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and Transforms ⊲ Revision Lecture The Basic Idea Real v Complex Series v Transform Fourier Power Conservation Gibbs Phenomenon Coefficient Decay Rate Periodic Extension and Transforms Dirac Delta Revision Lecture Correlation

E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 1 / 13 The Basic Idea

Fourier Series and Transforms Periodic can be written as a sum of sine and cosine : Revision Lecture ⊲ The Basic Idea Real v Complex a0 u(t) = + n∞ (an cos 2πnFt + bn sin 2πnFt) Series v Transform 2 =1 Fourier Analysis T π Power Conservation u(t) P T +0.65sin(2 Ft) Gibbs Phenomenon Coefficient Decay Rate = Periodic Extension -0.26sin(2πFt) T/2 Fourier Transform Convolution + Correlation +0.26sin(2πFt) T/3 + -0.13cos(2πFt) T/4 + Fundamental Period: the smallest T > 0 for which u(t + T ) = u(t). 1 th Fundamental : F = T . The n is at frequency nF . Some waveforms need infinitely many (countable infinity).

E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 2 / 13 Real versus Complex Fourier Series

Fourier Series and iωt Transforms All the algebra is much easier if we use e instead of cos ωt and sin ωt Revision Lecture The Basic Idea a 0 ∞ ⊲ Real v Complex u(t) = 2 + n=1 (an cos 2πnFt + bn sin 2πnFt) Series v Transform Fourier Analysis P Power Conservation Substitute: cos ωt = 1 eiωt + 1 e iωt sin ωt = i eiωt + i e iωt Gibbs Phenomenon 2 2 − −2 2 − Coefficient Decay Rate a0 1 i2πnF t 1 i2πnF t u(t) = + n∞ (an − ibn) e + (an + ibn) e− Periodic Extension 2 =1 2 2 Dirac Delta Function i πnF t ∞ 2 Fourier Transform = n=P Une  Convolution −∞ 1 1 Correlation • U+Pn = (an − ibn) and U n = (an + ibn) . 2 − 2 • U+n and U n are complex conjugates. − • U+n is half the equivalent phasor in Analysis of Circuits.

u(t) T 1

0.5 |U|

0 -4F -3F -2F -F 0 F 2F 3F 4F

Plot the magnitude spectrum

U 0

and spectrum: ∠ -pi -4F -3F -2F -F 0 F 2F 3F 4F

E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 3 / 13 Fourier Series versus Fourier Transform

Fourier Series and Transforms • Periodic signals → Fourier Series→ Discrete spectrum Revision Lecture T 1 The Basic Idea u(t) Real v Complex 0.5 ⊲ Series v Transform |U| Fourier Analysis 0 -4F -3F -2F -F 0 F 2F 3F 4F Power Conservation i2πnF t ∞ Gibbs Phenomenon u(t) = n= Une pi Coefficient Decay −∞ U 0 Rate ∠ Periodic Extension P -pi Dirac Delta Function -4F -3F -2F -F 0 F 2F 3F 4F Fourier Transform Convolution Correlation • Aperiodic signals → Fourier Transform→ Continuous Spectrum 1 0.5 a=2 0.4 0.5 0.3 u(t) 0.2 0 |U(f)| 0.1 -5 0 5 Time (s) -5 0 5 Frequency (Hz) pi/2 u(t) = ∞ U(f)ei2πftdf f= 0 −∞ U(f) (rad) U(f) -pi/2 R ∠ -5 0 5 Frequence (Hz) • Both types of spectrum are conjugate symmetric. • If u(t) is periodic, its Fourier transform consists of Dirac δ functions with {Un}.

E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 4 / 13 Fourier Analysis

Fourier Series and i2πnF t Transforms Fourier Series: u(t) = n∞= Une Revision Lecture −∞ The Basic Idea Fourier Analysis = “how do you work out the Fourier coefficients, Un ?” Real v Complex P Series v Transform ⊲ Fourier Analysis iωt 1 if ω = 0 Power Conservation Key idea: e = hcos ωt + i sin ωti = Gibbs Phenomenon (0 otherwise Coefficient Decay Rate m n Periodic Extension i2πnF t i2πmFt 1 for = ⇒ Orthogonality: e × e− = Dirac Delta Function m n Fourier Transform (0 for 6= Convolution Correlation So, to find a particular coefficient, Um, we work out i2πmFt i2πnF t i2πmFt u(t)e− = n∞= Une e− −∞ i2πnF t i2πmFt = Pn∞= Un e e− −∞ U = Pm [since all other terms are zero] Calculate the average by integrating over any integer number of periods i2πmFt 1 T i2πmFt Um = u(t)e− = T t=0 u(t)e− dt Notice the negative sign in FourierR analysis: in order to extract the term in +i2πmFt i2πmFt the series containing e we need to multiply by e− .

E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 5 / 13 Power Conservation

Fourier Series and i2πnF t Transforms Fourier Series: u(t) = n∞= Une Revision Lecture −∞ The Basic Idea u t P , u t 2 1 T u2 t dt u t Real v Complex Average power in ( ):P u | ( )| = T 0 ( ) [ ( ) real] Series v Transform Fourier Analysis Average power in Fourier componentD nE: R Power i2πnF t 2 2 i2πnF t 2 2 ⊲ Conservation Une = |Un| e = |Un| Gibbs Phenomenon Coefficient Decay Rate Power conservationD (Parseval’s E D Theorem): E Periodic Extension 2 2 Dirac Delta Function Pu = |u(t)| = n∞= |Un| Fourier Transform −∞ Convolution D E Correlation The average power in uP(t) is equal to the sum of the average powers in all the Fourier components. This is a consequence of orthogonality: 2 i2πnF t i2πmFt |u(t)| = n∞= Une m∞= Um∗ e− −∞ −∞ D E i2πnF t i2πmFt = P n∞= m∞= U nPUm∗ e e−  −∞ −∞ i2πnF t i2πmFt = Pn∞= Pm∞= UnUm∗ e e− −∞ −∞ 2 = Pn∞= |PUn| −∞

E1.10 Fourier Series and Transforms (2015-6200)P Revision Lecture: – 6 / 13 Gibbs Phenomenon

Fourier Series and N i2πnF t Transforms Truncated Fourier Series: uN (t) = n= N Une Revision Lecture − 1 max(u )=0.500 The Basic Idea 0 Approximation error: eN (t) = uN (t) − u(t) 0.5 Real v Complex P N=0 0 Series v Transform 2 0 5 10 15 20 Fourier Analysis Average error power PeN = n >N |Un| . 1 max(u )=1.137 | | 1 Power Conservation 0.5 PeN → 0 monotonically as N → ∞. N=1 ⊲ Gibbs Phenomenon P 0 Coefficient Decay 0 5 10 15 20 Rate 1 max(u )=1.100 Periodic Extension 3 Gibbs phenomenon 0.5 N=3 Dirac Delta Function 0 Fourier Transform If u(t0) has a discontinuity of height h then: 0 5 10 15 20 Convolution 1 max(u )=1.094 5 Correlation • u (t ) → the midpoint of the 0.5 N 0 N=5 0 discontinuity as N → ∞. 0 5 10 15 20 1 max(u )=1.089 41 • uN (t) overshoots by ≈ ±9% × h at 0.5 N=41 T 0 t ≈ t0 ± 2N+1 . 0 5 10 15 20 • For large N, the overshoots move

1 max(u )=1.089 closer to the discontinuity but do not 41 0.5 0 decrease in size. -1 -0.5 0 0.5 1 [Enlarged View: u41(t)]

E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 7 / 13 Coefficient Decay Rate

Fourier Series and i2πnF t Transforms Fourier Series: u(t) = n∞= Une Revision Lecture −∞ The Basic Idea Integration: P Real v Complex t 1 Series v Transform v(t) = 0 u(τ)dτ ⇒ Vn = i2πnF Un Fourier Analysis Power Conservation provided U0 = V0 = 0. Gibbs Phenomenon R Coefficient Decay Differentiation: ⊲ Rate du(t) Periodic Extension w(t) = dt ⇒ Wn = i2πnF × Un Dirac Delta Function Fourier Transform provided w(t) satisfies the Dirichlet conditions. Convolution Correlation Coefficient Decay Rate: 1 u(t) has a discontinuity ⇒ |Un| is O n for large |n| k d u(t) dtk is the lowest with a discontinuity  1 ⇒ |Un| is O nk+1 for large |n|

If the coefficients, Un, decrease rapidly then only a few terms are needed for a good approximation.

E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 8 / 13 Periodic Extension

Fourier Series and Transforms If u(t) is only defined over a finite , [0, B], we can make it periodic by Revision Lecture defining u(t ± B) = u(t). The Basic Idea 1 B i2πnF t Real v Complex • Coefficients are given by Un = u(t)e− dt. Series v Transform B 0 Fourier Analysis 2 Power Conservation Example: u(t) = t for 0 ≤ t < 2 R 4 4 4 Gibbs Phenomenon N=1 N=3 N=6 Coefficient Decay 2 2 2 Rate ⊲ Periodic Extension 0 0 0 Dirac Delta Function -2 0 2 4 -2 0 2 4 -2 0 2 4 Fourier Transform Convolution Symmetric extension: Correlation • To avoid a discontinuity at t = T , we can instead make the period 2B and define u(−t) = u(+t). 4 4 4 N=1 N=3 N=6

2 2 2

0 0 0 -2 0 2 4 -2 0 2 4 -2 0 2 4 • Symmetry around t = 0 coefficients are real-valued and symmetric (U n = Un∗ = Un). • Still have a first-derivative− discontinuity at t = B but now we have 2 1 no Gibbs phenomenon and coefficients ∝ n− instead of ∝ n− so approximation error power decreases more quickly.

E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 9 / 13 Dirac Delta Function

Fourier Series and 1 Transforms δ(x) is the limiting case as w → 0 of a pulse w wide and w high Revision Lecture The Basic Idea It is an infinitely thin, infinitely high pulse at x = 0 with unit area. Real v Complex δ (x) Series v Transform 4 0.2 1 Fourier Analysis δ(x) Power Conservation 2 0.5 Gibbs Phenomenon Coefficient Decay 0 Rate 0 Periodic Extension -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 Dirac Delta x x ⊲ Function Fourier Transform Convolution • Area: ∞ δ(x)dx = 1 Correlation −∞ 1 • Scaling:R δ(cx) = c δ(x) | | • Shifting: δ(x − a) is a pulse at x = a and is zero everywhere else • Multiplication: f(x) × δ(x − a) = f(a) × δ(x − a)

• Integration: ∞ f(x) × δ(x − a)dx = f(a) −∞ • Fourier Transform:R u(t) = δ(t) ⇔ U(f) = 1 • We plot hδ(x) as a pulse of height |h| (instead of its true height of ∞)

E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 10 / 13 Fourier Transform

Fourier Series and i2πft Transforms Fourier Transform: u(t) = ∞ U(f)e df Revision Lecture −∞ i2πft The Basic Idea U(f) =R ∞ u(t)e− dt Real v Complex −∞ Series v Transform 2 Fourier Analysis • An “Energy ” has finite energyR ⇔ Eu = ∞ |u(t)| dt < ∞ Power Conservation −∞ Gibbs Phenomenon ◦ Complex-valued spectrum, U(f), decays to zero as f → ±∞ Coefficient Decay R Rate 2 ∞ Periodic Extension ◦ Energy Conservation: Eu = EU where EU = |U(f)| df Dirac Delta Function −∞ 0.5 ⊲ Fourier Transform 1 0.4 a=2 0.3 R Convolution 0.5 0.2 u(t) |U(f)| 0.1 Correlation 0 -5 0 5 -5 0 5 Time (s) Frequency (Hz)

• Periodic Signals → Dirac δ functions at harmonics. Same complex-valued amplitudes as Un from Fourier Series 1

u(t) T 0.5 |U|

0 -4F -3F -2F -F 0 F 2F 3F 4F 2 2 ◦ Eu = ∞ but ave power is Pu = |u(t)| = n∞= |Un| −∞ D E P E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 11 / 13 Convolution

Fourier Series and Transforms Convolution: w(t) = u(t) ∗ v(t) ⇔ w(t) = ∞ u(τ)v(t − τ)dτ Revision Lecture −∞ The Basic Idea [In the , the arguments of u() and v() add up to t] Real v Complex R Series v Transform Fourier Analysis Power Conservation ∗ acts algebraically like × : Commutative, Associative, Distributive over +. Gibbs Phenomenon Identity element is δ(t): u(t) ∗ δ(t) = u(t) Coefficient Decay Rate Periodic Extension Dirac Delta Function Multiplication in either the time or Fourier Transform is equivalent to convolution in the other domain: ⊲ Convolution Correlation w(t) = u(t) ∗ v(t) ⇔ W (f) = U(f)V (f) y(t) = u(t)v(t) ⇔ Y (f) = U(f) ∗ V (f) Example application: • : [, y(t) for x(t) = δ(t)] t 1 RC h(t) = RC e− for t ≥ 0 1 • Frequency Response: H(f) = 1+i2πfRC • Convolution: y(t) = h(t) ∗ x(t) • Multiplication: Y (f) = H(f)X(f)

E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 12 / 13 Correlation

Fourier Series and Transforms Cross-correlation: Revision Lecture w(t) = u(t) ⊗ v(t) ⇔ w(t) = ∞ u (τ − t)v(τ)dτ The Basic Idea ∗ −∞ Real v Complex Series v Transform [In the integral, the argumentsR of u∗() and v() differ by t] Fourier Analysis Power Conservation ⊗ is not commutative or associative (unlike ∗) Gibbs Phenomenon Coefficient Decay Rate Periodic Extension Cauchy-Schwartz Inequality ⇒ Bound on |w(t)| Dirac Delta Function 2 Fourier Transform • For all values of t: |w(t)| ≤ EuEv Convolution 2 ⊲ Correlation • u(t − t0) is an exact multiple of v(t) ⇔ |w(t0)| = EuEv

Normalized cross-correlation: w(t) has a maximum absolute value of 1 √EuEv

• Cross-correlation is used to find the time shift, t0, at which two signals match and also how well they match. • Auto-correlation is the cross-correlation of a signal with itself: used to find the period of a signal (i.e. the time shift where it matches itself).

E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 13 / 13