HW7.2 (LTI Processing of Periodic Signals) Consider a periodic, discrete-time sig- nal x : Z → R having the discrete-time Fourier series (DTFS) expansion x[n] = Σ Xk ejkwon k=(p) where wo denotes the fundamental frequency of the signal; if p is the period of x, then wo= 2π/p. Suppose x is the input signal applied to a linear time-invariant (LTI) system charac- terized by the impulse response h: Z → R and corresponding frequency response H, where H(e¹w) = Σ h[n] e¯jwn, Vw. n=-∞ Let y be the corresponding output signal. (a) Prove that the output signal y is periodic; that is, show that if x[n+p] = x[n], then y[n +p] = y[n]. (b) Let the DTFS expansion of the output signal y be y[n] = [Yekwon k=(p) (i) Express the output-signal DTFS coefficients Yr in terms of the input sig- nal DTFS X and the frequency response H. (ii) Suppose the impulse response of the LTI system is given by h[n] [nno], where no € Z. Explicitly determine the output-signal DTFS coefficients Yk in terms of the input-signal DTFS coefficients Xk. =
HW7.2 (LTI Processing of Periodic Signals) Consider a periodic, discrete-time sig- nal x : Z → R having the discrete-time Fourier series (DTFS) expansion x[n] = Σ Xk ejkwon k=(p) where wo denotes the fundamental frequency of the signal; if p is the period of x, then wo= 2π/p. Suppose x is the input signal applied to a linear time-invariant (LTI) system charac- terized by the impulse response h: Z → R and corresponding frequency response H, where H(e¹w) = Σ h[n] e¯jwn, Vw. n=-∞ Let y be the corresponding output signal. (a) Prove that the output signal y is periodic; that is, show that if x[n+p] = x[n], then y[n +p] = y[n]. (b) Let the DTFS expansion of the output signal y be y[n] = [Yekwon k=(p) (i) Express the output-signal DTFS coefficients Yr in terms of the input sig- nal DTFS X and the frequency response H. (ii) Suppose the impulse response of the LTI system is given by h[n] [nno], where no € Z. Explicitly determine the output-signal DTFS coefficients Yk in terms of the input-signal DTFS coefficients Xk. =
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Publisher:Robert L. Boylestad
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![HW7.2 (LTI Processing of Periodic Signals) Consider a periodic, discrete-time sig-
nal x : Z → R having the discrete-time Fourier series (DTFS) expansion
x[n] = Σ Xk e³kwon,
k=(p)
where wò denotes the fundamental frequency of the signal; if p is the period of x,
then wo = 2π/p.
Suppose x is the input signal applied to a linear time-invariant (LTI) system charac-
terized by the impulse response h: Z → R and corresponding frequency response
H, where
∞
H(ew) = h[n] e¯jwn, Vw.
n=-∞
Let y be the corresponding output signal.
(a) Prove that the output signal y is periodic; that is, show that if x[n+p] = x[n],
then y[n+p] = y[n].
(b) Let the DTFS expansion of the output signal y be
y[n]
=
ΣYk jkwon
k=(p)
(i) Express the output-signal DTFS coefficients Yk in terms of the input sig-
nal DTFS Xk and the frequency response H.
=
(ii) Suppose the impulse response of the LTI system is given by h[n]
[nno], where no € Z. Explicitly determine the output-signal DTFS
coefficients Yk in terms of the input-signal DTFS coefficients Xk.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5ec99b66-acf3-492e-a494-05a1857e861e%2Fe49ff678-3330-42df-a898-ba7fefbd1850%2F1b1kiao_processed.png&w=3840&q=75)
Transcribed Image Text:HW7.2 (LTI Processing of Periodic Signals) Consider a periodic, discrete-time sig-
nal x : Z → R having the discrete-time Fourier series (DTFS) expansion
x[n] = Σ Xk e³kwon,
k=(p)
where wò denotes the fundamental frequency of the signal; if p is the period of x,
then wo = 2π/p.
Suppose x is the input signal applied to a linear time-invariant (LTI) system charac-
terized by the impulse response h: Z → R and corresponding frequency response
H, where
∞
H(ew) = h[n] e¯jwn, Vw.
n=-∞
Let y be the corresponding output signal.
(a) Prove that the output signal y is periodic; that is, show that if x[n+p] = x[n],
then y[n+p] = y[n].
(b) Let the DTFS expansion of the output signal y be
y[n]
=
ΣYk jkwon
k=(p)
(i) Express the output-signal DTFS coefficients Yk in terms of the input sig-
nal DTFS Xk and the frequency response H.
=
(ii) Suppose the impulse response of the LTI system is given by h[n]
[nno], where no € Z. Explicitly determine the output-signal DTFS
coefficients Yk in terms of the input-signal DTFS coefficients Xk.
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