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This blog describes a linear-phase comb filter having wider stopband notches than a traditional comb filter.

Background

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Let's first review the behavior of a traditional comb filter. Figure 1(a) shows a traditional comb filter comprising two cascaded recursive running sum (RRS) comb filters. Figure 1(b) shows the filter's co-located dual poles and dual zeros on the-plane, while Figure 1(c) shows the filter's positive-frequency magnitude response when, for example,= 9. The stopband notches (nulls) are located at integer multiples ofHz whereis the input signal sample rate measured in Hz.

In what follows we show a comb filter having wider stopband notches than are shown in Figure 1(c).

The Wide-Notch Comb Filter



The Figure 1(a) dual RRS network has a triangular time-domain impulse response. I have discovered that if we build a network whose impulse response is that of the dual RRS network but with a reduced-amplitude center sample then we'll have a wide-notch comb filter.



Figure 2(a) shows the block diagram of such a wide-notch comb filter. Figure 2(b) shows the filter's co-located dual poles and dual zeros on the-plane, while Figure 2(c) shows the filter's frequency magnitude response when, for example,= 9 and C = 0.05.

As described in Appendix A, the z-domain transfer function of the proposed wide-notch comb filter is:



Figure 2(a)'s feedforward signal path using coefficient C imparts the four following favorable characteristics to the wide-notch filter:

The filter's frequency-domain notch width and the peak of the mini-sidelobes within the notches depend on both the delay line length D and the C coefficient. In most comb filter applications variable D is a fixed value, so coefficient C becomes the primary control variable to influence the filter's notch characteristics.

Values in the range 0.01 ≤ C ≤ 0.1 are reasonable for coefficient C to start any software modeling of the wide-notch comb filter. What you will find is that when you double the value of the C coefficient the peak of the mini-sidelobes within the notches increases by roughly six dB.

The wide-notch comb filter has a high low-frequency gain, just as do cascaded integrator-comb filters (CIC) filters. So to avoid register overflow the filter should be implemented using two's complement arithmetic for the reasons described in Reference [2].

A Unity-Gain Wide-Notch Comb Filter

The lowpass gain (gain at zero Hz) of the Figure 2(a) wide-notch comb filter is:

gain = D2—C. (2)

Appendix B shows how to implement a unity-gain wide-notch comb filter.

When to Use the Wide-Notch Comb Filter

The wide-notch comb filter can be useful in reducing the computational workload of interpolated finite-impulse response filters [1,2] and specialized narrowband lowpass IIR filters [3,4], as well as reducing the aliasing error caused by sample rate change in even-ordered cascaded integrator-comb filters (CIC) filters [5]. Another potential application of a wide-notch comb filter is its use as a comb filter for attenuating noise contamination from AC power line harmonics.

Conclusion

I've discussed the development and performance of the wide-notch comb filter in Figure 2(a) that may prove beneficial in applications that have previously used traditional recursive running sum (RRS) comb filters. In addition, Appendix B shows how to implement a unity passband gain wide-notch comb filter.

References

[1] G. Dolecek, and V. Dolecek, "Multistage Digital Filter", 13th International Research/Expert Conference, Hammamet, Tunisia, Oct. 2009, Available online: http://www.tmt.unze.ba/zbornik/TMT2009/096-TMT09-147.pdf

[2] R. Lyons, "Understanding cascaded integrator-comb filters", Available online: https://www.embedded.com/understanding-cascaded-integrator-comb-filters/

[3] R. Lyons, "Improved Narrowband Lowpass IIR Filters in Fixed-Point Systems". IEEE Signal Processing Magazine. March, 2009. Available online: https://www.researchgate.net/publication/224397424_Improved_narrowband_low-pass_IIR_filters_in_fixed-point_systems_DSP_Tips_Tricks

[4] F. Harris and W. Lowdermilk, "Implementing Recursive Filters with Large Ratio of Sample Rate to Bandwidth", in Conference Record of the Forty-First Asilomar Conference on Signals, Systems and Computers, Pacific Grove, CA, Nov. 4–7, 2007, pp. 1149–1153. Available online: https://s3.amazonaws.com/embeddedrelated/user/124841/narrowband%20iir%20filters_2_40945.pdf

[5] R. Lyons, "Turbocharging Interpolated FIR Filters", IEEE Signal Processing Magazine, Vol. 24, No. 5, Sept. 2007. Available online: https://www.researchgate.net/publication/277702593_Turbocharging_Interpolated_FIR_Filters

[6] R. Lyons, "Controlling a DSP Network's Gain: A Note For DSP Beginners", available online at: https://www.dsprelated.com/showarticle/1249.php

Appendix A: Derivation of Eq. (1)

The derivation of this blog's Eq. (1) is described as follows: The foundation of the proposed wide-notch comb filter is two cascaded Recursive Running Sum filters as shown in Figure A-1(a).

The z-domain transfer functions of a single RRS network and the cascaded Figure A-1(a) network are given in the following two equations:

The initial structure of a wide-notch comb filter, with its three long delay lines, is shown in Figure A-1(b). That filter was inspired by a recursive filter proposed in Reference [5]. The Figure A-1(b) filter's-domain transfer function is:





We can eliminate one of Figure A-1(b)'s long delay lines using the proposed network given in this blog's Figure 2(a) whose z-domain transfer function is also given by Eq. (A-3).

Appendix B

To implement a unity passband (at zero Hz) gain wide-notch comb filter we must reduce the Figure 2(a) filter's gain by a factor of. The simplest method to do this is shown in Figure B-1(a). There we merely scaled the filter's output sequence by the reciprocal of the Figure 2(a) filter's passband gain of

In fixed-point implementations, to reduce the necessary word width of the 3rd and 4th accumulators (3rd and 4th adders) and the z‑D delay line, the distributed scaling method in Figure B-1(b) can be used. Equation (B-1) gives the transfer function for both filters in Figure B-1:



WARNING:

On the Internet I've seen digital filter gain reduction implemented with an attenuator at a filter's input as shown in Figure B-2.





Such a gain reduction method is a bad idea for the reasons discussed in Reference [6].













