Example 1: Low-Pass Filtering by FFT Convolution

In this example, we design and implement a length FIR lowpass 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

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 8.3 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) ]; X = fft(xzp); % signal H = fft(hzp); % filter

Figure 8.4 shows the input signal spectrum and the filter amplitude response overlaid. We see that only one sinusoidal component falls within the pass-band.

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.8.5.

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 8.6 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.^{9.5}

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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