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Modulo-N circular convolution
c = cconv(a,b,n)
c = cconv(gpuArrayA,gpuArrayB,n)
Circular convolution is used to convolve two discrete Fourier transform (DFT) sequences. For very long sequences, circular convolution may be faster than linear convolution.
c = cconv(a,b,n) circularly convolves vectors a and b. n is the length of the resulting vector. If you omit n, it defaults to length(a)+length(b)-1. When n = length(a)+length(b)-1, the circular convolution is equivalent to the linear convolution computed with conv. You can also use cconv to compute the circular cross-correlation of two sequences (see the example below).
c = cconv(gpuArrayA,gpuArrayB,n) returns the circular convolution of the input vectors of class gpuArray. See Establish Arrays on a GPU for details on gpuArray objects. Using cconv with gpuArray objects requires Parallel Computing Toolbox™ software and a CUDA-enabled NVIDIA GPU with compute capability 1.3 or above. See http://www.mathworks.com/products/parallel-computing/requirements.html for details. The output vector, c, is a gpuArray object. See Circular Convolution using the GPU for an example of using the GPU to compute the circular convolution.
The following example calculates a modulo-4 circular convolution.
a = [2 1 2 1]; b = [1 2 3 4]; c = cconv(a,b,4)
c = 14 16 14 16
The following example compares a circular correlation, where n uses the default value, and a linear convolution. The resulting norm is a value that is virtually zero, which shows that the two convolutions produce virtually the same result.
a = [1 2 -1 1]; b = [1 1 2 1 2 2 1 1]; c = cconv(a,b); % Circular convolution cref = conv(a,b); % Linear convolution dif = norm(c-cref)
dif = 9.7422e-16
The following example uses cconv to compute the circular cross-correlation of two sequences. The result is compared to the cross-correlation computed using xcorr.
a = [1 2 2 1]+1i; b = [1 3 4 1]-2*1i; c = cconv(a,conj(fliplr(b)),7); % Compute using cconv cref = xcorr(a,b); % Compute using xcorr dif = norm(c-cref)
dif = 3.3565e-15
The following example requires Parallel Computing Toolbox software and a CUDA-enabled NVIDIA GPU with compute capability 1.3 or above. See http://www.mathworks.com/products/parallel-computing/requirements.html for details.
Create two signals consisting of a 1 kHz sine wave in additive white Gaussian noise. The sampling rate is 10 kHz
Fs = 1e4; t = 0:1/Fs:10-(1/Fs); x = cos(2*pi*1e3*t)+randn(size(t)); y = sin(2*pi*1e3*t)+randn(size(t));
Put x and y on the GPU using gpuArray. Obtain the circular convolution using the GPU.
x = gpuArray(x); y = gpuArray(y); cirC = cconv(x,y,length(x)+length(y)-1);
Compare the result to the linear convolution of x and y.
linC = conv(x,y); norm(linC-cirC,2)
Return the circular convolution, cirC, to the MATLAB^{®} workspace using gather.
cirC = gather(cirC);