# The Swiss Army Knife of Digital Networks

This blog describes a general discrete-signal network that appears, in various forms, inside so many DSP applications.

Figure 1 shows how the network's structure has the distinct look of a digital filter—a comb filter followed by a 2nd-order recursive network. However, I do not call this useful network a filter because its capabilities extend far beyond simple filtering. Through a series of examples I've illustrated the fundamental strength of this ** Swiss Army Knife** of digital networks and its ability to be reconfigured to perform an astounding number of useful functions based on the values of its seven control coefficients.

**Figure
1.
**** General discrete-signal processing network.**

The Figure 1 network has a transfer function of

$$ H(z) = \frac{Y(z)}{X(z)} = (1-c_1z^{-N})\frac{b_0 + b_1z^{-1} + b_2z^{-2}}{1/a_0 - a_1z^{-1} - a_2z^{-2}} \tag{1}$$The tables in this blog list various
signal processing functions performed by the network based on the $ a_n,\ b_n\ and\ c_1 $ coefficients. Variable
*$ N $* is the order of the comb filter. Included in
the tables are depictions of the network's impulse response,
*z*-plane
pole/zero locations, as well as frequency-domain magnitude and phase responses.
The frequency axes in those tables is normalized such that a value of 0.5
represents a frequency of $ f_s / 2 $ where $ f_s $ is the
sample rate in Hz.

*To keep this blog manageable in size, detailed
explanations of the network Functions in the following tables are contained in
the downloadable PDF file associated with this blog. *

**[NOTE FROM AUTHOR LYONS (Oct. 2018): The downloadable PDF file has been updated since this blog’s original posting in June 2016.]**

- Comments
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I don't see any provisions for integrator anti-windup.

:)

May be better to explain the equaliser as an allpass filter.

Dear Rick,

Question regarding Table 4, Real FSF Type IV:

Could you please advise from your experience what maximum value can take N-comb filter order at practical implementation?

Best regards, Serhii Rybka

Hello SerhilRybka.

Keeping in mind that the order of the comb subfilter, used in a Real FSF Type IV lowpass digital filter, determines how many memory locations are required I don't know of any limit on the comb subfilter's order.

Dear Rick,

Thanks a lot for your reply. I have another question on this topic - why in the books you are not considering bandpass filter based on real FSF Type IV?

Best regards, Serhii

When I wrote about real FSF Type IV in the 3rd edition of my "Understanding DSP" book I concentrated on lowpass filters. But in my book's Table 7-3 I mentioned that such filters could be used for bandpass filtering.

If you're designing an FSF Type IV digital filter for some actual filtering implementation send me an e-mail at: R.Lyons@ieee.org

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