Suppose I have two vectors $x,y\in\mathbb{C}^n$. I am interested in efficient calculation of the convolution $$z(k) = \sum_{i=0}^{n-1}x(i)y(i+k)$$ for $k=0,...,n-1$. Note that $y$ is taken nearly periodic i.e. $y(n+i)=\alpha y(i)$. For the simple case $\alpha=1$, this can be calculated efficiently using a Fourier-Transform as $x*y=\mathcal{F}^{-1}(\mathcal{F}(x)\mathcal{F}(y))$. Is there a generalization of this method to other values of $\alpha$?
(If general $\alpha\in\mathbb{C}$ are too tricky, pure phase factors, i.e. $\alpha=e^{i\phi}$ are probably enough for my usecase)