电信研究所通信组.ppt

1、2000/5/18,1,Signal Processing for Communications Part II: Spectral Factorization and Its Applications,電信研究所通信組 李宇旼 教授,2000/5/18,2,Outline,Signal Processing for Communications Spectral Factorization Application: Whitening,2000/5/18,3,Signal Processing for Communications (1),“Classical” Signal Process

2、ing Digital Signal Processing Speech Signal Processing Image Processing Data Compression Signal Processing for Communications Processing of “communication signals” Equalization Synchronization Channel Estimation Channel coding/decoding Advanced modulation,2000/5/18,4,Signal Processing for Communicat

3、ions (2),A communication system communicates information from source to destination. Signal processing is necessary for making the communication as reliably as possible,2000/5/18,5,Signal Processing for Communications (3),2000/5/18,6,Signal Processing for Communications (4),2000/5/18,7,Signal Proces

4、sing for Communications (5),2000/5/18,8,Signal Processing for Communications (6),Purpose of receiver baseband signal processing: to recover the information in the distorted, noisy received signal. Typical processing: Filtering Sampling Synchronization Channel estimation Channel equalizaton Channel d

5、ecoding Source decoding,2000/5/18,9,Signal Processing for Communications (7),Simplified baseband processing Both whitening and equalization makes use of spectral factorization.,2000/5/18,10,Spectral Factorization (1),Problem Formulation (Time-Domain Description): Let hn be a discrete-time signal who

6、se discrete-time Fourier transform H(ej) is real and positive for all . Can we find h+n and h-n such that: h = h+ h-; h+n is stable causal, and has a causal inverse; and h-n is stable anti-causal and has an anti-causal inverse.,2000/5/18,11,Spectral Factorization (2),Problem Formulation (Freq-Domain

7、 Description): Let hn be a discrete-time signal whose discrete-time Fourier transform H(ej) is real and positive for all . Can we find H+(ej) and H-(ej) such that: H(ej) = H+(ej) H-(ej); H+(ej) represents a stable causal signal and H+(ej)-1 also represents a stable causal signal; and H-(ej) represen

8、ts a stable anti-causal signal and H-(ej)-1 also represents a stable anti-causal signal,2000/5/18,12,Spectral Factorization (3),Solution (General): We can find h+n and h-n provided that hn satisfies the certain conditions. The procedure for finding h+n and h-n is complicated in general. However, spe

9、ctral factorization is easy for signals with rational z-transform.,2000/5/18,13,Spectral Factorization (4),Quick review of z-transforms: Definition: , zC Rational z-transform: , where A(z) and B(z) are polynomials in z-1. Poles and Zeroes for rational z-transform: z0 is a zero if B(z0)=0. Similarly,

10、 p0 is a pole if A(p0)=0.,2000/5/18,14,Spectral Factorization (5),Stability and causality A signal is causal (anti-causal) if hn=0 for n0). A signal is stable if |hn| is bounded. All poles of a stable causal signal must be inside the unit circle. All poles of a stable anti-causal signal must be outs

11、ide the unit circle. Positions of the zeros do not matter.,X,X,X,O,O,2000/5/18,15,Spectral Factorization (6),Inverse of a signal The inverse of hn is a stable signal gn such that h g = . The z-transform of gn is H(z)-1 Poles (zeroes) of hn are zeroes (poles) of gn, and vice versa.,2000/5/18,16,Spect

12、ral Factorization (7),Minimum phase and maximum phase A signal is minimum (maximum) phase if all poles and zeroes are inside (outside) the unit circle. Minimum (maximum) phase signals are stable, causal (anti-causal), and has causal (anti-causal) inverses. Relationship to discrete-time Fourier trans

13、forms: H(ej) is equal to H(z) evaluated on the unit circle.,2000/5/18,17,Spectral Factorization (8),Spectral factorization for signals with rational z-transforms: Suppose that H(z) is rational and H(z) 0 when z is on the unit circle. We would like to find a minimum phase signal h+n and a maximum pha

14、se signal h-n such that h = h+ h- We want to find H+(z) and H-(z) such that: H(z) = H+(z) H-(z); All poles and zeroes of H+(z) are inside the unit circle; and All poles and zeroes of H-(z) are outside the unit circle.,2000/5/18,18,Spectral Factorization (9),Fact: Suppose that H(z) is rational and is

15、 real on the unit circle, then if z0 is a pole (zero) of H(z), (z0-1) * must also a pole (zero) of H(z). Poles (zeroes) of H(z) come in pairs if H(z) is real on the unit circle. If H(z) has a pole (zero) inside the unit circle, there must also be a pole (zero) outside the unit circle!,2000/5/18,19,S

16、pectral Factorization (10),Therefore, let H+(z) correspond to the poles and zeroes of H(z) inside the unit circle, and H-(z) correspond to the poles and zeroes of H(z) outside the unit circle.,2000/5/18,20,Spectral Factorization (11),Example:,2000/5/18,21,Spectral Factorization (12),Solution:,2000/5

17、/18,22,Spectral Factorization (13),Note that: H-(z) = H+(z-*)*, therefore h-n = (h+-n)* |H+(ej)| = |H-(ej)| = |H(ej)|,2000/5/18,23,Whitening Filter (1),Some definitions Random Signal: hn is a random signal (random process) if hn is a random variable for each n. A random process is an indexed set of

18、random variables. Mean function: Mhj Ehj Autocorrelation function Rhj,k Ehjh*k,2000/5/18,24,Whitening Filter (2),Wide-sense Stationary (WSS): hn is WSS if Mhj = constant (independent of time j) Rhj,k depends only on j-k. For a WSS random process, the autocorrelation function can be defined as Rhk =

19、Ehjh*j-k White: a WSS random process is white if Mhj = 0 Rhk = 2k hj and hk are uncorrelated if j k. A WSS random process is colored if it is not white.,2000/5/18,25,Whitening Filter (3),Properties of autocorrelation function Rh0 = Ehjh*j is real Rh-k = Ehjh*j+k = Eh*j+khj = Ehjh*j-k* = Rh*k Let Sh(

20、ej) = discrete-time Fourier transform of Rhk Sh(ej) is called the power spectral density of h Sh(ej) is real and positive.,2000/5/18,26,Whitening Filter (4),LTI Filtering a WSS random process If h is a WSS random process, then: g is also a WSS random process Rgn = Rhn f n (f -n)* Sg(z) = Sh(z)F(z)F

21、*(z-*) Sg(ej) = Sh(ej) |F(ej)|2,2000/5/18,27,Whitening Filter (5),The whitening problem: given a WSS random process hn whose autocorrelation function is known. Can we find a LTI stable causal filter such that when hn is the input, the output is white?,Stable Causal LTI filter f n = ?,hn,gn,(colored),(white),2000/5/18,28,Whitening Filter (6),Whitening filters are used in