The input to a causal LTI system isx[n] = u[-n – 1]+ G) «In).The z-transform of the output of this system isY(2) =(1– 4-1) (1 + =-')(a) Determine H(2), the z-transform of the system impulse response. Be sure to specifythe ROC.(b) What is the ROC for Y(z)?(c) Determine y[n].

Part a: Write the expression for the sum of the geometric progression of the infinite terms, Here,…

Answered: The input to a causal LTI system is…

Introductory Circuit Analysis (13th Edition)

13th Edition

ISBN:9780133923605

Author:Robert L. Boylestad

Publisher:Robert L. Boylestad

Chapter1: Introduction

Section: Chapter Questions

Problem 1P: Visit your local library (at school or home) and describe the extent to which it provides literature...

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2. The signal y[n] is the output of an LTI system with impulse response h[n] for a given input x[n]. Throughout problem, assume that y[n] is sable and has a z- transform Y(z) with the pole-zero diagram shown in Figure-1, and the signal x[n] is stable and has the pole-zero diagram shown Figure-2. (2)urz Im(z) 1.5 1 0.5 0 -0.5 -1 -1.5 1 0.5 0 -0.5 -1 -1 -1 ديا أحد 3 0 6-17 *-12 1 1 1 Re(z) ↑ Figure-1 11 4 2 0 Re(z) ↑ Figure-2 2 2 z-plane z-plane
1. Find the z-transform of the following signals: (a) a [n] = (-0.9)" u [n] (b) a [n] = (-0.9)"-'u [n – 1] (c) x [n] = -n (-0.9)" u [–n – (d) æ [n] = (-0.9)"-2 u [-n+ 1] + (0.8)"-lu [n – 1] (e) x [n] = (0.9)" cos (n) u [n] 2. Find the inverse z-transform of the following signals (a) X (2) = ; ROC = |2| > 1 (in closed form) +1: (b) X (z) = ; ROC = |2| > 1 (in power series form) (c) X (2) = , z3+1: ROC = |z| < 1 (in power series form) (d) X (z) = 23 –1 (z-0.5)²(z+1.5) ' ROC = (0.5 < |z| < 1.5) (in closed form) 3. x [n] = -6 [n + 2] + 28 [n + 1] + 8 [n] + & [n – 1] – 2 [n – 2] (a) Find the z-transform of x [n] (b) Find the z-transform of x [-n] (c) Find the Fourier-transform of x [n] (d) Find the Fourier-transform of x [-2n]
201 a) discrete time signal is given by x(n)=sin( 1024 What is the fundamental period (nyquest rate)? And what is the largest frequencies to avoid aliasing? b) Determine the I.D.F.T of the signal given by 1-4/14+4 cos sin w e) Find the Z.Transform of the following signal and its ROC x(n)=-a²w(-#-3)
Q5: Find the Z-transform of the following: Hint: akf(k) = F (4) a) x(k) = 2k ek , b) x(k) = e-k sin wk , c) x(k) = 3-k sin wk %3D d) x(k) = 2k sin wk ,e)x(k) = ek sin wk , f)x(k) = ek k ak.
4. (a) For the discrete-time signal given by x[n] = (1/6)*u[n-2] (i) Using the Z-transform definition, find its Z-transform, X(z). (ii) Plot the pole/zero diagram, shading the ROC. (b) For the two systems with impulse responses hi[n] = (0.5)" u[n] h2[n] = (2)ªu[n] Write down their transfer functions, H1(z) and H2(z). Don't forget to include the corresponding ROCS. and (i) If we connect H1(z) and H2(z) in cascade, the resulting cascade interconnection will have a combined transfer function calculated by (ii) H(z) = H1(z).H2(z),
Find Z Transform and ROC for each of the following discrete time signals using properties (Attempt only four) a) x(n)= u(n-2) b) x(n) = 3.2"u(-n) c) x(n)=nu(n-1) d) x(n)=(n+0.5)(0.3)" u(n) e) x(n)= u(n)u(n) f) x(n)=eu(n-1)
-4 -? C- Find the Z-transform of x(n)-D{...,.0,0,2,-1, 7,3,6.5,4,-8,0,0,..) Please resolvel quickly and with a clear and good line. Q2) A- Categorize the signal x(n)-{...,0,0,2,-1, 7,3,6.5,4,-8,0,0,...) as: 1- Deterministic or random signal. N 12- Periodic or non-periodic signal, and If periodic find its period, and its angular frequency. \ 3- Energy or power signal, and find the Energy or averaged power of it
3. A system defined by h(t) = 5e-5tu(t) is driven by a sinusoidal signal with frequency 5 rps and amplitude 4 V. How much is the amplitude of the output signal in volt? (hint: use L-transform of h(t))
Transfer function ? (?) = 2/1 + 2 ∗ 0.5?-1 + 0.52 ?-2, steady state response for unit step function input… ..
x[n] is given as shown below. What is the Z transform of x[n] What is the z-transform of the signal x(n)-na"u(n)? a) az (1+az 1)? (az) ! b) (1-(a2) ')? az 1 c) (1-az )2 az d) (1-(az) ')² e) a and c are correct a b O c d. O e
a) Determine the z-transform of the following signal. x[n] = alml + 8[n] b) What is the condition on a for the Z-transform to exist?
Question 2: Consider a Linear Time-Invariant (LTI) system with input x(n) and output y(n) related by the following difference equation 5 3 yln] -yln – 1] + yln – 2] = x[n] a) Determine the Z-transform. b) Propose the impulse response for the following system: i) ii) iii) causal stable neither stable nor causal Find the region of convergence (ROC) of the Z-transform in each of the above cases.

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The input to a causal LTI system is x[n] = u[-n – 1]+ G) «In). The z-transform of the output of this system is Y(2) = (1– 4-1) (1 + =-') (a) Determine H(2), the z-transform of the system impulse response. Be sure to specify the ROC. (b) What is the ROC for Y(z)? (c) Determine y[n].