Filter and downsample input signals
DSP System Toolbox / Filtering / Multirate Filters
DSP System Toolbox HDL Support / Filtering
The FIR Decimation block resamples vector or matrix inputs along the
first dimension. The FIR decimator (as shown in the schematic) conceptually consists of
an antialiasing FIR filter followed by a downsampler. To design an FIR antialiasing
filter, use the designMultirateFIR
function.
The FIR filter filters the data in each channel of the input using a directform FIR filter. The downsampler that follows downsamples each channel of filtered data by taking every Mth sample and discarding the M – 1 samples that follow. M is the value of the decimation factor that you specify. The resulting discretetime signal has a sample rate that is 1/M times the original sample rate.
The actual block algorithm implements a directform FIR polyphase structure, an efficient equivalent of the combined system depicted in the diagram. For more details, see Algorithms.
Under specific conditions, this block also supports SIMD code generation. For details, see Code Generation.
Data Types 

Direct Feedthrough 

Multidimensional Signals 

VariableSize Signals 

ZeroCrossing Detection 

The FIR decimation filter is implemented efficiently using a polyphase structure. For more details on polyphase filters, see Polyphase Subfilters.
To derive the polyphase structure, start with the transfer function of the FIR filter:
$$H(z)={b}_{0}+{b}_{1}{z}^{1}+\mathrm{...}+{b}_{N}{z}^{N}$$
N+1 is the length of the FIR filter.
You can rearrange this equation as follows:
$$H(z)=\begin{array}{c}\left({b}_{0}+{b}_{M}{z}^{M}+{b}_{2M}{z}^{2M}+\mathrm{..}+{b}_{NM+1}{z}^{(NM+1)}\right)+\\ {z}^{1}\left({b}_{1}+{b}_{M+1}{z}^{M}+{b}_{2M+1}{z}^{2M}+\mathrm{..}+{b}_{NM+2}{z}^{(NM+1)}\right)+\\ \begin{array}{c}\vdots \\ {z}^{(M1)}\left({b}_{M1}+{b}_{2M1}{z}^{M}+{b}_{3M1}{z}^{2M}+\mathrm{..}+{b}_{N}{z}^{(NM+1)}\right)\end{array}\end{array}$$
M is the number of polyphase components, and its value equals the decimation factor that you specify.
You can write this equation as:
$$H(z)={E}_{0}({z}^{M})+{z}^{1}{E}_{1}({z}^{M})+\mathrm{...}+{z}^{(M1)}{E}_{M1}({z}^{M})$$
E_{0}(z^{M}), E_{1}(z^{M}), ..., E_{M1}(z^{M}) are the polyphase components of the FIR filter H(z).
Conceptually, the FIR decimation filter contains a lowpass FIR filter followed by a downsampler.
Replace H(z) with its polyphase representation.
Here is the multirate noble identity for decimation.
Applying the noble identity for decimation moves the downsampling operation to before the filtering operation. This move enables you to filter the signal at a lower rate.
You can replace the delays and the decimation factor at the input with a commutator switch. The switch starts on the first branch 0 and moves in the counterclockwise direction as shown in this diagram. The accumulator at the output receives the processed input samples from each branch of the polyphase structure and accumulates these processed samples until the switch goes to branch 0. When the switch goes to branch 0, the accumulator outputs the accumulated value.
When the first input sample is delivered, the switch feeds this input to the branch 0 and the decimator computes the first output value. As more input samples come in, the switch moves in the counter clockwise direction through branches M−1, M−2, and all the way up to branch 0, delivering one sample at a time to each branch. When the switch comes to branch 0, the decimator outputs the next set of output values. This process continues as data keeps coming in. Every time the switch comes to the branch 0, the decimator outputs y[m]. The decimator effectively outputs one sample for every M samples it receives. Hence the sample rate at the output of the FIR decimation filter is fs/M.
[1] Fliege, N. J. Multirate Digital Signal Processing: Multirate Systems, Filter Banks, Wavelets . West Sussex, England: John Wiley & Sons, 1994.
[2] Orfanidis, Sophocles J. Introduction to Signal Processing . Upper Saddle River, NJ: PrenticeHall, 1996.