fourier series
DESCRIPTION
FOURIER SERIESTRANSCRIPT
SEE 2073 Section 01
SIGNAL & SYSTEM
DR NOOR ASMAWATI BINTI SAMSURI
P19a – Level 4
1
DR. NAS 20132014-2
2
CHAPTER 2
FOURIER SERIES
PART 1
TYPES OF SIGNALS
Signal expression
Time domain signal
Frequency domain spectrum
Complex exponential representation
Fourier series - Periodic signal
OUTLINE
SIGNAL EXPRESSION
Sinusoidal signal is a common type of signal used in communication system
Consider a time-varying voltage signal
Where A0 = amplitude of the sinusoid
ω0= angular frequency
θ0= phase shift
Can be written as sin(ω0t + θ0) but with a difference of π/2
Sinusoid signal can be presented in time domain and frequency domain.
5
)cos()( 000 tAtv
)2
2cos(
)2sin(
)sin()(
00
00
00
tfA
tfA
tAtv
Sinusoidal signal can be sketched in time domain graph.
Consider a time-varying voltage signal
Where A0 = amplitude of the sinusoid
ω0= angular frequency
θ0= phase shift
6
T = 1/f0 = 2π/ω0
A0
TIME DOMAIN SIGNAL
Each sinusoidal signal has frequency (f), amplitude (A) and phase (θ).
Amplitude and phase can be plot with frequency to produce spectrum signal.
Example: v(t) = A1cos(2πf1t + θ1) + A2cos(2πf2t - θ2)
7
A1
ω ω1 ω2
V
A2
Amplitude spectrum
θ1
ω ω1 ω2
V
θ 2
Phase spectrum
π
-π
Single sided spectrum
FREQUENCY DOMAIN SIGNAL
(SPECTRUM)
Example
Plot the amplitude spectrum and phase spectrum for signal below:
8
tCtBtAtv 000 3sin2coscos)(
Hint: -cos θ = cos (θ + π)
sin θ = cos (θ - π/2)
Amplitude spectrum Phase spectrum
V
A B
C
ω0 2ω0 3ω0
ω
φ
-/2
ω
ω0 2ω0 3ω0
)2
3cos()2cos(cos)( 000 tCtBtAtv
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Example
Consider this condition:
Spectrum
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Example
Sketch the magnitude and phase spectrum of a signal, which is
given in a form of Trigonometric Fourier series below:
)
42cos(2cos2sin1)( 000
ttttv
Answer:
How???
PHASOR REPRESENTATION
Sinusoidal signal can be shown as x-axis or y-axis projection by
an anti clockwise rotational phasor:
11
0t + 0
Imaginary
Real
A0
)()(
)cos()(
000
00
tAtv
tAtv
Complex Exponential Representation
Is used to analyze a signal
In general:
Using Euler’s theorem:
In exponential representation:
12
}{2
)(
)cos()(
)()(0
00
tjtj eeA
tv
tAtv
)cos()( 00 tAtv
sincos jej
2cos
tjtj eet
j
eet
tjtj
2sin
Time-domain/polar form
Complex- exponential
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In exponential representation:
V0/2
V0cos(t + ) t +
-(t + )
V0/2
}{2
)cos()( )()(000
tjtj eeA
tAtv
0t + 0
Imaginary
Real
A0
)cos()( 00 tAtv }{2
)( )()(0 tjtj eeA
tv
In time-domain/polar representation:
This signal can be seen as one function which is generated by 2
complex components.
This signal can be plotted in amplitude and phase spectrum as
below:
Exponential form - Double Sided Spectrum
}{2
)( )()(0 tjtj eeV
tv
Amplitude
ω
V0/2
Freq.
-ω
V0/2
Amplitude vs Frequency
Phase
ω
Freq. -ω
-
Phase vs Frequency
Example:
t-domain signal vs exponential form
Amplitude
ω
V0/2
Freq.
-ω
A/2
Amplitude spectrum
Phase
ω
Freq. -ω
-
Phase spectrum
}{2
)cos()( )()( tjtj eeA
tAtv
A
ω ω
V
Amplitude spectrum
θ1
ω ω
V
Phase spectrum
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Example
For the continuous-time periodic signal
a) Represent the signal x(t) in the form of complex exponential
Fourier Series.
b) Sketch the double sided frequency spectrum (magnitude and
phase).
tttx
3
5sin4
3
2cos2)(
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FOURIER SERIES – PERIODIC SIGNAL
Repeats itself at equal intervals of time, T0. v(t)=v(t ± nT0) Where T0=1/ f0 n = integer
Representation of a
periodic signal in the
frequency domain by its
frequency components
(spectrum)
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FOURIER SERIES REPRESENTATION
Trigonometric Complex-
exponential
FS can be represented in two forms:
s(t) volt
2
-20 200 40 t(ms)
Periodic signal can be written as infinite cosine and sine function
General form of Fourier Series:
Where;
a0/2 = d.c. current
an and bn = Fourier coefficient
ω0 = fundamental frequency
nω0 = harmonic frequencies
n = 1, 2, 3, 4, … 19
1
000 )sincos(
2)(
n
nn tnbtnaa
tv
TRIGONOMETRIC FS
tnbtbtbtb
tnatatataa
tv
n
n
0030201
00302010
sin....3sin2sinsin
cos....3cos2coscos2
)(
To plot frequency spectrum, FS should be represented by:
(single sided spectrum)
Where and
Fourier coefficient an and bn can be obtained by integration:
20
1
00 )cos(
2)( nn tnA
atv
n
n
n
b
a tan ( )1
)(22
nnn baA
T
dttvT
a0
0 )(2
T
n tdtntvT
a0
0cos)(2
T
n tdtntvT
b0
0sin)(2
TRIGONOMETRIC FS
FS Symmetrical Properties
21
a0 is merely the time average of v(t). It is zero if it is symmetry in the period of time T.
bn = 0 if v(t) has cosine-like even symmetry along the time axis about t = 0 (i.e. v has same value at t and –t)
an = 0 if v(t) has sine-like odd symmetry along the time axis about t = 0. (i.e. v has same magnitude but opposite sign at t and –t)
an = bn = 0 for all even values of n (except a0) if v(t) has skew (rotational) symmetry (i.e. successive half periods have mirror image shapes)
If signal v(t) is an even symmetry in the period of time T the equation for
Fourier coefficient can be simplified and become:
If signal v(t) is an odd symmetry in the period of time T the equation for
Fourier coefficient can be simplified and become:
...3,2,1 ,cos)(4
2
0
0 ntdtntvT
a
T
n
...3,2,1 ,0 nbn
bn =4
Tv(t)sinnw0t dt
0
T 2
ò , n =1,2,3...
...3,2,1 ,0 nan
FS Symmetrical Properties - example
1
000 )sincos(
2)(
n
nn tnbtnaa
tv
s(t) volt
2
-20 200 40 t(ms)
020
a
020
a
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v(t)
20 mV
-20 mV
t (ms)0 1 2-1
Example – Trigonometric FS
Find the trigonometric Fourier Series for v(t) until the 5th harmonics.
COMPLEX EXPONENTIAL FS
Using Euler’s relation, we can express Fourier Series in complex exponential form.
Original FS:
Complex FS:
24
2cos
00
0
tjntjn eetn
sinnw0t =e jnw0t - e- jnw0t
2 j
1
0
1
0
00
0000
222
)22
(2
)(
n
tjnnntjnnn
n
tjntjn
n
tjntjn
n
ejba
ejbaa
j
eeb
eea
atv
C0 Cn C-n
1
000 )sincos(
2)(
n
nn tnbtnaa
tv
Now, v(t) has been simplified to:
Where cn is complex conjugate of c-n.
Hence, complex FS can be represented by:
where
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1
000)(
n
tjn
n
tjn
n eCeCCtv
n
tjn
neCtv 0)(
dtetvT
C tjnT
n0
0
1
COMPLEX EXPONENTIAL FS