# Does Circular Convolution Correspond to Periodically Expanded Linear Convolution?

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

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 $$y$$.