Consider a signal $x(t)$, which is input to a pulse shaping filter with transfer function $g_t(t)$:
$$x(t) = \sum_n d_n \delta(t-nT_s)$$
with $n$ an index from negative to positive infinity and $d_n$ random, equiprobable binary symbols such as 1 and -1 (antipodal) or 0 and 1 (on-off keying).
Well, the power spectral density of the filtered output is given by: $$S_{ss} = \frac{1}{T_s} E\left[|d_n|^2 \right] |G_t(f)|^2 $$ where $E[\cdot]$ denotes the expected value.
I'm confused by this formula since up to now I thought the power spectral density is given by the signal mutliplied with the absolute value squared of the transfer function. But why do I have to consider the expectation value in this case and why am I scaling by the sampling time?
Thx for any help!