Drill Problem $6\mathrm{.}4$ Equation $(6\mathrm{.}17)$ defines the raised$-$cosine pulse spectrum $P(f)$ as realvalued and therefore zero delay. In practice, every transmission system experiences some finite delay. To accommodate this practicality, we may associate with $P(f)$ a linear phase characteristic over the frequency band $0|f|2B_0-f_1.$

$($a$)$ Show that this modification of $P(f)$ introduces a finite delay into its inverse Fourier transform, namely, the pulse shape $p(t).$

$($b$)$ According to Eq$.(6\mathrm{.}19),p(t)$ represents a non$-$causal time response. The delay introduced into $p(t)$ through the modification of $P(f)$ has also a beneficial effect, tending to make $p(t)$ essentially causal. For this to happen, however, the delay must not be less than a certain value dependent on the roll$-$off factor $.$ Suggest suitable values for the delay for $=0,\frac{1}{2},$ and $1.$