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LECTURE 09: THE TRIGONOMETRIC FOURIER SERIES. Objectives: The Trigonometric Fourier Series Pulse Train Example Symmetry (Even and Odd Functions) Line Spectra Power Spectra More Properties More Examples - PowerPoint PPT Presentation
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ECE 8443 – Pattern RecognitionEE 3512 – Signals: Continuous and Discrete
• Objectives:The Trigonometric Fourier SeriesPulse Train ExampleSymmetry (Even and Odd Functions)Line SpectraPower SpectraMore PropertiesMore Examples
• Resources:CNX: Fourier Series PropertiesCNX: SymmetryAM: Fourier Series and the FFTDSPG: Fourier Series ExamplesDR: Fourier Series Demo
LECTURE 09: THE TRIGONOMETRIC FOURIER SERIES
URL:
EE 3512: Lecture 09, Slide 2
The Trigonometric Fourier Series Representations• For real, periodic signals:
complex"")(
,
series"Fourierrictrigonomet"sincos)(
0
01
00
k
tjkk
kk
k
ectx
or
tkbtkaatx
• The analysis equations for ak and bk are:
• Note that a0 represents the average, or DC, value of the signal.
• We can convert the trigonometric form to the complex form:
...,2,1,)sin()(2
...,2,1,)cos()(2
)(1
00
00
00
kdttktxT
b
kdttktxT
a
dttxT
a
T
k
T
k
T
...,2,1)(21)(
21
00 kjbacjbacac kkkkkk
EE 3512: Lecture 09, Slide 3
Example: Periodic Pulse Train (Complex Fourier Series)
kTk
TkTTkj
eeTk
jke
jke
Tjke
Tdtetx
Tdtetx
Tc
TjkTjk
TjkTjkT
T
tjkT
T
tjkT
T
tjkk
10
100
0
)(
00
2/
2/
sin
sin)/2(
22
2
11)(1)(1
1010
10101
1
01
1
00
EE 3512: Lecture 09, Slide 4
Example: Periodic Pulse Train (Trig Fourier Series)
TT
TTT
tT
dtT
dttxT
dttxT
aT
T
T
T
T
T
T1
11
2/
2/00
2))((11)1(1)(1)(1 1
1
1
1
• This is not surprising because a0 is the average value (2T1/T).
• Also,
TTTTTTkT
kkTkk
kTk
ckTk
c
ca
k
kk
1110
0
1010
0k
10
0
100
10
00
2)/2()cos()(/
)sin(/
:RulesHopital'L'Use
sinsin
EE 3512: Lecture 09, Slide 5
Example: Periodic Pulse Train (Cont.)
• Check this with our result for the complex Fourier series (k > 0):TkTk
kTk
kTk
T
ktk
Tdttktx
T
dttktxT
a
T
T
T
T
T
Tk
0
10
0
10
0
10
0
00
2/
2/0
)sin(4
))(sin()sin(2
)sin(2)cos()(2
)cos()(2
1
1
1
1
0
)cos(2)sin()(2
)sin()(2
1
1
1
1 0
00
2/
2/0
T
T
T
T
T
Tk
ktk
Tdttktx
T
dttktxT
b
kTk
TkTk
TkTk
jbac kkk)sin()sin(2
)0)sin(4
(21)(
21 10
0
10
0
10
EE 3512: Lecture 09, Slide 6
Even and Odd Functions
• Was this result surprising? Note: x(t) is an even function: x(t) = x(-t)
• If x(t) is an odd function: x(t) = –x(-t)
0,)sin(4
0
10 kk bTkTk
a
0)sin()(2
)cos()(22)cos()(2
2/
2/0
2/
00
2/
2/0
T
Tk
TT
Tk
dttktxT
b
dttktxT
dttktxT
a
])sin()(2[2)sin()(2
0)cos()(2
2/
00
2/
2/0
2/
2/0
TT
Tk
T
Tk
dttktxT
dttktxT
b
dttktxT
a
EE 3512: Lecture 09, Slide 7
Line Spectra
• Recall:
• From this we can show:
kTk
ck10sin
00 )2()2( fkT
kkk
...,2,1)(21)(
21
00 kjbacjbacac kkkkkk
kkkkkk cbajbac 22
21)(
21
kk
kk
kk
k
kk
k
k cab
ca
ab
aab
c
1
1
1
tan0tan
0tan
EE 3512: Lecture 09, Slide 8
Energy and Power Spectra• The energy of a CT signal is:
• The power of a signal is defined as:
Think of this as the power of a voltage across a 1-ohm resistor.
• Recall our expression for the signal:
• We can derive an expression for the power in terms of the Fourier series coefficients:
• Hence we can also think of the line spectrum as a power spectral density:
2/
2/
2 )(1 T
T
dttxT
P
k
tjkkectx 0)(
0
2220
22/
2/
22/
2/
2
211)(1
0
kkk
kk
T
T k
tjkk
T
T
baacdtecT
dttxT
P
2102 sinkTk
ck
dttxE )(2
EE 3512: Lecture 09, Slide 9
Properties of the Fourier Series
EE 3512: Lecture 09, Slide 12
• Reviewed the Trigonometric Fourier Series.
• Worked an example for a periodic pulse train.
• Analyzed the impact of symmetry on the Fourier series.
• Introduced the concept of a line spectrum.
• Discussed the relationship of the Fourier series coefficients to power.
• Introduced our first table of transform properties.
• Next: what do we do about non-periodic signals?
Summary
