"Circular Convolution Explained Mathematically with DFT"
Understanding Circular Convolution Using DFT Convolution is a foundational tool in signal processing, crucial for systems analysis, filtering, and more. While linear convolution is commonly taught, circular convolution is particularly important when working with the Discrete Fourier Transform (DFT) and periodic signals. What is Circular Convolution? Circular convolution (also called cyclic convolution ) assumes that signals are periodic. This means that when the index exceeds the length of the signal, it wraps around due to modulo operation. It is defined as: y[n] = Σ k=0 N-1 x[k] · h[(n - k) mod N] The modulo operation introduces the "circular" behavior. DFT and the Convolution Theorem According to the Convolution Theorem : Circular Convolution in Time Domain ⟺ Pointwise Multiplication in Frequency Domain That is: x[n] ⊛ h[n] ↔ DFT ↔ X[k] · H[k] This property enables fast computation using FFT. How to Perform Circular Co...