f Sketch the pole zero diagram of the system g Is the system a low pass filter

F sketch the pole zero diagram of the system g is the

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(f) Sketch the pole-zero diagram of the system. (g) Is the system a low-pass filter, high-pass filter, or neither? 1.9.9 A causal discrete-time LTI system is implemented with the di erence equation y ( n ) = 3 x ( n ) + 3 2 y ( n - 1) . (a) Find the output signal when the input signal is x ( n ) = 3 (2) n u ( n ) . Show your work. (b) Is the system stable or unstable? (c) Sketch the pole-zero diagram of this system. 1.9.10 A causal discrete-time LTI system is described by the equation y ( n ) = 1 3 x ( n ) + 1 3 x ( n - 1) + 1 3 x ( n - 2) where x is the input signal, and y the output signal. 63
(a) Sketch the impulse response of the system. (b) What is the dc gain of the system? (Find H f (0).) (c) Sketch the output of the system when the input x ( n ) is the constant unity signal, x ( n ) = 1. (d) Sketch the output of the system when the input x ( n ) is the unit step signal, x ( n ) = u ( n ). (e) Find the value of the frequency response at ! = . (Find H f ( ).) (f) Find the output of the system produced by the input x ( n ) = ( - 1) n . (g) How many zeros does the transfer function H ( z ) have? (h) Find the value of the frequency response at ! = 2 3 . (Find H f (2 / 3).) (i) Find the poles and zeros of H ( z ); and sketch the pole/zero diagram. (j) Find the output of the system produced by the input x ( n ) = cos ( 2 3 n ) . 1.9.11 Two-Point Moving Average. A discrete-time LTI system has impulse response h ( n ) = 0 . 5 δ ( n ) + 0 . 5 δ ( n - 1) . (a) Sketch the impulse response h ( n ). (b) What di erence equation implements this system? (c) Sketch the pole-zero diagram of this system. (d) Find the frequency resposnse H f ( ! ). Find simple expressions for | H f ( ! ) | and \ H f ( ! ) and sketch them. (e) Is this a lowpass, highpass, or bandpass filter? (f) Find the output signal y ( n ) when the input signal is x ( n ) = u ( n ). Also, x ( n ) = cos( ! o n ) u ( n ) for what value of ! o ? (g) Find the output signal y ( n ) when the input signal is x ( n ) = ( - 1) n u ( n ). Also, x ( n ) = cos( ! o n ) u ( n ) for what value of ! o ? 1.9.12 A causal discrete-time LTI system is described by the equation y ( n ) = 1 4 3 X k =0 x ( n - k ) where x is the input signal, and y the output signal. (a) Sketch the impulse response of the system. (b) How many zeros does the transfer function H ( z ) have? (c) What is the dc gain of the system? (Find H f (0).) (d) Find the value of the frequency response at ! = 0 . 5 . (Find H f (0 . 5 ).) (e) Find the value of the frequency response at ! = . (Find H f ( ).) (f) Based on (b), (d) and (e), find the zeros of H ( z ); and sketch the pole/zero diagram. (g) Based on the pole/zero diagram, sketch the frequency response magnitude | H f ( ! ) | . 1.9.13 Four-Point Moving Average. A discrete-time LTI system has impulse response h ( n ) = 0 . 25 δ ( n ) + 0 . 25 δ ( n - 1) + 0 . 25 δ ( n - 2) + 0 . 25 δ ( n - 3) (a) Sketch h ( n ). (b) What di erence equation implements this system? (c) Sketch the pole-zero diagram of this system. (d) Find the frequency resposnse H f ( ! ). Find simple expressions for | H f ( ! ) | and \ H f ( ! ) and sketch them. (e) Is this a lowpass, highpass, or bandpass filter?

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