I would like to design an FIR filter to decimate a high rate 1 bit signal to a low rate high bit signal. Due to implementation constraints I have fixed filter length constraints. 

My setup is as follows: I decimate by 32 followed by a second stage of 6. My constraints are: every 32 one bit input samples I can apply a 256 tap 16 bit FIR (for the first stage), and every 8 first stage outputs I apply a 128 tap 16 bit FIR.

I believe this gives me an effective filter length of 32 * 128 + (256-32) = 4,320 input samples (or something close to this).

If I design a single stage FIR with some spec using 4,320 taps I get a filter(using remez or firls, etc), F. I think that, due to the constraints applied by the above, I cannot (necessarily) implement F as a two staged cascade as F might not be able to be expressed as a linear combination of the first stage.

How then do I design the two stages to be jointly as close to the filter spec as possible? In the past I've manually designed two filters and they've been good enough. Thanks

Hello andrew,

I have looked at this problem a number of times and have reminded paeople tht when you are filting the output of a 1-bit sigm-delta, with a fir filter, ther are no multiplies because the input samples are 1-bit wide. there are only adds! If you fold the resampler into the FIR filter you have a polyphase filter with no multiplies that down samples as well as reduces BW. I presented a paper at a conference that was dedicated to CIC filters and my message was, don't use a CIC for 1-bit data, use the polyphase filter... A whole bunch of paper were written later presenting the same argument. I did the same thing for the sigma delta ADCs in cell phones and reduced power consumption while improving spectral envelopes! see attached paper

ISCAS_CIC_3.pdf

fred h

Thanks a lot, I'll go through your paper. In reality I won't be performing any multiplies in the first stage but I tried to simplify my question as it was getting long.

I have also done this before and, likewise, concluded that a polyphase FIR was the way for the architecture I was targetting.

If your focus is multipliers then a single filter with polyphase structure for 1 bit input might be better than the multiplier-less CIC. But memory requirements for full filter polyphases are still needed and the adders may be wide or very wide. So it depends on available resources and platforms. 

Hi Kaz,

The adders in the polyphases are quite narrow, nowhere close to the width of the accumultors of a multistage CIC filter. The width of the coefficient sets of the polyphase filtesa are also quite small. look at my paper! I convinced folks that make smart phone chips to replace their 120 dB dynamic range CIC filters with the polyphase filter option. They were quite pleased and marketed the higher perormance of their filters relative to the competition's filter esponses. 

fred 

Thank you both, in my specific situation I have hardware that can "multiply" and accumulate 256 one bit values by 16 bit values producing a single multi bit output. After that I use an integer Macc to do the second stage. Given my constraints, what is left for me to do is choose the specific coefficient values to implement a filter in two stages rather than one.