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.