In the system shown in Figure P5.63-1, assume that the input can be expressed in the for...

In the system shown in Figure P5.63-1, assume that the input can be expressed in the form

x[n] = s[n] cos(ω₀n).

Assume also that s[n] is lowpass and relatively narrowband; i.e., S(e^jω) = 0 for |ω| > Δ, with Δ very small and Δ << ω₀, so that X(e^jω) is narrowband around ω = ±ω₀.

(a) If |H(e^jω)| = 1 and ∠H(e^jω) is as illustrated in Figure P5.63-2, show that y[n] = s[n]cos(ω₀n − Φ₀).

(b) If |H(e^jω)| = 1 and ∠H(e^jω) is as illustrated in Figure P5.63-3, show that y[n] can be expressed in the form

y[n] = s[n − nd] cos(ω₀n − Φ₀ − ω₀n_d).

Show also that y[n] can be equivalently expressed as

y[n] = s[n − n_d] cos(ω₀n − Φ₁),

where −Φ₁ is the phase of H(e^jω)at ω = ω₀.

(c) The group delay associated with H(e^jω) is defined as

and the phase delay is defined as τ_ph(ω) = −(1/ω) ∠H(e^jω). Assume that |H(e^jω)| is unity over the bandwidth of x[n]. Based on your results in parts (a) and (b) and on the assumption that x[n] is narrowband, show that if τ_gr(ω0) and τ_ph(ω₀) are both integers, then

This equation shows that, for a narrowband signal x[n], ∠H(e^jω) effectively applies a delay of τ_gr(ω₀) to the envelope s[n] of x[n] and a delay of τ_ph(ω0) to the carrier cosω₀n.

(d) Referring to the discussion in Section 4.5 associated with noninteger delays of a sequence, how would you interpret the effect of group delay and phase delay if τ_gr(ω0)orτ_ph(ω₀) (or both) is not an integer?