Here's an interesting technique for improving the stopband attenuation of a digital under the condition that we're unable, for whatever reason, to modify that filter's coefficients. Actually, we can a filter's double stopband attenuation by cascading the filter with itself. This works, as shown in Figure 13-32(a), where the frequency magnitude response of a single filter is a dashed curve |H(m)| and the response of the filter cascaded with itself is represented by solid curve |H2(m)|. The problem with this simple cascade idea is that it also doubles the passband peak-to-peak ripple as shown in Figure 13-32(b). The frequency axis in Figure 13-32 is normalized such that a value of 0.5 represents half the signal sample rate.

Figure 13-32. Frequency magnitude responses of a single filter and that filter cascaded with itself: (a) full response; (b) passband detail.

Well, there's a better scheme for improving the stopband attenuation performance of a filter and avoiding passband ripple degradation without actually changing the filter's coefficients. The technique is called filter sharpening[34], and is shown as Hs in Figure 13-33.

Figure 13-33. Filter sharpening process.

The delay element in Figure 13-33 is equal to (N–1)/2 samples where N is the number of h(k) coefficients, the unit-impulse response length, in the original H(m) FIR filter. Using the sharpening process results in the improved |Hs(m)| filter performance shown as the solid curve in Figure 13-34, where we see the increased stopband attenuation and reduced passband ripple beyond that afforded by the original H(m) filter. Because of the delayed time-alignment constraint, filter sharpening is not applicable to filters having non-constant group delay, such as minimum-phase FIR filters or IIR filters.

Figure 13-34. |H(m)| and |Hs(m)| performance: (a) full frequency response; (b) passband detail.

If perhaps more stopband attenuation is needed then the process shown in Figure 13-35 can be used, where again the delay element is equal to (N–1)/2 samples.

Figure 13-35. Improved filter sharpening FIR process.

The filter sharpening procedure is straightforward and applicable to lowpass, bandpass, and highpass FIR filters having symmetrical coefficients and an odd number of taps. Filter sharpening can be used whenever a given filter response cannot be modified, such as an unchangeable software subroutine, and can even be applied to cascaded integrator-comb (CIC) filters to flatten their passband responses, as well as FIR fixed-point multiplierless filters where the coefficients are constrained to be powers of two[35,36].

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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

Understanding Digital Signal Processing
Understanding Digital Signal Processing (2nd Edition)
ISBN: 0131089897
EAN: 2147483647
Year: 2004
Pages: 183 © 2008-2020.
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