How to create a distortion compensation filter

Question

I am creating a channelizer that consists of a complex mixer, CIC decimator, and a FIR compensation/decimation filter. The final FIR filter may, if it matters, be implemented as more than one filter.

My question is, how do I design a filter such that it compensates for the very non-flat frequency response of the CIC filter? Do you create the frequency response that you want by calculating the reciprocal of the CIC's response and then run it through an inverse FFT to get the impulse response?

As you can see, though my particular problem pertains to CIC filters, my question is really about how you create any kind of distortion compensation filter.

Thanks for your time.

Answer 1

The following addresses the CIC compensation and is not a general "distortion" technique. But it is a straight-forward method to "compensate" non-varying "distortion". If the frequency response is known the inverse of the frequency can be used to compensate. Examples like the CIC filter, where a poor filter might be used because of reduced complexity, are compensated later the signal chain. In this example the frequency response is know and the inverse can be used. Note, with multi-rate filters you only want to use the "usable" spectrum after decimation.

Generally, to compensate a CIC filter the inverse of the CIC filters response can be used to generate the compensation filter. The CIC has a response of (see reference [r2] or [r3])

$$ H(\omega) = \left| \frac{sin(\omega D/2)}{sin(\omega M/2)} \right|^N $$

Where D is the differentiate delay, M is the decimation rate, and N is the filter order (number of cascaded filters). The inverse can be specified as

$$ H(\omega) = \left| \frac{sin(\omega M/2)}{sin(\omega D/2)} \right|^N $$

Once we have the frequency response of the compensation filter, we can simply choose the length of FIR filter that we desire. The length of the FIR is application specific. Obviously the longer the FIR filter the better compensation.

The following are plots of this straight forward compensation.

The following is the Python code to create the frequency responses and plots.

import numpy as np
from numpy import sin, abs, pi
import pylab

D = 1; M = 7; N = 3

Hfunc = lambda w : abs( (sin((w*M)/2)) / (sin((w*D)/2.)) )**N
HfuncC = lambda w : abs( (sin((w*D)/2.)) / (sin((w*M)/2.)) )**N

w = np.arange(1024) * pi/1024

G = (M*D)**N
H = np.array(map(Hfunc, w))
Hc = np.array(map(HfuncC, w))
# only use the inverse (compensation) roughly to the first null.
Hc[int(1024*pi/M/2):] = 1e-8
plot(w, 20*log10(H/G))
plot(w, 20*log10(Hc*G))
grid('on')

See [r1] for other approaches and $sinc^{-1}$ approximation.

[r1] Altera, "Understanding CIC compensation filters"

[r2] R. Lyons, "Understanding Digital Signal Processing", 2nd ed., Prentice Hall, Upper Saddle River, New Jersey, 2004

[r3] R. Lyons, "Understanding Cascaded Integrator Comb Filters"

Answer 2

I think Christopher's answer is a good one. I thought that I would add one for us lazy/cheater types.

While digging around Matlab's fdatool (Filter Design & Analysis Tool) I discovered that it can design and model both CIC and inverse sinc filters, where the inverse sinc filter is the CIC's distortion compensation filter.

You generate the CIC filter by going straight to the "Create a multi-rate filter" page (button on the left in version R2011b) and specifying the CIC filter. You can then set all the parameters that you want on it, such as interpolation/decimation rates, number of cascades (Matlab calls it "sections"), etc.

You create the inverse sinc filter by going to the "Design filter" page (the page that the tool defaults to when it starts), and specifying "Inverse Sinc Lowpass" from the Lowpass drop down menu. You can then set the various parameters for it. It has a couple of unique parameters in the Options part of the dialog, including "p", which corresponds to the number of cascades (sinc exponent).

The inverse sinc filter design was not completely satisfying, though, since getting it right seemed less an instance of doing the math than figuring out where my pass band was, what the CIC droop was at that point, and then designing a filter that roughly canceled out that droop. Since my CIC filter had four cascades, I would have thought that I would need to set "p" to four, but it way overcompensated when I did that. I ended up leaving "p" at 1, the default value.

The tool really shone when I saved both filters in my session, and then went into the filter manager and cascaded them to see the overall frequency response. This was very helpful, and easy to do.

Answer 3

Without duplicating the response here, please see

how to make CIC compensation filter

which shows a very simple 3 tap solution for CIC compensation.

This is specific to CIC or any application for an inverse Sinc is required (such as prior to D/A conversion).

To compensate for any general transversal distortion (that which could be replicated with a transversal filter as opposed to dynamic noise distortion) I would take the approach of using the Wiener Hopf equations to come up with a compensation filter using the Least Squared Error approach, ideally using a training sequence if available to compare desired to actual response and train the equalizer. For an application showing this with source code used see:

Compensating Loudspeaker frequency response in an audio signal