Signals and Systems
Assignment 1
Question 1
A continuous-time signal is defined as
x(t) = 1 +
1
cos(2π 1000t − φ )
2
=0
−
T
T
≤ t ≤ , T = 100m sec
2
2
elsewhere
a) Calculate the frequency response for x(t) for φ=0 and φ =π/2.
b) Use the results in the Fourier transform properties table to calculate the frequency
response for x(t) for φ=0 and φ =π/2.
c) What is the minimum and maximum frequency present in signal x(t)?
d) If the x(t) is the input signal and y(t) is the output signal, the input-output relationship
is a convolution and the frequency response of the system is given as
H ( f ) = 1,
=0
− 2000 ≤ f ≤ 2000
elsewhere
Calculate the power spectrum of the system and the output signal y(t) if φ=π/2.
Question 2
A signal is defined as
x(t) = cos(2π 1000t ) +
1
cos(2π 2000t )
4
=0
−
T
T
≤ t ≤ , T = 10m sec
2
2
elsewhere
a) Calculate the frequency response of the signal x(t).
b) Using the Fourier transform property table, calculate the effect on the frequency
representation if the signal is delayed such that y(t)=x(t-5e-3)
c) Using the Fourier transform property table, calculate the effect on the frequency
response if the signal is scaled such that y(t)=x(at) for a=1/10 and a=1/20.
Question 3
The zeros and poles for continuous-time linear time-invariant system are defined as follows
s=0
s=-1+j1,-1-j1
a) Draw the pole-zero plot for the system.
b) Explain if the system is stable.
c) Calculate the transfer function H(s) for the system.
d) Calculate the frequency response |H(f)| for the system.
e) Calculate the system impulse response h(t) for the system.
f) The input-output relationship expressed in terms of the transfer function is
Y (s) = H (s) X (s)
where X(s) is the input, Y(s) is the output, and H(s) is the transfer function. If the input
defined in time domain x(t) is a step function, calculate the output in time domain y(t).
Question 4
A continuous-time signal is defined as
x(t) = 1 +
=0
1
cos(2π 1000t)
2
0 ≤ t ≤ T , T = 100m sec
elsewhere
Sampling frequency is 8000 Hz.
a) Calculate the discrete-time signal x(n).
b) Calculate the DTFT for the signal x(n).
c) Sketch the spectrum of the discrete-time signal calculated in b).
d)
The discrete-time signal x(n) is multiplied with a complex exponential such that
y(n) = x(n) exp( j2πf c nTs )
If fc is 2000 Hz, use the frequency shift property of the DTFT determine the spectrum
representation of y(n).
e) Calculate and sketch the power spectrum of the results in part c) and part d).
Question 5
The frequency response of discrete-time system defined within − 4000 ≤ f ≤ 4000 is
H (exp( j2πf )) = 1
− 1000 ≤ f ≤ 1000,2000 ≤ f ≤ 4000,−2000 ≤ f ≤ −4000
=0
elsewhere
Sampling frequency is 8000 Hz.
f) Draw the frequency response of the discrete-time system.
g) Calculate the system impulse response h(n).
h) If the input signal to the system is
x(n) = 1 + cos 2π 3000nTs
=0
0 ≤ n ≤ 99
elsewhere
Calculate the output signal y(n) and its frequency response |Y(exp(j2πf))|.
Question 6
The amplitude spectrum and phase spectrum of discrete-time system defined within
− 4000 ≤ f ≤ 4000 is
| H (exp( j2πf )) |= 1
=0
φ (exp( j2πf )) = − π f
2000
− 2000 ≤ f ≤ 2000
elsewhere
− 2000 ≤ f ≤ 2000
= -π
f ≥ 2000
=π
f ≤ −2000
Sampling frequency is 8000 Hz.
a) Draw the amplitude and phase spectrum of the discrete-time system.
b) If the input signal to the system is
x(n) = 1 + cos 2π 1000nTs
=0
Calculate frequency response |X(exp(j2πf))|.
c) Based on part b), calculate the
output signal y(n) and its frequency
response
|Y(exp(j2πf))|.
0 ≤ n ≤ 99
Elsewhere