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 = .