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FIR Decimation

This example shows how to decrease the sampling rate of a signal using FIR decimators from the DSP System Toolbox™.

Creating FIR Decimators

The DSP System Toolbox supports different structures to perform decimation including different FIR based structures and CICs. Given an decimation factor M, a common design of an FIR decimation filter is a Nyquist filter with a cutoff frequency of pi/M. See the FIRHALFBAND, FIRNYQUIST, FIREQINT and INTFILT functions as well as FDESIGN.DECIMATOR and FDESIGN.NYQUIST for more on the design of decimation filters.

M  = 3; % Decimation factor
Hf = fdesign.decimator(M,'Nyquist',M);
Hd = design(Hf,'SystemObject',true); % Polyphase FIR Decimator

A more efficient implementation is the transposed FIR polyphase structure. This structure allows for sharing of the delays and multipliers

Hd.Structure = 'Direct form transposed';

To decimate by a fractional factor, you can use a Direct-Form FIR Polyphase Sample-Rate Converter. This structure uses L polyphase subfilters.

L   = 2; % Interpolation factor
Hf  = fdesign.rsrc(L,M,'Nyquist',max(L,M));
Hd2 = design(Hf,'SystemObject',true); % Polyphase FIR Fractional Decimator

Filtering with FIR Decimators

The input signal x[n] is a 1 kHz sinusoid sampled at 44.1 kHz.

N = 159;
Fs = 44.1e3;
n = (0:N-1)';
x = sin(2*pi*n*1e3/Fs);

Filter with a Direct-Form FIR Polyphase Decimator.

y = step(Hd,x);

The length of the transient response of the decimator is equal to half the order of a polyphase subfilter. This is also the group-delay of the filter.

delay = mean(grpdelay(Hd)); % Constant group delay equal to its mean
tx = delay+(1:length(x));
ty = 1:M:M*length(y);

Display the output of the Direct-Form FIR Polyphase Decimator and overlay a shifted version of the original signal.

stem(tx,x,'k');hold on;stem(ty,y,'filled');
axis([0 90 -Inf Inf])
legend('Original signal','Decimated signal')
xlabel('Samples'); ylabel('Amplitude');

Frequency-Domain Analysis of the Interpolated Signal

We compute the power spectral densities of both input and decimated signal.

% Create an audio file reader and point to an audio file with sound sampled
% at 48 kHz
Ha = dsp.AudioFileReader('audio48kHz.wav');

% Create a spectrum analyzer to view the spectrum of the input and
% decimated audio.
Hs = dsp.SpectrumAnalyzer('SampleRate',48e3,'ShowLegend',true,...
  'SpectralAverages',10);

% Design an decimate-by-2 filter to decimate the signal from 48 kHz
% to 24 kHz
M  = 2;
Hf = fdesign.decimator(M,'Halfband');
Hd = design(Hf,'SystemObject',true);

The output will be upsampled independently to 48 kHz by inserting a zero between every sample in order to plot the input and output spectrum simultaneously. To maintain the same power level, the upsampled signal is multiplied by the upsampling factor.

while ~isDone(Ha)
    x  = step(Ha);        % Original 48 kHz audio
    y  = step(Hd,x);      % Decimated 24 kHz audio
    yu = M*upsample(y,M); % Insert a zero every other sample to compare
    step(Hs,[x,yu]);
end

% Release Audio File Reader
release(Ha);

As expected, the decimated signal has spectral replicas centered at multiples of the low sampling frequency (24 kHz)

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