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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 RecognitionECE 3163 – Signals and Systems
• 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
Audio:URL:
ECE 3163: 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
0
0
0
0
0
0
kdttktxT
b
kdttktxT
a
dttxT
a
T
k
T
k
T
...,2,1)(2
1)(
2
100 kjbacjbacac kkkkkk
ECE 3163: Lecture 09, Slide 3
Example: Periodic Pulse Train (Complex Fourier Series)
k
Tk
TkTTkj
ee
Tk
jk
e
jk
e
Tjk
e
Tdtetx
Tdtetx
Tc
TjkTjk
TjkTjkT
T
tjkT
T
tjkT
T
tjkk
10
100
0
)(
00
2/
2/
sin
sin)/2(
2
2
2
11)(
1)(
1
1010
10101
1
01
1
00
ECE 3163: Lecture 09, Slide 4
Example: Periodic Pulse Train (Trig Fourier Series)
T
TTT
Tt
Tdt
Tdttx
Tdttx
Ta
T
T
T
T
T
T
T1
11
2/
2/0
0
2))((
11)1(
1)(
1)(
1 1
1
1
1
• This is not surprising because a0 is the average value (2T1/T).
• Also,
T
TTTTTkT
kk
Tkk
k
Tkc
k
Tkc
ca
k
k
k
1110
0
1010
0k
10
0
100
10
00
2)/2()cos(
)(/
)sin(/
:RulesHopital'L'Use
sinsin
ECE 3163: Lecture 09, Slide 5
Example: Periodic Pulse Train (Cont.)
• Check this with our result for the complex Fourier series (k > 0):
Tk
Tk
k
Tk
k
Tk
T
k
tk
Tdttktx
T
dttktxT
a
T
T
T
T
T
T
k
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
T
k
k
tk
Tdttktx
T
dttktxT
b
k
Tk
Tk
Tk
Tk
Tkjbac kkk
)sin()sin(2)0
)sin(4(2
1)(
2
1 10
0
10
0
10
ECE 3163: 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 bTk
Tka
0)sin()(2
)cos()(2
2)cos()(2
2/
2/
0
2/
0
0
2/
2/
0
T
T
k
TT
T
k
dttktxT
b
dttktxT
dttktxT
a
])sin()(2[2)sin()(
2
0)cos()(2
2/
0
0
2/
2/
0
2/
2/
0
TT
T
k
T
T
k
dttktxT
dttktxT
b
dttktxT
a
ECE 3163: Lecture 09, Slide 7
Line Spectra
• Recall:
• From this we can show:
k
Tkck
10sin
00 )2()2( fkT
kkk
...,2,1)(2
1)(
2
100 kjbacjbacac kkkkkk
kkkkkk cbajbac 22
2
1)(
2
1
kk
kk
kk
k
kk
k
k ca
bc
aa
b
aa
b
c
1
1
1
tan
0tan
0tan
ECE 3163: 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
2
11)(
10
kkk
kk
T
T k
tjkk
T
T
baacdtecT
dttxT
P
2102 sin
k
Tkck
dttxE )(2
ECE 3163: Lecture 09, Slide 9
Properties of the Fourier Series
ECE 3163: 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
