A causal discrete-time LTI system is described by the equation
A causal discrete-time LTI system is described by the equation
where z is the input signal, and y the output signal
y(n) = 1/3x(n) + 1/3x(n -1) + 1/3x(n - 2)
(a) Sketch the impulse response of the system.
(b) What is the dc gain of the system? (Find Hf(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 w = T. (Find Hf(π).)
(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 w = - π. (Find Hf(2π/3).)
(i) Find the poles and zeros of H(z); and sketch the pole/zero diagram.
(i) Find the output of the system produced by the input x(n) = cos (-πn).
Homework Answers
For a causal LTI discrete-time system described by the difference equation:
For a causal LTI discrete-time system described by the difference equation: y[n] + y[n – 1] = x[n] a) Find the transfer function H(z).b) Find poles and zeros and then mark them on the z-plane (pole-zero plot). Is this system BIBO? c) Find its impulse response h[n]. d) Draw the z-domain block diagram (using the unit delay block z-1) of the discrete-time system. e) Find the output y[n] for input x[n] = 10 u[n] if all initial conditions are 0.
1. A causal LTI system is implemented by the difference equation y(n) = 2r(n) - 0.5...
1. A causal LTI system is implemented by the difference equation y(n) = 2r(n) - 0.5 y(n-1). (a) Find the frequency response H/(w) of the system. (b) Plot the pole-zero diagram of the system. Based on the pole zero diagram, roughly sketch the frequency response magnitude |H'(w). (c) Indicate on your sketch of H w , its exact values at w=0, 0.5, and . (d) Find the output signal y(n) produced by the input signal (n) = 3 + cos(0.5...
2.6.1 Consider a causal continuous-time LTI system described by the differential equation u"(t) +...
2.6.1-2.6.62.6.1 Consider a causal contimuous-time LTI system described by the differential equation$$ y^{\prime \prime}(t)+y(t)=x(t) $$(a) Find the transfer function \(H(s)\), its \(R O C\), and its poles.(b) Find the impulse response \(h(t)\).(c) Classify the system as stable/unstable.(d) Find the step response of the system.2.6.2 Given the impulse response of a continuous-time LTI system, find the transfer function \(H(s),\) the \(\mathrm{ROC}\) of \(H(s)\), and the poles of the system. Also find the differential equation describing each system.(a) \(h(t)=\sin (3 t) u(t)\)(b)...
a causal discrete time LTI system is implemented using the difference equation y(n)-0.5y(n-1)=x(n)+x(n-1) where x(n) is...
a causal discrete time LTI system is implemented using the difference equation y(n)-0.5y(n-1)=x(n)+x(n-1) where x(n) is the input signal and y(n) the output signal. Find and sketch the impulse response of the system
A discrete-time LTI system with a cumpling frequency of 8kHz is shown in Figure 4.1. The...
A discrete-time LTI system with a cumpling frequency of 8kHz is shown in Figure 4.1. The rectangular boxes with the label z provide one sample period delay to their input signals. The circular components are adders or subtractors. The triangular components provide linear gain factors of or by, where i is 0, 1 or 2 a) Derive the system transfer function H(z). b) Find the difference equation relating the output y[n) and input x[n). c) Given that the gain values...
(2) Consider the causal discrete-time LTI system with an input r (n) and an output y(n)...
(2) Consider the causal discrete-time LTI system with an input r (n) and an output y(n) as shown in Figure 1, where K 6 (constant), system #1 is described by its impulse response: h(n) = -36(n) + 0.48(n- 1)+8.26(n-2), and system # 2 has the difference equation given by: y(n)+0.1y(n-1)+0.3y(n-2)- 2a(n). (a) Determine the corresponding difference equation of the system #1. Hence, write its fre- quency response. (b) Find the frequency response of system #2. 1 system #1 system #2...
please show steps 4. (25 points) Laplace and LCCDE Systems Consider an LTI system with input-output...
please show steps
4. (25 points) Laplace and LCCDE Systems Consider an LTI system with input-output relation described by the LCCDE: -2y(t) - y0) + 3x(t) + deco (O) = (a) (5 pts) Find the transfer function H(s) and write it in factored form. (b) (5 pt) Sketch the ROC corresponding to H(s) if it is known the system is causal. Mark the poles and zeros. (c) (5 pts) Sketch the ROC corresponding to H(s) if it is known the...
The frequency response Hf(w) of a discrete-time LTI system is as shown.
The frequency response Hf(w) of a discrete-time LTI system is as shown. Hf(w) is real-valued so the phase is 0. Find the output y(n) when the input x(n) is x(n) = 1+cos(0.3πn). Put y(n) in simplest real form (your answer should not contain j)
Bonus Question) A discrete-time LTI system with a sampling frequency of Ukm2 is shown in the...
Bonus Question) A discrete-time LTI system with a sampling frequency of Ukm2 is shown in the following Figure. The rectangular boxes with the label z provide one sample period delay to their input signals. The circular components are adders or subtractors. The triangular components provide linear vain factors of ar or bi where i is 0,1 or 2. i) Derive the system transfer function H(2). ü) Find the difference equation relating the output y[n] and input x[n]. iii) Given that...
4. (5 pts) Consider a discrete-time LTI system T that generates an output y[n] according to...
4. (5 pts) Consider a discrete-time LTI system T that generates an output y[n] according to a2 y[n] bx[n] – ay[n – 1] - *[n – 2] where a, b are non-negative real constants. (a) (2 pts) Find the poles of the z-transform of the impulse response h[n] of T. (b) (3 pts) Let H(ejl be the frequency response of T. Find a, b so that the system is causal and stable, |H(1)| = |H(ejº)] = 0.04, and |H(-1)] =...
1. A causal LTI system is implemented by the difference equation y(n) = 2r(n) - 0.5 y(n-1). (a) Find the frequency response H/(w) of the system. (b) Plot the pole-zero diagram of the system. Based on the pole zero diagram, roughly sketch the frequency response magnitude |H'(w). (c) Indicate on your sketch of H w , its exact values at w=0, 0.5, and . (d) Find the output signal y(n) produced by the input signal (n) = 3 + cos(0.5...
2.6.1-2.6.62.6.1 Consider a causal contimuous-time LTI system described by the differential equation$$ y^{\prime \prime}(t)+y(t)=x(t) $$(a) Find the transfer function \(H(s)\), its \(R O C\), and its poles.(b) Find the impulse response \(h(t)\).(c) Classify the system as stable/unstable.(d) Find the step response of the system.2.6.2 Given the impulse response of a continuous-time LTI system, find the transfer function \(H(s),\) the \(\mathrm{ROC}\) of \(H(s)\), and the poles of the system. Also find the differential equation describing each system.(a) \(h(t)=\sin (3 t) u(t)\)(b)...
A discrete-time LTI system with a cumpling frequency of 8kHz is shown in Figure 4.1. The rectangular boxes with the label z provide one sample period delay to their input signals. The circular components are adders or subtractors. The triangular components provide linear gain factors of or by, where i is 0, 1 or 2 a) Derive the system transfer function H(z). b) Find the difference equation relating the output y[n) and input x[n). c) Given that the gain values...
(2) Consider the causal discrete-time LTI system with an input r (n) and an output y(n) as shown in Figure 1, where K 6 (constant), system #1 is described by its impulse response: h(n) = -36(n) + 0.48(n- 1)+8.26(n-2), and system # 2 has the difference equation given by: y(n)+0.1y(n-1)+0.3y(n-2)- 2a(n). (a) Determine the corresponding difference equation of the system #1. Hence, write its fre- quency response. (b) Find the frequency response of system #2. 1 system #1 system #2...
please show steps
4. (25 points) Laplace and LCCDE Systems Consider an LTI system with input-output relation described by the LCCDE: -2y(t) - y0) + 3x(t) + deco (O) = (a) (5 pts) Find the transfer function H(s) and write it in factored form. (b) (5 pt) Sketch the ROC corresponding to H(s) if it is known the system is causal. Mark the poles and zeros. (c) (5 pts) Sketch the ROC corresponding to H(s) if it is known the...
Bonus Question) A discrete-time LTI system with a sampling frequency of Ukm2 is shown in the following Figure. The rectangular boxes with the label z provide one sample period delay to their input signals. The circular components are adders or subtractors. The triangular components provide linear vain factors of ar or bi where i is 0,1 or 2. i) Derive the system transfer function H(2). ü) Find the difference equation relating the output y[n] and input x[n]. iii) Given that...
4. (5 pts) Consider a discrete-time LTI system T that generates an output y[n] according to a2 y[n] bx[n] – ay[n – 1] - *[n – 2] where a, b are non-negative real constants. (a) (2 pts) Find the poles of the z-transform of the impulse response h[n] of T. (b) (3 pts) Let H(ejl be the frequency response of T. Find a, b so that the system is causal and stable, |H(1)| = |H(ejº)] = 0.04, and |H(-1)] =...
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