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Introduction
1
ECE 5650/4650 MATLAB Project 2
This project is to be treated as a takehome exam, meaning each student is to due his/her own
work. The project due date target is 5:00 PM Friday, November 19, 2010. To work the project you
will need access to M
ATLAB
and at minimum the signal processing toolbox.
Introduction
In this second Matlab computer simulation project we will consider:
•
Sampling theory
–
Basic modeling to study aliasing and reconstruction
–
Quantization effects
•
Filter design
–
Filter design basics
–
Group delay
–
Quantization effects in filters
•
Applications
•
Problems
•
Sampling Theory
In the first few problems we deal with basic sampling theory and modeling in M
ATLAB
. Consider
the A/D
D/A DSP system shown in Figure 1. In lowpass sampling the continuoustime or
analog signal,
, is first passed through an antialiasing filter to remove signals that lie above
the Nyquist frequency
. The A/D converter produces a sampled version of its input, in this
case the sequence
. A digital filter, or some discretetime system then processes the signal to
form the output
. Assuming the signal needs to be returned to the continuoustime domain, a
D/A converter converts the signal samples to an analog waveform, which needs further analog sig
nal processing to return it to a signal containing just the fundamental spectral translate.
To model this system in Matlab, we will make some assumptions and simplifications. To
Hz
x
a
t
f
s
2
xn
yn
H
(
z
)
A/D
Antialiasing
Filter
Reconstruction
Filter
D/A
f
s
f
s
x
a
(
t
)
x
aa
(
t
)
y
a
(
t
)
x
[
n
]
y
[
n
]
y
ZOH
(
t
)
Figure 1:
A basic A/D
D/A DSP system.
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View Full Document ECE 5650/4650 MATLAB Project 2
Introduction
2
begin with we will approximate
as a highly over sampled discretetime signal. The sam
pling rate used to approximate the analog input signal will be denoted
. For all of the prob
lems in this section
kHz. The analog antialiasing filter will be a third order Butterworh
lowpass digital filter of the form
>> [ba,aa] = butter(3,2*(fs/2)/96000);
where
fs
is the sampling rate in Hz used in the A/D. Note the 3dB cutoff frequency of the filter is
at the
folding
or Nyquist frequency,
.
Next we model the A/D as a downsampler followed by a
bit quantizer. The quantizer is
a nonlinearity represented as
. Since all signals in the model are discrete, the input to the
antialiasing filter is really
and the output of the antialiasing filter is
, so the A/D output is
(1)
The block diagram of the A/D model is shown in Figure 2. In M
ATLAB
the upsampling operation
performed by the
Signal Processing Toolbox
(SPTB) function
>> y = downsample(x,M);
The sequence
is the true discretetime signal in the DSP system. It nominally enters a signal
processing function block, e.g.,
, and returns the sequence
. In the sampling theory
problems we will assume that the DSP function block is simply a through connection, i.e.,
(2)
The output signal
(in this case also
) next enters the D/A model. This model is com
posed of an upsampler followed by a filter having a rectangular impulse response. A digital filter
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This document was uploaded on 02/25/2012.
 Fall '09

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