z-Transform z-Transform The z-transform generalizes the Discrete-Time Fourier Transform (DTFT) for analyzing infinite-length signals and systems Very useful for designing and analyzing signal processing systems Properties are very similar to the DTFT with a few caveats The theme this week is less linear algebra (vectors spaces) and more polynomial algebra 2 Recall: DTFT Discrete-time Fourier transform (DTFT) 1 X X(!) = x[n] e−j!n; π ! < π − ≤ n=−∞ Z π d! x[n] = X(!) ej!n ; < n < −π 2π 1 1 The core \basis functions" of the DTFT are the sinusoids ej!n with arbitrary frequencies ! The sinusoids ej!n are eigenvectors of LTI systems for infinite-length signals (infinite Toeplitz matrices) 3 Recall: Complex Sinusoids are Eigenvectors of LTI Systems Fact: The eigenvectors of a Toeplitz matrix (LTI system) are the complex sinusoids s [n] = ej!n = cos(!n) + j sin(!n); π ! < π; < n < ! − ≤ − 1 1 λω cos(ωn) 4 cos(ωn) 2 1 0 0 −1 −8 −6 −4 −2 0 2 4 6 −2 n −4 −8 −6 −4 −2 0 2 4 6 n s! λ!s! H λω sin(ωn) 4 sin(ωn) 2 1 0 0 −1 −8 −6 −4 −2 0 2 4 6 −2 n −4 −8 −6 −4 −2 0 2 4 6 n 4 Recall: Eigenvalues of LTI Systems The eigenvalue λ! C corresponding to the sinusoid eigenvector s! is called the frequency 2 response at frequency ! since it measures how the system \responds" to sk 1 X λ = h[n] e−!n = h; s = H(!)(DTFT of h) ! h !i n=−∞ λω cos(ωn) 4 Recall propertiescos(ωn) of the inner product: λ! grows/shrinks as h2 and s! become more/less similar 1 0 0 −1 −8 −6 −4 −2 0 2 4 6 −2 n −4 −8 −6 −4 −2 0 2 4 6 n λω sin(ωn) s! λ!s! 4 H sin(ωn) 2 1 0 0 −1 −8 −6 −4 −2 0 2 4 6 −2 n −4 −8 −6 −4 −2 0 2 4 6 n 5 Eigendecomposition and Diagonalization of an LTI System x y H 1 X y[n] = x[n] h[n] = h[n m] x[m] ∗ − m=−∞ While we can't explicitly display the infinitely large matrices involved, we can use the DTFT to \diagonalize" an LTI system Taking the DTFTs of x and h 1 1 X X X(!) = x[n] e−!n;H(!) = h[n] e−!n m=−∞ m=−∞ we have that Y (!) = X(!)H(!) 6 Recall: Complex Exponential n zn = z ej!n = z n ej!n = z n (cos(!n) + j sin(!n)) j j j j j j z n is a real exponential envelope (an with a = z ) j j j j ej!n is a complex sinusoid z < 1 z > 1 j j j j Re(zn), z < 1 Re(zn), z > 1 4 | | | | 2 2 0 0 −2 −2 −4 −4 −15 −10 −5 0 5 10 15 −15 −10 −5 0 5 10 15 n n Im(z n), z < 1 Im(z n), z > 1 | | | | 4 2 2 0 0 −2 −2 −4 −15 −10 −5 0 5 10 15 −15 −10 −5 0 5 10 15 n n 7 Complex Exponentials are Eigenvectors of LTI Systems Fact: A more general set of eigenvectors of a Toeplitz matrix (LTI system) are the complex exponentials zn, z C 2 n Re(λzz ), z < 1 n | | Re(z ), z < 1 5 4 | | 2 0 0 −2 −4 −15 −10 −5 0 5 10 15 −5 n −15 −10 −5 0 5 10 15 n n n z λzz n H Im(λzz ), z < 1 10 | | Im(z n), z < 1 4 | | 5 2 0 0 −2 −15 −10 −5 0 5 10 15 n −5 −15 −10 −5 0 5 10 15 n 8 Proof: Complex Exponentials are Eigenvectors of LTI Systems n n z h λzz Prove that complex exponentials are the eigenvectors of LTI systems simply by computing the convolution with input zn 1 1 X X zn h[n] = zn−m h[m] = zn z−m h[m] ∗ m=−∞ m=−∞ 1 ! X = h[m] z−m zn m=−∞ n = λz z X 9 Eigenvalues of LTI Systems n The eigenvalue λz C corresponding to the complex exponential eigenvector z is called the 2 transfer function; it measures how the system \transfers" the input zn to the output 1 X λ = h[n] z−n = h[n]; zn = H(z) z h i n=−∞ n Re(λzz ), z < 1 n | | Re(z ), z < 1 5 n 4 Recall properties| | of the inner product: λz grows/shrinks as h[n] and z become more/less 2 0 0 −2 similar −4 −15 −10 −5 0 5 10 15 −5 n −15 −10 −5 0 5 10 15 n n Im(λzz ), z < 1 10 | | Im(z n), z < 1 4 | | 5 2 n n 0 λ z z z 0 −2 H −15 −10 −5 0 5 10 15 n −5 −15 −10 −5 0 5 10 15 n 10 z-Transform Define the (bilateral) forward z-transform of x[n] as 1 X X(z) = x[n] z−n n=−∞ The core \basis functions" of the z-transform are the complex exponentials zn with arbitrary z C; these are the eigenvectors of LTI systems for infinite-length signals 2 X(z) = x[n]; zn measures the similarity of x[n] to zn (analysis) h i Notation abuse alert: We use X( ) to represent both the DTFT X(!) and the z-transform · X(z); they are, in fact, intimately related j! XDTFT(!) = X (z) j! = X (e ) z jz=e z (Note how this elegantly produces a 2π-periodic DTFT) 11 z-Transform as a Function 1 X X(z) = x[n] z−n n=−∞ X(z) is a complex-valued function of a complex variable: X(z) C; z C 2 2 12 Eigendecomposition and Diagonalization of an LTI System x y H 1 X y[n] = x[n] h[n] = h[n m] x[m] ∗ − m=−∞ While we can't explicitly display the infinitely large matrices involved, we can use the z-transform to \diagonalize" an LTI system Taking the z-transforms of x and h 1 1 X X X(z) = x[n] z−n;H(z) = h[n] z−n n=−∞ n=−∞ we have that Y (z) = X(z) H(z) 13 Proof: Eigendecomposition and Diagonalization of an LTI System x h y Compute the z-transform of the result of the convolution of x and h (Note: we are a little cavalier about exchanging the infinite sums below) 1 1 1 ! X X X Y (z) = y[n] z−n = x[m] h[n m] z−n − n=−∞ n=−∞ m=−∞ 1 1 ! X X = x[m] h[n m] z−n (let r = n m) − − m=−∞ n=−∞ 1 1 ! 1 ! 1 ! X X X X = x[m] h[r] z−r−m = x[m]z−m h[r] z−r m=−∞ r=−∞ m=−∞ r=−∞ = X(z) H(z) X 14 Summary Complex exponentials zn are the eigenfunctions of LTI systems for infinite-length signals (Toeplitz matrices) Forward z-transform 1 X X(z) = x[n] z−n n=−∞ Transfer function H(z) equals the z-transform of the impulse response h[n] Diagonalization by eigendecomposition implies Y (z) = X(z) H(z) DTFT is a special case of the z-transform (values on the unit circle in the complex z-plane) j! XDTFT(!) = X (z) j! = X (e ) z jz=e z (Note how this elegantly produces a 2π-periodic DTFT) 15 z-Transform Region of Convergence z-Transform The forward z-transform of the signal x[n] 1 X X(z) = x[n] z−n n=−∞ We have been a bit cavalier in our development regarding the convergence of the infinite sum that defines X(z) Work out two examples to expose the issues 2 Example 1: z-Transform of αnu[n] n Signal x1[n] = α u[n], α C (causal signal) 2 Example for α = 0:8 n x1[n] = α u[n], α = 0.8 1 0.5 0 −8 −6 −4 −2 0 2 4 6 8 n The forward z-transform of x1[n] 1 1 1 X X X 1 z X (z) = x [n] z−n = αn z−n = (α z−1)n = = 1 1 1 α z−1 z α n=−∞ n=0 n=0 − − Important: We can apply the geometric sum formula only when α z−1 < 1 or z > α j j j j j j 3 Region of Convergence Given a time signal x[n], the region of convergence (ROC) of its z-transform X(z) is the set of z C such that X(z) converges, that is, the set of z C such 2 2 that x[n] z−n is absolutely summable 1 DEFINITION X x[n] z−n < j j 1 n=−∞ n 1 Example: For x1[n] = α u[n], α C, the ROC of X1(z) = −1 is all z such that z > α 2 1−α z j j j j 4 Example 2: z-Transform of αnu[ n 1] − − − n Signal x2[n] = α u[ n 1], α C (anti-causal signal) − − − 2 Example for α = 0:8 −n x2[n] = α u[ n 1], α = 0.8 0 − − − −5 −8 −6 −4 −2 0 2 4 6 8 n The forward z-transform of x2[n] 1 −1 1 1 X X X X X (z) = x [n] z−n = αn z−n = α−n zn = (α−1 z)n + 1 2 2 − − − n=−∞ n=−∞ n=1 n=0 1 1 1 α−1 z α−1z 1 z = − + 1 = − + − = − = = 1 α−1 z 1 α−1 z 1 α−1 z 1 α−1 z 1 α z−1 z α − − − − − − ROC: α−1 z < 1 or z < α j j j j j j 5 Example Summary x [n] = αnu[n] x [n] = αnu[ n 1] 1 2 − − − causal signal anti-causal signal 1 0 0.5 −5 0 −8 −6 −4 −2 0 2 4 6 8 −8 −6 −4 −2 0 2 4 6 8 n n 1 X (z) = = X (z) 1 1 α z−1 2 − ROC ROC z > α z < α j j j j j j j j 6 ROC: What We've Learned So Far The ROC is important • A z-transform converges only for certain values of z and does not exist for other values of z • z-transforms are non-unique without it You must always state the ROC when you state a z-transform 7 Example 3: z-Transform of αnu[n] αnu[ n 1] 1 − 2 − − n n Signal x3[n] = α u[n] α u[ n 1], α1; α2 C 1 − 2 − − 2 Example for α1 = 0:8 and α2 = 1:2 x3[n] 1 0 −1 −8 −6 −4 −2 0 2 4 6 8 n The forward z-transform of the signal x3[n] 1 1 1 X X X X (z) = x [n] z−n = αnu[n] z−n αnu[ n 1] z−n 3 3 1 − 2 − − n=−∞ n=−∞ n=−∞ 1 1 = + 1 α z−1 1 α z−1 − 1 − 2 ROC: z > α and z < α j j j 1j j j j 2j 8 Example 3: A Tale of Two ROCs n n Signal x3[n] = α u[n] α u[ n 1], α1; α2 C 1 − 2 − − 2 z-transform 1 1 X (z) = + ; α < z < α 3 1 α z−1 1 α z−1 j 1j j j j 2j − 1 − 2 Case 1: α < α j 1j j 2j Case 2: α > α j 1j j 2j 9 Properties of the ROC The ROC is a connected annulus (doughnut) in the z-plane centered on the origin z = 0; that is, the ROC is of the form r < z < r 1 j j 2 If x[n] has finite duration, then the ROC is the entire z-plane (except possibly z = 0 or z = ) 1 If x[n] is causal, then the ROC is the outside of a disk If x[n] is anti-causal, then the ROC is the inside of a disk If x[n] is two-sided (neither causal nor anti-causal), then either the ROC is the inside of an annulus or the z-transform does not converge The ROC is a connected region of the z-plane 10 Summary The region of convergence (ROC) of a z-transform is the set of z C such that it converges 2 The ROC is important • A z-transform converges for only certain values of z and does not exist for other values of z • z-transforms are non-unique without it • Always state the ROC when you state a z-transform The ROC is a connected annulus (doughnut) in the z-plane centered on the origin z = 0; that is, the ROC is of the form r < z < r 1 j j 2 • If x[n] has finite duration, then the ROC is the entire z-plane (except possibly z = 0 or z = 1) • If x[n] is causal, then the ROC is the outside of a disk • If x[n] is anti-causal, then the ROC is the inside of a disk 11 z-Transform Transfer Function Poles and Zeros Table of Contents Lecture in two parts: • Part 1: Transfer Function • Part 2: Poles and Zeros 2 z-Transform Transfer Function A General Class of LTI Systems The most general class of practical causal LTI systems (for infinite-length