Analysis of circular convolution

Circular convolution can be efficiently computed using the Fast Fourier Transform (FFT) algorithm.

The convolution of two signals is a fundamental operation in signalThe methodology of circular convolution involves several steps. Let's break it down:

1.Input Sequences: Start with two finite-length sequences, 𝑥[𝑛]x[n] and ℎ[𝑛]h[n], where 𝑛nis the index ranging from 00 to 𝑁−1N−1, where 𝑁Nis the length of the sequences.

2.Periodic Extension: Extend both sequences to make them the same length, which is typically chosen as 𝑁N. This extension is done by wrapping the sequences around themselves. For example, if 𝑥[𝑛]x[n] and ℎ[𝑛]h[n] are of length 𝑁N, their periodic extensions can be represented as 𝑥~[𝑛]x~[n] and ℎ~[𝑛]h~[n], respectively.

3.𝑥~[𝑛]=𝑥[𝑛mod  𝑁]x~[n]=x[nmodN] ℎ~[𝑛]=ℎ[𝑛mod  𝑁]h~[n]=h[nmodN]

4.Element-wise Multiplication: Multiply corresponding elements of the extended sequences 𝑥~[𝑛]x~[n] and ℎ~[𝑛]h~[n].

5.𝑦[𝑛]=𝑥~[𝑛]⋅ℎ~[𝑛]y[n]=x~[n]⋅h~[n]

6.Summation: Sum up all the products obtained from the element-wise multiplication.

7.𝑦[𝑛]=∑𝑘=0𝑁−1𝑥~[𝑘]⋅ℎ~[𝑛−𝑘]y[n]=∑k=0N−1​x~[k]⋅h~[n−k]

8.Wraparound: If necessary, perform wraparound to maintain the original length of the sequences. This is often necessary because the summation index 𝑘k can exceed the range 00 to 𝑁−1N−1.

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Circular convolution is often implemented using the Fast Fourier Transform (FFT) algorithm due to its computational efficiency. The circular convolution of two sequences in the time domain is equivalent to the pointwise multiplication of their corresponding Discrete Fourier Transform (DFT) representations in the frequency domain. After the pointwise multiplication, the result can be obtained by applying the inverse DFT to the product.

This methodology allows for efficient computation of convolution, especially for long sequences, and finds applications in various fields such as signal processing, digital filtering, and communications.