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Noninvasive Detection of
Coronary Artery Disease
John Semmlow and John Kostis
Laboratory for Noninvasive Medical Instrumentation
Rutgers University and Robert Wood Johnson Medical School
Noninvasive Detection of Coronary Artery
Disease Basic Approaches
1) Electro cardiogram (ECG) Techniques:
Resting ECG, Exercise ECG (Stress test), cardio-integram.
2) Flow-based Techniques:
Thallium 201 myocardial scintigraphy (Thallium stress test),
Pharmacoloical stress imaging, gated blood pool scanning
3) Direct Imaging Techniques:
Positron emission tomography (PET), Magnetic reasonance imaging
(MRI), Digital subtraction angiography, computer-assisted
tomography (CAT)
4. Wall Motion:
Stress ecochardiology, apex cardiology, cardiokymography,
seismocardiography.
5) Acoustic method.
Signal Processing Algorithms
Editing/
Diastolic
Window
Signal
Signal
Detector
Diastolic
Signal
Classifier
Disease
Vector
Disease
State
Software Processing Components
Preprocessing
S2 Detection
Find Diastolic Window
Edit Data
Spectral Estimation (FFT)
Model-Based Analysis
Save Data
S2
Diastolic
Window
S1
FFT Spectra
Averaged spectra of
the diastolic portion
of 10 heart cycles
from a normal and
diseased patient.
Normal
Frequency
Diseased
Short-Term Fourier Transform
(Spectogram)

X(t, f) =
 x( ) w(t -  ) e
- j f 
d
Eq. 6 -1
-
where w(t-τ) it is the window and t slides the
window across the function.
In discrete form:
N
X(m, k) =
 x(n) W(n - k) e - jnm / N 
n=1
Eq 6 - 2
Time-Frequency Limitation
• To increase time resolution you need a shorter
window
• A shorter window decreases the frequency
resolution
• This leads to a time-frequency uncertainty:
BT 
1
4
Response of the Short-Term Fourier Transform
100
1.5
Contour Plot
90
1
80
70
Frequency (Hz)
x(t)
0.5
0
60
50
40
-0.5
30
20
-1
10
-1.5
0.4
0
0.6
0.8
1
1.2
1.4
1.6
0
0.2
0.4
0.6
0.8
1
Time (sec)
Time (sec)
Step-change in Frequency
1.2
1.4
1.6
1.8
2
Chirp Signal
1.5
Chirp Signal
1
x(t)
0.5
0
-0.5
-1
-1.5
0
0.05
0.1
0.15
0.2
0.25
Time (sec)
Linear increase in frequency with time
0.3
STFT Response to a Chirp
To overcome time-frequency limitations
of STFT, there are two different
approaches:
• Cohen class of distributions: Wigner-Ville,
Choi-Williams and many others. These are all
based on the “instantaneous autocorrelation function.
• Time-scale approaches: The Wavelet
Transform
The Wavelet Transform
There are two basic type of Wavelet Transform:
• The Continuous Wavelet Transform
(CWT). Similar to the STFT except scale is
changed.
• The Discrete Wavelet Transform (DWT).
Also known as the Dyadic Wavelet
Transform. Non-redundant, used bilaterally,
best described with filter banks
Continuous Wavelet Transform
Recall the Short Term Fourier Transform

STFT(t, f) =  x( ) (w(t -  ) e - 2jf ) d

In the STFT a family of windowed, harmonically
related sinusoids ‘slides’ across the signal function,
x(τ).
In the CWT, the family is a series of functions at
different scales (sizes) that slide across the signal
function

W(a,b) =  x(t)

 t - b
 *
 dt
 a 
a
1
Where: a scale the
function and b does
the sliding.
Wavelet Functions
• A wide variety of functions can be used as long
as they are finite.
• For example, the Morlet Wavelet is a popular
function
 (t) = e
- t2

cos 
2
ln 2
t

The Morlet Wavelet at Four different scales. The
wavelet at a = 1, is the baseline, or “mother” wavelet
1
1
a = 0.5
a=1
0.5
Wavelet
Wavelet
0.5
0
0
-0.5
-0.5
-1
-1
-1
-0.5
0
Time (sec)
0.5
1
-1
1
0.5
1
0
Time (sec)
0.5
1
a=4
0.5
Wavelet
0.5
Wavelet
0
Time (sec)
1
a=2
0
0
-0.5
-0.5
-1
-1
-1
-0.5
-0.5
0
Time (sec)
0.5
1
-1
-0.5
Different scales (values of a) produce a different
time-frequency trade-off.
12
10
a = 0.5
Frequency (rad)
8
6
a=1
4
a=2
a=3
2
0
0
0.2
0.4
0.6
0.8
1
Time (sec)
1.2
Δω Δt = constant
1.4
1.6
1.8
CWT to a step change in frequency
Cohen’s Class of Distributions
Basic equation:
(t, f) =
j2v ( u   )
   g(v, ) e
x(u   ) x *(u   ) e
1
2
1
2
- j2f
dv du d
While this equation is quite complicated, it breaks down
into three components.
•Two dimensional filter
•Instantaneous autocorrelation function
•Sinusoids that take the Fourier Transform
Wigner-Ville Distribution
The Wigner-Ville Distribution has now filter so it
consists only of the Fourier Transform of the
instantaneous autocorrelation function.
W(t, f) =


-
x(t -

2
) x( t -

2
) e - j2 f d
In discrete notation:

W(n, m) =
- 2 nm / N
e
Rx(n, k) = FFTk [Rx(n, k)]

m = -
where Rx(n,k) is the instantaneous autocorrelation function
Instantaneous Autocorrelation Function
• In the regular autocorrelation function, time is
integrated out of the result, so is is only a
function of the shift.
• In the instantaneous autocorrelation function,
no integration is performed and time remains in
the function. The function becomes a function
of both time and shift R(t,τ).
Rxx(t,τ) = x(t + τ/2) x*(t - τ/2)
Cosine Wave
Instantaneous Autocorrelation
0.5
0.45
0.4
Time (sec)
0.35
0.3
0.25
0.2
0.15
0.1
0.05
-0.25
-0.2
Cosine wave
Double frequency
-0.15
-0.1
-0.05
0
tau (sec)
0.05
0.1
0.15
0.2
0.25
Wigner-Ville to a chirp function (analytic signal)
Wigner-Ville to a step change in frequency (analytic
signal). Note the cross products