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BME 310 Biomedical Computing -J.Schesser
118
Spectrum Representation
Lecture #5Chapter 3
BME 310 Biomedical Computing -J.Schesser
119
Mathematical Forms of a Decomposed Signal
2 2
1 1
2 2 2
1
Two Sided Spectrum
( ) cos(2 ) { }2 2
*{ }2 2
Using the subsitution ( ) for the second summation to combi
k k k k
k k k
N Nj j f t j j f tk k
o k k k ok k
N Nj f t j f t j f tk k
o kk k N
A Ax t A A f t X e e e e
X XX e e a e
k k
ne into a single summation
1where for 0; for 02 2
kjko o k k
Xa A k a A e k
2
1 1
Single Sided Spectrum
( ) cos(2 ) { }
Where DC component and is the phasor for frequency
k
k
N Nj f t
o k k k o kk k
jo k k
x t A A f t X e X e
X X A e k
BME 310 Biomedical Computing -J.Schesser
120
Periodic Waveforms• A periodic waveform must satisfy this condition:
x(t+To) = x(t)where To is the period of x(t).
– It can be shown that periodical waveforms are made up of sinusoidal functions which are harmonically related frequencies.
– Harmonic frequencies are frequencies which are multiples of each other.
• If a periodic signal, x(t), has a period of To then it has a fundamental frequency fo = 1/ To.
• Furthermore, we say that x(t) has frequencies which are harmonics of fo.– That is, the kth harmonic of x(t) is k fo
BME 310 Biomedical Computing -J.Schesser
121
Periodic Vs Nonperiodic Signals• We say that is nonperiodic
since there is no assumption about theindividual frequencies.
• We say that is periodic since
there is a fundamental period associated withthe signal
• The fundamental frequency fo is the greatest common divisor of fk
N
Nk
tfjk
keatx 2)(
N
Nk
tkfjk
oeatx 2)(
BME 310 Biomedical Computing -J.Schesser
122
Periodic Vs Nonperiodic Signals
-2-1.5
-1-0.5
00.5
11.5
22.5
33.5
0 0.05 0.1
f0 100 200
21
-100-200
21
f0 165 233.33
21
-165-233.33
21
Spectrum of a Periodic Signal Fundamental Frequency = 100Hz
Spectrum of a Nonperiodic Signal No Fundamental Frequency
-4
-3
-2
-1
0
1
2
3
4
0 0.05 0.1
BME 310 Biomedical Computing -J.Schesser
123
Fourier Analysis• We have shown that a signal can be formulated in
terms of a sum of sinusoids which defines its spectrum
• What we like to find are the ak’s so we can completely determine an signal’s spectrum
• If a signal is periodic, (i.e., the fk’s are multiples of fo) then the determination of these the ak’s are easily achieved using Fourier Analysis.
N
Nk
tfjk
keatx 2)(
N
Nk
tkfjk
oeatx 2)(
BME 310 Biomedical Computing -J.Schesser
124
Another Mathematical Form of a Decomposed Signal
Two Sided Spectrum with N
x(t) Ao Ak cos(2 fkt k )k1
X o {Ak
2e jk e j2 fkt
k1
Ak
2e jk e j2 fkt}
akej2 fkt
k
where fk kfo; ak Ao for k 0;ak X k
2 1
2Ake
jk for k 0
ao akej2kfot ak *e j2kfot
k1
ao 2e{akej2kfot}
k1
since s s* 2e{s}
ao 2e{ ak e jk e j2kfot}k1
ao 2 ak cos(2kfot k )k1
BME 310 Biomedical Computing -J.Schesser
125
Summary of Mathematical Forms of a Decomposed Signal
2
1 1
1
Single Sided Spectrum with
( ) cos(2 ) { }
Where DC component and is the phasor for frequency
Two Sided Spectrum with
( ) cos(2 )
k
k
j f to k k k o k
k kj
o k k
o k k kk
N
x t A A f t X e X e
X X A e k
N
x t A A f t
2 2
1
2
2
1
1
{ }2 2
1 where for 0; for 02 2
2 { }
2 cos(2 )
k k k k
k k
o
j j f t j j f tk ko
k
j f t jkk k o k k
k
j kf to k
k
o k o kk
A AX e e e e
Xa e a A k a A e k
a e a e
a a kf t
BME 310 Biomedical Computing -J.Schesser
126
Fourier Analysis and Synthesis
• Given a Fourier Series (note the limits go to -∞ to ∞ )
• Fourier Analysis is the technique to determine the ak’sfrom x(t).
• Fourier Synthesis is the (opposite) technique to determine x(t) from the ak’s.
2
1
( ) cos(2 )
1 1Where ; ; *;2
o
k
j kf tk o k o k
k k
jo o k k k k o
o
x t a e A A f kt
A a a A e a a fT
BME 310 Biomedical Computing -J.Schesser
127
Fourier Analysis• The way to perform Fourier Analysis is by using the
Fourier Series Integral:
• This states to obtain the ak’s1. Take the periodic function x(t) and multiply it by e-j2π/Tokt
2. integrate the product over 1 period, To, of x(t) 3. Divide the result by To
o
o
T ktT
j
o
k dtetxT
a0
)2(
)(1
BME 310 Biomedical Computing -J.Schesser
128
Derivation of the Fourier Integral
• Some background: The Orthogonality Property• This is a property of certain functions• Orthogonality means perpendicular• So if two functions are orthogonal they are
perpendicular as defined by the following:
lkT
dttvtv
lk
o
T
lk
o
if
)(*)(
if 0
0
BME 310 Biomedical Computing -J.Schesser
129
Sinusoids and Orthogonality• Note that sinusoidal and complex exponential
functions are orthogonal.• First recall:
kT
j
e
kT
j
edte
o
kTT
j
T
oo
ktT
jT kt
Tj
oo
o
ooo
21
2 )1
2
2
0
2
1012sin2cos since
0for ;021
21
2 )1
2
22
2
0
2
jkjke
kk
Tj
e
kT
j
e
kT
j
edte
kjo
kj
o
kTT
j
T
oo
ktT
jT ktT
j
oo
o
oo
o
kT
j
e
kT
j
e
kT
j
edte
o
kj
o
kTT
j
T
oo
ktT
jT kt
Tj
oo
o
ooo
21
21
2 )1
22
2
0
2
o
oo
o
T
oo
ktT
jT kt
Tj
kT
j
edte
2 )1
2
0
2
o
o
T ktT
j
dte0
2
)1
BME 310 Biomedical Computing -J.Schesser
130
Sinusoids and Orthogonality• Note that sinusoidal and complex exponential
functions are orthogonal.• First recall:
1012sin2cos since
0for ;021
21
2 )1
2
22
2
0
2
jkjke
kk
Tj
e
kT
j
e
kT
j
edte
kjo
kj
o
kTT
j
T
oo
ktT
jT ktT
j
oo
o
oo
o
BME 310 Biomedical Computing -J.Schesser
131
Sinusoids and Orthogonality• Note that sinusoidal and complex exponential
functions are orthogonal.• For k = 0:
oo
TTT t
TjT kt
Tj
TT
tdtdtedte o
oo
o
o
o
0
|1 000
02
0
2
BME 310 Biomedical Computing -J.Schesser
132
Sinusoids and Orthogonality• Another way
kT
kT
kTTj
kT
kT
kTT
kT
ktTj
kT
ktT
ktdtT
jktdtT
dte
o
o
o
o
o
o
o
o
T
o
o
T
o
o
T
o
T
o
T ktT
j
oo
oooo
2
02cos2cos
2
02sin2sin
2
2cos
2
2sin
2sin2cos )2
00
000
2
0for 0211
200
20cos2cos
20sin2sin
kk
T
jk
T
kT
kjk
T
k
oo
oo
oT
TTTT
o
T
o
T
o
T
o
T ktT
jTtdtdtjdttdt
Tjtdt
Tktdt
Tjktdt
Tdte
k
o
oooooooo
o
000000000
2
|10102sin02cos2sin2cos
0for
BME 310 Biomedical Computing -J.Schesser
133
Sinusoids and Orthogonality
• Now let’s check sinusoids and complex exponential function for orthogonality:
lkT
dtdtj
dtjdte
dtedte
dteedttvtv
o
TT
TTj
T tllT
jT tlkT
j
T ltT
jktT
jT
lk
oo
oo
oo
oo
ooo
o
if
)01(
)0sin0(cos
)(*)(
00
00
0
0
)(2
0
)(2
0
22
0
lk
dte
dteedttvtv
oo
o
oo
o
T tlkT
j
T ltT
jktT
jT
lk
if0
)(*)(
0
)(2
0
22
0
BME 310 Biomedical Computing -J.Schesser
134
Proof of Fourier Series Integral
oo
oo
oo
T tllT
j
l
T tllT
j
l
T tllT
j
l dteadteadtea0
)1(2
10
)(2
0
)1(2
1
olTa
00 11 loll aTaa
k
T tlkT
j
k
o
o dtea0
)(2
dteadtetxo
o
o
o
T
k
tlkT
j
k
T ltT
j
0
)(2
0
2
}{)(
k
tlkT
j
kk
ltT
jktT
j
k
ltT
joooo eaeeaetx
)(2222
)(
k
ktT
j
koeatx2
)(
o
o
T ktT
j
o
k dtetxT
a0
)2(
)(1
BME 310 Biomedical Computing -J.Schesser
135
Fourier Series
• Fourier Analysis:
• Fourier Synthesis:
o
o
T ktT
j
o
k dtetxT
a0
)2(
)(1
k
ktT
j
koeatx2
)(
BME 310 Biomedical Computing -J.Schesser
136
Example• Find the spectrum of: sin3(3πt)
32;
23
23;3
)833(1
)833(1
)2
(1
)3(sin1
2
0
9339
2
0
963369
2
0
333
2
0
3
ooo
ktT
jT tjtjtjtj
o
ktT
jT tjtjtjtjtjtj
o
ktT
jT tjtj
o
ktT
jT
ok
Tf
dtej
eeeeT
dtej
eeeeeeT
dtejee
T
dtetT
a
o
o
o
o
o
o
o
o
BME 310 Biomedical Computing -J.Schesser
137
Example• Find the spectrum of: sin3(3πt)
3/2
0
3)3(3/2
0
3)1(3/2
0
3)1(3/2
0
3)3(
3/2
0
)39()33()33()39(
33/2
0
9339
3/223/2
0
9339
163
169
169
163
)833(
23
)833(
23
)833(
23
dtej
dtej
dtej
dtej
dtj
eeee
dtej
eeee
dtej
eeeea
tkjtkjtkjtkj
tkjtkjtkjtkj
ktjtjtjtjtj
ktjtjtjtjtj
k
BME 310 Biomedical Computing -J.Schesser
138
Example• Find the spectrum of: sin3(3πt)
2/3 2/3(3 )3 2
30 0
2/3 2/3(1 )3 2
10 0
2/3 2/3(1 )3
0 0
3 3 3 2 1 1; for 3;16 16 16 3 8 8 8 8
9 9 9 2 3 3 3 3; for 1;16 16 16 3 8 8 8 8
9 9 9 216 16 16 3
jj k tk
jj k t
j k t
j ja e dt dt k a ej j j j
j je dt dt k a ej j j j
e dt dtj j j
21
2/3 2/3(3 )3 2
30 0
3 3 3 3; for 1;8 8 8 8
3 3 3 2 1 1; for 3;16 16 16 3 8 8 8 8
j
jj k t
j jk a ej
j je dt dt k a ej j j j
BME 310 Biomedical Computing -J.Schesser
139
Fourier Series of a Square Wave
To.5To-.5To-To
1
s(t)
oo
o
TtT
ts
Tt
21for 0
)(210for 1
t
BME 310 Biomedical Computing -J.Schesser
140
Fourier Series of a Square WaveAnalysis
kkkj
k
kj
o
kT
jTkT
j
o
T
o
ktT
j
o
T ktT
j
o
T
T
ktT
j
o
T ktT
j
o
T ktT
j
o
k
jkjke
kkj
kje
kT
j
eeT
kT
j
eT
dteT
dteT
dteT
dtetsT
a
o
o
o
o
ooo
o
o
oo
o
oo
)1(0)1()sin()cos(
0;2
)1(1
21
)2(
1
|)2(
111
0111
)(1
0)2(2
)2(
2/0
)2(2/
0
)2(
2/
)2(2/
0
)2(
0
)2(
2
/2
00 0
1 ( 1) ; 02
1 ( 1) 1 1 ; odd2
1 (1) 0; even2
1 1 1( ) 12
o o
k
k
j
k
T T
o o
a kj k
a e kj k j k k
kj k
a s t dt dtT T
BME 310 Biomedical Computing -J.Schesser
141
Fourier Series of a Square WaveSynthesis
-0.5
0
0.5
1
1.5
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
-0.3
-0.1
0.1
0.3
0.5
0.7
0.9
1.1
1.3
1.5
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
2 terms To=4
-0.3
-0.1
0.1
0.3
0.5
0.7
0.9
1.1
1.3
1.5
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
3 terms To=458 terms To=4
2 2
01
22
2( )2
( ) 2 { }
1 2 { }2
1 12 { }21 1 22 cos( )2 2
o o
o
o
j kt j ktT T
k kk k
jj kt
T
k odd
j ktT
k odd
k odd o
x t a e a e a e
ee ek
e ek
ktk T
t
t t
BME 310 Biomedical Computing -J.Schesser
142
Spectrum of a Square Wave2
0
1 1 ; odd
0; even12
j
k
k
a e kj k k
a k
a
0
To=4
fo=0.25 Hz
0.25 0.75 1.25-1.25 -0.75 -0.25
k=1
a1=-j/π
|a1|=1/π k=3
a3=-j/3π
|a3|=1/3π
k=5
a5=-j/5π
|a5|=1/5π
k=-1
a1=j/π
|a-1|=1/πk=-3
a3=j/3π
|a-3|=1/3π
k=-5
a-5=j/5π
|a-5|=1/5π
-0.5-1.00 0.5 1.00
k=0
a0=1/2
f
BME 310 Biomedical Computing -J.Schesser
143
Fourier Series of a Sawtooth Wave
To-To-2To
1s(t)
oo TtTtts 0for /)(
2To 3To-3Tot
BME 310 Biomedical Computing -J.Schesser
144
Fourier Series of a Sawtooth WaveAnalysis
01)2sin()2cos( since
}1{32
1)(32
1(}03{32
1[91
])32
1(|32
[91
331
321,
31,
3
331)(
31
2
22
3
0
3230
32
3
0
32
32
32
3
0
323
0
32
jkjke
ekjkj
ekj
dtekj
ekj
t
dteta
ekj
vdtedv
dtdutu
dtetdtetsa
kj
kjkj
ktjktj
ktj
k
ktjktj
ktjtkjk
BME 310 Biomedical Computing -J.Schesser
145
Fourier Series of a Sawtooth WaveAnalysis
2
3 3
00 0
1
1 1 1 1[ {3 1} ( )( {1 1}9 2 3 2 3 2 31 1[ {3 1} 09 2 3
1 1 1 1 ; 03 ( 2 3) 2 2
1 1 1 9( ) ( ) 0.53 3 3 9 21 1 2( ) 2 cos( )2 2 3 2
k
j
aj k j k j k
j k
j e kj k k k
ta f t dt dt
s t ktk
BME 310 Biomedical Computing -J.Schesser
146
-0.5
0
0.5
1
1.5
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
-0.5
-0.3
-0.10.1
0.3
0.5
0.7
0.91.1
1.3
1.5
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
-0.5
0
0.5
1
1.5
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
-0.5
0
0.5
1
1.5
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Fourier Series of a Sawtooth WaveSynthesis
2 terms To=3
3 terms To=358 terms To=3
11 terms To=3
t t
t t
BME 310 Biomedical Computing -J.Schesser
147
Spectrum of a Sawtooth Wave2
0
1 1 ; 02 2
12
j
ka j e kk k
a
0
To=3
fo=0.33 Hz
0.33 1.00 1.66-1.66 -1.00 -0.33
k=1
a1=j/2π
|a1|=1/2π k=2
a3=j/4π
|a3|=1/4π
k=-1
a1=-j/2π
|a-1|=1/2πk=-2
a3=-j/4π
|a-3|=1/4π
-0.66-1.33 0.66 1.33
k=0
a0=1/2
f
BME 310 Biomedical Computing -J.Schesser
148
Time-Frequency Spectrum• Up till now we have seen signals whose frequencies
do not change with time.• However, in reality, signals can produce different
frequencies at different times– Music– Voice– Frequency Modulation
• To plot the spectrum of such signals, we can use a 3-D plot called a spectrogram.– Plots time on the x-axis, frequency on the y-axis and
magnitude on the z-axis (out of the plane of the paper)
BME 310 Biomedical Computing -J.Schesser
149
Example of a Spectrogram• Three sinusoids (0.1 Hz, 0.2 Hz, 0.4Hz) in sequence
BME 310 Biomedical Computing -J.Schesser
150
Frequency Modulation
• We have signals of the type:
where the angle (of the cosine) varies linearly with time and the time derivative of the angle is o, constant
• However, we can general this signal such that the angle varies with time such that its derivative is not constant.
( )( ) { } cos( )oj tox t e Ae A t
)()(
))(cos(}{)( )(
tdtdt
tAAeetx tj
Instantaneous Frequency
BME 310 Biomedical Computing -J.Schesser
151
FM Radio
• Frequency Modulation is the scheme used in FM broadcast.
• For example, if ω(t)=ωc+m(t) where ωc is called the carrier frequency and m(t) contains the “information” (voice or music), then FM broadcast is:
dttmtdttmt cc )(})({)(
BME 310 Biomedical Computing -J.Schesser
152
Homework
• Exercises:– 3.4 – 3.8
• Problems:
3.5 Instead use x(t) 10 20cos(2 (200)t 14 ) 10cos(2 (250)t)
3.8,3.9 Use Matlab for plotting the spectrum; submit your code 0 for 0 t 2.53.12 Instead use x(t) Use Matlab for plotting the spectrum; submit your code 2 for 2.5 t 5
3.13,3.14