we showed that an LTI system is stable if its impulse response {h[n]}n. is absolutely summable,...

 

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we showed that an LTI system is stable if its impulse response {h[n]}n. is absolutely summable, i.e. if Enez h[n]<co. Prove that this condition is also necessary for the system to be stable. (Hint: Assume the system's impulse response h[n] does not satisfy the condition you found in the previous part. Consider input signal¹ x[n] = h*[-n]/h-n] for n such that h[n] #0 otherwise. What can you say about the output of the system at n = 0?). (b) (1 point) Show that the frequency response of an LTI system exists provided that the system is stable. (Hint: Recall the definition of frequency response H(e) in terms of the impulse response. What condition is needed so that H(ejw)| <∞ ?). (c) (2 points) Find a sufficient condition on the impulse response of an LTI system such that for any input signal {x[n]}n, the output signal {y[n]}n is such that max ly[n] ≤ max|x[n]. ΠΕΖ nEZ (d) (2 points) Show that the condition you found above is also necessary. (Hint: Assume the system's. impulse response h[n] does not satisfy the condition you found in the previous part. Consider the same input signal as in Part (a). What can you say about the output of the system at n = 0?). ¹h*[n] denotes complex conjugate since in the general case we assume the impulse response is a complex signal. If that confuses you, you may assume for simplicity that the impulse response is reals so that h*[-n] =h[-n]. we showed that an LTI system is stable if its impulse response {h[n]}n. is absolutely summable, i.e. if Enez h[n]<co. Prove that this condition is also necessary for the system to be stable. (Hint: Assume the system's impulse response h[n] does not satisfy the condition you found in the previous part. Consider input signal¹ x[n] = h*[-n]/h-n] for n such that h[n] #0 otherwise. What can you say about the output of the system at n = 0?). (b) (1 point) Show that the frequency response of an LTI system exists provided that the system is stable. (Hint: Recall the definition of frequency response H(e) in terms of the impulse response. What condition is needed so that H(ejw)| <∞ ?). (c) (2 points) Find a sufficient condition on the impulse response of an LTI system such that for any input signal {x[n]}n, the output signal {y[n]}n is such that max ly[n] ≤ max|x[n]. ΠΕΖ nEZ (d) (2 points) Show that the condition you found above is also necessary. (Hint: Assume the system's. impulse response h[n] does not satisfy the condition you found in the previous part. Consider the same input signal as in Part (a). What can you say about the output of the system at n = 0?). ¹h*[n] denotes complex conjugate since in the general case we assume the impulse response is a complex signal. If that confuses you, you may assume for simplicity that the impulse response is reals so that h*[-n] =h[-n].

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a Suppose the impulse response hn does not satisfy the absolute summability condition Consider the i... View the full answer