信号系统代写 | Systems and Computer Engineering SYSC 3500

[1] (10 marks) 𝑥[𝑛] = [1 -1 0 0 0 1] is a periodic sequence, with period length N=6.

(a) Find the DFT of this periodic sequence, 𝑋(𝜔)

(b) Sketch the magnitude and phase of 𝑋(𝜔).

[2] (10 marks) x(t) is a continuous time (CT), non periodic signal as shown in the figure.

time

(a) Is x(t) energy or power signal, or neither?

(b) If x(t) is an energy signal, compute the energy and if it is power signal

compute the average power.

(c) Find the Fourier transform of x(t) when A=T=1.

[3] (10 marks) x(t) is a continuous time (CT), Periodic signal, with a period T. The

equation for a single period is: 𝑥(𝑡) = {

𝐴 cos (

2𝜋𝑡

𝑇

) ; −

𝑇

4

≤ 𝑡 ≤

𝑇

4

0 ; 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

Find an expression for the fifth harmonic frequency when A=1 and T=8

(Hint: to evaluate the integration express the cosine term as two exponential terms

using Euler’s rule)

[4] (10 marks) x(t) is the input to a LTI system with an impulse response h(t).

𝑥(𝑡) = 𝑢(𝑡) − 𝑢(𝑡 − 1)

ℎ(𝑡) = 𝑢(𝑡 − 1) − 𝑢(𝑡 − 2)

Use the convolution integral to evaluate and sketch the output y(t).

[5] (10 marks) The input to an LTI system is 𝑥(𝑡) = sin (8𝑡 +

𝜋

3

) + cos (12𝑡) . The

system transfer function is 𝐻(𝜔) =

100

8+𝑗𝜔

(a) What is the output y(t)?

(b) What is the ratio between the powers of the input signal, x(t), and the output

signal, y(t)?

[6] (10 marks) x(t) is real, band-limited CT signal with maximum frequency equals B

rad/sec. The signal y(t) is formed by multiplying x(t) by [cos (𝜔1𝑡) ∙ cos (𝜔2𝑡)].

Assuming that 𝜔2 = 3𝜔1 = 12𝐵, sketch the frequency spectrum of y(t) (assume that

the spectrum of x(t) is a square shape with constant amplitude A).

[7] (10 marks) x(t) is a band-limited signal with bandwidth B=3000 rad/sec. Its

spectrum is assumed to have a square shape with unity amplitude. We sample x(t) at

a rate of one sample every Ts second.

(a) What is the minimum sampling rate 𝜔𝑠 =

2𝜋

𝑇𝑠

required to avoid the Aliasing effect?

(b) Draw the spectrum of the sampled signal, when the sampling frequency is 1500,

3000 and 7500 rad/sec.

(c) If we pass the sampled signal through an ideal LPF with bandwidth equals B, draw

the spectrum of the reconstructed signal in each of the three cases described in (b).

[8] (10 marks) x(t) is a band-limited signal with bandwidth B=1500 rad/sec. We want to

design an analog to digital convertor (ADC) which converts the signal so that its

relative error is less than 0.01%. What will be the number of bits required and the

resultant bit rate of the ADC, if aliasing is to be avoided? Use an oversampling

factor of 0.20.