Digital Signal Processing Reference
In-Depth Information
N
−
1
N
−
1
1
∑
∑
2
E
=
gmn
(,)
−
g mn
′
(,)
mse
2
N
n
=
0
m
=
0
In this project, two compression techniques will be attempted:
•
Low pass filtering
: One common property of all DCT transform
matrices is that the major part of the energy of
t
(
i,j
) is concen-
trated in one corner (0
≤
m, n < N
1
) of the matrix.
d.
Create new transformed matrices
t'
(
i,j
), choosing
N
1
= N
/2 and
N
/4,
putting the remaining values of the matrix
t
(
i,j
) to zero.
e.
Plot the reconstructed image and determine the
E
mse
in each case.
•
Selective matrix truncation
: In this compression method, the
original transform matrix
t
(
i,j
) is truncated by setting individual
matrix elements to zero, one at a time. The pixels with the least
variance are removed first, and then the pixels with increasing
variance. The variance of the element
t
(
i,j
) is defined as follows:
<=
∑
8
2
⎣
⎦
=
Var
ti j
(, )
ti j
(, )
−
t i j
k
(, )
k
=
1
where
t
k
(
i,j
) is one of the eight neighboring pixels. For elements
on the borders of the matrix, the neighboring elements will be
less than eight.
f.
Implement this truncation method, one pixel at a time, and deter-
mine the
E
mse
between the reconstructed image and the original
image in each case. Make a plot of the
E
mse
versus the number of
pixels removed. At which stage of the process is the image just
visible? Plot the image at that stage only.
6.3.6
Time-Frequency Analysis of Practical Signals
Exercise 7: Spectral analysis of the electrocardiogram (EKG) signal
The record of the potential fluctuations during the cardiac cycle of the heart
is called the electrocardiogram (EKG or ECG).
4
Most EKG machines record
these fluctuations on a moving strip of paper. The EKG is a very useful
means of diagnosing abnormalities in the heart by analyzing the EKG wave-
form directly on a time axis, or by analyzing the Fourier spectrum of the
EKG recording on a frequency axis.
The EKG is a periodic time signal, as shown in
Figure 6.8
,
for a normal
heart. It has a period of ~1 second, corresponding to a fundamental frequency
of ~1 Hz. It is characterized by the PQRSTU peaks as shown in Figure 6.8.
In this exercise, we consider the following two types of heart conditions.
