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c) Find the impulse response h[n] for the following system and decide if it is causal 1. y[n] = a₁x[n] + a₂x[n − 1] − by[n 1] with the initial condition y[n] = 0,n<0 2. y[n] = (x[n+ 1] + x[n])

c) Find the impulse response h[n] for the following system and decide if it is causal 1. y[n] = a₁x[n] + a₂x[n − 1] − by[n 1] with the initial condition y[n] = 0,n<0 2. y[n] = (x[n+ 1] + x[n])

Introductory Circuit Analysis (13th Edition)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
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c) Find the impulse response h[n] for the following system and decide if it is causal d) e) 1. y[n] = a₁x[n] + a₂x[n − 1] − by[n 1] with the initial condition y[n] = 0,n<0 2. y[n] = (x[n+ 1] + x[n]) Determine the output y[n] of the system with impulse response 0≤n≤4 elsewhere h[n] = { to the input (an,-3≤n≤5 10, elsewhere Consider designing a DT inverse system to eliminate the distortion associated with an undesired echo in a data transmission problem. The echo causes the input signal x[n] to be attenuated by a constant factor a and delayed by one time unit. The attenuated signal is represented by the following equation y[n] = x[n] + ax[n 1] If H{x[n]} = y[n], find a causal system H-¹ such that H-¹{y[n]} = x[n]. That is, if we receive a distorted signal y[n], we shall be able to recover the original transmitted signal x[n]. Hint: Let the impulse response of H be denoted h[n] and the impulse response of H-1 be denoted h¹ [n]. The following relationship must hold x[n] = h[n] h¹[n] = 8[n] Then the problem of finding H-¹ reduces to finding h¹ [n] such that the above mentioned equation holds. Try to find h[n] first.
Transcribed Image Text:c) Find the impulse response h[n] for the following system and decide if it is causal d) e) 1. y[n] = a₁x[n] + a₂x[n − 1] − by[n 1] with the initial condition y[n] = 0,n<0 2. y[n] = (x[n+ 1] + x[n]) Determine the output y[n] of the system with impulse response 0≤n≤4 elsewhere h[n] = { to the input (an,-3≤n≤5 10, elsewhere Consider designing a DT inverse system to eliminate the distortion associated with an undesired echo in a data transmission problem. The echo causes the input signal x[n] to be attenuated by a constant factor a and delayed by one time unit. The attenuated signal is represented by the following equation y[n] = x[n] + ax[n 1] If H{x[n]} = y[n], find a causal system H-¹ such that H-¹{y[n]} = x[n]. That is, if we receive a distorted signal y[n], we shall be able to recover the original transmitted signal x[n]. Hint: Let the impulse response of H be denoted h[n] and the impulse response of H-1 be denoted h¹ [n]. The following relationship must hold x[n] = h[n] h¹[n] = 8[n] Then the problem of finding H-¹ reduces to finding h¹ [n] such that the above mentioned equation holds. Try to find h[n] first.
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