Confusion understanding phase shift/delay? I am reading proakis, as shown highlighted in attached snap shot,there is '-'sign along with j,but still it is written (underlined red) implies shift in positive n direction,why and how positive?when we have negative sign with j?
Delay is the negative derivative of phase with respect to frequency. Phase that increases linearly negative as frequency increases will result in a positive (later) delay in time. This is given by the Fourier Transform property of time delay:
$$\mathscr{F} \{f(t-\tau)\} = e^{-j\tau \omega}F(\omega)$$
And is easy to prove from the Fourier Transform:
$$\mathscr{F} \{f(t-\tau)\} = \int_{-\infty}^\infty f(t-\tau)e^{-j\omega t}dt$$
$$ = \int_{-\infty}^\infty f(t-\tau)e^{-j\tau \omega}e^{j\tau \omega}e^{-j\omega t}dt$$
$$ = e^{-j\tau \omega}\int_{-\infty}^\infty f(t-\tau)e^{-j\omega (t-\tau)}dt$$
If we substitute $u=t-\tau$, then $du = dt$ and we get:
$$ = e^{-j\tau \omega}\int_{-\infty}^\infty f(u)e^{-j\omega u}du =e^{-j\tau \omega}F(\omega) $$
Also without going into the details of the Fourier Transform, consider how we subtract phase in the time domain to shift a sine wave in time to the right: