# PSD of linearly modulated signal using autocorrelation?

Consider a signal $$v(t)$$ given by $$v(t)=\sum_{n=-\infty}^{\infty} b[n]p(t-nT).$$ Assume that $$b[n]$$ is uncorrelated with zero mean, i.e. $$\mathbb{E}[b[n]b^*[m]]=\mathbb{E}[|b[n]|^2]\delta[n-m]$$ and $$\mathbb{E}[b[n]]=0$$.

How can one show, using the autocorrelation of $$v(t)$$, that the PSD of $$v(t)$$ is given by the following?

$$S_{v}(f)=\frac{\mathbb{E}[|b[n]|^2]}{T}|P(f)|^2$$

• The derivation of this result is given in almost any textbook on digital communications. – Matt L. Nov 4 '20 at 17:12