Wavelets in DSP (excerpts)

©2006-2010 ATICOURSES
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©2010 D.L. Fugal

WAVELETS: ANOTHER DIMENSION
IN DIGITAL SIGNAL PROCESSING

D. Lee Fugal, Chairman, IEEE Signal Processing Society

IEEE San Diego Section Talk 11/18/09

1
©2006 Spac e & Signals Te chnologies, LLC. All Rights Re serve d. www.Conc eptualWave le ts.com 877-845-6459

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WHAT IS A WAVELET? ©2010 D.L. Fugal

Cosine Wave Db4 Wavelet
• Sinusoids extend from minus to plus infinity.
• Wavelet is waveform of limited duration (Starts & Stops)
• Sinusoids are smooth and predictable.
• Wavelets tend to be irregular and asymmetric.
• Wavelets have an average value of zero
• Wavelets are compared (correlated) with signals that have
“events” in time like heartbeat, stock market, pulses.
• Jargon Alert: This type of signal called “Non-Stationary”
2

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EXAMPLES OF WAVELETS ©2010 D.L. Fugal

• WAVE for Frequency, LET indicates Compact Support.
• Jargon Alert*: Compact Support = having start & stop time
• Some more localized in time, some more localized in freq.

Haar Shannon or Sinc Daubechies 4 Daubechies 20

Gaussian or Spline Biorthogonal Mexican Hat Custom (arbitrary) 3

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USES OF PARTICULAR WAVELETS ©2010 D.L. Fugal

• Haar: Good for edge detection in images, for
matching binary pulses, for very short
phenomenon.
• Shannon: Dual of Haar wavelet. Good
frequency resolution and signal identification
using frequency. Poor time resolution.
• Daubechies: Robust, fast for identifying signals
with both time and freq characteristics (use
longer filters for better frequency resolution).
Used in speech, fractals, non-symmetrical
transients. Identifies polynomial signals or noise
• Biorthogonal (2 wavelets). Symmetry and
Linear Phase. Used extensively in Image
Processing because human vision more tolerant
of symmetrical errors and because images can
be extended. Chosen by FBI and for JPEG.
4

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APPLICATIONS OF WAVELETS ©2010 D.L. Fugal

• Signal and Image Compression and Denoising. JPEG, FBI
• Geology, Oceanography, Astronomy, Electrical Systems.
• MRIs and similar non-invasive procedures. Mammogram
enhancement to distinguish Tumors from calcifications.
• EEG/EKG detection of transient “events”.
• Finance for stock market patterns, quick variations of
value. Internet Traffic. Biology. Metallurgy. Speech.
• Radar and Sonar. Pulse detection by both time and
frequency. Automatic signal and target recognition.*
• Study of short-time phenomena as transient processes.
• Non-Destructive Testing, SAR imagery
• Motion Pictures (e.g. “A Bug’s Life”)
• Rupture and Edge
Detection (airport baggage screening). 5

©2006-2010
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TRANSFORMS AND COMPARISONS ATICOURSES
©2010 D.L. Fugal

COMPARISON OF WAVELET
TRANSFORMS TO FOURIER
TRANSFORMS (DFT/FFT)
AND SHORT-TIME FOURIER
TRANSFORMS (STFT)

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FFT CLASSIC EXAMPLE ©2010 D.L. Fugal

• Noise in signal can be identified using FFT.
• Can be removed using conventional filtering methods.
• Here we remove 60-Hz noise “spike” or hum.
• For this signal, the FFT is a better choice than Wavelets
2 HZ SIG WITH 60 HZ HIGH FREQ NOISE 2 HZ SIG WITH 60HZ HIGH FREQ NOISE
1.5 140

120
1

100

MAGNITUDE -->
0.5
MAGNITUDE -->

80
0
60

-0.5
40

-1 20

-1.5 0
0 50 100 150 200 250 0 50 100 150
FREQUENCY -->

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FFT SIGNAL + NOISE ©2010 D.L. Fugal

• De-noised signal shown below. Wavelets refers to this as
the low-freq “Approximation” of the original signal.
• Noise is also shown. Wavelets nomenclature refers to as
high-frequency “Details”.
• Note “Approximation” + “Details” = Original signal.
DENOISED (LOW FREQ "APPROXIMATION") NOISE (HIGH FREQ "DETAILS")
1.5 1.5

1 1

0.5 0.5

0 0

-0.5 -0.5

-1 -1

-1.5 -1.5
0 50 100 150 200 250 0 50 100 150 200 250
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FFT SHORTCOMINGS ©2010 D.L. Fugal

LOW FREQ SIGNAL WITH HIGH FREQ NOISE
2 180

1.5 160
140
1

MAGNITUDE -->
AMPLITUDE -->

120
0.5
100
0
80
-0.5
60
-1 40
-1.5 20
-2 0
0 100 200 300 400 500 600 0 100 200 300 400 500 600
FREQ -->

LOW FREQ SIGNAL THEN HIGH FREQ SIGNAL LOW FREQ SIGNAL THEN HIGH FREQ SIGNAL
1 120

100
0.5
AMPLITUDE -->

MAGNITUDE -->
80

0 60

40
-0.5
20

-1 0
0 100 200 300 400 500 600 0 100 200 300 400 500 600
TIME --> FREQ -->

• Signal characteristics not seen in the FFT
• Why wavelets are needed. Show both time & freq. 9

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TIME/FREQ RESPONSE DEMO ©2010 D.L. Fugal

• Compromise between the time- and frequency-based
views of a signal. Provides some information about both.
• Example of Discrete Fourier Transform (DFT) with
piano strings and the word “Hello” heard in the 88
resonating piano-string frequencies (an “Audio-Based
Discrete Fourier Transform”).
• Example of Short Time Fourier Transform (STFT) by
hearing the the piano-string DFT for the time-sequential
words “Heh” and “Low” in succession.
• Next look at Heisenberg Cells (boxes).
• Jargon Alert: The Heisenberg Uncertainty Principle says
you can’t know an exact frequency at an exact time* (it
takes some time to oscillate--more for low notes, less for
high). Thus a cell or box has the same area (next slide). 10

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SHORT TIME FOURIER TRANSFORM ©2010 D.L. Fugal

Amplitude 2 LONG WINDOWS PRECISION

Frequency
Time Time Heisenberg
4 SHORT WINDOWS PRECISION Cells (boxes).
Note same
Amplitude

Frequency
area in both
shapes.

Time Time
• Looking at signal for long times (integration time) gives
better frequency precision, but poorer time precision
(when did it occur?).
• Looking at signal for short times gives better time
precision , but poorer frequency precision (what was it’s
frequency at that time?). 11

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WAVELET WINDOWING PATTERN ©2010 D.L. Fugal

PRECISION
“NATURAL”

FREQUENCY
PATTERN (don’t
Amplitude

need as much time to
identify high freqs)

Time TIME

• Windowing technique with variable-sized regions.
• Allows the use of long time intervals where we need
more precise frequency information (low freqs), and
shorter regions where we need precise time information.
• An example of this “natural” pattern has been around for
hundreds of years: Sheet Music
12

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TIME/FREQ EXAMPLE (MUSIC SCORE) ©2010 D.L. Fugal

ff
tempo 60
Frequency -->

Frequency (inverse of Scale)
4
4
mf
4
4
pp
Time -->
Time
• Frequency of musical notes are factors of 2 apart (octaves)
• “Digital” in time (tempo, “4/4”), frequency, magnitude (ff)
• “Low Notes” (lower frequency notes) need longer times to
be correctly generated (tuba vs. piccolo).
• Human ear requires longer time to determine frequency
(pitch) and overtones of low notes.
• (Display here is inverted from most wavelet displays). 13

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MUSICAL EXAMPLE ©2010 D.L. Fugal

• Top level of display has shorter times for higher
frequencies (which don’t need as much time for good
resolution).
• Demonstration of Piccolo solo from John Philip Sousa’s
“Stars and Stripes Forever” shows capability rapid changes
at higher frequencies (lower scales).
• Demonstration of Piccolo solo played on tuba shows not
enough “integration time” for the lower frequency (higher
scale) notes to be formed correctly (even if the musician
does a perfect job of valve fingering).

Piccolo solo from “Stars and Stripes Forever”
14

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FFT-TYPE PULSE COMPARISON ©2010 D.L. Fugal

Pulse Signal. 5 cycles in
A 1/4 second = 20 Hz.
Centered at 3/8 second.

40 Cycle per Second
B (40 Hz) Sinusoid
for comparison with pulse
signal A. Poor correlation.

Sinusoid stretched to 20 Hz for
comparison. Good correlation.
C Same frequency as pulse so
peaks and valleys can align.

Sinusoid stretched to
10 Cycles/Sec (10 Hz)
D for comparison. Poor
correlation again.
Time (seconds)
0 1/4 1/2 3/4 1
15

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ACTUAL FFT (DFT) OF PULSE ©2010 D.L. Fugal

D C B

EQUATION INDICATES THAT
Magnitude

THE SIGNAL IS MADE UP OF
CONSTITUENT SINUSOIDS

Frequency (Hz)
0 10 20 30 40 50
NOTE: Only frequency information is given by the FFT
N −1
Χ ( k ) = ∑ x(n)e − j ( 2 π / N ) nk OR USING THE EULER IDENTITY
n =0
N −1 N −1
= ∑ x (n) cos( 2 πnk / N ) − j ∑ x (n) sin( 2πnk / N )
n=0 n=0
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CWT-TYPE PULSE COMPARISON (1) ©2010 D.L. Fugal

1/4 second = 20 Hz.
A Centered at 3/8 second.

Roughly 40 Hz Daubechies
20 (Db20) Wavelet
B for comparison with pulse
signal A. Poor correlation.

Roughly 40 Hz Db20 Wavelet
shifted in time to line up with
the pulse. Still a poor
C comparison because the
frequencies don’t match.

Db20 Wavelet stretched
D (“scaled”) by 2 to roughly 20
Hz and shifted for comparison.
Good comparison (correlation).
Time (seconds)
0 1/4 1/2 3/4 1

(If energy of wavelet and signal are both unity, values are correlation coefficients) 17

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CWT-TYPE PULSE COMPARISON (2) ©2010 D.L. Fugal

1/4 second = 20 Hz.
Centered at 3/8 second.
Db20 Wavelet stretched to
roughly 20 Hz and shifted to
where peaks begin to line up
with peaks (or valleys).
Weak correlation just past
1/4 second.
Db20 Wavelet stretched to
roughly 20 Hz and shifted to
where more peaks line up.
Stronger correlation just
before 3/8 second.

Db20 Wavelet stretched by
2 to roughly 20 Hz and
shifted for comparison.
Strongest correlation at
3/8 second.
Time (seconds)
0 1/4 1/2 3/4 1 18

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ACTUAL CWT OF PULSE ©2010 D.L. Fugal

wavelet shifted
to right by 3/8
second wavelet
stretched
Stretching or “scaling”

to approx.
(inverse of frequency)

20 Hz.

unstretched
basic
wavelet at
low scale.
(poor
results)
0 1/4 1/2 3/4 1
Time (seconds)

NOTE: Both time AND frequency information of pulse given by the CWT!
Also repeating this with various wavelets indicates the SHAPE of event!
(Equation indicates that the signal is made up of constituent wavelets).

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DEMO OF CWT CAPABILITY ©2010 D.L. Fugal

TIME PLOT OF SIGNAL WITH SMALL DISCONTINUITY FFT PLOT OF SIGNAL WITH SMALL DISCONTINUITY
1
160
0.8 Hidden High frequency
140
0.6 discontinuity segments of
0.4
at time = 180
120 discontinuity too small
0.2
not visible on 100 to see on this
0
Amplitude vs. 80 Magnitude vs.
-0.2
Time plot. 60
Frequency FFT plot,
-0.4

-0.6
and would give no
40
-0.8
indication as to when
20
-1
they occurred anyway.
0 50 100 150 200 250 300
0
Time 0 50 100 150 200 250 300
Frequency
WAVELET PLOT OF SIGNAL & DISCONTINUITY
20
Stretching (“Scaling” or “Level”)

18

16
Stretched “low frequency” Db4
14 wavelet compares better to
12 sinusoidal (wave) signal. It
10 “finds” peaks and valleys.
8

6
Small “high frequency” wavelet
4
compares well to discontinuity.
2
50 300
It “finds” it’s location at 180.
Time 20

©2006-2010
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FILTERS FROM WAVELETS ATICOURSES
©2010 D.L. Fugal

OBTAINING REAL-WORLD
DISCRETE FILTERS FROM
WAVELETS WITH EXPLICIT
MATHEMATICAL
EXPRESSIONS
(“CRUDE WAVELETS”)
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WAVELET “LENGTH” FOR MEXH ©2010 D.L. Fugal

• Wavelets in the real world of digital computers are also
filters. First look at the the Mexican Hat “crude” wavelet:.
• mexh(t) = 2/(sqrt(3)∗pi^0.25)∗exp(-t^2/2) ∗ (1-t^2)
• Jargon Alert: 1
MEXICAN HAT WAVELET

effective length
“Crude” means
0.8
generated from
explicit math 0.6

equation.
AMPLITUDE -->

0.4

• Effective 0.2
Length
from -8 to +8 0

(e.g. value at -0.2

time 5.1 = -0.4
-8 -6 -4 -2 0 2 4 6 8
3.6939e-06) (Relative) TIME -->
22

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MEXH 17 POINTS ©2010 D.L. Fugal

• Jargon Alert: Effective Length is often referred to as
“Effective Support”.
• Even with explicit mathematical expressions, we still must
treat them as 17 POINTS ON MEXICAN HAT WAVELET
1
digital filters in
convolving with 0.8
the signal in the 0.6

time domain.
AMPLITUDE -->

0.4

• For the CWT, 0.2
start with short,
0
unstretched, HF
filter. Values at -0.2

integers produce17 -0.4
-8 -6 -4 -2 0 2 4 6 8
points from -8 to +8. (Relative) TIME -->
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MEXH 33 POINTS ©2010 D.L. Fugal

• After comparing 17-point “filter” with signal (scale = a =
1), CWT software “stretches” it to 33 points corresponding
to values of MEXH wavelet at the 1/2 integer points from -
8 to +8 (-8, -7.5, -7 . . . 0 . . . +8). This is scale = 2.
• The next stretching (scale =3) is the 49 points
corresponding to 1/3 integer values in the same interval.
33 POINTS ON MEXICAN HAT WAVELET 49 POINTS ON MEXICAN HAT WAVELET
1 1

0.8 0.8

0.6 0.6
AMPLITUDE -->

AMPLITUDE -->
0.4 0.4

0.2 0.2

0 0

-0.2 -0.2

-0.4 -0.4
-5 0 5 -5 0 5
TIME --> TIME --> 24

©2006-2010
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WAVELETS FROM FILTERS ATICOURSES
©2010 D.L. Fugal

WAVELET FILTERS OF
SPECIFIC LENGTH THAT
BUILD APPROXIMATIONS
TO A “CONTINUOUS”
WAVELET FUNCTION
WHICH IN TURN CAN THEN
PRODUCE FILTERS OF ANY
DESIRED LENGTH.
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BASIC WAVELET & SCALING FUNCT. ©2010 D.L. Fugal

• Here is the basic Db4 wavelet filter
-0.1294 -0.2241 0.8365 -0.4830
and the lowpass filter or scaling function filter
0.4830 0.8365 0.2241 -0.1294
• Note similarities in the filter values. PRQMFs.
4 PT BASIC HP DB4 WAVELET FILTER 4 PT BASIC LP SCALING FUNCTION FILTER
1 1

0.8
AMPLITUDE -->

AMPLITUDE -->
0.5 0.6

0.4

0 0.2

0

-0.5 -0.2
1 2 3 4 1 2 3 4
NUMBER OF POINTS (n) --> NUMBER OF POINTS (n) -->
26

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DB4 UPSAMPLED AND LPF (STRETCH) ©2010 D.L. Fugal

• Here is the basic wavelet filter upsampled with zeros
between the existing points.
• After lowpass filtering we have a “stretched” 10-point
filter (length = 7 pts of upsampled filter + 4 pts LPF -1 =
10)
UPSAMPLED STRETCHED
1 1.5

1

0.5
AMPLITUDE -->

AMPLITUDE -->
0.5

0
0

-0.5

-0.5 -1
0 2 4 6 8 0 2 4 6 8 10
27

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DB4 STRETCHED TO MORE POINTS ©2010 D.L. Fugal

• We continue the process of upsampling and lowpass
filtering to produces increasingly stretched wavelet filters
with 22, 46 (shown below), 94, 190 (shown below), 382
and finally 766 points.
• We now have an approximation of a Db4 “continuous”
wavelet function built from the original 4 points.
1.5 1.5

1 1

AMPLITUDE -->
AMPLITUDE -->

0.5 0.5

0 0

-0.5 -0.5

-1 -1
0 10 20 30 40 50 0 50 100 150 200
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4 + 2 FILTER PTS SUPERIMPOSED ©2010 D.L. Fugal

• We superimpose the original 4 Db4 filter points used to
build this wavelet function. As we convert our 766 point
“continuous” function to a “length” of 0 to 3, the points
-0.1294 -0.2241 0.8365 -0.4830
are found at 2/6, 5/6, 8/6 and 11/6 or 1/2 integer apart
starting at 1/3. They are overplotted on the wavelet
function along with the zero values at 14/6 and 17/6.
WAVELET FUNCTION PSI
• Like the “crude” 1.5

wavelet filters, can be 1

used with a CWT.
AMPLITUDE -->

0.5

• Unlike the crude filters 0
they can be used with
-0.5
a Discrete Wavelet
Transform (DWT) -1
0 0.5 1 1.5 2 2.5 3
"TIME" (t) --> 29

©2006-2010
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DISCRETE WAVELET TRANSFORMS ATICOURSES
©2010 D.L. Fugal

THE UNDECIMATED
DISCRETE WAVELET
TRANSFORM (UDWT).
Also called Stationary, Shift
Invariant, “A’ Trous”, or
“redundant” (but not near as
redundant as the CWT)
30

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1 LEVEL UDWT SYSTEM ©2010 D.L. Fugal

• The Scaling Funtion filters L and L’ produce a Halfband
Lowpass Filter while the Wavelet Filters H and H’
produce a Halfband Highpass Filter.
• Jargon Alert: Halfband filters cut the frequency band in
half as shown below--with some symmetrical overlap.
• Summing the results of the highpass and lowpass halfband
filters produces a constant in the frequency domain
• Final result, S’, is the H H’
same as the original
signal, S, except for a D1
delay and/or a scaling
constant
S S’

D1 L L’
A1 A1
Frequency Spectrum
31

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2-LEVEL UDWT ©2010 D.L. Fugal

• The filter in the RED oval is essentially the original
Lowpass “Scaling Function Filter” stretched (“scaled”) in
time by a factor of 2.
• Recall the Haar Scaling Function (lowpass) Filter
L = [1 1] convolved with the upsampled Wavelet
(highpass) Filter Hup = [1 0 -1] produced [1 1 -1 -1]
H cD1 H
D1

S S’
Hup cD2 Hup

cA1 cA1 A1
L L
cA2

Lup Lup 32

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2-LEVEL EQUIVALENT UDWT ©2010 D.L. Fugal

• The filters in the GREEN oval are also stretched versions
of the highpass and lowpass filters. The output is D2 and
A2. Added together they produce D2+A2=A1.
• Thus the reconstructed signal, S’, is given by
S’ = D1+A1 = D1+D2+A2

H cD1 H
D1

S S’
Hup cD2 Hup
D2
L
cA1
A1
L
cA2
A2
Lup Lup L 33

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FREQUENCY ALLOCATION ©2010 D.L. Fugal

• Freq. allocation for a 2 and a 4-level UDWT shown here
• Beginning to see utility and flexibility. Remember that,
unlike the FFT we can adjust Details and Approximations
for any desired part of the total time.
2-level S
MAGNITUDE

UDWT A1
freq. NORMALIZED
FREQUENCY
bands A2 D2 D1 (NYQUIST = 1)

FREQUENCY

S
A1
4-level A2
UDWT A3
MAGNITUDE

freq. Nyquist
Frequency
bands
A4 D4 D3 D2 D1

FREQUENCY
34

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DISCRETE WAVELET TRANSFORMS ATICOURSES
©2010 D.L. Fugal

•

THE DISCRETE
(Conventional, Decimated)
WAVELET TRANSFORM
(Usually called “The DWT”)

35

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1 LEVEL DWT SYSTEM ©2010 D.L. Fugal

H H’
D1
cD1 cD1

S S’
L cA1 cA1 L’
A1

“ANALYSIS” “SYNTHESIS”

A1 D1
Frequency Spectrum
A = Approximation or lower frequency components
D = Details or higher frequency components

• Problem: Downsampling can produce aliasing!
• Solution: Proper design of filters can eliminate aliasing
under certain conditions (see downsampling/aliasing 101)
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DWT APPROXIMATIONS, DETAILS ©2010 D.L. Fugal

500 pts Cd1 d1
250 pts Cd2
1000 pts S’
Cd3 d2
25 Ca2 d3 a1
0 Ca3 125 pts a2
Ca1
500 pts a3 S’ = d1+a1
a1= d2+a2
a2=d3+a3
• Signal, S, can be decomposed into various
Approximations and Details using the Analysis portion
• Signal can then be reconstructed from these
Approximations and Details at the end of the Syntheses
portion. S’ = d1 + a1 = d1 + d2 + a2 = d1 + d2 + d3 + a3
• Same relationship can be seen in frequency domain below
A1
A2
A3 D3 D2 D1
Frequency Spectrum of Signal, S 39

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2-LEVEL WAVELET PACKET SYS (1) ©2010 D.L. Fugal

• 2-Level Wavelet Packet Analysis System
• Jargon Alert: Analysis portion is the left half of the DWT
that decomposes the signal into coefficients. Synthesis
portion is the right half of DWT that rebuilds the signal.
• In Wavelet Packets, Details and Approximations are split.

Underline shows 1 of
4 ways signal can be
S decomposed

A1 D1

AA2 AD2 DA2 DD2
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TIME DEPENDENT THRESHOLD - 1 ©2010 D.L. Fugal

• Pure Binary Signal (BPSK PNRZ) with no noise and its
FFT

PURE SIGNAL FFT OF PURE SIGNAL
10 1200

8
1000
6

4
800

MAGNITUDE-->
AMPLITUDE-->

2

0 600

-2
400
-4

-6
200
-8

-10 0
2000 4000 6000 8000 0 2000 4000 6000 8000 10000
TIME--> FREQ-->

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• Binary Signal buried in 10000x chirp noise at left.
• 100 times closeup shows bit pattern overplotted, but bits
are actually buried in the 1x104 noise.

x 10
4 CHIRP JAMMER WITH BURRIED SIGNAL 200xCLOSEUP
1 100

0.8 80

0.6 60

0.4 40

AMPLITUDE-->
0.2 20

0 0

-0.2 -20

-0.4 -40

-0.6 -60

-0.8 -80

-1 -100
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 2000 2500 3000 3500 4000 4500 5000
TIME-->

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• 4 Level DWT (frequency allocation shown here) used
to decompose noisy signal
• Signal is 2 13 points long. 7 levels of decomposition
plenty.
S
A1
A2
AMPLTITUDE

A3

Nyquist
Frequency

A4 D4 D3 D2 D1

FREQUENCY
46

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Highest frequency portion, D1, of noisy signal as a function
of time. Noise is 10000 x as large as signal but is confined
to end of time sequence. Note scale is +/- 15,000
x 10
4

x 10 4
d1 d1
1.5
1.5

1
1

0.5
0.5

0
0

-0.5
-0.5

-1
-1

-1.5
0
-1.5
1000 2000 3000
0 20004000 5000 6000
4000 7000 8000
6000 9000
8000 10000
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D1 “scrap” with noisy portion zeroed out. Note change of
scale from +/- 15000 to +/- 0.6. Signal “remnants” can now
be clearly seen.
We use Time-Dependant Thresholding on the other levels
as well and then reconstruct the signal from the remnants
d1
0.8
Note that the
0.6
time/scale property
0.4
of wavelet analysis
0.2
allows us to do
0

this. FFT would -0.2

not work. -0.4

-0.6

-0.8
0 1000 2000 3000 4000 5000 6000 7000 8000
48

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• Main portion of original noiseless binary signal (top)
• Wavelet Time-Dependant Thresholding de-noised signal
(from 10000x or 80 dB noise) shown superimposed on
noiseless signal (bottom).
• We have exploited DWT knowledge of both time AND
frequency

49

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BPSK PNRZ CWT EXAMPLE - 2 ©2010 D.L. Fugal

• First look at
DWT of
noiseless signal
• Note there is no
information to
be had by
adding Details
from levels 1
through 4 on
noiseless test
case.
• Can thus
threshold out
levels 1 through
4 on noisy
Signal.
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BPSK PNRZ CWT EXAMPLE - 3 ©2010 D.L. Fugal

As • Levels 1 - 4
thresholded out
and now don’t
contribute to the
reconstructed
signal at all.
• Note thresholded
coeffs. exist only
for levels >= 5.
• Reconstructed
signal seen in
upper left graph.
• Knowing signal
is +/- 1, we can
reconstruct signal
exactly
51

Wavelets in DSP (excerpts)

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Wavelets in DSP (excerpts)