Question 7.5: Frequency response of a system Graph the magnitude and phas......
Frequency response of a system
Graph the magnitude and phase of the frequency response of the system in Figure 7.17. If the system is excited by a signal x[n] = sin(Ω_0n), find and graph the response y[n] for Ω_0 = π/4, π/2, 3π/4.
The difference equation describing the system is y[n] + 0.7 y[n − 1] = x[n] and the impulse response is h[n] = (−0.7)^n u[n]. The frequency response is the Fourier transform of the impulse response. We can use the DTFT pair
\alpha ^{n} u[u] \xleftrightarrow[]{\mathcal{F} } \frac{1}{1-\alpha e^{-j\Omega }}to get
h[n]=(-0.7)^n u[n] \xleftrightarrow[]{\mathcal{F} } H(e^{j\Omega })\frac{1}{1+0.7 e^{-j\Omega }}Since the frequency response is periodic in Ω with period 2π, a range −π ≤ Ω < π will show all the frequency-response behavior. At Ω= 0 the frequency response is H(e ^{j0} ) = 0.5882. At Ω= ±π the frequency response is H(e^{± jπ}) = 3.333. The response at Ω= Ω_0 is
y[n]=\left|H(e^{j\Omega _o})\right| sin(\Omega _on+\measuredangle H(e^{j\Omega _o})),
(Figure 7.18).
