QUIZ 2
ECSE 304
QUIZ 2
SIGNALS AND SYSTEMS II
ECSE304 Signals and Systems 2 Winter Semester 2012 Monday, March 19 McGill University Department of Electrical and Computer Engineering
QUIZ 2
NAME: _____________________________
STUDENT NUMBER: _______________________
Instructions: The Quiz consists of three problems. No notes or crib sheets are permitted during the quiz. You will be given 30 minutes to complete all the questions. Please show all your work on the pages provided. Problem 1: The following system is used for discrete time processing of a continuous time signal: x (t )
x p (t )
C/D
xd [n]
H d ( e j )
y d [n ]
y p (t )
D/C
Hc ( j)
y (t )
p (t )
The box marked "C/D" is a continuous to discrete converter, or an ideal analog-to-digital converter. The box marked "D/C" is an ideal discrete to continuous converter. Suppose that is band-limited with for as shown in the figure below, X(j) 1
2 1000
2 1000
and the discrete time system is an ideal low-pass filter with cutoff frequency . Moreover, the reconstruction filter is an ideal low-pass filter with gain and cutoff frequency , where is the sampling period. a) Find the minimum sampling frequency (and sampling period) so that can be exactly reconstructed from . b) Assume sampling is performed at twice the minimum rate determined in part a), sketch the frequency domain representations (the continuous time and discrete time Fourier Transforms) of , , , , and . Carefully label the important points on the frequency and amplitude axis.
ECSE 304
QUIZ 2
SIGNALS AND SYSTEMS II
Problem 2 : The discrete time signal x[n] was obtained by sampling a continuous time signal at a rate of 12 KHz. However, to reduce the amount of disk space required for storing the signal, we would like to resample the signal at a rate of 8 KHz. For performing this sampling rate conversion we use the setup as shown below:
Choose L=2 and M=3 to realize the re-sampling ratio for the problem. The blocks labeled L and M in the diagram above perform up-sampling and down-sampling, respectively. x[n] has the following discrete time Fourier transform. The plot for H (e j ) is also shown below:
Draw and label the discrete time Fourier transforms of sequences v[n], w[n] and y[n]. Carefully label the important points on the frequency and amplitude axis.
QUIZ 2
SIGNALS AND SYSTEMS II
ECSE304 Signals and Systems 2 Winter Semester 2012 Monday, March 19 McGill University Department of Electrical and Computer Engineering
QUIZ 2
NAME: _____________________________
STUDENT NUMBER: _______________________
Instructions: The Quiz consists of three problems. No notes or crib sheets are permitted during the quiz. You will be given 30 minutes to complete all the questions. Please show all your work on the pages provided. Problem 1: The following system is used for discrete time processing of a continuous time signal: x (t )
x p (t )
C/D
xd [n]
H d ( e j )
y d [n ]
y p (t )
D/C
Hc ( j)
y (t )
p (t )
The box marked "C/D" is a continuous to discrete converter, or an ideal analog-to-digital converter. The box marked "D/C" is an ideal discrete to continuous converter. Suppose that is band-limited with for as shown in the figure below, X(j) 1
2 1000
2 1000
and the discrete time system is an ideal low-pass filter with cutoff frequency . Moreover, the reconstruction filter is an ideal low-pass filter with gain and cutoff frequency , where is the sampling period. a) Find the minimum sampling frequency (and sampling period) so that can be exactly reconstructed from . b) Assume sampling is performed at twice the minimum rate determined in part a), sketch the frequency domain representations (the continuous time and discrete time Fourier Transforms) of , , , , and . Carefully label the important points on the frequency and amplitude axis.
ECSE 304
QUIZ 2
SIGNALS AND SYSTEMS II
Problem 2 : The discrete time signal x[n] was obtained by sampling a continuous time signal at a rate of 12 KHz. However, to reduce the amount of disk space required for storing the signal, we would like to resample the signal at a rate of 8 KHz. For performing this sampling rate conversion we use the setup as shown below:
Choose L=2 and M=3 to realize the re-sampling ratio for the problem. The blocks labeled L and M in the diagram above perform up-sampling and down-sampling, respectively. x[n] has the following discrete time Fourier transform. The plot for H (e j ) is also shown below:
Draw and label the discrete time Fourier transforms of sequences v[n], w[n] and y[n]. Carefully label the important points on the frequency and amplitude axis.
