Does Circular Convolution Correspond to Periodically Expanded Linear Convolution?

Question

I have a question whether circular convolution and periodically expanded linear convolution corresponds in following case or why it does not?

Think about a signal $x[t]$ and a signal $y[t]$ both of length $N$. We expand the singal $x[t]$ periodically to a signal $x'[t]=[x[t],x[t]]$ with length $2N$. We zeropadd the signal $y[t]$ to a signal $y'[t] = [y[t],0]$ to obtain the same length $2N$.

My question is does the circular convolution result $(x[t] \star y[t] )_{mod_N}$, correspond to the bins $[N...2N]$ in the linear convolution result $x'[t] * y'[t]$?

Best

Answer 1

Yes, they are the same.

Let us take the linear convolution of $x[t]$ and $y[t]$ as 2 portions $H_1$ of length $N$ corresponding to first $N$ samples, and $H_2$ of remaining $N-1$ samples. Circular convolution between $x$ and $y$ causes $H_2$ to overlap over $H_1$ because of time-aliasing. So the first $N-1$ samples of the result is $H_1 + H_2$ with only the last sample being correct (corresponding to linear convolution).

Now, for the linear convolution between $\tilde{x}$ and $\tilde{y}$, the samples from $N$ to $2N-1$ will have additional component of first $N$ samples of $\textrm{conv}(x,y)$. So these samples will correspond circular convolution of $x$ and $y$.