If the impulse response of a linear time-invariant system is e−tu(t + 1), then which of the following most accurately resembles the output of the system when the input is δ(t+1) − δ(t−1) ? (a) (b)
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If the impulse response of a linear time-invariant system is , then which of the following most accurately resembles the output of the system when the input is ?
(a)
(b)
![A length- M finite impulse response (FIR) can be described by the input-output relation y[n] = ∑m = 0 M−1 bmx[n−m], where bm are the filter coefficients. The coefficients can be optimized for a variety of applications, which makes FIR filters widely popular in digital signal processing. (a) Show that FIR filters are LTI systems. (b) Find the impulse response of an FIR filter using the input-output relationship above. (c) Use the impulse response to find the output of the system for the input x[n] = δ[n]+4 δ[n−1]+δ[n−2]. (d) Plot the impulse response of the system in MATLAB for M = 4 and bm = (1 /4)m. Using these same parameter values, plot the output obtained from (c) in MATLAB.](https://www.doubtrix.com/uploads/editor/3611692880sjsFQSNRwo.png)
![30 points. Determine the impulse response and frequency response of the system defined by the difference equation y[n] = ay[n−1] + x[n−1] assuming |a| < 1. Your answers will be a function of a. 3.30 points. Suppose you have an LTI system with impulse response h[n] = {1 n = −1, 0, 10 otherwise (a) Determine whether this LTI system is causal and/or stable. (b) Determine the output of this system to the input x[n] = cos(2πn/3 + π/6) for all n.](https://www.doubtrix.com/uploads/editor/4603478099LZJFvHUsCR.png)