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

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 | 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 | Signals and Systems - Electronics and Communication Engineering (ECE)

By taking the IDFT of the above we get

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

After solving the above equation, finally, we get

Circular Convolution - Discrete Fourier Transform | 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.

The document Circular Convolution - Discrete Fourier Transform | Signals and Systems - Electronics and Communication Engineering (ECE) is a part of the Electronics and Communication Engineering (ECE) Course Signals and Systems.
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FAQs on Circular Convolution - Discrete Fourier Transform - Signals and Systems - Electronics and Communication Engineering (ECE)

1. What is circular convolution in the context of the Discrete Fourier Transform?
Ans. Circular convolution refers to the operation of convolving two sequences using the Discrete Fourier Transform (DFT) in a circular manner. It involves multiplying the corresponding frequency components of the two sequences and then applying the inverse DFT to obtain the circularly convolved output.
2. How is circular convolution different from linear convolution?
Ans. Circular convolution differs from linear convolution in that it assumes the sequences to be periodic and of finite length. Unlike linear convolution, circular convolution does not involve any zero-padding or cropping of the sequences. This makes it suitable for processing signals with periodic properties.
3. Can circular convolution be performed using the Fast Fourier Transform (FFT)?
Ans. Yes, circular convolution can be efficiently computed using the Fast Fourier Transform (FFT) algorithm. By applying the DFT to the input sequences, multiplying their frequency components, and then applying the inverse DFT, the circular convolution can be computed with a complexity of O(N log N), where N is the length of the sequences.
4. What are the applications of circular convolution in signal processing?
Ans. Circular convolution has various applications in signal processing. It is commonly used for filtering and equalization in audio and image processing. It is also utilized in linear systems analysis, modulation techniques, and spectral analysis. Additionally, circular convolution plays a crucial role in implementing circular buffers and circularly shifting sequences.
5. Are there any limitations or drawbacks of circular convolution?
Ans. Yes, circular convolution has a few limitations. One limitation is the presence of circular artifacts or time-domain aliasing, which can distort the convolved output. Another limitation is the assumption of periodicity, which may not hold true for all signals. Furthermore, circular convolution can introduce spectral leakage and phase wrapping effects. These limitations should be considered when applying circular convolution in practical signal processing applications.
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