What is circular convolution?

[dexterdev]

Hi PF,
What is circular convolution? Why do we need such an operation if we have linear convolution, What is its basic difference of both convolutions. Is circular convolution used only in frequency domain?

-Devanand T

 

[atyy]

http://ocw.mit.edu/resources/res-6-008-digital-signal-processing-spring-2011/study-materials/MITRES_6_008S11_lec10.pdf

According to these notes, we usually want linear convolution. But there's a fast algorithm for circular convolution, so we adapt that to do linear convolution.

 

[rbj]

that's a good pdf.

one note about meaning:

$$((n))_N \ \triangleq \ n\,\bmod\,N \ = \ n - N \left\lfloor \frac{n}{N} \right\rfloor$$

this just makes the index $n$ wrap around so that it is always $0 \le n < N$ . that's what makes it circular.

 

[DSRadin]

I would offer that we don't particularly want circular convolution, but it is a necessary by-product of the finite-length DFT operations.

Circular convolution also drives the need for windowing and filtering to remove all of the translated spectral images. Learning to mitigate the negative effects of circular convolution is important.

 

[alphy]

Circular convolution is a mathematical operation that combines two signals or functions by multiplying them and summing the results over a periodic interval. It is similar to linear convolution, but with the key difference being that the interval over which the signals are multiplied and summed is periodic, meaning it repeats infinitely.

We need circular convolution because it allows us to efficiently process signals that have periodic properties, such as periodic signals in a communication system or signals in a rotating system. In comparison, linear convolution is used for non-periodic signals.

The basic difference between circular and linear convolution is that circular convolution assumes that the signals are periodic, while linear convolution does not. This means that circular convolution can exploit the periodicity of signals to simplify calculations and reduce computational complexity.

Circular convolution is not limited to the frequency domain and can be used in both time and frequency domains. In the time domain, circular convolution can be used to convolve two periodic signals, while in the frequency domain, it can be used to multiply two signals with different spectra.

In conclusion, circular convolution is a powerful tool in signal processing that allows us to efficiently handle periodic signals. Its use is not limited to the frequency domain, and it offers advantages over linear convolution when dealing with periodic signals.