Circular vs Linear Convolution

Linear convolution is the basic operation to calculate the output for any linear time invariant system given its input and its impulse response. Circular convolution is the same thing but considering that the support of the signal is periodic (as in a circle, hence the name).

1. Why is circular convolution better than linear?
2. What is the advantage of circular convolution over linear convolution?
3. What is relation between linear and circular convolution?
4. Why do we use circular convolution?

Why is circular convolution better than linear?

Linear convolution may or may not result in a periodic output signal. The output of a circular convolution is always periodic, and its period is specified by the periods of one of its inputs.

What is the advantage of circular convolution over linear convolution?

This holds in continuous time, where the convolution sum is an integral, or in discrete time using vectors, where the sum is truly a sum. It also holds for functions defined from -Inf to Inf or for functions with a finite length in time.

What is relation between linear and circular convolution?

The linear convolution of an N-point vector, x , and an L-point vector, y , has length N + L - 1. For the circular convolution of x and y to be equivalent, you must pad the vectors with zeros to length at least N + L - 1 before you take the DFT.

Why do we use circular convolution?

Although DTFTs are usually continuous functions of frequency, the concepts of periodic and circular convolution are also directly applicable to discrete sequences of data. In that context, circular convolution plays an important role in maximizing the efficiency of a certain kind of common filtering operation.