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Impulse response of CIC filter

Started by ChuckMcM 1 week ago7 replieslatest reply 3 days ago97 views

So this is a curious one. I am continuing investigating #CIC filters now that I have run @Rick Lyons tests and proven to myself that they are now correctly implemented. It involves the impulse response (or what I was looking for, the frequency response) of these filters.

One of the things I like about FIR filters is that you can send the unit impulse through them, and take the FFT of the result and see the frequency response of the filter. So naturally I plotted the FFT of the impulse response of the 3 stage, decimate by 5, CIC filter. Not surprisingly, depending on where in the decimation cycle the impulse comes in, the outputs of the filter change. So I wrote some code to send an impulse through at offset 0, 1, 2, 3, and 4 (since it is a decimate by 5 the sequence repeat). These are the first 10 values for each offset

Simple Impulse test, first 10 values:
Offset  0   1   2   3   4
--------------------------
[0] - 15, 10,  6,  3,  1
[1] - 10, 15, 18, 19, 18
[2] -  0,  0,  1,  3,  6
[3] -  0,  0,  0,  0,  0
... (all the rest of the values are 0

When you plot the FFTs of these you get this:

multi-cic-plot_7690.png

There are essentially three distinct plots for the three unique sequences from the output. I am at a loss to explain what this value would change. The spectral content shouldn't be different as far as I can tell.

Curious.

[ - ]
Reply by gretzteamJanuary 15, 2020

Those are the frequency responses of each individual 'polyphase' of the full FIR filter. The frequency response you're most likely interested in is that of the full FIR filter impulse, which is the interleaved version of those 'polyphase'. Try FFTing:

[1 3 6 10 15 18 19 18 15 10 6 3 1]

Should look more like what you're expecting.

[ - ]
Reply by ChuckMcMJanuary 15, 2020

Ok, that is pretty awesome!

with-polyphase_39506.png

[ - ]
Reply by gretzteamJanuary 15, 2020

if the yaxis was in 'dB' it would even look like a normal frequency response! 

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Reply by ChuckMcMJanuary 15, 2020

Your right it does. Now nothing I've read about CIC filters talks about polyphase. That is something I need to understand, can you point me to a reference?

And as an update I've downloaded half a dozen papers that refer to implementing CIC filters using polyphase techniques. It will take me a bit to digest them.

My first reading is that each 'stage' of the CIC filter is one FIR phase. Thus a 3 stage CIC has 3 phases which is what I found for my example above. I will get to test this by tweaking to code a bit. 

[ - ]
Reply by gretzteamJanuary 15, 2020

I would ignore the CIC filter part and read up about standard FIR filters and their polyphase implementation when used for decimation.

CIC is just a clever recursive way to implement a certain family of FIR filters,essentially an implementation detail. 

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Reply by ChuckMcMJanuary 22, 2020

How did you choose how to do the interleaving?

[ - ]
Reply by gretzteamJanuary 22, 2020

I'm not sure what you mean...

Start from the full FIR filter and pick every N samples...?