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# Decimation filters

The mic array unit provided by this library uses a two-stage decimation process, implemented in `TwoStageDecimator`, to convert a high sample rate stream of (1-bit) PDM samples into a lower sample rate stream of (32-bit) PCM samples. This is shown in Simplified Decimator Model.

Fig. 4 Simplified Decimator Model

The first stage filter is a decimating FIR filter with a fixed tap count (`S1_TAP_COUNT`) of `256` and a fixed decimation factor (`S1_DEC_FACTOR`) of `32`.

The second stage decimator is a fully configurable FIR filter with tap count `S2_TAP_COUNT` and a decimation factor of `S2_DEC_FACTOR` (this can be `1`).

## Filters provided as part of `lib_mic_array`

`lib_mic_array` provides first and second stage decimation filter coefficients for filters targeting output sampling rates of 16 kHz, 32 kHz and 48 kHz from a starting input PDM frequency of 3.072 MHz. The first stage decimation filters have a fixed decimation factor of `32` and a fixed tap count of `256`.

The second stage filters decimation factors vary based on the output sampling rate.

These filters are available to include in the application, through the header `mic_array/etc/filters_default.h`

Note

Although the first stage has a fixed decimation factor of 32, its coefficients must differ for 16 kHz, 32 kHz, and 48 kHz output sampling rate paths. The passband must be extended appropriately so sufficient bandwidth is preserved before the second stage decimates by 6 (output sampling rate 16 kHz), 3 (output sampling rate 32 kHz), or 2 (output sampling rate 48 kHz).

The increased sample rate will place a higher MIPS burden on the processor. The typical MIPS usage (see section Resource usage) is in the order of 11 MIPS per channel using a 16 kHz output decimator.

Increasing the output sample rate to 32 kHz using the same length filters will increase processor usage per channel to approximately 13 MIPS rising to 15.6 MIPS for 48 kHz.

Increasing the filter lengths to 148 and 96 for stages 1 and 2 respectively at 48 kHz will increase processor usage per channel to around 20 MIPS.

### Stage 1 filters

Table 4 Provided first stage decimation filters
Output PCM sampling rate | Decimation factor | Tap count | Coeffs
--- | --- | --- | ---
16 kHz | 32 | 256 | `stage1_coef`
32 kHz | 32 | 256 | `stage1_32k_coefs`
48 kHz | 32 | 256 | `stage1_48k_coefs`

### Stage 2 filters

Table 5 Provided second stage decimation filters
Output PCM sampling Rate | Decimation factor | Tap count | Coeffs | Right shift
--- | --- | --- | --- | ---
16 kHz | 6 | 256 | `stage2_coef` | `stage2_shr`
32 kHz | 3 | 96 | `stage2_32k_coefs` | `stage2_32k_shift`
48 kHz | 2 | 96 | `stage2_48k_coefs` | `stage2_48k_shift`

### Filter characteristics

This section provides characteristics of the filters provided as part of `lib_mic_array`.

### 16 kHz output PCM sampling rate filter

16 kHz output sampling rate filter freq response shows the frequency response of the first and second stages of the provided 16 kHz sampling rate filters as well as the cascaded overall response.

Fig. 5 16 kHz output sampling rate filter freq response

### 32 kHz output PCM sampling rate filter

32 kHz output sampling rate filter freq response shows the frequency response of the first and second stages of the provided 32 kHz sampling rate filters as well as the cascaded overall response. Note that the overall combined response provides a nice flat passband.

Fig. 6 32 kHz output sampling rate filter freq response

### 48 kHz output PCM sampling rate filter

48 kHz output sampling rate filter freq response shows the frequency response of the first and second stages of the provided 48 kHz sampling rate filters as well as the cascaded overall response. Note that the overall combined response provides a nice flat passband.

Fig. 7 48 kHz output sampling rate filter freq response

The following sections provide more details about the first and second stage decimation filters, implemented in `TwoStageDecimator`.

## Decimator stage 1

For the first stage decimating FIR filter, the actual filter coefficients used are configurable, so an application is free to use a custom first stage filter, as long as the tap count is `256` and the decimation factor is `32`.

### Filter implementation (Stage 1)

The input to the first stage decimator (here called “Stream A”) is a stream of 1-bit PDM samples with a sample rate of `PDM_FREQ`. Rather than each PDM sample representing a value of `0` or `1`, each PDM sample represents a value of either `+1` or `-1`. Specifically, on-chip and in-memory, a bit value of `0` represents `+1` and a bit value of `1` represents `-1`.

The output from the first stage decimator, Stream B, is a stream of 32-bit PCM samples with a sample rate of `PDM_FREQ/S1_DEC_FACTOR = PDM_FREQ/32`. For example, if `PDM_FREQ` is 3.072 MHz, then Stream B’s sample rate is 96.0 kHz.

The first stage filter is structured to make optimal use of the XCore XS3 vector processing unit (VPU), which can compute the dot product of a pair of 256-element 1-bit vectors in a single cycle. The first stage uses 256 16-bit coefficients for its filter taps.

The signature of the filter function is

    int32_t fir_1x16_bit(uint32_t signal[8], uint32_t coeff_1[]);

Each time 32 PDM samples (1 word) become available for an audio channel, those samples are shifted into the 8-word (256-bit) filter state, and a call to `fir_1x16_bit` results in 1 Stream B sample element for that channel.

The actual implementation for the first stage filter can be found in `src/fir_1x16_bit.S`. Additional usage details can be found in `api/etc/fir_1x16_bit.h`.

Note that the 256 16-bit filter coefficients are not stored in memory as a standard coefficient array (i.e. `int16_t filter[256] = {b[0], b[1], ... };`). Rather, in order to take advantage of the VPU, the coefficients must be rearranged bit-by-bit into a block form suitable for VPU processing.

The filter state (delay line) consists of 256 one-bit PDM samples (equal to the number of filter taps) and requires a buffer of 8 unsigned 32-bit words for storage.

### Filter Conversion Script

Taking a set of floating-point coefficients, quantizing them into 16-bit coefficients and ‘boggling’ them into the correct memory layout can be a tricky business. To simplify this process, this library provides a Python3 script that does this process automatically.

The script can be found in this repository at `python/stage1.py`.

## Decimator stage 2

An application is free to supply its own second stage filter.

### Filter implementation (Stage 2)

The input to the second stage decimator (here called “Stream B”) is the stream of 32-bit PCM samples emitted from the first stage decimator with a sample rate of `PDM_FREQ/32`.

The output from the second stage decimator, Stream C, is a stream of 32-bit PCM samples with a sample rate of `PDM_FREQ/(32*S2_DEC_FACTOR)`. For example, if `PDM_FREQ` is 3.072 MHz, and `S2_DEC_FACTOR` is `6`, then Stream C’s sample rate (the sample rate received by the main application code) is

> 3.072 MHz / (32*6) = 16 kHz

The second stage filter uses the 32-bit FIR filter implementation from lib_xcore_math. See `xs3_filter_fir_s32()` in that library for more implementation details.

The filter state (delay line) consists of as many 32-bit samples as there are taps in the stage-2 filter, and requires that many 32-bit words for storage.