Wavelets & Wavelet Algorithms
Vladimir Kulyukin
www.vkedco.blogspot.comwww.vkedco.blogspot.com
1D Discrete Fourier Transform&
Inverse Discrete Fourier Transform
Outline
● Review● Complex Numbers & Euler's Identity● Nth Roots of Unity● Roots of Unity & Complex Sinusoids● Discrete Fourier Transform (DFT) & Inverse DFT● DFT & Inverse DFT as Matrix Multiplication
Review
Sinusoids
(radians). phase ousinstantane theis
(radians); phase initial theis
seconds; 0.01 1/100 is sampleeach ofduration thethus
cond,samples/se 100 means 100Hz e.g. (Hz);frequency theis
(sec); timeis
(rad/sec);frequency radian theis
amplitude;peak negative-non theis
constants. are ,, variable;realt independenan is
.sin:form theoffunction a is sinusoidA
t
f
t
A
At
tAtx
Harmonic Function Forms
b
abaA
tAtxtbtatx
122 tan,
where,sinsincos
Definition of Function Orthogonality
b
a
dxxgxfxgxf 0 if orthogonal are , Functions
Orthogonality of Basic Trigonometric System
.2, interval over the
0 is system tric trigonomebasic theof functionsdifferent
any two of integral that theshow formulasn integratio The
,...sin,cos,...,2sin,2cos,sin,cos,1
functions ofset infinite theis system tric trigonomebasic The
aa
nxnxxxxx
Fourier Coefficients
,...3,2,1,sin
1ndxnxxfbn
,...3,2,1,cos
1ndxnxxfan
Fourier Series
. of seriesFourier the
called is sincos2
series tric trigonomeThe
,...3,2,1,sin1
and cos1
where,sincos2
:expansion series
tric trigonomefollowing thehas and 2 period offunction a is If
1
0
1
0
xf
kxbkxaa
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1D Fourier Analysis Algorithm
1D Fourier Analysis
● Given a 1D data array, determine the frequency range [W
lower, W
upper]
● Compute the cosine and sine coefficients for each frequency value in the frequency range
● Once the sine and cosine coefficients are computed, determine the amplitude and phase for every constituent harmonic
● Optional: If the signal function is known, recombine the reconstructed harmonics and compute how closely their sum approximates the signal function
1D Fourier Analysis: Harmonic Recovery
.sin
harmonicth - therepresents ,, tuple-3 The
}
;under indexed
isit that socontainer map ain ,, Save
;tan
;sin1
;cos1
{in every For
., e.g., range,frequency a be Let
.:001.0: e.g., interval, timea be Let
function. signal some from valuesofarray 1D a be Let
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1D Fourier Analysis: Harmonic Recovery
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Frequency Reconstructed Harmonics
Example
0.001. of incrementsin dconstructe ,on of analysisFourier 1D Do
.57.30,98.7,73.1,32.12,5.0,5.4,42
where
,20sin573030cos987
20sin73120cos3212
10sin5010cos544
Let
3322110
tf
bababaa
t.t.
t.t.
t.t.π
tf
Solution Steps
● Generate 1D data (this step is unnecessary if the data array is given)
● Approximate sine & cosine coefficients● Reconstruct 10th, 20th, and 30th harmonics● Combine reconstructed harmonics to reconstruct
the original sinusoid● Compute approximate error b/w reconstructed &
original sinusoids
Plotting Computed Cosine Coeffs
>> cosine_coeff_map(0)=1.5886
>> cosine_coeff_map(10)=4.5178
>> cosine_coeff_map(20)=12.3378
>> cosine_coeff_map(30) = 38.0054
>> cosine_coeff_map(40)=5.9178
>> cosine_coeff_map(50)=7.3578
Plotting Computed Sine Coeffs
>> sine_coeff_map(10)=0.5000
>> sine_coeff_map(20)=1.7300
>> sine_coeff_map(30)=2.4350
>> sine_coeff_map(40)=10.7799
>> sine_coeff_map(50)=3.4779
Complex Numbers &
Euler's Identity
Square Roots of Negative Numbers
.551515 :Example
.11
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j
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Exponents of J
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...
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Complex Numbers: Imaginary & Real Parts
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.; Formally,
. ofpart imaginary theis and ofpart real theis and reals are
, where, as written becan number complex Every
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Complex Plane
sin
;cos
;,tan
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;
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ry
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xy
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function. continuous a is .0,1
Consider
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and 112
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Euler's Identity
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.1sincos then , If
.sincos:Identity sEuler'
.sincosThen .Let
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Rectangular & Polar Form of Complex Numbers
form.polar theis
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.sincossincosThen
.Let
.sincos:identity sEuler'
j
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Complex Number Multiplication
.
Then . and Let 212121
21
21212121
2211
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erzerz
Euler's Formula & Conjugate Multiplication
.
Then . and
. and Let
22 rerrerezz
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Expressing Conjugates with Euler's Identity
.sincos
sincossincos
Then .Let
.sinsin i.e.,odd, is sin
while,coscos i.e.,even, is cos that Recall
j
j
j
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x
x
Sine & Cosine with Euler's Formula
.2
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So, .sincos
sincossincossin2
.22
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So, .sincossincoscos2
j
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Complex Sinusoid
.sincos then ,1 and 0 If
.sincos
as defined is sinusoidcomplex then offset, the
is and seconds,in timeis (rad/sec),frequency
radian theis where, and 0Let
.sincos :identity sEuler' Recall
tjtetsA
tjAtAAets
t
tA
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tj
tj
j
Real & Imaginary Parts of Complex Sinusoid
. sinusoidcomplex theofpart imaginary theis sin
; sinusoidcomplex theofpart real theis cos
.sincosLet
tsteim
tstere
tjtets
tj
tj
tj
Circular Motion in 3D
Let us assume that a point is moving in circles in 3D, as shown on the right. If we agree that time can be measured along the three axes (x, y, & z), then we can project the point's circular motion along the three axes (lines): x, y, and z vs. time
Projection of 3D Circular Motion on X-Axis
The X-axis projection is obtained from an observer who looks at the trace of the 3D circular motion from the x-y plane.
.cos i.e., sinusoid,complex the
ofpart real theis axis-X on the projection The
tere tj
Projection of 3D Circular Motion on Y-Axis
The Y Axis projection is obtained from an observer who looks at the trace of the 3D circular motion from the y-z plane.
.sin i.e.,
sinusoid,complex theofpart imaginary
theis axis-Y on the projection The
teim tj
Projection of 3D Circular Motion on Z
The Z Axis projection is obtained from the observer who looks at the trace of the 3D circular motion from the x-z plane.
.by defined is circle This tje
Circular Motion
increases. as
planecomplex in the circleunit thealongmotion
circular clockwise a of trace thedefines
increases. as plane
complex in the circleunit thealongmotion circular
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t
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ets
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tj
Positive & Negative Frequency Sinusoids
sinusoidfrequency -negative a is
sinusoidfrequency -positive a is
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ets
ets
Positive & Negative Frequencies
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Modulus of Complex Sinusoid
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Nth Roots of Unity
Finding Nth Roots
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Commonly Used Formulas: Cheat Sheet
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A Different Notation for Roots of Unity
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Computing Nth Roots of Unity: Example
.,,
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Computing Nth Roots of Unity
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Computing Nth Roots of Unity
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Computing Nth Roots of Unity
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Computing Nth Roots of Unity
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Computing Nth Roots of Unity
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Computing Nth Roots of Unity
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Computing Nth Roots of Unity
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Computing Nth Roots of Unity
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Roots of Unity &
Complex Sinusoids
Roots of Unity & Sinusoids
unity. ofroot th -
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,by defined sinusoids
complex generateunity of rootsth - The/2
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Real Sinusoid at k = 0
Imaginary Sinusoid at k = 0
Sinusoid at k=1: Real Sinusoid
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Sinusoid at k=1: Real & Imaginary Sinusoids
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Real Sinusoid at k = 1 7,...,0,4
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nn
t=0:.1:8;y = cos(pi/4*t);plot(t, y);
t=0:.1:8;y = sin(pi/4*t);plot(t, y);
Sinusoid at k=3: Real & Imaginary Sinusoids
7,...,0,4
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4
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.2
2
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4
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4
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4
33
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4
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Imaginary Sinusoid at k = 3
t=0:.1:8;y = cos(3*pi/4*t);plot(t, y);
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7,...,0,4
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nn
Complex Sinusoid Basis Set
Complex Sinusoid Basis Set
.1,...,2,1,0for
,,...,,,
,..., , ,
isset basis sinusoidcomplex Then the .1,2Let
/121
/222
/121
/020
1210
Nn
ensensensens
WWWW
iN
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n
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i
Inner Product of 2 Sinusoids
1
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/2/21
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.,
as defined is and ofproduct inner The
.1,0for ,
and Let
N
n
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nmkmk
mk
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eessss
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.01
1
1
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/2
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0
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n
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k
Periodicity of Complex Sinusoids
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n
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An Orthonormal Sinusoidal Set
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e
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Signal Projection
.in present is signal much the how
i.e., ,projection oft coefficien thecomputesproduct inner The
.,product inner theas computed is onto of projection The
signals. twobe ,Let
yx
xyxy
xy
Discrete Fourier Transform
Human Ear & Sinusoids
● The human ear is a spectrum analyzer● The cochlea of the inner ear splits sound into its
sinusoidal components● A spectrum that displays the amount of each
sinusoid frequency present in sound is likely to be similar to the representation that the brain receives from the ear
● The human ear can be construed as a Fourier spectrum analyzer of sorts
DFT Definition
1
0
21
0
2
1
0
.1,...,1,0,,
is definition equivalent The
.1
,2,, where
,1,...,1,0,,
:follows as defined is of (DFT) Transform
Fourier Discrete the, spans that sinusoidscomplex of basis
orthogonal thebe and signalcomplex a be Let
N
n
N
knjN
n
tfN
kj
kk
sskntj
k
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0
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0
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,
:follows as computed is DFT the
Then signal.input thebe Let ./2 as computed
are sfrequencie that theAssume sinusoid.complex amplitude
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N
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n
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n
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Continuous Time Fourier Transform
.,
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0
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110
110
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s
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Computing Spectrum Elements
., If
. if ,1
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1D Fourier Analysis: Harmonic Recovery
0l
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2l
u
002
02
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112
12
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222
22
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upperlower WW ,
Frequency Reconstructed Harmonics
1D Discrete Fourier Transform: Spectral Analysis
0
1
2
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Frequency Presence of complex basic sinusoid in signal x
1
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k
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N
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Proportionality of DFT to Projection Coefficients
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t coefficien The
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2
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pure.ally mathematic more isit However,
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DFT as a Digital Filter
2/sin
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as computed isfilter thisof response
frequency theof magnitude The . isoutput whoseand
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T
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DFT as a Digital Filter
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Source: J. O. Smith III, Mathematics of the DFT with Audio Applications, 2nd Edition
DFT Example for N=2
DFT Example for N=2
.11cos1sin1cos1 and 10 because
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.1,11,0 Thus, .1,0for ,1
., compute usLet
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Recall ., :sinusoids basis twohave then we2, have weIf
11
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DFT Example for N=2
.41216,
and
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11
00
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Review: Projection of Signals on Sinusoids
k
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DFT Example: Projection Coefficients
2,21,1222
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DFT Example:Projections on Two Sinusoids
DFT & Inverse DFT as
Matrix Multiplication
DFT Matrix & Inverse DFT Matrix
1 ... 1 0
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111
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DFT as Matrix Manipulation
xS
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DFT as Matrix Manipulation: DFT Matrix
... 1
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/112/122/12
/122/8/4
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Inverse DFT as Matrix Manipulation
1
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1
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1
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1
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1
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1
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110
110
110
110
1/11
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1/11
0/10
...
1 ... 1 1
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2 2 2
1 1 1
0 0 0
11
well.astion multiplicamatrix as DFT inverse theformulatecan We
N
k kk
N
k kk
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k kk
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k kk
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Back to DFT Example for N=2
.41216,
and
81216,
.components twohasIt .2,6 signal theof spectrum thecompute usLet
11
00
sxX
sxX
XXxSxSx
SxSX
NNNN
x
x
ss
ss
X
X
4
8
21 61
21 61
2
6
1 1
1 1
2
6
1 1
1 1
2
6
1 1
1 1
1
0
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tion.multiplicamatrix as computed DFT above The
****
11
00
1
0
Inverse DFT Example for N=2
2
6
2
41 812
4181
4
8
1 1
1 1
2
1
4
8
1 1
0 0
2
11
.4,8 spectrum thefrom signal original erecover th usLet
10
10
ss
ssXS
Nx
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N
References● J. O. Smith III, Mathematics of the Discrete Fourier Transform with
Audio Applications, 2nd Edition.
● G. P. Tolstov. Fourier Series.