Lecture 4: FT Pairs, Random Signals and z-transform
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1 EE518 Digital Signal Processing University of Washington Autumn 2001 Dept. of Electrical Engineering Lecture 4: T Pairs, Rom Signals z-transform Wed., Oct. 10, 2001 Prof: J. Bilmes <bilmes@ee.washington.edu> TA: Mingzhou Song <msong@u.washington.edu> 4.1 ourier Transform Theorems Let the T of x[n] y[n] be X(e jω ) Y (e jω ), respectively. Linearity Delay Parseval s Theorem ax[n] + by[n] ax(e jω ) + by (e jω ) (4.1) x[n n d ] e jωn d X(e jω ) (4.2) x[n] 2 1 π X(e jω ) 2 dω (4.3) n 2π π Convolution (important, so we spend more time on it) x[n] y[n] X(e jω )Y (e jω ) (4.4) Convolutoin in time corresponds to multiplication in frequency. Ex: Why this must hold in terms of LTI systems? (eigenfunctions) When x[n] e jωn, y[n] H(e jω )e jωn. H(e jω ) is x[n] LTI h y[n] x[n] * h[n] igure 4.1: An LTI system. an eigenvalue e jωn is an eigenfunction. Consider a weighted complex sinusoid, so the output will be x[n] ae jωn X(e jω )e jωn y[n] H(e jω )X(e jω )e jωn where H(e jω ) is a new weight from the LTI system X(e jω ) is the original weight. Consider x[n] k k a k e jω kn X(e jω k )e jω kn 4-1
2 4-2 j ω j ω X( e ) W( e ) π π 0 π 2π 2π π 0 π 2π igure 4.2: Periodic convolution in frequency. lip entire W(e jω ), but take sweeping only over one period. Then by linearity of the system y[n] h[n] x[n] k H(e jω k )X(e jω k )e jω kn inally, in the limits of an uncountably infinite number of weights complex exponentials, x[n] 1 2π So H(e jω )X(e jω ) must be the ourier transform of y[n]. Time Reversal If x[n] X(e jω ), π π X(e jω )e jωn dω y[n] 1 π H(e jω )X(e jω )e jωn dω 2π π x[ n] X(e jω ) If x[n] is real for all n, then X(e jω ) X (e jω ), then it follows It means x[ n] has the same magnitude of T with x[n]. Differential in requency If x[n] X(e jω ), then x[ n] X (e jω ) nx[n] j dx(e jω ) dω So by multiplying in time domain by n, we get differential in frequency. Modulation (Windowing) Theorem If then x[n] X(e jω ) y[n] x[n]w[n] 1 2π w[n] W(e jω ) π π X(e jθ )W(e j(ω θ) )dθ So pointwise multiplication in time corresponds to periodic convolution in frequency. Periodic convolution is calculated across one period of the periodic signals, as shown in ig 4.2. Ex: ind the frequency response H(e jω ) of the LTI system with input/output relationship given by the following difference equation: y[n] 1 y[n 1] x[n] + 2x[n 1] + x[n 2] 2
3 4-3 Take T of both sides: Y (e jω ) 1 2 Y (e jω )e jω X(e jω ) + 2X(e jω )e jω + X(e jω )e j2ω Note: By y[n] h[n] x[n] Y (e jω ) H(e jω )X(e jω ), H(e jω ) Y (e jω ) X(e jω ) So just divide the T of output by the T of input (under certain conditions we will discuss later, assuming the T exists.) So for this example H(e jω ) Y (e jω ) X(e jω ) 1 + 2e jω + e j2ω 1 1 2e jω Ex: Consider system with complex response h[n] ( j 2 )n u[n] j 1 with input x[n] cos(πn)u[n]. ind y[n]. y[n] x[n] h[n] k k n k0 h[k]x[n k] [ ( j ] 2 )k u[k] [cos(π(n k))u[n k]] ( j 2 )k cos(π(n k)) ( 1) n 1 ( j 2 )n j 2 Steady State Response (what happens when n gets big) Since ( j 2 )n 0 as n ( 1)n lim y[n] n 1 + j cos(πn) j 2 So T doesn t always help us here, because T of the steady state response does not exist. 4.2 Rom Processes Before x[n] was a single sequence. Now x[n] might be rom. The sequence might take on different values with different probabilities. Question: Why use rom processes? Why not just use deterministic signal modeling? Answer 1: don t have good model (e.x., linear constant coefficient difference equations) for all signals. Instead, we say things are rom, the process has certain properties. Types of properties 1. amily of probability distribution used for the process, e.g., Gaussian, Poisson, Types of dependencies in processes
4 Values of average of proportions, e.g., the mean or variance, covariance,... Answer 2: They are very useful for studying the efficient quantization. Note: Rom processes are not absolute or square summable, so we can t find T for them in general. But certain properties (average) of processes are square summable, so we can find T of that. Here we give some loose definitions (sufficing for our purposes) Definition 4.1 (Rom Process). x[n] is a rom process if each of x[n] is a rom variable, there exists a function such that x[n],x[n+n1 ],,x[n+n M ](v 0,v 1,,v M ) P(x[n] < v 0,x[n + n 1 ] < v 1,,x[n + n M ] < v M ) (4.5) where P( ) is a probability function. Definition 4.2 (Stationary). x[n] is stationary if k,m,v 0,v 1,,v M,n, x[n],x[n+n1 ],,x[n+n M ](v 0,v 1,,v M ) x[n+k],x[n+k+n1 ],,x[n+k+n M ](v 0,v 1,,v M ) (4.6) i.e., shifting the time axis does not change any of the statistical properties of the process. Definition 4.3 (Mean). where p( ) is probability density function. or a stationary process, m x[n] m x[n+k] m x, k m x[n] E[x[n]] xp x[n] (x,n)dx (4.7) Definition 4.4 (Autocorrelation). φ xx [m] E[x[n + m]x [n]] (4.8) Φ xx (e jω ) {φ xx [m]} (4.9) Definition 4.5 (Autocovariance). γ xx [m] E[(x[n + m] m x )(x[n] m x ) ] (4.10) Definition 4.6 (Cross-Correlation). Γ xx (e jω ) {γ xx [m]} (4.11) φ xy [m] E[x[n + m]y [n]] (4.12) Φ xy (e jω ) {φ xy [m]} (4.13) Definition 4.7 (Cross-Covariance). γ xy [m] E[(x[n + m] m x )(y[n] m y ) ] (4.14) Γ xy (e jω ) {γ xy [m]} (4.15) Note: Properties of r.v. (rom variable) describe general properties. Process is rom, so these might display average energy at different frequencies.
5 4-5 When rom processes are applied to an LTI system, what happens? if x[n] is a rom process, so does y[n]. if x[n] is stationary, then y[n] k m y [n] E[y[n]] m y [n] m x k h[k]x[n k] k h[k]e{x[n k]} h[k] a constant m x H(e j0 ) since T of h[n] is H(e jω ) k h[k]e jωk, the autocorrelation of the output is 1 where φ yy [n,n + m] E[y[n]y[n + m]] { } E h[k]h[r]x[n k]x[n + m r] k k l l k r r h[k] h[k]h[r]e {x[n k]x[n + m r]} r φ xx [m l] h[r]φ xx [m + k r] k φ xx [m l]c hh [l] c hh [l] is deterministic autocorrelation sequence. Note if so, Also, note convolution above: φ yy [m] k ( x[n] is stationary) h[k]h[l + k] ( Let l r k) h[k]h[l + k] h[l] h[ l] (4.16) h[n] H(e jω ) h[ n] H(e jω ) H (e jω ) (if h[n] is real.) h[n] h[ n] H(e jω )H (e jω ) H(e jω ) 2 l so, this tells us how to interprete Φ xx (e jω ). Question: Why? φ xx [m l]c hh [l] Φ yy (e jω ) Φ xx (e jω )C hh (e jω ) Answer: Let H(e jω ) be an ideal b pass filter, { H(e jω 1 ωa < ω < ω ) b 0 else then C hh (e jω ) H(e jω )H (e jω ) is b pass filter also. Φ yy (e jω ) H(e jω ) 2 Φ xx (e jω ) so it tells us, if ω a ω b is small, roughly the spectral content of the rom signal x[n]. Therefore Φ xx (e jω ) tells us about spectral content. 1 Assume x[n] h[n] are real.
6 z-transform The ourier transform of x[n] is Definition 4.8 (z-transform). where z is a complex continuous variable. Notation X(e jω ) X(z) n n Z x[n] X(z) x[n]e jωn x[n]z n (4.17) Definition 4.9 (One-sided, Unilateral z-transform). Note Consider X(z) n0 (ω) X(z) ze jω X(e jω ) when the sum exists for these values of z at z 1. In general So X(z) X(re jω ) n z re jω x[n](re jω ) n x[n]z n (4.18) n n x[n]e jωn (x[n]r n )e jωn T {x[n]r n } X(z) is complex function of a complex variable defined on z-plane, shown in ig 4.3. Note X(e jω ) is evaluation of z-transform around unit circle. X(e jω ) ω0 X(z) z1, X(e jω ) ω π 2 X(z) z j. So ourier transform is unwrapping of the unit circle periodically into a line. Im unit circle ω ze j ω Re igure 4.3: z-plane
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