Circular Convolution - Discrete Fourier Transform

Circular Convolution - Discrete Fourier Transform Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

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Let us take two finite duration sequences x1(n) and x2(n), having integer length as N. Their DFTs are X1(K) and X2(K) respectively, which is shown below −

Now, we will try to find the DFT of another sequence x3(n), which is given as X3(K)

By taking the IDFT of the above we get

After solving the above equation, finally, we get

Comparison pointsLinear ConvolutionCircular Convolution
ShiftingLinear shiftingCircular shifting
Samples in the convolution resultN1+N2−1Max(N1,N2)
Finding response of a filterPossiblePossible with zero padding

Methods of Circular Convolution

Generally, there are two methods, which are adopted to perform circular convolution and they are −

• Concentric circle method,
• Matrix multiplication method.

Concentric Circle Method

Let x1(n) and x2(n) be two given sequences. The steps followed for circular convolution of x1(n) and x2(n) are

• Take two concentric circles. Plot N samples of x1(n) on the circumference of the outer circle (maintaining equal distance successive points) in anti-clockwise direction.

• For plotting x2(n), plot N samples of x2(n) in clockwise direction on the inner circle, starting sample placed at the same point as 0thsample of x1(n)

• Multiply corresponding samples on the two circles and add them to get output.

• Rotate the inner circle anti-clockwise with one sample at a time.

Matrix Multiplication Method

Matrix method represents the two given sequence x1(n) and x2(n) in matrix form.

• One of the given sequences is repeated via circular shift of one sample at a time to form a N X N matrix.

• The other sequence is represented as column matrix.

• The multiplication of two matrices give the result of circular convolution.

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