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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 −

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

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

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

By taking the IDFT of the above we get

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

After solving the above equation, finally, we get

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

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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