Example 1: Low-Pass Filtering by FFT Convolution

In this example, we design and implement a length FIR low-pass filter having a cut-off frequency at Hz. The filter is tested on an input signal consisting of a sum of sinusoidal components at frequencies Hz. We'll filter a single input frame of length , which allows the FFT to be samples (no wasted zero-padding).

% Signal parameters:
f = [ 440 880 1000 2000 ];      % frequencies   
M = 256;                        % signal length 
Fs = 5000;                      % sampling rate

% Generate a signal by adding up sinusoids:
x = zeros(1,M); % pre-allocate 'accumulator'
n = 0:(M-1);    % discrete-time grid 
for fk = f; 
    x = x + sin(2*pi*n*fk/Fs); 
end

A plot of the synthesized input signal is shown in Fig.8.3. Next we design the lowpass filter using the window method:

% Filter parameters:
L = 257;    % filter length 
fc = 600;   % cutoff frequency 

% Design the filter using the window method:
hsupp = (-(L-1)/2:(L-1)/2);
hideal = (2*fc/Fs)*sinc(2*fc*hsupp/Fs);
h = hamming(L)' .* hideal; % h is our filter

Figure 8.4 plots the impulse response and amplitude response of our FIR filter designed by the window method. Next, the signal frame and filter impulse response are zero padded out to the FFT size and transformed:

% Choose the next power of 2 greater than L+M-1 
Nfft = 2^(ceil(log2(L+M-1))); % or 2^nextpow2(L+M-1)

% Zero pad the signal and impulse response:
xzp = [ x zeros(1,Nfft-M) ];
hzp = [ h zeros(1,Nfft-L) ];

% Transform the signal and the filter:
X = fft(xzp);
H = fft(hzp);

Figure 8.5 shows the input signal spectrum and the filter amplitude response overlaid. We see that only one sinusoidal component falls within the passband.

Now we perform cyclic convolution in the time domain using pointwise multiplication in the frequency domain:

Y = X .* H;

The modified spectrum is shown in Fig.8.6.

The final acyclic convolution is the inverse transform of the pointwise product in the frequency domain. The imaginary part is not quite zero as it should be due to finite numerical precision:

y = ifft(Y);
relrmserr = norm(imag(y))/norm(y) % check... should be zero
y = real(y);

Figure 8.7 shows the filter output signal in the time domain. As expected, it looks like a pure tone in steady state. Note the equal amounts of ``pre-ringing'' and ``post-ringing'' due to the use of a linear-phase FIR filter.^9.1

For an input signal approximately samples long, this example is 2-3 times faster than the conv function in Matlab (which is precompiled C code implementing time-domain convolution).

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.