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By Bellanger

This article emphasizes the complicated courting among adaptive filtering and sign research - highlighting stochastic techniques, sign representations and homes, analytical instruments, and implementation equipment. This moment version contains new chapters on adaptive options in communications and rotation-based algorithms. It offers functional functions in details, estimation, and circuit theories.

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Extra resources for Adaptive Digital Filters, 2nd Edition (Signal Processing and Communications)

Example text

Several natural signal exam- 54 Chapter 2 FIG. 14 Image signal: three lines of a black-and-white picture. ples, namely speech, radar, and image samples, have been selected to illustrate the theory. EXERCISES 1. Calculate the z-transform YR ðzÞ of the damped cosinusoid ( 0; n<0 yR ðnÞ ¼ eÀ0:1n cos n ; n 5 0 2 and show the poles in the complex plane. Give the signal energy spectrum and verify the energy relationship Z 1 X 1 2 yR ðnÞ ¼ Y ðzÞYR ðzÀ1 ÞzÀ1 dz Ey ¼ 2j jzj¼1 R n¼0 2. Give the coefficients, initial conditions, and diagram of the secondorder section which generates yR ðnÞ.

Rð2NÞ rð0Þ rð1Þ .. ð2:79Þ For p > N, the sequence rðpÞ is again generated recursively from the N preceding terms. 76) gives xðnÞ ¼ N X bi eðn À iÞ þ i¼0 N X i¼1 ai 1 X hj eðn À i À jÞ j¼0 and xðnÞ ¼ N X bi eðn À iÞ þ i¼0 1 X eðn À kÞ N X ð2:81Þ ai hkÀi i¼1 k¼1 Clearly, the impulse response coefficients can be computed recursively: h0 ¼ b0 ; hk ¼ 0 for k < 0 hk ¼ bk þ N X ai hkÀi ; ð2:82Þ k51 i¼1 In matrix form, for the N þ 1 first terms we have 32 2 h0 0 0 1 0 0 ÁÁÁ 0 76 h1 6 Àa1 1 0 Á Á Á 0 h 0 0 76 6 7 6 Àa2 6 Àa 1 Á Á Á 0 h h h 1 1 0 76 2 6 6 ..

48), but another approach is to use HðzÞ, since it is available, via the equation 2 N X 2 j2if Sð f Þ ¼ e hi e ð2:61Þ i¼0 An infinite impulse response (IIR) filter corresponds to an autoregressive (AR) model. 6 for the filter transfer function: ð2:66Þ 30 Chapter 2 FIG. 6 Spectrum of an AR signal. HðzÞ ¼ 1 ð1 þ 0:80zÀ1 þ 0:64zÀ2 Þð1 À 1:23zÀ1 þ 0:64zÀ2 Þ Since the spectrum of a real signal is symmetric about the zero frequency, only the band ½0; fs=2 Š, where fs is the sampling frequency, is represented.

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