Foundations of Signal Processing Tables

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1 Foundations of Signal Processing Tables Martin Vetterli École Polytechnique Fédérale de Lausanne and University of California, Berkeley Jelena Kovačević Carnegie Mellon University Vivek K Goyal Massachusetts Institute of Technology August 8, 2012 Copyright (c) 2012 Martin Vetterli, Jelena Kovačević, and Vivek K Goyal. These materials are protected by copyright under the Attribution-NonCommercial-NoDerivs 3.0 Unported License from Creative Commons.

2 iisignal Processing: Foundations Cover photograph by Christiane Grimm, Geneva, Switzerland. Experimental set up by Prof. Libero Zuppiroli and Philippe Bugnon, Laboratory of Optoelectronics Molecular Materials, EPFL, Lausanne, Switzerland. The photograph captures an experiment first described by Isaac Newton in Opticks in 1730, explaining how the white light can be split into its color components and then resynthesized. It is a physical implementation of a white light decomposition into its Fourier components the colors of the rainbow, followed by a synthesis to recover the original. This experiment graphically summarizes the major theme of the book many signals can be split into essential components, and the signal s characteristics can be better understood by looking at its components; the process called analysis. These components can be used for processing the signal; more importantly, the signal can be perfectly recovered from those same components, through the process called synthesis.

3 Fundamental subspaces associated with matrix A The four fundamental subspaces associated with a real matrix A R M N. The matrix determines an orthogonal decomposition of R N into the row space of A and the null space of A; and an orthogonal decomposition of R M into the column space (range) of A and the left null space of A. The column and row spaces of A have the same dimension, which equals the rank of A. (Figure inspired by the cover of G. Strang. Introduction to Linear Algebra. 4th edition, 2009.) R N A R M row space R(A T ) column space R(A) A T 0 A T A 0 null space N(A) left null space N(A T ) R(A T ) N(A) dim(r(a T )) + dim(n(a)) = N R(A) N(A T ) dim(r(a)) + dim(n(a T )) = M Space Symbol Definition Dimension Column space (range) R(A) y C M y = Ax for some x C N } rank(a) Left null space N(A ) y C M A y = 0} M rank(a) dim(r(a))+dim(n(a )) = M Row space R(A ) x C N x = A y for some y C M } rank(a) Null space (kernel) N(A) x C N Ax = 0} N rank(a) dim(r(a ))+dim(n(a)) = N

4 Discrete-time processing concepts Concept Notation Infinite-length sequences Finite-length sequences Shift δ n 1 linear circular Sequence x n n Z n 0,1,...,N 1} vector LSI system h n n Z n 0,1,...,N 1} filter, impulse response operator Convolution h x x k h n k k Z k=0 x k h N,(n k) mod N Eigensequence v e jωn e j(2π/n)kn satisfies h v λ = λv λ h v ω = H(e jω )v ω h v k = H k v k invariant space S λ S ω = αe jωn } S k = αe j(2π/n)kn } α C,ω R α C,k Z Frequency response λ λ ω = H(e jω ) λ k = H k eigenvalue h ne jωn h ne j(2π/n)kn Fourier transform X DTFT DFT spectrum X(e jω ) = n Zx ne jωn X k = n Z n=0 x ne j(2π/n)kn n=0

5 Multirate discrete-time processing concepts Concept Expression Sampling factor 2 Input x n,x(z),x(e jω ) Downsampling by 2 y n = x 2n y = D 2 x [ ] Y(z) = (1/2) X(z 1/2 )+X( z 1/2 ) [ ] Y(e jω ) = (1/2) X(e jω/2 )+X(e j(ω 2π)/2 ) x Upsampling by 2 y n = n/2, n even; y = U 2 x Y(z) = X(z 2 ) Y(e jω ) = X(e j2ω ) Filtering with h y = D 2 H x [ ] & downsampling by 2 Y(z) = (1/2) 1 H(z 1/2 )X(z 1/2 )+H( z 1/2 )X( z 1/2 ) 2 [ ] Y(e jω ) = (1/2) H(e jω/2 )X(e jω/2 )+H(e j(ω 2π)/2 )X(e j(ω 2π)/2 ) Upsampling by 2 y = GU 2 x & filtering with g Y(z) = G(z)X(z 2 ) Y(e jω ) = G(e jω )X(e j2ω ) Sampling factor N Input x n,x(z),x(e jω ) Downsampling by N y n = x Nn y = D N x Upsampling by N y n = Y(z) = (1/N) X(ω k N z1/n ) k=0 Y(e jω ) = (1/N) X(e j(ω 2πk/N) ) k=0 x n/n, n/n Z; y = U N x Y(z) = X(z N ) Y(e jω ) = X(e jnω )

6 Kronecker delta sequence properties Kronecker delta sequence Normalization δ n = 1 Sifting n Z n0 nδ n = n Zx δ n0 nx n = x n0 n Z Shifting x n n δ n n0 = x n n0 Sampling x nδ n = x 0 δ n Restriction x nδ n = 1 0} x Dirac delta function properties Dirac delta function Normalization δ(t) dt = 1 Sifting x(t 0 t) δ(t) dt = x(t) δ(t 0 t) dt = x(t 0 ) Shifting x(t) t δ(t t 0 ) = x(t t 0 ) Sampling x(t) δ(t) = x(0) δ(t) Scaling δ(t/a) = a δ(t) for any nonzero a R

7 Ideal filters with unit-norm impulse response Ideal filters Time domain DTFT domain Ideal lowpass filter Ideal Nth-band filter Ideal halfband lowpass filter ω0 2π sinc(ω 0n/2) (1/ N) sinc(πn/n) (1/ 2) sinc(πn/2) 2π/ω0, 0, N, 0, 2, 0, ω ω 0 /2; otherwise. ω π/n; otherwise. ω π/2; otherwise. Unit-norm box and sinc functions/sequences Functions on the real line x(t), t R, x = 1 1/ t 0, t t 0 /2; Box ω0 Sinc 2π sinc(ω 0t/2) Periodic functions x(t), t [ T/2,T/2), x = 1 1/ t 0, t t 0 /2; Box k0 sinc(πk 0 t/t) Sinc T sinc(πt/t) Infinite-length sequences x n, n Z, x = 1 Box Sinc 1/ n 0, n (n 0 1)/2; ω0 2π sinc(ω 0n/2) Finite-length sequences x n, n 0,1,...,N 1}, x = 1 1/ n 0, n N/2 (n 0 1)/2; Box k0 sinc(πnk 0 /N) Sinc N sinc(πn/n) FT X(ω), ω R, X = 2π t0 sinc(ωt 0 /2) 2π/ω0, ω ω 0 /2; FS X k, k Z, X = 1/ T t0 T sinc(πkt 0/T) 1/ k 0 T, k (k 0 1)/2; DTFT X(e jω ), ω [ π,π), X = 2π sinc(ωn 0 /2) n0 sinc(ω/2) 2π/ω0, ω ω 0 /2; DFT X k, k 0,1,...,N 1}, X = N sinc(πn 0 k/n) n0 sinc(πk/n) N/k0, k N/2 (k 0 1)/2;

8 Energy and power concepts for sequences Deterministic sequences Energy spectral density A(e jω ) = X(e jω ) 2 WSS discrete-time stochastic processes Power spectral density A(e jω ) = E[x nx n k ]e jωk k Z Energy Power E = 1 π A(e jω )dω P = 1 π A(e jω )dω 2π π 2π π E = a 0 = n Z x n 2 P = a 0 = E[ x n 2 ] Orthogonality concepts for sequences Deterministic sequences WSS discrete-time stochastic processes Time c x,y,k = x n, y n k n = 0 c x,y,k = E[x nyn k ] = 0 Frequency C x,y(e jω ) = X(e jω )Y (e jω ) = 0 C x,y(e jω ) = 0

9 Deterministic discrete-time autocorrelation/crosscorrelation properties Domain Autocorrelation/Crosscorrelation Properties Sequences x n, y n Time a n k Z x kx k n a n = a n c n k Z x kyk n c x,y,n = c y,x, n DTFT A(e jω ) X(e jω ) 2 A(e jω ) = A (e jω ) C(e jω ) X(e jω )Y (e jω ) C x,y(e jω ) = C y,x (ejω ) z-transform A(z) X(z)X (z 1 ) A(z) = A (z 1 ) C(z) X(z)Y (z 1 ) C x,y(z) = C y,x (z 1 ) DFT A k X k 2 A k = A k mod N C k X k Yk C k = Cy,x, k mod N Real sequences x n, y n Time a n k Z x kx k n a n = a n c n k Z x ky k n c x,y,n = c y,x, n DTFT A(e jω ) X(e jω ) 2 A(e jω ) = A(e jω ) C(e jω ) X(e jω )Y(e jω ) C x,y(e jω ) = C y,x(e jω ) z-transform A(z) X(z)X(z 1 ) A(z) = A(z 1 ) C(z) X(z)Y(z 1 ) C x,y(z) = C y,x(z 1 ) DFT A k X k 2 A k = A k mod N C k X k Y k C k = C y,x, k mod N Vector of sequences [x 0,n ] T x 1,n Time A n a 0,n c 0,1,n c 1,0,n a 1,n [ ] [ ] DTFT A(e jω ) A 0 (e jω ) C 0,1 (e jω ) C 1,0 (e jω ) A 1 (e jω ) [ ] z-transform A(z) A 0 (z) C 0,1 (z) C 1,0 (z) A 1 (z) DFT A k A 0,k C 0,1,k C 1,0,k A 1,k [ ] A n = A n A n = A T n A(e jω ) = A (e jω ) A(e jω ) = A T (e jω ) A(z) = A (z 1 ) A(z) = A T (z 1 ) A k = A k mod N A k = A T k mod N

10 Discrete/continuous Fourier transform summary Transform Forward/Inverse Duality/Periodicity X(ω) = x(t)e jωt dt Fourier transform x(t) = 1 X(ω)e jωt dω 2π Fourier series X k = 1 T/2 x(t)e j(2π/t)kt dt T T/2 x(t) = X k e j(2π/t)kt k Z dual with DTFT x(t+t) = x(t) Discrete-time Fourier transform Discrete Fourier transform X(e jω ) = x ne jωn n Z x n = 1 π X(e jω )e jωn dω 2π π X k = x ne j(2π/n)kn n=0 x n = 1 X k e j(2π/n)kn N k=0 dual with Fourier series X(e j(ω+2π) ) = X(e jω )

11 Discrete Fourier transform (DFT) properties DFT properties Time domain DFT domain Basic properties Linearity αx n +βy n αx k +βy k Circular shift in time x (n n0 ) mod N W kn 0 N X k Circular shift in frequency W k 0n N x n X (k k0 ) mod N Circular time reversal x n mod N X k mod N Circular convolution in time (h x) n H k X k Circular convolution in frequency h nx n 1 N (H X) k Circular deterministic autocorrelation a n = x k x (k n) mod N A k = X k 2 k=0 N 1 Circular deterministic crosscorrelation c n = x k y(k n) mod N C k = X k Yk k=0 Parseval s equality x 2 = x n 2 = 1 X k 2 = 1 N N X 2 n=0 k=0 Related sequences Conjugate x n X k mod N Conjugate, time reversed x n mod N Xk Real part R(x n) (X k +X k mod N )/2 Imaginary part I(x n) (X k X k mod N )/(2j) Conjugate-symmetric part (x n +x n mod N )/2 R(X k) Conjugate-antisymmetric part (x n x n mod N )/(2j) I(X k) Symmetries for real x X conjugate symmetric X k = X k mod N Real part of X even R(X k ) = R(X k mod N ) Imaginary part of X odd I(X k ) = I(X k mod N ) Magnitude of X even X k = X k mod N Phase of X odd argx k = argx k mod N Common transform pairs Kronecker delta sequence δ n 1 Shifted Kronecker delta sequence δ (n n0 ) mod N W kn 0 N Constant sequence 1 Nδ k s Geometric sequence α n (1 αwn kn )/(1 αwk N ) Periodic sinc sequence k0 sinc(πnk 0 /N) N k, k0 N 2 k 0 1 ; 2 (ideal lowpass filter) N sinc(πn/n) 1 Box sequence n0, n N 2 n 0 1 ; sinc(πn 2 0 k/n) n0 sinc(πk/n)

12 Fourier series properties FS properties Time domain FS domain Basic properties Linearity αx(t)+βy(t) αx k +βy k Shift in time x(t t 0 ) e j(2π/t)kt 0 X k Shift in frequency e j(2π/t)k0t x(t) X(k k 0 ) Time reversal x( t) X k Differentiation d n x(t)/dt n t (j2πk/t) n X k Integration x(τ)dτ (T/(j2πk))X k, X 0 = 0 T/2 Circular convolution in time (h x)(t) T H k X k Convolution in frequency h(t)x(t) (H X) k T/2 Circular deterministic autocorrelation a(t) = x(τ)x (τ t)dτ A k = T X k 2 T/2 T/2 Circular deterministic crosscorrelation c(t) = x(τ)y (τ t)dτ C k = T X k Yk T/2 T/2 Parseval s equality x 2 = T/2 x(t) 2 dt = T X k 2 = T X 2 k Z Related functions Conjugate x (t) X k Conjugate, time reversed x ( t) Xk Real part R(x(t)) (X k +X k )/2 Imaginary part I(x(t)) (X k X k )/(2j) Conjugate-symmetric part (x(t)+x ( t))/2 R(X k ) Conjugate-antisymmetric part (x(t) x ( t))/(2j) I(X k ) Symmetries for real x X conjugate symmetric X k = X k Real part of X even R(X k ) = R(X k ) Imaginary part of X odd I(X k ) = I(X k ) Magnitude of X even X k = X k Phase of X odd argx k = argx k Common transform pairs Dirac comb Periodic sinc function (ideal lowpass filter) Box function (one period) Square wave (one period with T = 1) Triangle wave (one period with T = 1) Sawtooth wave (one period with T = 1) δ(t nt) 1/T n Z k0 sinc(πk 0 t/t) T sinc(πt/t) 1/ t0, t t 0 /2; 0, t 0 /2 < t T/2. 1, t [ 1/2,0); 1, t [0,1/2). 1 t, t 1/2. 2 2t, t 1/2. 1/ k0 T, k (k 0 1)/2; t0 T sinc(πkt 0/T) 2j/(πk), k odd; 0, k even. 1/4, k = 0; 1/(πk) 2, k odd; 0, k 0 even. 0, k = 0; j( 1) k /(πk), k 0.

13 z-transform properties z-transform properties Time domain z domain ROC ROC properties General seq. 0 r 1 < z < r 2 Finite-length seq. all z, except possibly 0, Right-sided seq. z > largest pole Left-sided seq. z < smallest pole BIBO stable z = 1 Basic properties Linearity αx n +βy n αx(z)+βy(z) ROC x ROC y Shift in time x n n0 z n 0 X(z) ROC x Scaling in time N 1 1 Downsampling x Nn X(WN k N ) (ROC x) 1/N k=0 xn/n, n/n Z; Upsampling X(z N ) (ROC x) N Scaling in z α n x n X(α 1 z) α ROC x Time reversal x n X(z 1 ) 1/ROC x Differentiation n k x n ( 1) k z k k X(z)/ z k ROC x m k = n k x n = ( 1) k k X(z)/ z k z=1 n Z Convolution in time (h x) n H(z)X(z) ROC h ROC x Deterministic autocorrelation a n = k Zx k x k n A(z) = X(z)X (z 1 ) ROC x 1/ROC x Deterministic crosscorrelation c n = k Zx k y k n C(z) = X(z)Y (z 1 ) 1/ROC x ROC y Related sequences Conjugate x n X (z ) ROC x Conjugate, time reversed x n X (z 1 ) 1/ROC x Real part R(x n) (X(z)+X (z ))/2 ROC x Imaginary part I(x n) (X(z) X (z ))/(2j) ROC x Symmetries for real x X conjugate symmetric X(z) = X (z ) Common transform pairs Kronecker delta sequence δ n 1 all z Shifted Kronecker delta sequence δ n n0 z n 0 all z, except possibly 0, Geometric sequence α n u n 1/(1 αz 1 ) z > α α n u n 1 z < α Arithmetic-geometric sequence nα n u n (αz 1 )/(1 αz 1 ) 2 z > α nα n u n 1 z < α

14 DTFT: Discrete-time Fourier transform properties DTFT properties Time domain DTFT domain Basic properties Linearity αx n +βy n αx(e jω )+βy(e jω ) Shift in time x n n0 e jωn 0 X(e jω ) Shift in frequency e jω0n x n X(e j(ω ω0) ) Scaling in time N 1 1 Downsampling x Nn X(e j(ω 2πk)/N ) N k=0 Upsampling x n/n, n = ln; 0, otherwise X(e jnω ) Time reversal x n X(e jω ) Differentiation in freq. ( jn) k k X(e jω ) x n ω k Moments m k = n k x n = ( j) k X(ejω ) ω n Z ω=0 Convolution in time (h x) n H(e jω )X(e jω ) 1 Circular convolution in frequency h nx n 2π (H X)(ejω ) Deterministic autocorrelation a n = x k x k n A(e jω ) = X(e jω ) 2 k Z Deterministic crosscorrelation c n = k Zx k yk n C(e jω ) = X(e jω )Y (e jω ) Parseval s equality x 2 = x n 2 = 1 π X(e jω ) 2 dω = 1 2π n Z π 2π X 2 Related sequences Conjugate x n X (e jω ) Conjugate, time reversed x n X (e jω ) Real part R(x n) (X(e jω )+X (e jω ))/2 Imaginary part I(x n) (X(e jω ) X (e jω ))/(2j) Conjugate-symmetric part (x n +x n)/2 R(X(e jω )) Conjugate-antisymmetric part (x n x n )/(2j) I(X(ejω )) Symmetries for real x X conjugate symmetric X(e jω ) = X (e jω ) Real part of X even R(X(e jω )) = R(X(e jω )) Imaginary part of X odd I(X(e jω )) = I(X(e jω )) Magnitude of X even X(e jω ) = X(e jω ) Phase of X odd argx(e jω ) = argx(e jω ) Common transform pairs Kronecker delta sequence δ n 1 Shifted Kronecker delta sequence δ n n0 e jωn 0 Constant sequence 1 2π δ(ω 2πk) k= Geometric sequence α n u n 1/(1 αe jω ), α < 1 Arithmetic-geometric sequence nα n u n αe jω /(1 αe jω ) 2, α < 1 Sinc sequence ω0 (ideal lowpass filter) 2π sinc(ω 2π/ω0, ω ω 0 /2; 0n/2) 1/ n0, n (n Box sequence 0 1)/2; sinc(ωn 0 /2) n0 sinc(ω/2)

15 Fourier transform properties FT properties Time domain FT domain Basic properties Linearity αx(t) + βy(t) αx(ω) + βy(ω) Shift in time x(t t 0 ) e jωt 0 X(ω) Shift in frequency e jω0t x(t) X(ω ω 0 ) Scaling in time and frequency x(αt) (1/α)X (ω/α) Time reversal x( t) X( ω) Differentiation in time d n x(t)/dt n (jω) n X(ω) Differentiation in frequency ( jt) n x(t) t d n X(ω)/dω n Integration in time x(τ)dτ X(ω)/jω, X(0) = 0 Moments m k = t k x(t)dt = (j) kdk X(ω) dω k ω=0 Convolution in time (h x)(t) H(ω) X(ω) Convolution in frequency 1 h(t) x(t) (H X)(ω) 2π Deterministic autocorrelation a(t) = x(τ)x (τ t)dτ A(ω) = X(ω) 2 Deterministic crosscorrelation c(t) = x(τ)y (τ t)dτ C(ω) = X(ω)Y (ω) Parseval s equality x 2 = x(t) 2 dt = 1 X(ω) 2 dω = 1 2π 2π X 2 Related functions Conjugate x (t) X ( ω) Conjugate, time reversed x ( t) X (ω) Real part R(x(t)) (X(ω) +X ( ω))/2 Imaginary part I(x(t)) (X(ω) X ( ω))/(2j) Conjugate-symmetric part (x(t)+x ( t))/2 R(X(ω)) Conjugate-antisymmetric part (x(t) x ( t))/(2j) I(X(ω)) Symmetries for real x X conjugate symmetric Real part of X even Imaginary part of X odd Magnitude of X even Phase of X odd X(ω) = X ( ω) R(X(ω)) = R(X( ω)) I(X(ω)) = I(X( ω)) X(ω) = X( ω) argx(ω) = argx( ω) Common transform pairs Dirac delta function δ(t) 1 Shifted Dirac delta function δ(t t 0 ) e jω 0t Dirac comb n Zδ(t nt) (2π/T) δ(ω (2π/T)k) k Z Constant function 1 2π δ(ω) Exponential function e α t (2α)/(ω 2 +α 2 ) Gaussian function e αt2 π/αe ω 2 /α Sinc function ω0 (ideal lowpass filter) 2π sinc(ω 2π/ω0, ω ω 0 /2; 0t/2) 1/ t0, t t Box function 0 /2; t0 sinc(ωt 0 /2) 1 t, t < 1; Triangle function (sinc(ω/2))