Convolution of Short Signals
where and are arbitrary real or complex sequences, and and are the DTFTs of and , respectively. The convolution of and is defined by
In practice, we always use the DFT (preferably an FFT) in place of the DTFT, in which case we may write
where now (length complex sequences). It is important to remember that the specific form of convolution implied in the DFT case is cyclic (also called circular) convolution :
where means `` modulo .'' Another way to look at convolution is as the inner product of , and , where , i.e.,
This form describes graphical convolution in which the output sample at time is computed as an inner product of the impulse response after flipping it about time 0 and shifting time 0 to time . See [264, p. 105] for an illustration of graphical convolution.
Cyclic FFT ConvolutionThanks to the convolution theorem, we have two alternate ways to perform cyclic convolution in practice:
- Direct calculation in the time domain using (8.13)
- Frequency-domain convolution:
Acyclic FFT ConvolutionIf we add enough trailing zeros to the signals being convolved, we can obtain acyclic convolution embedded within a cyclic convolution. How many zeros do we need to add? Suppose the signal consists of contiguous nonzero samples at times 0 to , preceded and followed by zeros, and suppose is nonzero only over a block of samples starting at time 0. Then the acyclic convolution of with reduces to
which is zero for and . Thus,
The number is easily checked for signals of length 1 since , where is 1 at time zero and 0 at all other times. Similarly,
and so on. When or is infinity, the convolution result can be as small as 1. For example, consider , with , and . Then . This is an example of what is called deconvolution. In the frequency domain, deconvolution always involves a pole-zero cancellation. Therefore, it is only possible when or is infinite. In practice, deconvolution can sometimes be accomplished approximately, particularly within narrow frequency bands . We thus conclude that, to embed acyclic convolution within a cyclic convolution (as provided by an FFT), we need to zero-pad both operands out to length , where is at least the sum of the operand lengths (minus one). convolution:
octave:1> conv([1 2],[3 4]) ans = 3 10 8Note that it returns an output vector which is long enough to accommodate the entire result of the convolution, unlike the filter primitive, which always returns an output signal equal in length to the input signal:
octave:2> filter([1 2],1,[3 4]) ans = 3 10 octave:3> filter([1 2],1,[3 4 0]) ans = 3 10 8
8.2 shows schematically the result of convolving two zero-padded signals and . In this case, the signal starts some time after , say at . Since begins at time 0 , the output starts promptly at time , but it takes some time to ``ramp up'' to full amplitude. (This is the transient response of the FIR filter .) If the length of is , then the transient response is finished at time . Next, when the input signal goes to zero at time , the output reaches zero samples later (after the filter ``decay time''), or time . Thus, the total number of nonzero output samples is . If we don't add enough zeros, some of our convolution terms ``wrap around'' and add back upon others (due to modulo indexing). This can be called time-domain aliasing. Zero-padding in the time domain results in more samples (closer spacing) in the frequency domain, i.e., a higher `sampling rate' in the frequency domain. If we have a high enough spectral sampling rate, we can avoid time aliasing. The motivation for implementing acyclic convolution using a zero-padded cyclic convolution is that we can use a Cooley-Tukey Fast Fourier Transform (FFT) to implement cyclic convolution when its length is a power of 2.
Acyclic FFT Convolution in MatlabThe following example illustrates the implementation of acyclic convolution using a Cooley-Tukey FFT in matlab:
x = [1 2 3 4]; h = [1 1 1]; nx = length(x); nh = length(h); nfft = 2^nextpow2(nx+nh-1) xzp = [x, zeros(1,nfft-nx)]; hzp = [h, zeros(1,nfft-nh)]; X = fft(xzp); H = fft(hzp); Y = H .* X; format bank; y = real(ifft(Y)) % zero-padded result yt = y(1:nx+nh-1) % trim and print yc = conv(x,h) % for comparisonProgram output:
nfft = 8 y = 1.00 3.00 6.00 9.00 7.00 4.00 0.00 0.00 yt = 1.00 3.00 6.00 9.00 7.00 4.00 yc = 1 3 6 9 7 4
FFT versus Direct ConvolutionUsing the Matlab test program in ,9.1FFT convolution was found to be faster than direct convolution starting at length (looking only at powers of 2 for the length ).9.2 FFT convolution was also never significantly slower at shorter lengths for which ``calling overhead'' dominates. Running the same test program in 2011,9.3 FFT convolution using the fft function was found to be faster than conv for all (power-of-2) lengths. The speed of FFT convolution divided by that of direct convolution started out at 14 for , fell to a minimum of at , above which it started to climb as expected, reaching at . Note that this comparison is unfair because the Octave fft function is a dynamically linked, separately compiled module, while conv is written in the matlab language and thus suffers more overhead from the matlab interpreter. An analysis reported in Strum and Kirk [279, p. 521], based on the number of real multiplies, predicts that the fft is faster starting at length , and that direct convolution is significantly faster for very short convolutions (e.g., 16 operations for a direct length-4 convolution, versus 176 for the fft function). See 9.4for further discussion of FFT algorithms and their applications. In digital audio, FIR filters are often hundreds of taps long. For such filters, the FFT method is much faster than direct convolution in the time domain on single CPUs. On GPUs, FFT convolution is faster than direct convolution only for much longer FIR-filter lengths (in the thousands of taps ); this is because massively parallel hardware can perform an algorithm (direct convolution) faster than a single CPU can perform an algorithm (FFT convolution).
Audio FIR FiltersFIR filters shorter than the ear's ``integration time'' can generally be characterized by their magnitude frequency response (no perceivable ``delay effects''). The nominal ``integration time'' of the ear can be defined as the reciprocal of a critical bandwidth of hearing. Using Zwicker's definition of critical bandwidth , the smallest critical bandwidth of hearing is approximately 100 Hz (below 500 Hz). Thus, the nominal integration time of the ear is 10ms below 500 Hz. (Using the equivalent-rectangular-bandwidth (ERB) definition of critical bandwidth [179,269], longer values are obtained). At a 50 kHz sampling rate, this is 500 samples. Therefore, FIR filters shorter than the ear's ``integration time,'' i.e., perceptually ``instantaneous,'' can easily be hundreds of taps long (as discussed in the next section). FFT convolution is consequently an important implementation tool for FIR filters in digital audio applications.
Example 1: Low-Pass Filtering by FFT ConvolutionIn 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); endNext 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 filter8.3 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); % filterFigure 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.
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);
Example 2: Time Domain AliasingFigure 8.7 shows the effect of insufficient zero padding, which can be thought of as undersampling in the frequency domain. We will see aliasing in the time domain results. The lowpass filter length is and the input signal consists of an impulse at times and , where the data frame length is . To avoid time aliasing (i.e., to implement acyclic convolution using an FFT), we must use an FFT size at least as large as . In the figure, the FFT sizes , , and are used. Thus, the first case is heavily time aliased, the second only slightly time aliased (involving only some of the filter's ``ringing'' after the second pulse), and the third is free of time aliasing altogether.
Convolving with Long Signals