Fourier Series and Transforms ⊲ Revision Lecture The Basic Idea Real v Complex Series v Transform Fourier Analysis Power Conservation Gibbs Phenomenon Coefficient Decay Rate Periodic Extension Fourier Series and Transforms Dirac Delta Function Fourier Transform Revision Lecture Convolution Correlation E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 1 / 13 The Basic Idea Fourier Series and Transforms Periodic signals can be written as a sum of sine and cosine waves: 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) Dirac Delta Function 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 Frequency: F = T . The n harmonic is at frequency nF . Some waveforms need infinitely many harmonics (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 pi U 0 and phase 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 amplitudes {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∞= P|Un| −∞ 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 derivative 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 range, [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 means 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 Signal” 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.