Modulo-n circular convolution
Generate two signals of different lengths. Compare their circular convolution and their linear convolution. Use the default value for
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 resulting norm is virtually zero, which shows that the two convolutions produce the same result to machine precision.
Generate two vectors and compute their modulo-4 circular convolution.
a = [2 1 2 1]; b = [1 2 3 4]; c = cconv(a,b,4)
c = 1×4 14 16 14 16
Generate two complex sequences. Use
cconv to compute their circular cross-correlation. Flip and conjugate the second operand to comply with the definition of cross-correlation. Specify an output vector length of 7.
a = [1 2 2 1]+1i; b = [1 3 4 1]-2*1i; c = cconv(a,conj(fliplr(b)),7);
Compare the result to the cross-correlation computed using
cref = xcorr(a,b); dif = norm(c-cref)
dif = 3.3565e-15
Generate two signals: a five-sample triangular waveform and a first-order FIR filter with response .
x1 = conv([1 1 1],[1 1 1])
x1 = 1×5 1 2 3 2 1
x2 = [-1 1]
x2 = 1×2 -1 1
Compute their circular convolution with the default output length. The result is equivalent to the linear convolution of the two signals.
ccnv = cconv(x1,x2)
ccnv = 1×6 -1.0000 -1.0000 -1.0000 1.0000 1.0000 1.0000
lcnv = conv(x1,x2)
lcnv = 1×6 -1 -1 -1 1 1 1
The modulo-2 circular convolution is equivalent to splitting the linear convolution into two-element arrays and summing the arrays.
ccn2 = cconv(x1,x2,2)
ccn2 = 1×2 -1 1
nl = numel(lcnv); mod2 = sum(reshape(lcnv,2,nl/2)')
mod2 = 1×2 -1 1
Compute the modulo-3 circular convolution and compare it to the aliased linear convolution.
ccn3 = cconv(x1,x2,3)
ccn3 = 1×3 0 0 0
mod3 = sum(reshape(lcnv,3,nl/3)')
mod3 = 1×3 0 0 0
If the output length is smaller than the convolution length and does not divide it exactly, pad the convolution with zeros before adding.
c = 5; z = zeros(c*ceil(nl/c),1); z(1:nl) = lcnv; ccnc = cconv(x1,x2,c)
ccnc = 1×5 0.0000 -1.0000 -1.0000 1.0000 1.0000
modc = sum(reshape(z,c,numel(z)/c)')
modc = 1×5 0 -1 -1 1 1
If the output length is equal to or larger than the convolution length, pad the convolution and do not add.
d = 13; z = zeros(d*ceil(nl/d),1); z(1:nl) = lcnv; ccnd = cconv(x1,x2,d)
ccnd = 1×13 -1.0000 -1.0000 -1.0000 1.0000 1.0000 1.0000 0.0000 -0.0000 0.0000 0.0000 0.0000 -0.0000 -0.0000
modd = z'
modd = 1×13 -1 -1 -1 1 1 1 0 0 0 0 0 0 0
The following example requires Parallel Computing Toolbox™ software. Refer to GPU Support by Release (Parallel Computing Toolbox) to see what GPUs are supported.
Create two signals consisting of a 1 kHz sine wave in additive white Gaussian noise. The sample 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));
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)
ans = 1.4047e-08
Return the circular convolution,
cirC, to the MATLAB® workspace using
cirC = gather(cirC);
b— Input arrays
Input array, specified as vectors or
gpuArray objects. See
Run MATLAB Functions on a GPU (Parallel Computing Toolbox) for details
gpuArray objects. Using
gpuArray objects requires Parallel
Computing Toolbox™ software. Refer to GPU Support by Release (Parallel Computing Toolbox) to see what GPUs
sin(2*pi*(0:9)/10) + randn([1 10])/10 specifies a noisy
sinusoid as a row vector.
gpuArray(sin(2*pi*(0:9)/10) + randn([1 10])/10) specifies
a noisy sinusoid as a
Complex Number Support: Yes
n— Convolution length
Convolution length, specified as a positive integer. If you do not specify
n, then the convolution has length
c— Circular convolution
Circular convolution of input vectors, returned as a vector or
For long sequences, circular convolution can be faster than linear convolution.
 Orfanidis, Sophocles J. Introduction to Signal Processing. Englewood Cliffs, NJ: Prentice-Hall, 1996, pp. 524–529.
This function fully supports GPU arrays. For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).