Introduction

In the past 5 episodes of this series, I discussed sigma-delta converters, how to read the datasheet of a microphone, different types of filters, and how to come up with the right coefficients of these filters.

Finally, it’s time to bring all the pieces together and create a full signal processing pipeline to convert the output of a PDM microphone into standard PCM samples.

The Design Requirements

Let’s quickly repeat the overall design requirements that I came up with in a previous blog post:

  • PDM input sample rate: 2.304MHz

  • PCM output sample rate: 48kHz

    In other words, a decimation ratio of 48.

  • Pass band: 0 to 6kHz
  • Stop band: 10kHz
  • Pass band ripple: 0.1dB
  • Stop band attenuation: 89dB

The Canonical Multi-Stage Decimation Pipeline

There is an overwhelming amount of literature out there about the pipeline to decimate an incoming signal at very high sample rate down to something much lower. And while there are an almost infinite amount of subtle variations, the fundamentals can almost always be reduced to the following basic building blocks:

  • a CIC filter at the front
  • one or more half-band filters in the middle
  • a generic FIR filter in the back

Canonical Decimation Pipeline

The reason for this arrangement is obvious when you consider typical design constraints such as clock speeds and resource usage.

  1. A CIC filter only requires adders and delay stages, but no multipliers. This makes them extremely area efficient.
  2. The lack of multipliers also makes it easier to run CIC filters at high clock speeds.
  3. A half-band filter requires only around 50% of the multiplications of an equivalent generic FIR filter. This reduction is at all times large enough to overcome the inefficiency of splitting a larger generic FIR filter into a cascaded half-band filter and a FIR filter that has half the order of the original one.

With the core pipeline in place, the next step is figuring out the exact filter parameters for each stage.

Pass Band Ripple, Stop Band Attenuation and Multi-Stage Decimation

The requirements call for an overall pass band ripple of 0.1dB and a stop band attenuation of 89dB.

When there’s only 1 filter, meeting that goal is a matter of specifying these numbers as the filter design parameter.

But what when multiple filters are cascaded?

Pass Band Ripple

In their 1975 paper “Optimum FIR Digital Filter Implementations for Decimation, Interpolation, and Narrow-Band Filtering”, Crochiere and Rabiner write the following:

As it is desired that the overall pass band ripple for the composite of K stages be maintained within 1±δ, it is necesary to require more severe frequency constraints on the individual filters in the cascade. A convenient choice which will satisfy this requirement is to specify the pass band ripple constraints for each stage i to be within 1±δ/K.

In other words: if you split the filter into 3 stages, they suggest to split the joint passband ripple of 0.1dB into 3 smaller pass band ripples.

Unfortunately, this advise doesn’t work very well for our case: the referenced paper assumes that all stages are using similar filter types, with the ability to indepdently choose the boundaries for pass band and stop band ripple for each filter stage.

This is not the case for us: a CIC filter has very little flexibility wrt pass and stop band attenuation, and half-band filters have only 1 parameter, the filter order, to control the ripple of both pass band and stop band.

What we can take away, however, is the knowledge that we need to divide the overall pass band into smaller values and uses those as design parameter for the individual stages.

Stop Band Attenuation

The same paper has the following to say about the stop band:

In the stop band the ripple constraint for the composite filter must be δ and this constraint must be imposed on each of the individual low-pass filter as well, in order to suppress the effects of aliasing.

This is much easier: I calculated a stop band attenuation of 89dB. We can just to use that attenuation for each filter stage.

CIC Filter Configuration and Pass Band Ripple

Since they don’t require any multiplications at all, it’s desirable to shift as much as of the decimation to the front-end CIC filter.

One problem with CIC filters, however, is that the pass band ripple gets progressively worse with increasing frequency and with an increasing number of cascade stages. Without using a counter measure in the form of a compensation filter, the frequency response in the pass band can easily exceed the maximum ripple requirements of the overall filter specification.

Pass band ripple and decimation ratio

Our overall filter has a decimation ratio of 48, and thus 9 possible CIC decimation ratios: 2, 3, 4, 6, 8, 12, 16, 24, 48.

For a 5 stage configuration, these 9 options give the following magnitude frequency plots:

All decimation ratios for a 5 stage CIC filter

That tiny rectangle in the top left corner shows the frequency range from 0 to 24kHz, our final output bandwidth. Let’s zoom in on that:

All decimation ratios for a 5 stage CIC filter - Zoom

The rectangle now shows the pass band: a frequency range from 0 to 6kHz and a maximum pass band ripple boundary of 0.1dB.

You can see how a decimation ratio of 16 and higher violates the maximum ripple requirements, and that’s before taking into account that the filters that follow the CIC stage need to get their share of the overall pass band ripple!

Pass band ripple and number of cascaded stages

In the graphs above, I use an example of 5 stages with variable decimation rate. I can do the same with a fixed decimation rate and a variable number of stages:

Fixed Decimation Rate, Variable Number of Stage CIC filter - Zoom

The result is more predictable, but similar: a higher number of stages increases the pass band ripple.

One way to solve this issue is to add a compensation filter with a gain in the pass band region that is the exactly opposite of the chosen CIC filter, thus resulting in a flat pass band behavior. But that’s something for a future blog post.

Pass band ripple table: decimation ratio vs number of stages

For simplicity, we will just need to limit decimation ratio and/or the number of stages such that the pass band ripple doesn’t exceed a certain maximum value. This maximum value is than the 0.1dB pass band ripple of the overall filter to leave something for for the filter stages that follow the CIC filter.

Let’s choose a maximum ripple of 0.05dB.

The table below lists all the combinations of decimation ratio and number of CIC stages, and their ripple. The entries in green are the ones that don’t violate the 0.05dB maximum:

Pass Band Ripple (dB)
Decimation Ratio / Nr of CIC Stages
1 2 3 4 5 6
2 -0.0003 -0.0006 -0.0009 -0.0012 -0.0015 -0.0018
3 -0.0008 -0.0016 -0.0024 -0.0032 -0.0039 -0.0047
4 -0.0015 -0.0030 -0.0044 -0.0059 -0.0074 -0.0089
6 -0.0034 -0.0069 -0.0103 -0.0138 -0.0172 -0.0207
8 -0.0062 -0.0124 -0.0186 -0.0248 -0.0310 -0.0372
12 -0.0141 -0.0282 -0.0423 -0.0564 -0.0705 -0.0846
16 -0.0251 -0.0502 -0.0753 -0.1004 -0.1256 -0.1507
24 -0.0568 -0.1135 -0.1703 -0.2271 -0.2839 -0.3406
48 -0.2190 -0.4381 -0.6571 -0.8761 -1.0952 -1.3142

If pass band ripple were the only design criterium, the best choice would be a solution with the highest decimation ratio, because that reduces the decimation ratio of the downstream non-CIC filter blocks.

But pass band ripple is not the only consideration. There’s also the anti-aliasing performance requirement as specified by the stop band attenuation.

CIC Filter Configuration and Anti-Aliasing Performance

In this section of my blog post about CIC filters, I show how the frequency of the pass band determines the amount of aliasing of higher frequencies into the pass band after decimation: the closer the pass band frequency to the overall output bandwidth, the more stages are needed to achieve a certain anti-aliasing performance.

For example, at frequency of 2000Hz, the CIC filter below attenuates aliased frequencies by at least 92.4dB. If we were to reduce this frequency to 1000Hz, this attenuation would go up to ~115dB.

Anti-Alias Performance

This attenuation number also depends on the number of CIC stages and the decimation ratio. The design requirement of our filter requires a stop band attenuation of 89dB. Just like with the pass band ripple, we need a table that shows which CIC parameters have sufficient attenuation at the stop band frequency of 10kHz:

Stop Band Attenuation (dB)
Decimation Ratio / Nr of CIC Stages
1 2 3 4 5 6
2 -37.3 -74.6 -112.0 -149.3 -186.6 -223.9
3 -36.0 -72.0 -108.0 -144.0 -180.0 -216.0
4 -34.2 -68.4 -102.6 -136.8 -171.0 -205.2
6 -31.1 -62.2 -93.3 -124.4 -155.5 -186.6
8 -28.7 -57.4 -86.1 -114.8 -143.5 -172.2
12 -25.2 -50.3 -75.5 -100.6 -125.8 -150.9
16 -22.6 -45.2 -67.7 -90.3 -112.9 -135.5
24 -18.8 -37.7 -56.5 -75.3 -94.2 -113.0
48 -12.2 -24.4 -36.6 -48.8 -61.0 -73.2

From all the green entries above, it’s best to choose those with the lowest number of CIC stages: a higher number of stages increases the width of the accumulation and delay registers, which also increases the time delay through the adders.

Final CIC Filter Configuration

Let’s combine these 2 tables to find the CIC parameters that satisfy both pass band and stop band requirements:

Pass Band/Stop Band Attenuation (dB)
Decimation Ratio / Nr of CIC Stages
1 2 3 4 5 6
2 -0.0003
-37.3		 -0.0006
-74.6		 -0.0009
-112.0		 -0.0012
-149.3		 -0.0015
-186.6		 -0.0018
-223.9
3 -0.0008
-36.0		 -0.0016
-72.0		 -0.0024
-108.0		 -0.0032
-144.0		 -0.0039
-180.0		 -0.0047
-216.0
4 -0.0015
-34.2		 -0.0030
-68.4		 -0.0044
-102.6		 -0.0059
-136.8		 -0.0074
-171.0		 -0.0089
-205.2
6 -0.0034
-31.1		 -0.0069
-62.2		 -0.0103
-93.3		 -0.0138
-124.4		 -0.0172
-155.5		 -0.0207
-186.6
8 -0.0062
-28.7		 -0.0124
-57.4		 -0.0186
-86.1		 -0.0248
-114.8		 -0.0310
-143.5		 -0.0372
-172.2
12 -0.0141
-25.2		 -0.0282
-50.3		 -0.0423
-75.5		 -0.0564
-100.6		 -0.0705
-125.8		 -0.0846
-150.9
16 -0.0251
-22.6		 -0.0502
-45.2		 -0.0753
-67.7		 -0.1004
-90.3		 -0.1256
-112.9		 -0.1507
-135.5
24 -0.0568
-18.8		 -0.1135
-37.7		 -0.1703
-56.5		 -0.2271
-75.3		 -0.2839
-94.2		 -0.3406
-113.0
48 -0.2190
-12.2		 -0.4381
-24.4		 -0.6571
-36.6		 -0.8761
-48.8		 -1.0952
-61.0		 -1.3142
-73.2

From all the green cells above, the best solutions are those with the lowest number of CIC stages and highest decimation ratio:

  • 6x decimation ratio with 3 stages

    This option requires 3 half-band filters to decimate by 8 and 1 generic FIR filter to satisfy final pass-band and stop-band attenuation.

  • 8x decimation ratio with 4 stages

    This requires the remaining 6x decimation ratio to be done with 1 half-band filter to decimate by 2x, followed by some generic FIR configuraton for the final 3x decimation.

    There are 2 options for this generic FIR configuraiton:

    • 1 generic FIR filter to decimate by 3x followed by 1 generic FIR filter for the final pass-band and stop-band attenuation
    • 1 generic FIR filter that immediately applies the final pass-band and stop-band attenuation

It is not at all obvious which of the 3 options above is most optimal in terms of number of multiplications! The best way forward is to just try all 3 and see what comes out as optimal.

However, a closer look at the table above shows that the CIC configuration with 12x decimation ratio and 4 stages comes very close to passing the overall requirements, with a pass band ripple of 0.056dB, only 0.06dB higher than your 0.05dB limit.

And the thing is: that 0.05dB was chosen rather arbitrarily. We can just as arbitrarily increase that limit a bit, make this case pass, and end up with a overall filter architecture that might very well be better than the 3 earlier ones:

  • 12x decimation ratio with 4 stages

    This requires 2 half-band filters to decimate by 4 and 1 generic FIR filter for the final pass-band and stop-band requirements.

Design of the Half-Band and Generic FIR Filters

Once the properties of the CIC filter has been decided, the half-band and generic FIR filter design is mostly a matter of just filling in the numbers and running the Parks-McClellan/Remez algorithm to come up with the exact filter coefficients.

The table below shows the resul for all 4 filter architecture candidates:

CIC Config HB1 mul/s HB2 mul/s HB3 mul/s Decim FIR mul/s Final FIR mul/s Total mul/s
Decim:6
Stages:3		 4 x 192k = 768k 6 x 96k = 576k 10 x 48k = 480k 47 x 48k = 2256k 4080k
Decim:8
Stages:4		 6 x 144k = 864k 21 x 48k = 1008k 51 x 48k = 2448k 4320k
Decim:8
Stages:4		 6 x 144k = 864k 141 x 48k = 6768k 7632k
Decim:12
Stages:4		 6 x 96k = 576k 10 x 48k = 480k 51 x 48k = 2448k 3504k

It turns out that the solution with 12 stages is indeed the optimal one.

The naive single-stage FIR filter that I used as initial stake in the ground required 106M mul/s. This number has been reduced to 3.5M multiplications per second, an improvement of more than 30x!

Here are some conclusions based on the table above:

  • whenever possible, try to use half-band filters for all decimation operations. Using a generic FIR filter as decimation stage has a significant cost.
  • if you have to use a generic FIR filter for decimation (e.g. because the overall decimation ratio doesn’t have a factor of 2), don’t merge it with the final FIR filter
  • play around with the pass band ripple portion that’s assigned to the CIC filter to find the optimal solution

This drawing shows all the operations of the final pipeline:

All Filter Operations

And here are the magnitude frequency plot and impluse response of all the filter components:

All Filter Plots

Coming Up

The architecture of decimation pipeline is now complete, but there are still things to come:

  • convert theory to practise, implement everything to RTL, and run things on an FPGA
  • check if a CIC compensation filter would reduce the number of multiplications even more

References

My Blog Posts in this Series

Code

  • NumPy Code used to create plots and graphs of this blog post.

Decimation