You can cancel the nonlinear phase effects of an IIR filter by following the process shown in Figure 13-31(a). The y(n) output will be a filtered version of x(n) with no filter-induced phase distortion. The same IIR filter is used twice in this scheme, and the time reversal step is a straight left-right flipping of a time-domain sequence. Consider the following. If some spectral component in x(n) has an arbitrary phase of a degrees, and the first filter induces a phase shift of –b degrees, that spectral component's phase at node A will be a–b degrees. The first time reversal step will conjugate that phase and induce an additional phase shift of –q degrees. (Appendix C explains this effect.) Consequently, the component's phase at node B will be –a+b–q degrees. The second filter's phase shift of –b degrees yields a phase of –a–q degrees at node C. The final time reversal step (often omitted in literary descriptions of this zero-phase filtering process) will conjugate that phase and again induce an additional phase shift of –q degrees. Thankfully, the spectral component's phase in y(n) will be a+q–q = a degrees, the same phase as in x(n). This property yields an overall filter whose phase response is zero degrees over the entire frequency range.

Figure 13-31. Two, equivalent, zero-phase filtering techniques.

An equivalent zero-phase filter is presented in Figure 13-31(b). Of course, these methods of zero-phase filtering cannot be performed in real time because we can't reverse the flow of time (at least not in our universe). This filtering is a block processing, or off-line process, such as filtering an audio sound file on a computer. We must have all the time samples available before we start processing. The initial time reversal in Figure 13-31(b) illustrates this restriction.

There will be filter transient effects at the beginning and end of the filtered sequences. If transient effects are bothersome in a given application, consider discarding L samples from the beginning and end of the final y(n) time sequence, where L is 4 (or 5) times the order of the IIR filter.

By the way, the final peak-to-peak passband ripple (in dB) of this zero-phase filtering process will be twice the peak-to-peak passband ripple of the single IIR filter. The final stopband attenuation will also be double that of the single filter.

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Chapter One. Discrete Sequences and Systems

Chapter Two. Periodic Sampling

Chapter Three. The Discrete Fourier Transform

Chapter Four. The Fast Fourier Transform

Chapter Five. Finite Impulse Response Filters

Chapter Six. Infinite Impulse Response Filters

Chapter Seven. Specialized Lowpass FIR Filters

Chapter Eight. Quadrature Signals

Chapter Nine. The Discrete Hilbert Transform

Chapter Ten. Sample Rate Conversion

Chapter Eleven. Signal Averaging

Chapter Twelve. Digital Data Formats and Their Effects

Chapter Thirteen. Digital Signal Processing Tricks

Appendix A. The Arithmetic of Complex Numbers

Appendix B. Closed Form of a Geometric Series

Appendix C. Time Reversal and the DFT

Appendix D. Mean, Variance, and Standard Deviation

Appendix E. Decibels (dB and dBm)

Appendix F. Digital Filter Terminology

Appendix G. Frequency Sampling Filter Derivations

Appendix H. Frequency Sampling Filter Design Tables

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Understanding Digital Signal Processing
Understanding Digital Signal Processing (2nd Edition)
ISBN: 0131089897
EAN: 2147483647
Year: 2004
Pages: 183
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