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.
![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)
![3.41. In Figure P3.41-1, h[n] is the impulse response of the LTI system within the inner box. The input to system h[n] is v[n], and the output is w[n]. The z-transform of h[n], H(z), exists in the following region of convergence: . 0 < rmin < |z| < rmax < ∞. (a) Can the LTI system with impulse response h[n] be BIBO stable? If so, determine inequality constraints on rmin and rmax such that it is stable. If not, briefly explain why. (b) Is the overall system (in the large box, with input x[n] and output y[n] ) LTI? If so, find its impulse response g[n]. If not, briefly explain why. (c) Can the overall system be BIBO stable? If so, determine inequality constraints relating α, rmin, and rmax such that it is stable. If not, briefly explain why. Figure P3.41-1](https://www.doubtrix.com/uploads/editor/8252444907vePtflDPFT.png)
![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)
![Consider the open-loop circuit (on the left) and closed-loop circuit (on the right) shown in Figure 1 below. In this problem, the open-loop circuit (on the left) will be used to analyze the open-loop behavior of the closed-loop circuit (on the right). closed - loof ciruit Figure 1. Open-loop (left) and closed-loop (right) circuits for Problem 1. Here are the problem parameters:The NMOS devices have gmn = 6 mA/V and rdsn = 25 kΩ. The PMOS devices have gmp = 3 mA/V and rdsp = 15 kΩ. Resistor values are R1 = 1 kΩ and R2 = 3 kΩ. a) Referring to the open-loop circuit of Figure 1, determine the numerical open-loop gain Av. OL = Vout /Vin . For the purposes of this analysis, as we have done a million times before (reminder: this is the 2 nd semester of electronics, so this should not be your first electronics rodeo), you should deactivate all sources in the circuit, except Vin. . b) Referring to the open-loop circuit of Figure 1, determine the numerical open-loop output resistance Rout, oL. , i. e. , the resistance seen looking in at the output node Vout. . You should deactivate all sources in the circuit; feel free to add an appropriate test source(s), if you think you need to do that. c) Referring to the open-loop circuit of Figure 1, determine the numerical loop gain from the test source to the output node: LG = |Vout /Vtest | For the purposes of this analysis, you should deactivate all sources in the circuit, except Vtest. . Also, note that the way the loop gain is defined, it will be a positive-valued quantity. Explain why we want to use a positive loop gain for determining the closed-loop performance [see, e. g. , parts (d) and (e) below]. Your answer for this part should be both i) specific to this problem and ii) generalized to any closed-loop circuit with negative feedback. You are strongly encouraged to consider incorporating a generalized block diagram to help strengthen your argument(s). d) Referring to the closed-loop circuit of Figure 1, determine the numerical closed-loop gain Av, CL = Vout /Vin . Rather than solving the closed-loop circuit of Figure 1 directly for the closed-loop gain, you should instead utilize the results of the previous parts to determine Av, cL. Here, I need to remind you that you should do as you have been (hopefully) doing on the previous parts: first, state the equation you will use to determine Av, cL in terms of, e. g. , Av. ol. and/or Rout, ol and/or LG (i. e. , resist the temptation to dive straight into the numerical values, but feel free to write your equation in terms of Av, ol and/or Rout, ol. and/or LG instead of gmn, gmp, rdsn, rdpp , R1, R2 ), and then plug the numbers in to your equation, and then box your numerical answer along with all the things that we have expected of your numerical quantities since day one. Make it really easy for me to find the equation you used (same story for all of the parts of this problem). e) Referring to the closed-loop circuit of Figure 1, determine the numerical closed-loop output resistance Rout, CL. Rather than solving the closed-loop circuit of Figure 1 directly for the closed-loop output resistance, you should instead follow the same guidance that was given for part (d) of this problem.](https://www.doubtrix.com/uploads/editor/5672518564OTwrDIyJcJ.jpg)