Solution of Week12 Quiz Question 4
a. Linearity
Given $ v[n]=ax[n]+by[n] $ ,then
$ \begin{align} V(\omega ,n) &= \sum_k v[k]w[n-k]e^{-j\omega k} \\ &= \sum_k (ax[k]+by[k])w[n-k]e^{-j\omega k} \\ &= \sum_k ax[k]w[n-k]e^{-j\omega k}+\sum_k by[k]w[n-k]e^{-j\omega k} \\ &= aX(\omega ,n)+bY(\omega ,n) \end{align} $
b. Modulation
Given $ v[n]=x[n]e^{j\omega_0n} $ ,then
$ \begin{align} V(\omega ,n) &= \sum_k v[k]w[n-k]e^{-j\omega k} \\ &= \sum_k (x[k]e^{j\omega_0 k})w[n-k]e^{-j\omega k} \\ &= \sum_k x[k]w[n-k]e^{-j(\omega -\omega_0)k} \\ &= X(\omega -\omega_0 ,n) \end{align} $
