ECGR4124/5124 Digital Signal Processing
Projects will NOT be accepted late. It is due at the beginning of class.
Project 1
Matched Filters: Applications of Cross-Correlation
One possible application of cross-correlation is the detection of a signal embedded within noise. The
signal can be estimated by cross-correlating a template signal with the input signal. The template signal,
h[n]
, is considered to be known
a-priori
. Detection is accomplished by time-reversing the template and
convolving it with the input signal. Maxima observed in the output signal indicate lag indices where the
template and the shape of interest in the noisy signal are aligned.
h[n]
is referred to as a
matched lter
, i.e.,
a lter which is custom-designed to detect the presence of a specic structure within the input signal, recall
from class that
x[n]? h[−n] = rxh[n]
.
ECGR 4124/5124 Task 1
This task has students detect the phase and period of a sinusoid embedded within a noisy signal. To
do so, we generate a sinusoidal with known frequency but unknown phase and corrupt this sinusoid with
additive random noise (steps (a,b)). We then compute a zero phase template of the sinusoid consisting of
K
samples using the known frequency of the sinusoid (step (c)). We then attempt to detect the presence of
the unknown sinusoid by cross-correlating the template and the noisy signal (step (d)) by detecting peaks,
i.e., local maxima in the output signal from the cross-correlation.
1. Generate a sinusoidal sequence
x[n] = cos( π
15 n+φ) 1 ≤n≤200
and
φ= 0
.
2. Corrupt this signal with additive uniformly distributed noise
e[n]
of amplitude
±15
with distribution
U∼[−0.5,0.5]
. For a length
N
sequence, you can generate the noise with the following MATLAB
code:
noisevals = 30*(rand(N,1)-0.5);
3. Compute matched lters of varying length
hK[n] = cos( π
15 n)
where
0≤n<K
and
K= 10,20,40,80
.
4. Compute the cross-correlation of the noisy signal,
x[n] + e
, and
h[n]
.
ECGR 4124/5124 Task 1 (30 pts)
To Turn in:
You are responsible for turning in the following objects IN THE ORDER LISTED BELOW.
Deductions
will be made for reports which are messy and dicult to comprehend.
Using your MATLAB
program for steps 1-4 provide the following:
1. (5pts) Turn in your MATLAB code for computing steps 1-4. NO MATLAB CODE IMPLIES NO
CREDIT FOR ANY OF YOUR SOLUTIONS. These points are awarded for code that is clearly
written and follows the guidelines of the project.
2. (5pts) Using subplot(), plot the cross-correlation signals for each value of
K
as a 4x1 (4-row,1-column)
matrix of plots.
3. (5pts) Use the location of peaks in the output to estimate the period of the noisy sinusoidal input
x[n]
.
State the estimated value of the sinusoid period.
4. (5pts) Provide a 1-paragraph explanation of how the variable
K
impacts the estimation of the period.
5. (5pts) If the noise amplitude were increased, how would you change
h[n]
to get a better estimate?
6. (5pts) Is it also possible to estimate the phase of the function if
φ6= 0
? If so, give an idea of how you
would do this?
Hint :
From the program above, let
φ=π
or
φ=π
2
and observe how the value of the indices of your
peaks change.
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