scaling bδ(x) has an area of b sinceR • Gaussian Pulse ∞ (bδ(x)) dx = b ∞ δ(x)dx = b • Summary −∞ −∞ b canR be a complex numberR (on a graph, we then plot only its magnitude) Time Scaling: δ(cx) dy : ∞ ∞ [sub ] c > 0 x= δ(cx)dx = y= δ(y) c y = cx −∞ −∞ 1 ∞ 1 = c y= δ(y)dy = c R R −∞ dy : ∞ −∞ [sub ] c < 0 x= δ(cx)dx = yR=+ δ(y) c y = cx −∞ +∞ 1 ∞ R = R−c y= δ(y)dy −∞ R

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 6 / 12 Dirac Delta Function: Scaling and Translation

6: Fourier Transform • Fourier Series as Translation: δ(x − a) δ δ T → ∞ 1 (x) (x-2) • Fourier Transform δ(x) is a pulse at x = 0 • Fourier Transform 0 Examples δ(x − a) is a pulse at x = a -0.5δ(x+2) • Dirac Delta Function -1 • -3 -2 -1 0 1 2 3 Dirac Delta Function: x Scaling and Translation Amplitude Scaling: bδ(x) • Dirac Delta Function: Products and Integrals δ x has an area of ∞ δ x dx • ( ) 1 ⇔ ( ) = 1 Periodic Signals −∞ • Duality • Time Shifting and Scaling bδ(x) has an area of b sinceR • Gaussian Pulse ∞ (bδ(x)) dx = b ∞ δ(x)dx = b • Summary −∞ −∞ b canR be a complex numberR (on a graph, we then plot only its magnitude) Time Scaling: δ(cx) dy : ∞ ∞ [sub ] c > 0 x= δ(cx)dx = y= δ(y) c y = cx −∞ −∞ 1 ∞ 1 = c y= δ(y)dy = c R R −∞ dy : ∞ −∞ [sub ] c < 0 x= δ(cx)dx = yR=+ δ(y) c y = cx −∞ +∞ 1 ∞ 1 R = R−c y= δ(y)dy = −c −∞ R

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 6 / 12 Dirac Delta Function: Scaling and Translation

6: Fourier Transform • Fourier Series as Translation: δ(x − a) δ δ T → ∞ 1 (x) (x-2) • Fourier Transform δ(x) is a pulse at x = 0 • Fourier Transform 0 Examples δ(x − a) is a pulse at x = a -0.5δ(x+2) • Dirac Delta Function -1 • -3 -2 -1 0 1 2 3 Dirac Delta Function: x Scaling and Translation Amplitude Scaling: bδ(x) • Dirac Delta Function: Products and Integrals δ x has an area of ∞ δ x dx • ( ) 1 ⇔ ( ) = 1 Periodic Signals −∞ • Duality • Time Shifting and Scaling bδ(x) has an area of b sinceR • Gaussian Pulse ∞ (bδ(x)) dx = b ∞ δ(x)dx = b • Summary −∞ −∞ b canR be a complex numberR (on a graph, we then plot only its magnitude) Time Scaling: δ(cx) dy : ∞ ∞ [sub ] c > 0 x= δ(cx)dx = y= δ(y) c y = cx −∞ −∞ 1 ∞ 1 1 = c y= δ(y)dy = c = c R R −∞ | | dy : ∞ −∞ [sub ] c < 0 x= δ(cx)dx = yR=+ δ(y) c y = cx −∞ +∞ 1 ∞ 1 1 R = R−c y= δ(y)dy = −c = c −∞ | | R

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 6 / 12 Dirac Delta Function: Scaling and Translation

6: Fourier Transform • Fourier Series as Translation: δ(x − a) δ δ T → ∞ 1 (x) (x-2) • Fourier Transform δ(x) is a pulse at x = 0 • Fourier Transform 0 Examples δ(x − a) is a pulse at x = a -0.5δ(x+2) • Dirac Delta Function -1 • -3 -2 -1 0 1 2 3 Dirac Delta Function: x Scaling and Translation Amplitude Scaling: bδ(x) • Dirac Delta Function: Products and Integrals δ x has an area of ∞ δ x dx • ( ) 1 ⇔ ( ) = 1 Periodic Signals −∞ • Duality • Time Shifting and Scaling bδ(x) has an area of b sinceR • Gaussian Pulse ∞ (bδ(x)) dx = b ∞ δ(x)dx = b • Summary −∞ −∞ b canR be a complex numberR (on a graph, we then plot only its magnitude) Time Scaling: δ(cx) dy : ∞ ∞ [sub ] c > 0 x= δ(cx)dx = y= δ(y) c y = cx −∞ −∞ 1 ∞ 1 1 = c y= δ(y)dy = c = c R R −∞ | | dy : ∞ −∞ [sub ] c < 0 x= δ(cx)dx = yR=+ δ(y) c y = cx −∞ +∞ 1 ∞ 1 1 R = R−c y= δ(y)dy = −c = c −∞ | | 1 In general, δ(cx) = c δ(x) forRc 6= 0 | |

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 6 / 12 Dirac Delta Function: Scaling and Translation

6: Fourier Transform • Fourier Series as Translation: δ(x − a) δ δ T → ∞ 1 (x) (x-2) • Fourier Transform δ(x) is a pulse at x = 0 • Fourier Transform 0 Examples δ(x − a) is a pulse at x = a -0.5δ(x+2) • Dirac Delta Function -1 • -3 -2 -1 0 1 2 3 Dirac Delta Function: x Scaling and Translation Amplitude Scaling: bδ(x) • Dirac Delta Function: Products and Integrals δ x has an area of ∞ δ x dx 1 • ( ) 1 ⇔ ( ) = 1 Periodic Signals δ(4x) = 0.25δ(x) −∞ 0 • Duality • Time Shifting and Scaling bδ(x) has an area of b sinceR -1 • Gaussian Pulse -3 -2 -1 0 1 2 3 ∞ (bδ(x)) dx = b ∞ δ(x)dx = b x • Summary −∞ −∞ b canR be a complex numberR (on a graph, we then plot only its magnitude) Time Scaling: δ(cx) dy : ∞ ∞ [sub ] c > 0 x= δ(cx)dx = y= δ(y) c y = cx −∞ −∞ 1 ∞ 1 1 = c y= δ(y)dy = c = c R R −∞ | | dy : ∞ −∞ [sub ] c < 0 x= δ(cx)dx = yR=+ δ(y) c y = cx −∞ +∞ 1 ∞ 1 1 R = R−c y= δ(y)dy = −c = c −∞ | | 1 In general, δ(cx) = c δ(x) forRc 6= 0 | |

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 6 / 12 Dirac Delta Function: Scaling and Translation

6: Fourier Transform • Fourier Series as Translation: δ(x − a) δ δ T → ∞ 1 (x) (x-2) • Fourier Transform δ(x) is a pulse at x = 0 • Fourier Transform 0 Examples δ(x − a) is a pulse at x = a -0.5δ(x+2) • Dirac Delta Function -1 • -3 -2 -1 0 1 2 3 Dirac Delta Function: x Scaling and Translation Amplitude Scaling: bδ(x) • Dirac Delta Function: Products and Integrals δ x has an area of ∞ δ x dx 1 • ( ) 1 ⇔ ( ) = 1 Periodic Signals δ(4x) = 0.25δ(x) −∞ 0 • Duality • Time Shifting and Scaling bδ(x) has an area of b sinceR -3δ(-4x-8) = -0.75δ(x+2) -1 • Gaussian Pulse -3 -2 -1 0 1 2 3 ∞ (bδ(x)) dx = b ∞ δ(x)dx = b x • Summary −∞ −∞ b canR be a complex numberR (on a graph, we then plot only its magnitude) Time Scaling: δ(cx) dy : ∞ ∞ [sub ] c > 0 x= δ(cx)dx = y= δ(y) c y = cx −∞ −∞ 1 ∞ 1 1 = c y= δ(y)dy = c = c R R −∞ | | dy : ∞ −∞ [sub ] c < 0 x= δ(cx)dx = yR=+ δ(y) c y = cx −∞ +∞ 1 ∞ 1 1 R = R−c y= δ(y)dy = −c = c −∞ | | 1 In general, δ(cx) = c δ(x) forRc 6= 0 | |

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 6 / 12 Dirac Delta Function: Products and Integrals

6: Fourier Transform 6 • Fourier Series as If we multiply δ(x − a) by a function of x: y=x2 T → ∞ 2 4 • Fourier Transform y = x × δ(x − 2) • Fourier Transform 2 Examples 0 • Dirac Delta Function • -1 0 1 2 Dirac Delta Function: x Scaling and Translation 6 • Dirac Delta Function: y=δ(x-2) Products and Integrals 4 • Periodic Signals • Duality 2 • Time Shifting and Scaling 0 • Gaussian Pulse -1 0 1 2 x • Summary

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 7 / 12 Dirac Delta Function: Products and Integrals

6: Fourier Transform 6 • Fourier Series as If we multiply δ(x − a) by a function of x: y=x2 T → ∞ 2 4 • Fourier Transform y = x × δ(x − 2) • Fourier Transform 2 Examples The product is 0 everywhere except at x = 2. 0 • Dirac Delta Function • -1 0 1 2 Dirac Delta Function: x Scaling and Translation 6 • Dirac Delta Function: y=δ(x-2) Products and Integrals 4 • Periodic Signals • Duality 2 • Time Shifting and Scaling 0 • Gaussian Pulse -1 0 1 2 x • Summary

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 7 / 12 Dirac Delta Function: Products and Integrals

6: Fourier Transform 6 • Fourier Series as If we multiply δ(x − a) by a function of x: y=x2 T → ∞ 2 4 • Fourier Transform y = x × δ(x − 2) • Fourier Transform 2 Examples The product is 0 everywhere except at x = 2. 0 • Dirac Delta Function • -1 0 1 2 Dirac Delta Function: So δ(x − 2) is multiplied by the value taken by x Scaling and Translation 2 6 • Dirac Delta Function: x at x = 2: y=δ(x-2) Products and Integrals 2 2 4 • Periodic Signals x × δ(x − 2) = x × δ(x − 2) • Duality x=2 2 • Time Shifting and Scaling 0 • Gaussian Pulse   -1 0 1 2 x • Summary

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 7 / 12 Dirac Delta Function: Products and Integrals

6: Fourier Transform 6 • Fourier Series as If we multiply δ(x − a) by a function of x: y=x2 T → ∞ 2 4 • Fourier Transform y = x × δ(x − 2) • Fourier Transform 2 Examples The product is 0 everywhere except at x = 2. 0 • Dirac Delta Function • -1 0 1 2 Dirac Delta Function: So δ(x − 2) is multiplied by the value taken by x Scaling and Translation 2 6 • Dirac Delta Function: x at x = 2: y=δ(x-2) Products and Integrals 2 2 4 • Periodic Signals x × δ(x − 2) = x × δ(x − 2) • Duality x=2 2 • Time Shifting and Scaling = 4 × δ(x − 2) 0 • Gaussian Pulse   -1 0 1 2 x • Summary 6 y=22×δ(x-2) = 4δ(x-2) 4

2

0 -1 0 1 2 x

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 7 / 12 Dirac Delta Function: Products and Integrals

6: Fourier Transform 6 • Fourier Series as If we multiply δ(x − a) by a function of x: y=x2 T → ∞ 2 4 • Fourier Transform y = x × δ(x − 2) • Fourier Transform 2 Examples The product is 0 everywhere except at x = 2. 0 • Dirac Delta Function • -1 0 1 2 Dirac Delta Function: So δ(x − 2) is multiplied by the value taken by x Scaling and Translation 2 6 • Dirac Delta Function: x at x = 2: y=δ(x-2) Products and Integrals 2 2 4 • Periodic Signals x × δ(x − 2) = x × δ(x − 2) • Duality x=2 2 • Time Shifting and Scaling = 4 × δ(x − 2) 0 • Gaussian Pulse   -1 0 1 2 x • Summary 6 y=22×δ(x-2) = 4δ(x-2) 4

2

0 -1 0 1 2 x

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 7 / 12 Dirac Delta Function: Products and Integrals

6: Fourier Transform 6 • Fourier Series as If we multiply δ(x − a) by a function of x: y=x2 T → ∞ 2 4 • Fourier Transform y = x × δ(x − 2) • Fourier Transform 2 Examples The product is 0 everywhere except at x = 2. 0 • Dirac Delta Function • -1 0 1 2 Dirac Delta Function: So δ(x − 2) is multiplied by the value taken by x Scaling and Translation 2 6 • Dirac Delta Function: x at x = 2: y=δ(x-2) Products and Integrals 2 2 4 • Periodic Signals x × δ(x − 2) = x × δ(x − 2) • Duality x=2 2 • Time Shifting and Scaling = 4 × δ(x − 2) 0 • Gaussian Pulse   -1 0 1 2 x • Summary 6 In general for any function, f(x), that is y=22×δ(x-2) = 4δ(x-2) continuous at x = a, 4 2

f(x)δ(x − a) = f(a)δ(x − a) 0 -1 0 1 2 x

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 7 / 12 Dirac Delta Function: Products and Integrals

6: Fourier Transform 6 • Fourier Series as If we multiply δ(x − a) by a function of x: y=x2 T → ∞ 2 4 • Fourier Transform y = x × δ(x − 2) • Fourier Transform 2 Examples The product is 0 everywhere except at x = 2. 0 • Dirac Delta Function • -1 0 1 2 Dirac Delta Function: So δ(x − 2) is multiplied by the value taken by x Scaling and Translation 2 6 • Dirac Delta Function: x at x = 2: y=δ(x-2) Products and Integrals 2 2 4 • Periodic Signals x × δ(x − 2) = x × δ(x − 2) • Duality x=2 2 • Time Shifting and Scaling = 4 × δ(x − 2) 0 • Gaussian Pulse   -1 0 1 2 x • Summary 6 In general for any function, f(x), that is y=22×δ(x-2) = 4δ(x-2) continuous at x = a, 4 2

f(x)δ(x − a) = f(a)δ(x − a) 0 -1 0 1 2 x Integrals: ∞ f(x)δ(x − a)dx = ∞ f(a)δ(x − a)dx −∞ −∞ R R [if f(x) continuous at a]

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 7 / 12 Dirac Delta Function: Products and Integrals

6: Fourier Transform 6 • Fourier Series as If we multiply δ(x − a) by a function of x: y=x2 T → ∞ 2 4 • Fourier Transform y = x × δ(x − 2) • Fourier Transform 2 Examples The product is 0 everywhere except at x = 2. 0 • Dirac Delta Function • -1 0 1 2 Dirac Delta Function: So δ(x − 2) is multiplied by the value taken by x Scaling and Translation 2 6 • Dirac Delta Function: x at x = 2: y=δ(x-2) Products and Integrals 2 2 4 • Periodic Signals x × δ(x − 2) = x × δ(x − 2) • Duality x=2 2 • Time Shifting and Scaling = 4 × δ(x − 2) 0 • Gaussian Pulse   -1 0 1 2 x • Summary 6 In general for any function, f(x), that is y=22×δ(x-2) = 4δ(x-2) continuous at x = a, 4 2

f(x)δ(x − a) = f(a)δ(x − a) 0 -1 0 1 2 x Integrals: ∞ f(x)δ(x − a)dx = ∞ f(a)δ(x − a)dx −∞ −∞ = f(a) ∞ δ(x − a)dx R R −∞ [if f(x) continuous at a] R

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 7 / 12 Dirac Delta Function: Products and Integrals

6: Fourier Transform 6 • Fourier Series as If we multiply δ(x − a) by a function of x: y=x2 T → ∞ 2 4 • Fourier Transform y = x × δ(x − 2) • Fourier Transform 2 Examples The product is 0 everywhere except at x = 2. 0 • Dirac Delta Function • -1 0 1 2 Dirac Delta Function: So δ(x − 2) is multiplied by the value taken by x Scaling and Translation 2 6 • Dirac Delta Function: x at x = 2: y=δ(x-2) Products and Integrals 2 2 4 • Periodic Signals x × δ(x − 2) = x × δ(x − 2) • Duality x=2 2 • Time Shifting and Scaling = 4 × δ(x − 2) 0 • Gaussian Pulse   -1 0 1 2 x • Summary 6 In general for any function, f(x), that is y=22×δ(x-2) = 4δ(x-2) continuous at x = a, 4 2

f(x)δ(x − a) = f(a)δ(x − a) 0 -1 0 1 2 x Integrals: ∞ f(x)δ(x − a)dx = ∞ f(a)δ(x − a)dx −∞ −∞ = f(a) ∞ δ(x − a)dx R R = f(a) −∞ [if f(x) continuous at a] R

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 7 / 12 Dirac Delta Function: Products and Integrals

6: Fourier Transform 6 • Fourier Series as If we multiply δ(x − a) by a function of x: y=x2 T → ∞ 2 4 • Fourier Transform y = x × δ(x − 2) • Fourier Transform 2 Examples The product is 0 everywhere except at x = 2. 0 • Dirac Delta Function • -1 0 1 2 Dirac Delta Function: So δ(x − 2) is multiplied by the value taken by x Scaling and Translation 2 6 • Dirac Delta Function: x at x = 2: y=δ(x-2) Products and Integrals 2 2 4 • Periodic Signals x × δ(x − 2) = x × δ(x − 2) • Duality x=2 2 • Time Shifting and Scaling = 4 × δ(x − 2) 0 • Gaussian Pulse   -1 0 1 2 x • Summary 6 In general for any function, f(x), that is y=22×δ(x-2) = 4δ(x-2) continuous at x = a, 4 2

f(x)δ(x − a) = f(a)δ(x − a) 0 -1 0 1 2 x Integrals: ∞ f(x)δ(x − a)dx = ∞ f(a)δ(x − a)dx −∞ −∞ = f(a) ∞ δ(x − a)dx R R = f(a) −∞ [if f(x) continuous at a] R Example: ∞ x2 x δ x dx x2 x 3 − 2 ( − 2) = 3 − 2 x=2 = 8 −∞

E1.10 Fourier Series and Transforms (2014-5559)R
 Fourier Transform: 6 – 7 / 12 Periodic Signals

6: Fourier Transform i2πft • Fourier Series as Fourier Transform: u(t) = ∞ U(f)e df [Fourier Synthesis] T → ∞ −∞ i2πft • Fourier Transform U(f) = ∞ u(t)e− dt [Fourier Analysis] • Fourier Transform Rt= Examples −∞ • Dirac Delta Function R • Dirac Delta Function: Scaling and Translation • Dirac Delta Function: Products and Integrals • Periodic Signals • Duality • Time Shifting and Scaling • Gaussian Pulse • Summary

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 8 / 12 Periodic Signals

6: Fourier Transform i2πft • Fourier Series as Fourier Transform: u(t) = ∞ U(f)e df [Fourier Synthesis] T → ∞ −∞ i2πft • Fourier Transform U(f) = ∞ u(t)e dt [Fourier Analysis] • t= − Fourier Transform R −∞ Examples 1.5 1.5δ(f+2) 1.5δ(f-2) • Dirac Delta Function Example: U f . δ f . δ f ( ) = 1 5 ( +R 2) + 1 5 ( − 2) 1 • Dirac Delta Function: Scaling and Translation 0.5 • Dirac Delta Function: 0 Products and Integrals -3 -2 -1 0 1 2 3 Frequency (Hz) • Periodic Signals • Duality • Time Shifting and Scaling • Gaussian Pulse • Summary

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 8 / 12 Periodic Signals

6: Fourier Transform i2πft • Fourier Series as Fourier Transform: u(t) = ∞ U(f)e df [Fourier Synthesis] T → ∞ −∞ i2πft • Fourier Transform U(f) = ∞ u(t)e dt [Fourier Analysis] • t= − Fourier Transform R −∞ Examples 1.5 1.5δ(f+2) 1.5δ(f-2) • Dirac Delta Function Example: U f . δ f . δ f ( ) = 1 5 ( +R 2) + 1 5 ( − 2) 1 • Dirac Delta Function: Scaling and Translation ∞ i2πft 0.5 • u(t) = U(f)e df Dirac Delta Function: 0 Products and Integrals −∞ -3 -2 -1 0 1 2 3 Frequency (Hz) • Periodic Signals R • Duality • Time Shifting and Scaling • Gaussian Pulse • Summary

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 8 / 12 Periodic Signals

6: Fourier Transform i2πft • Fourier Series as Fourier Transform: u(t) = ∞ U(f)e df [Fourier Synthesis] T → ∞ −∞ i2πft • Fourier Transform U(f) = ∞ u(t)e dt [Fourier Analysis] • t= − Fourier Transform R −∞ Examples 1.5 1.5δ(f+2) 1.5δ(f-2) • Dirac Delta Function Example: U f . δ f . δ f ( ) = 1 5 ( +R 2) + 1 5 ( − 2) 1 • Dirac Delta Function: Scaling and Translation ∞ i2πft 0.5 • u(t) = U(f)e df Dirac Delta Function: 0 Products and Integrals −∞ i2πft -3 -2 -1 0 1 2 3 ∞ Frequency (Hz) • Periodic Signals = 1.5δ(f + 2)e df R−∞ • Duality + ∞ 1.5δ(f − 2)ei2πftdf • Time Shifting and Scaling R • Gaussian Pulse −∞ • Summary R

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 8 / 12 Periodic Signals

6: Fourier Transform i2πft • Fourier Series as Fourier Transform: u(t) = ∞ U(f)e df [Fourier Synthesis] T → ∞ −∞ i2πft • Fourier Transform U(f) = ∞ u(t)e dt [Fourier Analysis] • t= − Fourier Transform R −∞ Examples 1.5 1.5δ(f+2) 1.5δ(f-2) • Dirac Delta Function Example: U f . δ f . δ f ( ) = 1 5 ( +R 2) + 1 5 ( − 2) 1 • Dirac Delta Function: Scaling and Translation ∞ i2πft 0.5 • u(t) = U(f)e df Dirac Delta Function: 0 Products and Integrals −∞ i2πft -3 -2 -1 0 1 2 3 ∞ Frequency (Hz) • Periodic Signals = 1.5δ(f + 2)e df R−∞ • Duality + ∞ 1.5δ(f − 2)ei2πftdf • Time Shifting and Scaling R −∞ • Gaussian Pulse = 1.5 ei2πft + 1.5 ei2πft • Summary f= 2 f=+2 R −    

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 8 / 12 Periodic Signals

6: Fourier Transform i2πft • Fourier Series as Fourier Transform: u(t) = ∞ U(f)e df [Fourier Synthesis] T → ∞ −∞ i2πft • Fourier Transform U(f) = ∞ u(t)e dt [Fourier Analysis] • t= − Fourier Transform R −∞ Examples 1.5 1.5δ(f+2) 1.5δ(f-2) • Dirac Delta Function Example: U f . δ f . δ f ( ) = 1 5 ( +R 2) + 1 5 ( − 2) 1 • Dirac Delta Function: Scaling and Translation ∞ i2πft 0.5 • u(t) = U(f)e df Dirac Delta Function: 0 Products and Integrals −∞ i2πft -3 -2 -1 0 1 2 3 ∞ Frequency (Hz) • Periodic Signals = 1.5δ(f + 2)e df R−∞ • Duality + ∞ 1.5δ(f − 2)ei2πftdf • Time Shifting and Scaling R −∞ • Gaussian Pulse = 1.5 ei2πft + 1.5 ei2πft • Summary R f= 2 f=+2 i4πt −i4πt = 1.5 e + e−  

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 8 / 12 Periodic Signals

6: Fourier Transform i2πft • Fourier Series as Fourier Transform: u(t) = ∞ U(f)e df [Fourier Synthesis] T → ∞ −∞ i2πft • Fourier Transform U(f) = ∞ u(t)e dt [Fourier Analysis] • t= − Fourier Transform R −∞ Examples 1.5 1.5δ(f+2) 1.5δ(f-2) • Dirac Delta Function Example: U f . δ f . δ f ( ) = 1 5 ( +R 2) + 1 5 ( − 2) 1 • Dirac Delta Function: Scaling and Translation ∞ i2πft 0.5 • u(t) = U(f)e df Dirac Delta Function: 0 Products and Integrals −∞ i2πft -3 -2 -1 0 1 2 3 ∞ Frequency (Hz) • Periodic Signals = 1.5δ(f + 2)e df R−∞ • Duality + ∞ 1.5δ(f − 2)ei2πftdf • Time Shifting and Scaling R −∞ • Gaussian Pulse = 1.5 ei2πft + 1.5 ei2πft • Summary R f= 2 f=+2 i4πt −i4πt = 1.5 e + e− = 3 cos 4πt

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 8 / 12 Periodic Signals

6: Fourier Transform i2πft • Fourier Series as Fourier Transform: u(t) = ∞ U(f)e df [Fourier Synthesis] T → ∞ −∞ i2πft • Fourier Transform U(f) = ∞ u(t)e dt [Fourier Analysis] • t= − Fourier Transform R −∞ Examples 1.5 1.5δ(f+2) 1.5δ(f-2) • Dirac Delta Function Example: U f . δ f . δ f ( ) = 1 5 ( +R 2) + 1 5 ( − 2) 1 • Dirac Delta Function: Scaling and Translation ∞ i2πft 0.5 • u(t) = U(f)e df Dirac Delta Function: 0 Products and Integrals −∞ i2πft -3 -2 -1 0 1 2 3 ∞ Frequency (Hz) • Periodic Signals = 1.5δ(f + 2)e df R−∞ • Duality + ∞ 1.5δ(f − 2)ei2πftdf 4 3cos(4πt) • Time Shifting and Scaling 2 R −∞ • Gaussian Pulse = 1.5 ei2πft + 1.5 ei2πft 0 • Summary f= 2 f=+2 -2 R − i4πt i4πt 0 1 2 3 4 5 = 1.5 e + e− = 3 cos 4πt Time (s)

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 8 / 12 Periodic Signals

6: Fourier Transform i2πft • Fourier Series as Fourier Transform: u(t) = ∞ U(f)e df [Fourier Synthesis] T → ∞ −∞ i2πft • Fourier Transform U(f) = ∞ u(t)e dt [Fourier Analysis] • t= − Fourier Transform R −∞ Examples 1.5 1.5δ(f+2) 1.5δ(f-2) • Dirac Delta Function Example: U f . δ f . δ f ( ) = 1 5 ( +R 2) + 1 5 ( − 2) 1 • Dirac Delta Function: Scaling and Translation ∞ i2πft 0.5 • u(t) = U(f)e df Dirac Delta Function: 0 Products and Integrals −∞ i2πft -3 -2 -1 0 1 2 3 ∞ Frequency (Hz) • Periodic Signals = 1.5δ(f + 2)e df R−∞ • Duality + ∞ 1.5δ(f − 2)ei2πftdf 4 3cos(4πt) • Time Shifting and Scaling 2 R −∞ • Gaussian Pulse = 1.5 ei2πft + 1.5 ei2πft 0 • Summary f= 2 f=+2 -2 R − i4πt i4πt 0 1 2 3 4 5 = 1.5 e + e− = 3 cos 4πt Time (s) If u(t) is periodic then U(f) is
a sum of Dirac delta functions:

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 8 / 12 Periodic Signals

6: Fourier Transform i2πft • Fourier Series as Fourier Transform: u(t) = ∞ U(f)e df [Fourier Synthesis] T → ∞ −∞ i2πft • Fourier Transform U(f) = ∞ u(t)e dt [Fourier Analysis] • t= − Fourier Transform R −∞ Examples 1.5 1.5δ(f+2) 1.5δ(f-2) • Dirac Delta Function Example: U f . δ f . δ f ( ) = 1 5 ( +R 2) + 1 5 ( − 2) 1 • Dirac Delta Function: Scaling and Translation ∞ i2πft 0.5 • u(t) = U(f)e df Dirac Delta Function: 0 Products and Integrals −∞ i2πft -3 -2 -1 0 1 2 3 ∞ Frequency (Hz) • Periodic Signals = 1.5δ(f + 2)e df R−∞ • Duality + ∞ 1.5δ(f − 2)ei2πftdf 4 3cos(4πt) • Time Shifting and Scaling 2 R −∞ • Gaussian Pulse = 1.5 ei2πft + 1.5 ei2πft 0 • Summary f= 2 f=+2 -2 R − i4πt i4πt 0 1 2 3 4 5 = 1.5 e + e− = 3 cos 4πt Time (s) If u(t) is periodic then U(f) is
a sum of Dirac delta functions: i2πnF t u(t) = n∞= Une ⇒ U(f) = n∞= Unδ (f − nF ) −∞ −∞ P P

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 8 / 12 Periodic Signals

6: Fourier Transform i2πft • Fourier Series as Fourier Transform: u(t) = ∞ U(f)e df [Fourier Synthesis] T → ∞ −∞ i2πft • Fourier Transform U(f) = ∞ u(t)e dt [Fourier Analysis] • t= − Fourier Transform R −∞ Examples 1.5 1.5δ(f+2) 1.5δ(f-2) • Dirac Delta Function Example: U f . δ f . δ f ( ) = 1 5 ( +R 2) + 1 5 ( − 2) 1 • Dirac Delta Function: Scaling and Translation ∞ i2πft 0.5 • u(t) = U(f)e df Dirac Delta Function: 0 Products and Integrals −∞ i2πft -3 -2 -1 0 1 2 3 ∞ Frequency (Hz) • Periodic Signals = 1.5δ(f + 2)e df R−∞ • Duality + ∞ 1.5δ(f − 2)ei2πftdf 4 3cos(4πt) • Time Shifting and Scaling 2 R −∞ • Gaussian Pulse = 1.5 ei2πft + 1.5 ei2πft 0 • Summary f= 2 f=+2 -2 R − i4πt i4πt 0 1 2 3 4 5 = 1.5 e + e− = 3 cos 4πt Time (s) If u(t) is periodic then U(f) is
a sum of Dirac delta functions: i2πnF t u(t) = n∞= Une ⇒ U(f) = n∞= Unδ (f − nF ) −∞ −∞ Proof:Pu(t) = ∞ U(f)ei2πftdf P −∞ R

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 8 / 12 Periodic Signals

6: Fourier Transform i2πft • Fourier Series as Fourier Transform: u(t) = ∞ U(f)e df [Fourier Synthesis] T → ∞ −∞ i2πft • Fourier Transform U(f) = ∞ u(t)e dt [Fourier Analysis] • t= − Fourier Transform R −∞ Examples 1.5 1.5δ(f+2) 1.5δ(f-2) • Dirac Delta Function Example: U f . δ f . δ f ( ) = 1 5 ( +R 2) + 1 5 ( − 2) 1 • Dirac Delta Function: Scaling and Translation ∞ i2πft 0.5 • u(t) = U(f)e df Dirac Delta Function: 0 Products and Integrals −∞ i2πft -3 -2 -1 0 1 2 3 ∞ Frequency (Hz) • Periodic Signals = 1.5δ(f + 2)e df R−∞ • Duality + ∞ 1.5δ(f − 2)ei2πftdf 4 3cos(4πt) • Time Shifting and Scaling 2 R −∞ • Gaussian Pulse = 1.5 ei2πft + 1.5 ei2πft 0 • Summary f= 2 f=+2 -2 R − i4πt i4πt 0 1 2 3 4 5 = 1.5 e + e− = 3 cos 4πt Time (s) If u(t) is periodic then U(f) is
a sum of Dirac delta functions: i2πnF t u(t) = n∞= Une ⇒ U(f) = n∞= Unδ (f − nF ) −∞ −∞ Proof:Pu(t) = ∞ U(f)ei2πftdf P −∞ ∞ i2πft = R n∞= Unδ (f − nF ) e df −∞ −∞ R P

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 8 / 12 Periodic Signals

6: Fourier Transform i2πft • Fourier Series as Fourier Transform: u(t) = ∞ U(f)e df [Fourier Synthesis] T → ∞ −∞ i2πft • Fourier Transform U(f) = ∞ u(t)e dt [Fourier Analysis] • t= − Fourier Transform R −∞ Examples 1.5 1.5δ(f+2) 1.5δ(f-2) • Dirac Delta Function Example: U f . δ f . δ f ( ) = 1 5 ( +R 2) + 1 5 ( − 2) 1 • Dirac Delta Function: Scaling and Translation ∞ i2πft 0.5 • u(t) = U(f)e df Dirac Delta Function: 0 Products and Integrals −∞ i2πft -3 -2 -1 0 1 2 3 ∞ Frequency (Hz) • Periodic Signals = 1.5δ(f + 2)e df R−∞ • Duality + ∞ 1.5δ(f − 2)ei2πftdf 4 3cos(4πt) • Time Shifting and Scaling 2 R −∞ • Gaussian Pulse = 1.5 ei2πft + 1.5 ei2πft 0 • Summary f= 2 f=+2 -2 R − i4πt i4πt 0 1 2 3 4 5 = 1.5 e + e− = 3 cos 4πt Time (s) If u(t) is periodic then U(f) is
a sum of Dirac delta functions: i2πnF t u(t) = n∞= Une ⇒ U(f) = n∞= Unδ (f − nF ) −∞ −∞ Proof:Pu(t) = ∞ U(f)ei2πftdf P −∞ ∞ i2πft = R n∞= Unδ (f − nF ) e df −∞ −∞ ∞ i2πft = R n∞=P Un δ (f − nF ) e df −∞ −∞ P R

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 8 / 12 Periodic Signals

6: Fourier Transform i2πft • Fourier Series as Fourier Transform: u(t) = ∞ U(f)e df [Fourier Synthesis] T → ∞ −∞ i2πft • Fourier Transform U(f) = ∞ u(t)e dt [Fourier Analysis] • t= − Fourier Transform R −∞ Examples 1.5 1.5δ(f+2) 1.5δ(f-2) • Dirac Delta Function Example: U f . δ f . δ f ( ) = 1 5 ( +R 2) + 1 5 ( − 2) 1 • Dirac Delta Function: Scaling and Translation ∞ i2πft 0.5 • u(t) = U(f)e df Dirac Delta Function: 0 Products and Integrals −∞ i2πft -3 -2 -1 0 1 2 3 ∞ Frequency (Hz) • Periodic Signals = 1.5δ(f + 2)e df R−∞ • Duality + ∞ 1.5δ(f − 2)ei2πftdf 4 3cos(4πt) • Time Shifting and Scaling 2 R −∞ • Gaussian Pulse = 1.5 ei2πft + 1.5 ei2πft 0 • Summary f= 2 f=+2 -2 R − i4πt i4πt 0 1 2 3 4 5 = 1.5 e + e− = 3 cos 4πt Time (s) If u(t) is periodic then U(f) is
a sum of Dirac delta functions: i2πnF t u(t) = n∞= Une ⇒ U(f) = n∞= Unδ (f − nF ) −∞ −∞ Proof:Pu(t) = ∞ U(f)ei2πftdf P −∞ ∞ i2πft = R n∞= Unδ (f − nF ) e df −∞ −∞ ∞ i2πft = R n∞=P Un δ (f − nF ) e df −∞ −∞ i2πnF t = Pn∞= UneR −∞ E1.10 Fourier Series and Transforms (2014-5559) P Fourier Transform: 6 – 8 / 12 Duality

6: Fourier Transform i2πft • Fourier Series as Fourier Transform: u(t) = ∞ U(f)e df [Fourier Synthesis] T → ∞ −∞ i2πft • Fourier Transform U(f) = ∞ u(t)e− dt [Fourier Analysis] • Fourier Transform Rt= Examples −∞ • Dirac Delta Function R • Dirac Delta Function: Scaling and Translation • Dirac Delta Function: Products and Integrals • Periodic Signals • Duality • Time Shifting and Scaling • Gaussian Pulse • Summary

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 9 / 12 Duality

6: Fourier Transform i2πft • Fourier Series as Fourier Transform: u(t) = ∞ U(f)e df [Fourier Synthesis] T → ∞ −∞ i2πft • Fourier Transform U(f) = ∞ u(t)e− dt [Fourier Analysis] • Fourier Transform Rt= Examples −∞ • Dirac Delta Function R • Dirac Delta Function: Dual transform: Scaling and Translation • Dirac Delta Function: Suppose v(t) = U(t) Products and Integrals • Periodic Signals • Duality • Time Shifting and Scaling • Gaussian Pulse • Summary

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 9 / 12 Duality

6: Fourier Transform i2πft • Fourier Series as Fourier Transform: u(t) = ∞ U(f)e df [Fourier Synthesis] T → ∞ −∞ i2πft • Fourier Transform U(f) = ∞ u(t)e− dt [Fourier Analysis] • Fourier Transform Rt= Examples −∞ • Dirac Delta Function R • Dirac Delta Function: Dual transform: Scaling and Translation • Dirac Delta Function: Suppose v(t) = U(t), then Products and Integrals • Periodic Signals V (f) = ∞ v(t)e i2πftdτ • Duality t= − • Time Shifting and Scaling −∞ • Gaussian Pulse R • Summary

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 9 / 12 Duality

6: Fourier Transform i2πft • Fourier Series as Fourier Transform: u(t) = ∞ U(f)e df [Fourier Synthesis] T → ∞ −∞ i2πft • Fourier Transform U(f) = ∞ u(t)e− dt [Fourier Analysis] • Fourier Transform Rt= Examples −∞ • Dirac Delta Function R • Dirac Delta Function: Dual transform: Scaling and Translation • Dirac Delta Function: Suppose v(t) = U(t), then Products and Integrals • Periodic Signals V (f) = ∞ v(t)e i2πftdτ • Duality t= − • Time Shifting and Scaling −∞ i2πgt V (g) = ∞ U(t)e− dt [substitute f = g, v(t) = U(t)] • Gaussian Pulse t= R −∞ • Summary R

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 9 / 12 Duality

6: Fourier Transform i2πft • Fourier Series as Fourier Transform: u(t) = ∞ U(f)e df [Fourier Synthesis] T → ∞ −∞ i2πft • Fourier Transform U(f) = ∞ u(t)e− dt [Fourier Analysis] • Fourier Transform Rt= Examples −∞ • Dirac Delta Function R • Dirac Delta Function: Dual transform: Scaling and Translation • Dirac Delta Function: Suppose v(t) = U(t), then Products and Integrals • Periodic Signals V (f) = ∞ v(t)e i2πftdτ • Duality t= − • Time Shifting and Scaling −∞ i2πgt V (g) = ∞ U(t)e− dt [substitute f = g, v(t) = U(t)] • Gaussian Pulse t= R −∞ • Summary ∞ i2πgf [substitute ] = Rf= U(f)e− df t = f −∞ R

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 9 / 12 Duality

6: Fourier Transform i2πft • Fourier Series as Fourier Transform: u(t) = ∞ U(f)e df [Fourier Synthesis] T → ∞ −∞ i2πft • Fourier Transform U(f) = ∞ u(t)e− dt [Fourier Analysis] • Fourier Transform Rt= Examples −∞ • Dirac Delta Function R • Dirac Delta Function: Dual transform: Scaling and Translation • Dirac Delta Function: Suppose v(t) = U(t), then Products and Integrals • Periodic Signals V (f) = ∞ v(t)e i2πftdτ • Duality t= − • Time Shifting and Scaling −∞ i2πgt V (g) = ∞ U(t)e− dt [substitute f = g, v(t) = U(t)] • Gaussian Pulse t= R −∞ • Summary ∞ i2πgf [substitute ] = Rf= U(f)e− df t = f −∞ = Ru(−g)

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 9 / 12 Duality

6: Fourier Transform i2πft • Fourier Series as Fourier Transform: u(t) = ∞ U(f)e df [Fourier Synthesis] T → ∞ −∞ i2πft • Fourier Transform U(f) = ∞ u(t)e− dt [Fourier Analysis] • Fourier Transform Rt= Examples −∞ • Dirac Delta Function R • Dirac Delta Function: Dual transform: Scaling and Translation • Dirac Delta Function: Suppose v(t) = U(t), then Products and Integrals • Periodic Signals V (f) = ∞ v(t)e i2πftdτ • Duality t= − • Time Shifting and Scaling −∞ i2πgt V (g) = ∞ U(t)e− dt [substitute f = g, v(t) = U(t)] • Gaussian Pulse t= R −∞ • Summary ∞ i2πgf [substitute ] = Rf= U(f)e− df t = f −∞ = Ru(−g) So: v(t) = U(t) ⇒ V (f) = u(−f)

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 9 / 12 Duality

6: Fourier Transform i2πft • Fourier Series as Fourier Transform: u(t) = ∞ U(f)e df [Fourier Synthesis] T → ∞ −∞ i2πft • Fourier Transform U(f) = ∞ u(t)e− dt [Fourier Analysis] • Fourier Transform Rt= Examples −∞ • Dirac Delta Function R • Dirac Delta Function: Dual transform: Scaling and Translation • Dirac Delta Function: Suppose v(t) = U(t), then Products and Integrals • Periodic Signals V (f) = ∞ v(t)e i2πftdτ • Duality t= − • Time Shifting and Scaling −∞ i2πgt V (g) = ∞ U(t)e− dt [substitute f = g, v(t) = U(t)] • Gaussian Pulse t= R −∞ • Summary ∞ i2πgf [substitute ] = Rf= U(f)e− df t = f −∞ = Ru(−g) So: v(t) = U(t) ⇒ V (f) = u(−f)

Example: t u(t) = e−| |

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 9 / 12 Duality

6: Fourier Transform i2πft • Fourier Series as Fourier Transform: u(t) = ∞ U(f)e df [Fourier Synthesis] T → ∞ −∞ i2πft • Fourier Transform U(f) = ∞ u(t)e− dt [Fourier Analysis] • Fourier Transform Rt= Examples −∞ • Dirac Delta Function R • Dirac Delta Function: Dual transform: Scaling and Translation • Dirac Delta Function: Suppose v(t) = U(t), then Products and Integrals • Periodic Signals V (f) = ∞ v(t)e i2πftdτ • Duality t= − • Time Shifting and Scaling −∞ i2πgt V (g) = ∞ U(t)e− dt [substitute f = g, v(t) = U(t)] • Gaussian Pulse t= R −∞ • Summary ∞ i2πgf [substitute ] = Rf= U(f)e− df t = f −∞ = Ru(−g) So: v(t) = U(t) ⇒ V (f) = u(−f)

Example: t 2 u(t) = e−| | ⇒ U(f) = 1+4π2f 2 [from earlier]

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 9 / 12 Duality

6: Fourier Transform i2πft • Fourier Series as Fourier Transform: u(t) = ∞ U(f)e df [Fourier Synthesis] T → ∞ −∞ i2πft • Fourier Transform U(f) = ∞ u(t)e− dt [Fourier Analysis] • Fourier Transform Rt= Examples −∞ • Dirac Delta Function R • Dirac Delta Function: Dual transform: Scaling and Translation • Dirac Delta Function: Suppose v(t) = U(t), then Products and Integrals • Periodic Signals V (f) = ∞ v(t)e i2πftdτ • Duality t= − • Time Shifting and Scaling −∞ i2πgt V (g) = ∞ U(t)e− dt [substitute f = g, v(t) = U(t)] • Gaussian Pulse t= R −∞ • Summary ∞ i2πgf [substitute ] = Rf= U(f)e− df t = f −∞ = Ru(−g) So: v(t) = U(t) ⇒ V (f) = u(−f)

Example: t 2 u(t) = e−| | ⇒ U(f) = 1+4π2f 2 [from earlier] 2 v(t) = 1+4π2t2

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 9 / 12 Duality

6: Fourier Transform i2πft • Fourier Series as Fourier Transform: u(t) = ∞ U(f)e df [Fourier Synthesis] T → ∞ −∞ i2πft • Fourier Transform U(f) = ∞ u(t)e− dt [Fourier Analysis] • Fourier Transform Rt= Examples −∞ • Dirac Delta Function R • Dirac Delta Function: Dual transform: Scaling and Translation • Dirac Delta Function: Suppose v(t) = U(t), then Products and Integrals • Periodic Signals V (f) = ∞ v(t)e i2πftdτ • Duality t= − • Time Shifting and Scaling −∞ i2πgt V (g) = ∞ U(t)e− dt [substitute f = g, v(t) = U(t)] • Gaussian Pulse t= R −∞ • Summary ∞ i2πgf [substitute ] = Rf= U(f)e− df t = f −∞ = Ru(−g) So: v(t) = U(t) ⇒ V (f) = u(−f)

Example: t 2 u(t) = e−| | ⇒ U(f) = 1+4π2f 2 [from earlier] 2 f v(t) = 1+4π2t2 ⇒ V (f) = e−|− |

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 9 / 12 Duality

6: Fourier Transform i2πft • Fourier Series as Fourier Transform: u(t) = ∞ U(f)e df [Fourier Synthesis] T → ∞ −∞ i2πft • Fourier Transform U(f) = ∞ u(t)e− dt [Fourier Analysis] • Fourier Transform Rt= Examples −∞ • Dirac Delta Function R • Dirac Delta Function: Dual transform: Scaling and Translation • Dirac Delta Function: Suppose v(t) = U(t), then Products and Integrals • Periodic Signals V (f) = ∞ v(t)e i2πftdτ • Duality t= − • Time Shifting and Scaling −∞ i2πgt V (g) = ∞ U(t)e− dt [substitute f = g, v(t) = U(t)] • Gaussian Pulse t= R −∞ • Summary ∞ i2πgf [substitute ] = Rf= U(f)e− df t = f −∞ = Ru(−g) So: v(t) = U(t) ⇒ V (f) = u(−f)

Example: t 2 u(t) = e−| | ⇒ U(f) = 1+4π2f 2 [from earlier] 2 f f v(t) = 1+4π2t2 ⇒ V (f) = e−|− | = e−| |

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 9 / 12 Time Shifting and Scaling

6: Fourier Transform i2πft • Fourier Series as Fourier Transform: u(t) = ∞ U(f)e df [Fourier Synthesis] T → ∞ −∞ i2πft • Fourier Transform U(f) = ∞ u(t)e− dt [Fourier Analysis] • Fourier Transform Rt= Examples −∞ • Dirac Delta Function R • Dirac Delta Function: Scaling and Translation • Dirac Delta Function: Products and Integrals • Periodic Signals • Duality • Time Shifting and Scaling • Gaussian Pulse • Summary

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 10 / 12 Time Shifting and Scaling

6: Fourier Transform i2πft • Fourier Series as Fourier Transform: u(t) = ∞ U(f)e df [Fourier Synthesis] T → ∞ −∞ i2πft • Fourier Transform U(f) = ∞ u(t)e− dt [Fourier Analysis] • Fourier Transform Rt= Examples −∞ • Dirac Delta Function R • Dirac Delta Function: Time Shifting and Scaling: Scaling and Translation • Dirac Delta Function: Products and Integrals Suppose v(t) = u(at + b) • Periodic Signals • Duality • Time Shifting and Scaling • Gaussian Pulse • Summary

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 10 / 12 Time Shifting and Scaling

6: Fourier Transform i2πft • Fourier Series as Fourier Transform: u(t) = ∞ U(f)e df [Fourier Synthesis] T → ∞ −∞ i2πft • Fourier Transform U(f) = ∞ u(t)e− dt [Fourier Analysis] • Fourier Transform Rt= Examples −∞ • Dirac Delta Function R • Dirac Delta Function: Time Shifting and Scaling: Scaling and Translation • Dirac Delta Function: Products and Integrals Suppose v(t) = u(at + b), then • Periodic Signals i2πft • ∞ Duality V (f) = t= u(at + b)e− dt • Time Shifting and Scaling −∞ • Gaussian Pulse • Summary R

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 10 / 12 Time Shifting and Scaling

6: Fourier Transform i2πft • Fourier Series as Fourier Transform: u(t) = ∞ U(f)e df [Fourier Synthesis] T → ∞ −∞ i2πft • Fourier Transform U(f) = ∞ u(t)e− dt [Fourier Analysis] • Fourier Transform Rt= Examples −∞ • Dirac Delta Function R • Dirac Delta Function: Time Shifting and Scaling: Scaling and Translation • Dirac Delta Function: Products and Integrals Suppose v(t) = u(at + b), then • Periodic Signals i2πft • Duality ∞ [now sub ] V (f) = t= u(at + b)e− dt τ = at + b • Time Shifting and Scaling −∞ • Gaussian Pulse • Summary R

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 10 / 12 Time Shifting and Scaling

6: Fourier Transform i2πft • Fourier Series as Fourier Transform: u(t) = ∞ U(f)e df [Fourier Synthesis] T → ∞ −∞ i2πft • Fourier Transform U(f) = ∞ u(t)e− dt [Fourier Analysis] • Fourier Transform Rt= Examples −∞ • Dirac Delta Function R • Dirac Delta Function: Time Shifting and Scaling: Scaling and Translation • Dirac Delta Function: Products and Integrals Suppose v(t) = u(at + b), then • Periodic Signals i2πft • Duality ∞ [now sub ] V (f) = t= u(at + b)e− dt τ = at + b • Time Shifting and Scaling −∞ τ−b • Gaussian Pulse i2πf( ) 1 = sgnR (a) ∞ u(τ)e− a dτ • Summary τ= a −∞ R

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 10 / 12 Time Shifting and Scaling

6: Fourier Transform i2πft • Fourier Series as Fourier Transform: u(t) = ∞ U(f)e df [Fourier Synthesis] T → ∞ −∞ i2πft • Fourier Transform U(f) = ∞ u(t)e− dt [Fourier Analysis] • Fourier Transform Rt= Examples −∞ • Dirac Delta Function R • Dirac Delta Function: Time Shifting and Scaling: Scaling and Translation • Dirac Delta Function: Products and Integrals Suppose v(t) = u(at + b), then • Periodic Signals i2πft • Duality ∞ [now sub ] V (f) = t= u(at + b)e− dt τ = at + b • Time Shifting and Scaling −∞ τ−b • Gaussian Pulse i2πf( ) 1 = sgnR (a) ∞ u(τ)e− a dτ • Summary τ= a −∞ 1 a > 0 note that ±∞ Rlimits swap if a < 0 hence sgn(a) = (−1 a < 0

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 10 / 12 Time Shifting and Scaling

6: Fourier Transform i2πft • Fourier Series as Fourier Transform: u(t) = ∞ U(f)e df [Fourier Synthesis] T → ∞ −∞ i2πft • Fourier Transform U(f) = ∞ u(t)e− dt [Fourier Analysis] • Fourier Transform Rt= Examples −∞ • Dirac Delta Function R • Dirac Delta Function: Time Shifting and Scaling: Scaling and Translation • Dirac Delta Function: Products and Integrals Suppose v(t) = u(at + b), then • Periodic Signals i2πft • Duality ∞ [now sub ] V (f) = t= u(at + b)e− dt τ = at + b • Time Shifting and Scaling −∞ τ−b • Gaussian Pulse i2πf( ) 1 = sgnR (a) ∞ u(τ)e− a dτ • Summary τ= a −∞ 1 a > 0 note that ±∞ Rlimits swap if a < 0 hence sgn(a) = (−1 a < 0 2 1 i πfb i π f τ a ∞ 2 a = a e τ= u(τ)e− dτ | | −∞ R

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 10 / 12 Time Shifting and Scaling

6: Fourier Transform i2πft • Fourier Series as Fourier Transform: u(t) = ∞ U(f)e df [Fourier Synthesis] T → ∞ −∞ i2πft • Fourier Transform U(f) = ∞ u(t)e− dt [Fourier Analysis] • Fourier Transform Rt= Examples −∞ • Dirac Delta Function R • Dirac Delta Function: Time Shifting and Scaling: Scaling and Translation • Dirac Delta Function: Products and Integrals Suppose v(t) = u(at + b), then • Periodic Signals i2πft • Duality ∞ [now sub ] V (f) = t= u(at + b)e− dt τ = at + b • Time Shifting and Scaling −∞ τ−b • Gaussian Pulse i2πf( ) 1 = sgnR (a) ∞ u(τ)e− a dτ • Summary τ= a −∞ 1 a > 0 note that ±∞ Rlimits swap if a < 0 hence sgn(a) = (−1 a < 0 2 1 i πfb i π f τ a ∞ 2 a = a e τ= u(τ)e− dτ | | −∞ 2 1 i πfb f a R = a e U a | |  

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 10 / 12 Time Shifting and Scaling

6: Fourier Transform i2πft • Fourier Series as Fourier Transform: u(t) = ∞ U(f)e df [Fourier Synthesis] T → ∞ −∞ i2πft • Fourier Transform U(f) = ∞ u(t)e− dt [Fourier Analysis] • Fourier Transform Rt= Examples −∞ • Dirac Delta Function R • Dirac Delta Function: Time Shifting and Scaling: Scaling and Translation • Dirac Delta Function: Products and Integrals Suppose v(t) = u(at + b), then • Periodic Signals i2πft • Duality ∞ [now sub ] V (f) = t= u(at + b)e− dt τ = at + b • Time Shifting and Scaling −∞ τ−b • Gaussian Pulse i2πf( ) 1 = sgnR (a) ∞ u(τ)e− a dτ • Summary τ= a −∞ 1 a > 0 note that ±∞ Rlimits swap if a < 0 hence sgn(a) = (−1 a < 0 2 1 i πfb i π f τ a ∞ 2 a = a e τ= u(τ)e− dτ | | −∞ 2 1 i πfb f a R = a e U a | |   2πfb f 1 i a So: v(t) = u(at + b) ⇒ V (f) = a e U a | |  

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 10 / 12 Gaussian Pulse

2 6: Fourier Transform t • Fourier Series as 1 2 2 → ∞ Gaussian Pulse: u(t) = e− σ T √2πσ2 • Fourier Transform • Fourier Transform Examples • Dirac Delta Function • Dirac Delta Function: Scaling and Translation • Dirac Delta Function: Products and Integrals • Periodic Signals • Duality • Time Shifting and Scaling • Gaussian Pulse • Summary

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 11 / 12 Gaussian Pulse

2 6: Fourier Transform t • Fourier Series as 1 2 2 → ∞ Gaussian Pulse: u(t) = e− σ T √2πσ2 • Fourier Transform • Fourier Transform Examples • Dirac Delta Function • Dirac Delta Function: Scaling and Translation • Dirac Delta Function: Products and Integrals • Periodic Signals • Duality • Time Shifting and Scaling • Gaussian Pulse • Summary

0.4 σ=1

0.2 u(t)

0 -4 -2 0 2 4 Time (s)

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 11 / 12 Gaussian Pulse

2 6: Fourier Transform t • Fourier Series as 1 2 2 → ∞ Gaussian Pulse: u(t) = e− σ T √2πσ2 • Fourier Transform • Fourier Transform This is a Normal (or Gaussian) probability distribution, so ∞ u(t)dt = 1. Examples −∞ • Dirac Delta Function • Dirac Delta Function: R Scaling and Translation • Dirac Delta Function: Products and Integrals • Periodic Signals • Duality • Time Shifting and Scaling • Gaussian Pulse • Summary

0.4 σ=1

0.2 u(t)

0 -4 -2 0 2 4 Time (s)

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 11 / 12 Gaussian Pulse

2 6: Fourier Transform t • Fourier Series as 1 2 2 → ∞ Gaussian Pulse: u(t) = e− σ T √2πσ2 • Fourier Transform • Fourier Transform This is a Normal (or Gaussian) probability distribution, so ∞ u(t)dt = 1. Examples −∞ • Dirac Delta Function i2πft • Dirac Delta Function: U(f) = ∞ u(t)e− dt R Scaling and Translation −∞ • Dirac Delta Function: Products and Integrals R • Periodic Signals • Duality • Time Shifting and Scaling • Gaussian Pulse • Summary

0.4 σ=1

0.2 u(t)

0 -4 -2 0 2 4 Time (s)

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 11 / 12 Gaussian Pulse

2 6: Fourier Transform t • Fourier Series as 1 2 2 → ∞ Gaussian Pulse: u(t) = e− σ T √2πσ2 • Fourier Transform • Fourier Transform This is a Normal (or Gaussian) probability distribution, so ∞ u(t)dt = 1. Examples 2 t −∞ • Dirac Delta Function i2πft 1 2σ2 i2πft • Dirac Delta Function: U(f) = ∞ u(t)e− dt = 2 ∞ e− e− dt √2πσ R Scaling and Translation −∞ −∞ • Dirac Delta Function: Products and Integrals R R • Periodic Signals • Duality • Time Shifting and Scaling • Gaussian Pulse • Summary

0.4 σ=1

0.2 u(t)

0 -4 -2 0 2 4 Time (s)

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 11 / 12 Gaussian Pulse

2 6: Fourier Transform t • Fourier Series as 1 2 2 → ∞ Gaussian Pulse: u(t) = e− σ T √2πσ2 • Fourier Transform • Fourier Transform This is a Normal (or Gaussian) probability distribution, so ∞ u(t)dt = 1. Examples 2 t −∞ • Dirac Delta Function i2πft 1 2σ2 i2πft • Dirac Delta Function: U(f) = ∞ u(t)e− dt = 2 ∞ e− e− dt √2πσ R Scaling and Translation −∞ 1 2 2 −∞ • Dirac Delta Function: 1 2 (t +i4πσ ft) = ∞ e− 2σ dt Products and Integrals R√2πσ2 R • Periodic Signals −∞ • Duality R • Time Shifting and Scaling • Gaussian Pulse • Summary

0.4 σ=1

0.2 u(t)

0 -4 -2 0 2 4 Time (s)

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 11 / 12 Gaussian Pulse

2 6: Fourier Transform t • Fourier Series as 1 2 2 → ∞ Gaussian Pulse: u(t) = e− σ T √2πσ2 • Fourier Transform • Fourier Transform This is a Normal (or Gaussian) probability distribution, so ∞ u(t)dt = 1. Examples 2 t −∞ • Dirac Delta Function i2πft 1 2σ2 i2πft • Dirac Delta Function: U(f) = ∞ u(t)e− dt = 2 ∞ e− e− dt √2πσ R Scaling and Translation −∞ 1 2 2 −∞ • Dirac Delta Function: 1 2 (t +i4πσ ft) = ∞ e− 2σ dt Products and Integrals R√2πσ2 R • Periodic Signals −∞ 1 2 2 2 2 2 2 • Duality 2 t +i4πσ ft+(i2πσ f) (i2πσ f) = 1 R ∞ e− 2σ  − dt • Time Shifting and Scaling √2πσ2 • Gaussian Pulse −∞ • Summary R

0.4 σ=1

0.2 u(t)

0 -4 -2 0 2 4 Time (s)

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 11 / 12 Gaussian Pulse

2 6: Fourier Transform t • Fourier Series as 1 2 2 → ∞ Gaussian Pulse: u(t) = e− σ T √2πσ2 • Fourier Transform • Fourier Transform This is a Normal (or Gaussian) probability distribution, so ∞ u(t)dt = 1. Examples 2 t −∞ • Dirac Delta Function i2πft 1 2σ2 i2πft • Dirac Delta Function: U(f) = ∞ u(t)e− dt = 2 ∞ e− e− dt √2πσ R Scaling and Translation −∞ 1 2 2 −∞ • Dirac Delta Function: 1 2 (t +i4πσ ft) = ∞ e− 2σ dt Products and Integrals R√2πσ2 R • Periodic Signals −∞ 1 2 2 2 2 2 2 • Duality 2 t +i4πσ ft+(i2πσ f) (i2πσ f) = 1 R ∞ e− 2σ  − dt • Time Shifting and Scaling √2πσ2 2 2 • Gaussian Pulse 1 −∞2 1 2 2 i2πσ f 2 t+i2πσ f • 2σ ( ) 1 2σ ( ) Summary = e R 2 ∞ e− dt √2πσ −∞ R

0.4 σ=1

0.2 u(t)

0 -4 -2 0 2 4 Time (s)

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 11 / 12 Gaussian Pulse

2 6: Fourier Transform t • Fourier Series as 1 2 2 → ∞ Gaussian Pulse: u(t) = e− σ T √2πσ2 • Fourier Transform • Fourier Transform This is a Normal (or Gaussian) probability distribution, so ∞ u(t)dt = 1. Examples 2 t −∞ • Dirac Delta Function i2πft 1 2σ2 i2πft • Dirac Delta Function: U(f) = ∞ u(t)e− dt = 2 ∞ e− e− dt √2πσ R Scaling and Translation −∞ 1 2 2 −∞ • Dirac Delta Function: 1 2 (t +i4πσ ft) = ∞ e− 2σ dt Products and Integrals R√2πσ2 R • Periodic Signals −∞ 1 2 2 2 2 2 2 • Duality 2 t +i4πσ ft+(i2πσ f) (i2πσ f) = 1 R ∞ e− 2σ  − dt • Time Shifting and Scaling √2πσ2 2 2 • Gaussian Pulse 1 −∞2 1 2 2 i2πσ f 2 t+i2πσ f • 2σ ( ) 1 2σ ( ) Summary = e R 2 ∞ e− dt √2πσ −∞ (i) 1 2 2 2 (i2πσ f) = e 2σ R

0.4 σ=1

0.2 u(t)

0 -4 -2 0 2 4 Time (s)

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 11 / 12 Gaussian Pulse

2 6: Fourier Transform t • Fourier Series as 1 2 2 → ∞ Gaussian Pulse: u(t) = e− σ T √2πσ2 • Fourier Transform • Fourier Transform This is a Normal (or Gaussian) probability distribution, so ∞ u(t)dt = 1. Examples 2 t −∞ • Dirac Delta Function i2πft 1 2σ2 i2πft • Dirac Delta Function: U(f) = ∞ u(t)e− dt = 2 ∞ e− e− dt √2πσ R Scaling and Translation −∞ 1 2 2 −∞ • Dirac Delta Function: 1 2 (t +i4πσ ft) = ∞ e− 2σ dt Products and Integrals R√2πσ2 R • Periodic Signals −∞ 1 2 2 2 2 2 2 • Duality 2 t +i4πσ ft+(i2πσ f) (i2πσ f) = 1 R ∞ e− 2σ  − dt • Time Shifting and Scaling √2πσ2 2 2 • Gaussian Pulse 1 −∞2 1 2 2 i2πσ f 2 t+i2πσ f • 2σ ( ) 1 2σ ( ) Summary = e R 2 ∞ e− dt √2πσ −∞ (i) 1 2 2 2 (i2πσ f) = e 2σ R [(i) uses a result from complex analysis theory that: 1 2 2 1 2 1 2 (t+i2πσ f) 1 2 t ∞ e− 2σ dt = ∞ e− 2σ dt = 1] √2πσ2 √2πσ2 −∞ −∞ 0.4 R σ=1 R

0.2 u(t)

0 -4 -2 0 2 4 Time (s)

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 11 / 12 Gaussian Pulse

2 6: Fourier Transform t • Fourier Series as 1 2 2 → ∞ Gaussian Pulse: u(t) = e− σ T √2πσ2 • Fourier Transform • Fourier Transform This is a Normal (or Gaussian) probability distribution, so ∞ u(t)dt = 1. Examples 2 t −∞ • Dirac Delta Function i2πft 1 2σ2 i2πft • Dirac Delta Function: U(f) = ∞ u(t)e− dt = 2 ∞ e− e− dt √2πσ R Scaling and Translation −∞ 1 2 2 −∞ • Dirac Delta Function: 1 2 (t +i4πσ ft) = ∞ e− 2σ dt Products and Integrals R√2πσ2 R • Periodic Signals −∞ 1 2 2 2 2 2 2 • Duality 2 t +i4πσ ft+(i2πσ f) (i2πσ f) = 1 R ∞ e− 2σ  − dt • Time Shifting and Scaling √2πσ2 2 2 • Gaussian Pulse 1 −∞2 1 2 2 i2πσ f 2 t+i2πσ f • 2σ ( ) 1 2σ ( ) Summary = e R 2 ∞ e− dt √2πσ −∞ (i) 1 2 2 1 2 2 (i2πσ f) (2πσf) = e 2σ = e− 2R [(i) uses a result from complex analysis theory that: 1 2 2 1 2 1 2 (t+i2πσ f) 1 2 t ∞ e− 2σ dt = ∞ e− 2σ dt = 1] √2πσ2 √2πσ2 −∞ −∞ 0.4 R σ=1 R

0.2 u(t)

0 -4 -2 0 2 4 Time (s)

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 11 / 12 Gaussian Pulse

2 6: Fourier Transform t • Fourier Series as 1 2 2 → ∞ Gaussian Pulse: u(t) = e− σ T √2πσ2 • Fourier Transform • Fourier Transform This is a Normal (or Gaussian) probability distribution, so ∞ u(t)dt = 1. Examples 2 t −∞ • Dirac Delta Function i2πft 1 2σ2 i2πft • Dirac Delta Function: U(f) = ∞ u(t)e− dt = 2 ∞ e− e− dt √2πσ R Scaling and Translation −∞ 1 2 2 −∞ • Dirac Delta Function: 1 2 (t +i4πσ ft) = ∞ e− 2σ dt Products and Integrals R√2πσ2 R • Periodic Signals −∞ 1 2 2 2 2 2 2 • Duality 2 t +i4πσ ft+(i2πσ f) (i2πσ f) = 1 R ∞ e− 2σ  − dt • Time Shifting and Scaling √2πσ2 2 2 • Gaussian Pulse 1 −∞2 1 2 2 i2πσ f 2 t+i2πσ f • 2σ ( ) 1 2σ ( ) Summary = e R 2 ∞ e− dt √2πσ −∞ (i) 1 2 2 1 2 2 (i2πσ f) (2πσf) = e 2σ = e− 2R [(i) uses a result from complex analysis theory that: 1 2 2 1 2 1 2 (t+i2πσ f) 1 2 t ∞ e− 2σ dt = ∞ e− 2σ dt = 1] √2πσ2 √2πσ2 −∞ −∞ 0.4 R σ=1 1 R 1/(2πσ)=0.159

0.2 0.5 u(t) |U(f)|

0 0 -4 -2 0 2 4 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 Time (s) Frequency (Hz)

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 11 / 12 Gaussian Pulse

2 6: Fourier Transform t • Fourier Series as 1 2 2 → ∞ Gaussian Pulse: u(t) = e− σ T √2πσ2 • Fourier Transform • Fourier Transform This is a Normal (or Gaussian) probability distribution, so ∞ u(t)dt = 1. Examples 2 t −∞ • Dirac Delta Function i2πft 1 2σ2 i2πft • Dirac Delta Function: U(f) = ∞ u(t)e− dt = 2 ∞ e− e− dt √2πσ R Scaling and Translation −∞ 1 2 2 −∞ • Dirac Delta Function: 1 2 (t +i4πσ ft) = ∞ e− 2σ dt Products and Integrals R√2πσ2 R • Periodic Signals −∞ 1 2 2 2 2 2 2 • Duality 2 t +i4πσ ft+(i2πσ f) (i2πσ f) = 1 R ∞ e− 2σ  − dt • Time Shifting and Scaling √2πσ2 2 2 • Gaussian Pulse 1 −∞2 1 2 2 i2πσ f 2 t+i2πσ f • 2σ ( ) 1 2σ ( ) Summary = e R 2 ∞ e− dt √2πσ −∞ (i) 1 2 2 1 2 2 (i2πσ f) (2πσf) = e 2σ = e− 2R [(i) uses a result from complex analysis theory that: 1 2 2 1 2 1 2 (t+i2πσ f) 1 2 t ∞ e− 2σ dt = ∞ e− 2σ dt = 1] √2πσ2 √2πσ2 −∞ −∞ 0.4 R σ=1 1 R 1/(2πσ)=0.159

0.2 0.5 u(t) |U(f)|

0 0 -4 -2 0 2 4 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 Time (s) Frequency (Hz) Uniquely, the Fourier Transform of a Gaussian pulse is a Gaussian pulse. E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 11 / 12 Summary

6: Fourier Transform • Fourier Series as • Fourier Transform: → ∞ T ◦ Inverse transform (synthesis): u(t) = ∞ U(f)ei2πftdf • Fourier Transform • Fourier Transform −∞ Examples • Dirac Delta Function R • Dirac Delta Function: Scaling and Translation • Dirac Delta Function: Products and Integrals • Periodic Signals • Duality • Time Shifting and Scaling • Gaussian Pulse • Summary

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 12 / 12 Summary

6: Fourier Transform • Fourier Series as • Fourier Transform: → ∞ T ◦ Inverse transform (synthesis): u(t) = ∞ U(f)ei2πftdf • Fourier Transform • −∞ Fourier Transform i2πft Examples ◦ Forward transform (analysis): U(f) = ∞ u(t)e− dt • Dirac Delta Function R t= • Dirac Delta Function: −∞ Scaling and Translation R • Dirac Delta Function: Products and Integrals • Periodic Signals • Duality • Time Shifting and Scaling • Gaussian Pulse • Summary

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 12 / 12 Summary

6: Fourier Transform • Fourier Series as • Fourier Transform: → ∞ T ◦ Inverse transform (synthesis): u(t) = ∞ U(f)ei2πftdf • Fourier Transform • −∞ Fourier Transform i2πft Examples Forward transform (analysis): ∞ ◦ U(f) =R t= u(t)e− dt • Dirac Delta Function −∞ • Dirac Delta Function: ⊲ U(f) is the spectral density function (e.g. Volts/Hz) Scaling and Translation R • Dirac Delta Function: Products and Integrals • Periodic Signals • Duality • Time Shifting and Scaling • Gaussian Pulse • Summary

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 12 / 12 Summary

6: Fourier Transform • Fourier Series as • Fourier Transform: → ∞ T ◦ Inverse transform (synthesis): u(t) = ∞ U(f)ei2πftdf • Fourier Transform • −∞ Fourier Transform i2πft Examples Forward transform (analysis): ∞ ◦ U(f) =R t= u(t)e− dt • Dirac Delta Function −∞ • Dirac Delta Function: ⊲ U(f) is the spectral density function (e.g. Volts/Hz) Scaling and Translation R • Dirac Delta Function: Products and Integrals • Dirac Delta Function: • Periodic Signals • Duality ◦ δ(t) is a zero-width infinite-height pulse with ∞ δ(t)dt = 1 • Time Shifting and Scaling −∞ • Gaussian Pulse R • Summary

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 12 / 12 Summary

6: Fourier Transform • Fourier Series as • Fourier Transform: → ∞ T ◦ Inverse transform (synthesis): u(t) = ∞ U(f)ei2πftdf • Fourier Transform • −∞ Fourier Transform i2πft Examples Forward transform (analysis): ∞ ◦ U(f) =R t= u(t)e− dt • Dirac Delta Function −∞ • Dirac Delta Function: ⊲ U(f) is the spectral density function (e.g. Volts/Hz) Scaling and Translation R • Dirac Delta Function: Products and Integrals • Dirac Delta Function: • Periodic Signals • Duality ◦ δ(t) is a zero-width infinite-height pulse with ∞ δ(t)dt = 1 • Time Shifting and Scaling −∞ • Gaussian Pulse ◦ Integral: ∞ f(t)δ(t − a) = f(a) R • Summary −∞ R

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 12 / 12 Summary

6: Fourier Transform • Fourier Series as • Fourier Transform: → ∞ T ◦ Inverse transform (synthesis): u(t) = ∞ U(f)ei2πftdf • Fourier Transform • −∞ Fourier Transform i2πft Examples Forward transform (analysis): ∞ ◦ U(f) =R t= u(t)e− dt • Dirac Delta Function −∞ • Dirac Delta Function: ⊲ U(f) is the spectral density function (e.g. Volts/Hz) Scaling and Translation R • Dirac Delta Function: Products and Integrals • Dirac Delta Function: • Periodic Signals • Duality ◦ δ(t) is a zero-width infinite-height pulse with ∞ δ(t)dt = 1 • Time Shifting and Scaling −∞ • Gaussian Pulse ◦ Integral: ∞ f(t)δ(t − a) = f(a) R • Summary −∞ 1 ◦ Scaling: δR(ct) = c δ(t) | |

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 12 / 12 Summary

6: Fourier Transform • Fourier Series as • Fourier Transform: → ∞ T ◦ Inverse transform (synthesis): u(t) = ∞ U(f)ei2πftdf • Fourier Transform • −∞ Fourier Transform i2πft Examples Forward transform (analysis): ∞ ◦ U(f) =R t= u(t)e− dt • Dirac Delta Function −∞ • Dirac Delta Function: ⊲ U(f) is the spectral density function (e.g. Volts/Hz) Scaling and Translation R • Dirac Delta Function: Products and Integrals • Dirac Delta Function: • Periodic Signals • Duality ◦ δ(t) is a zero-width infinite-height pulse with ∞ δ(t)dt = 1 • Time Shifting and Scaling −∞ • Gaussian Pulse ◦ Integral: ∞ f(t)δ(t − a) = f(a) R • Summary −∞ 1 ◦ Scaling: δR(ct) = c δ(t) | | i2πnF t • Periodic Signals: u(t) = n∞= Une −∞ ⇒ U(f) = n∞= Unδ (f − nF ) P −∞ P

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 12 / 12 Summary

6: Fourier Transform • Fourier Series as • Fourier Transform: → ∞ T ◦ Inverse transform (synthesis): u(t) = ∞ U(f)ei2πftdf • Fourier Transform • −∞ Fourier Transform i2πft Examples Forward transform (analysis): ∞ ◦ U(f) =R t= u(t)e− dt • Dirac Delta Function −∞ • Dirac Delta Function: ⊲ U(f) is the spectral density function (e.g. Volts/Hz) Scaling and Translation R • Dirac Delta Function: Products and Integrals • Dirac Delta Function: • Periodic Signals • Duality ◦ δ(t) is a zero-width infinite-height pulse with ∞ δ(t)dt = 1 • Time Shifting and Scaling −∞ • Gaussian Pulse ◦ Integral: ∞ f(t)δ(t − a) = f(a) R • Summary −∞ 1 ◦ Scaling: δR(ct) = c δ(t) | | i2πnF t • Periodic Signals: u(t) = n∞= Une −∞ U f ∞ U δ f nF P ⇒ ( ) = n= n ( − ) • Fourier Transform Properties: −∞ ◦ v(t) = U(t) ⇒ V (f) = u(P−f)

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 12 / 12 Summary

6: Fourier Transform • Fourier Series as • Fourier Transform: → ∞ T ◦ Inverse transform (synthesis): u(t) = ∞ U(f)ei2πftdf • Fourier Transform • −∞ Fourier Transform i2πft Examples Forward transform (analysis): ∞ ◦ U(f) =R t= u(t)e− dt • Dirac Delta Function −∞ • Dirac Delta Function: ⊲ U(f) is the spectral density function (e.g. Volts/Hz) Scaling and Translation R • Dirac Delta Function: Products and Integrals • Dirac Delta Function: • Periodic Signals • Duality ◦ δ(t) is a zero-width infinite-height pulse with ∞ δ(t)dt = 1 • Time Shifting and Scaling −∞ • Gaussian Pulse ◦ Integral: ∞ f(t)δ(t − a) = f(a) R • Summary −∞ 1 ◦ Scaling: δR(ct) = c δ(t) | | i2πnF t • Periodic Signals: u(t) = n∞= Une −∞ U f ∞ U δ f nF P ⇒ ( ) = n= n ( − ) • Fourier Transform Properties: −∞ ◦ v(t) = U(t) ⇒ V (f) = u(P−f) 2πfb f 1 i a ◦ v(t) = u(at + b) ⇒ V (f) = a e U a | |  

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 12 / 12 Summary

6: Fourier Transform • Fourier Series as • Fourier Transform: → ∞ T ◦ Inverse transform (synthesis): u(t) = ∞ U(f)ei2πftdf • Fourier Transform • −∞ Fourier Transform i2πft Examples Forward transform (analysis): ∞ ◦ U(f) =R t= u(t)e− dt • Dirac Delta Function −∞ • Dirac Delta Function: ⊲ U(f) is the spectral density function (e.g. Volts/Hz) Scaling and Translation R • Dirac Delta Function: Products and Integrals • Dirac Delta Function: • Periodic Signals • Duality ◦ δ(t) is a zero-width infinite-height pulse with ∞ δ(t)dt = 1 • Time Shifting and Scaling −∞ • Gaussian Pulse ◦ Integral: ∞ f(t)δ(t − a) = f(a) R • Summary −∞ 1 ◦ Scaling: δR(ct) = c δ(t) | | i2πnF t • Periodic Signals: u(t) = n∞= Une −∞ U f ∞ U δ f nF P ⇒ ( ) = n= n ( − ) • Fourier Transform Properties: −∞ ◦ v(t) = U(t) ⇒ V (f) = u(P−f) 2πfb f 1 i a ◦ v(t) = u(at + b) ⇒ V (f) = a e U a 2 | | 1 t 1 2 1 2 ( σ ) 2 (2πσf)  ◦ v(t) = 2 e− ⇒ V (f) = e− √2πσ

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 12 / 12 Summary

6: Fourier Transform • Fourier Series as • Fourier Transform: → ∞ T ◦ Inverse transform (synthesis): u(t) = ∞ U(f)ei2πftdf • Fourier Transform • −∞ Fourier Transform i2πft Examples Forward transform (analysis): ∞ ◦ U(f) =R t= u(t)e− dt • Dirac Delta Function −∞ • Dirac Delta Function: ⊲ U(f) is the spectral density function (e.g. Volts/Hz) Scaling and Translation R • Dirac Delta Function: Products and Integrals • Dirac Delta Function: • Periodic Signals • Duality ◦ δ(t) is a zero-width infinite-height pulse with ∞ δ(t)dt = 1 • Time Shifting and Scaling −∞ • Gaussian Pulse ◦ Integral: ∞ f(t)δ(t − a) = f(a) R • Summary −∞ 1 ◦ Scaling: δR(ct) = c δ(t) | | i2πnF t • Periodic Signals: u(t) = n∞= Une −∞ U f ∞ U δ f nF P ⇒ ( ) = n= n ( − ) • Fourier Transform Properties: −∞ ◦ v(t) = U(t) ⇒ V (f) = u(P−f) 2πfb f 1 i a ◦ v(t) = u(at + b) ⇒ V (f) = a e U a 2 | | 1 t 1 2 1 2 ( σ ) 2 (2πσf)  ◦ v(t) = 2 e− ⇒ V (f) = e− √2πσ For further details see RHB Chapter 13.1 (uses ω instead of f)

E1.10 Fourier Series and Transforms (2014-5559) Fourier Transform: 6 – 12 / 12