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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} $


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