[SOLVED] CAN207 Continuous and Discrete Time Signals and Systems Assignment 1 SQL

CAN207

Continuous and Discrete Time Signals and Systems

Assignment 1

Assignment 1 :     CT Signals and Systems

Deadline:                           Nov. 11th, 9:00 a.m.

Submission:                      Submit the electronic version to Learning Mall.

Information:                     This assignment takes 15% in the total mark.

Late submission:              5% each day, less than 1 day is counted as 1 day.

Submissions later than 5 working days won’t be accepted.

Question 1 ( L3-4) 20 marks

(a)           For the signal x(t) shown below, plot 2x(2t + 2)

(b)           Express the signal shown below using scaled and time shifted unit step function u(t).

(c)           For each of the following signals, determine whether they are even, odd or neither.

I)            x(t) = sin(3t − 2/π)

II)           x(t) = u(t) − 0.5

(d)           For the given signals, if the signal is periodic, find its fundamental period

and its fundamental frequency; otherwise, prove that the signal is not periodic.

I)          x(t) = 4cos(4t + 40°) + 3e−j12t

II)         x(t) = cos(2πt) + sin(6t)

(e)           Determine whether the following signals are power signal, energy signal or neither:

I)          x(t) = e−2tu(t)

II)         x(t) = ej(2t+π/4)

Question 2 ( L5-6) 20 marks

(a)           For  the  systems given  below, decide whether they are causal, stable, linear and time-invariant? Conclusions only.

I)           Input-output relationship: y[n] = x [3 − 2n];

II)          Input-output relationship: y(t) = cos(πt)x(t);

III)          Impulse response: ℎ(t) = u(t + 3) − u(t − 3);

IV)          Impulse response: ℎ[n] =  5nu [ − n].

(b)          Suppose the following systems take x(t)  as the  input and y(t)  as the

corresponding output. Find the impulse response ℎ(t).

I)           y(t) = x(t − 7);

II)           y(t) = x(t − 7)d;

(c)           Consider the LTI system shown as below:

Express  the  system   impulse  response  as  a  function  of  the   impulse responses of the subsystems.

(d)          Suppose the systems with impulse response ℎ(t) take x(t) as the input.

Find the output y(t).

x(t) = u(1 − t) and ℎ(t) = e−tu(t − 2);

(e)           For the convolution between two time-domain signals f(t) and g(t), the

diferentiation property is:

Question 3 ( L7-9) 20 marks

(a)           Find the Fourier coefficients of the exponential form.

x(t) = 2sin24t + cos4t and

(b)            Calculate the Fourier coefficients for each signal:

(c)           A signal x(t) has a Fourier transform. X(w). 4

Calculate the Fourier transform. of x(at)cos(w0 t), with 0 < a < 1.

(d)          The magnitude and phase spectrum of a LTI system are plotted below:

If input signal is x(t) =  1 + 2cos(2πt), find the corresponding output.

Question 4 ( L10-11) 20 marks

(a)           A stable system is characterized by the transfer function: 10

I)           Draw the zero-pole plot of the system;

II)          Determine the ROC of the system;

III)         Find the impulse response of the system;

IV)        Decide   whether   the   system’s   magnitude   response   is lowpass, highpass, bandpass or bandstop.

(b)          The characteristic equation of a continuous-time causal system is given:

D(S) = S2  + 2S + a

For the system to be stable, decide the range of the real value a in the equation.

(c)           Given the relationships:

Use Fourier transform. properties to show that g(t)  has the form. like: g(t) = AY(Bt), and determine the values of A and B.

Question 5 ( L12-13) 20 marks

(a)           The following differential equation is used to model a RLC circuit whose input is x(t) = e−tu(t):

y” (t) + 5y’ (t) + 6y(t) = x(t)

With the initial conditions:

y(0− ) = 1   and   y’ (0− ) = 0

Solve the differential equation in time domain to get:

i) Zero-input response;

ii) Zero-state response;

iii) Overall response.

(b)          Solve sub-question c) in frequency domain.