Let $\Bbb{C}^N$ be the N-dimensional Euclidean space with its inner product defined as $$ \langle f,g\rangle=\sum_{n=0}^{N-1} f[n]g^*[n],\ \forall f,g \in \Bbb{C}^N$$ where $g^*[n]$ is the complex conjugate of $g[n]$. Given 2 signals $f,g \in \Bbb{C}^N$, define their corresponding circulant convolution $f*g$ by $$ (f*g)[n]=\sum_{n=0}^{N-1} f*[n]g^*[(n-k)\ mod\ N], \ 0\le n\lt N$$ For a signal $f\in \Bbb{C}^N$, define its periodic total variation by $$ ||f||_V = \sum_{n=1}^{N-1}|f[n]-f[n-1]| + |f[N-1]-f[0]| \ \ \ \ \ (equation\ 1)$$
Prove that $$ ||f||_V=\sum_{n=0}^{N-1} |(f*h)[n]|$$for some filter $h \in \Bbb{C}^N$
From the periodic nature of signal $f$, we can write $f[N]=f[0]$ so we can write equation 1 as $$ ||f||_V = \sum_{n=1}^{N}|f[n]-f[n-1]|$$ I don't see how does a convolution relate to differences in vector values between adjacent positions. I tried re-writing the RHS of equation 1 in terms of inner products and also expanding the RHS of the required identity but I don't see a way to reconcile the 2 expressions I got. Please help.