ch#3 fourier series and transform 2 nd semester 1434-1435 king saud university college of applied...
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CH#3
Fourier Series and Transform
2nd semester 1434-1435
King Saud University College of Applied studies and Community Service1301CTBy: Nour Alhariqi
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Outline Introduction Fourier Series Fourier Series Harmonics Fourier Series Coefficients Fourier Series for Some Periodic Signals Example Fourier Series of Even Functions Fourier Series of Odd Functions Fourier Series- complex form
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Introduction The Fourier analysis is the mathematical tool that shows
us how to deconstruct the waveform into its sinusoidal components.
This tool help us to changes a time-domain signal to a frequency-domain signal and vice versa.
•Time domain: periodic signal
•Frequency domain: discrete
Fourier Series
•Time domain: nonperiodic signal
•Frequency domain: continuous
Fourier Transform
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Fourier Series Fourier proved that a composite periodic signal with
period T (frequency f ) can be decomposed into the sum of sinusoidal functions ( or complex exponentials) .
A function is periodic, with fundamental period T, if the following is true for all t: f(t+T)=f(t)
0
T
f
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Fourier Series A periodic signal can be represented by a Fourier series
which is an infinite sum of sinusoidal functions (cosine and sine), each with a frequency that is an integer multiple of f = 1/T
tnbtmaatfn
nm
m sincos)(11
0
nftbmftaatfn
nm
m 2sin2cos)(11
0
T
ntb
T
mtaatf
nn
mm
2sin
2cos)(
110
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Fourier Series Harmonics
tkttt
tkttt
sin,3sin,2sin,sin
andcos,3cos,2cos,cos
ftftft
ftftft
6sin,4sin,2sin
and6cos,4cos,2cos
Fourier Series = a sum of harmonically related sinusoids
fundamental frequency the kth harmonic frequencythe 2nd harmonic frequency
fundamental the kth harmonic
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Fourier Series Harmonics
ωω ω
ω ω ω
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Fourier Series Coefficients
Are called the Fourier series coefficients, it determine the relative weights for each of the sinusoids and they can be obtained from
tnbtmaatfn
nm
m sincos)(11
0
dttfT
a T 001
,2,1cos2
0 mdttmtfT
a Tm
,2,1sin2
0 ndttntfT
b Tn
DC component or average value
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Function s(x) (in red) is a sum of six sine functions of different
amplitudes and harmonically related frequencies. Their
summation is called a Fourier series .
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Fourier Series for Some Periodic Signals
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Example
The Fourier series representation of the square wave
Single term representation of the periodic square wave
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Example
The two term representation of the Fourier series of the periodic square wave
The three term representation of the Fourier series of the periodic square wave
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Example
Fourier representation to contain up to the eleventh harmonic
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Example
Since the continuous time periodic signal is the weighted sum of sinusoidal signals, we can obtain the frequency spectrum of the periodic square-wave as shown below
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Example
From the above figure we see the effect of compression in time domain, results in expansion in frequency domain. The converse is true, i.e., expansion in time domain results in compression in frequency domain.
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Even functions can solely be represented by cosine waves because, cosine waves are even functions. A sum of even functions is another even function.
10 0 105
0
5
q
Fourier Series of Even Functions
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Odd functions can solely be represented by sine waves because, sine waves are odd functions. A sum of odd functions is another odd function.
10 0 105
0
5
q
Fourier Series of Odd Functions
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The Fourier series of an even function tf
is expressed in terms of a cosine series .
10 cos
nn tnaatf
The Fourier series of an odd function tf
is expressed in terms of a sine series .
1sin
nn tnbtf
Fourier Series of Even/Odd Functions
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Fourier Series- complex form The Fourier series can be expressed using complex
exponential function
n
tjnnectf
T tjnn dtetf
Tc 0
1
The coefficient cn is given as
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Fourier Transform
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Outline Fourier transform Inverse Fourier transform Basic Fourier transform pairs Properties of the Fourier transform Fourier transform of periodic signal
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Fourier Transform Fourier Series showed us how to rewrite any periodic
function into a sum of sinusoids. The Fourier Transform is the extension of this idea for non-periodic functions.
the Fourier Transform of a function x(t) is defined by:
The result is a function of ω (frequency).
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Inverse Fourier Transform We can obtain the original function x(t) from the function
X(ω ) via the inverse Fourier transform.
As a result, x(t) and X(ω ) form a Fourier Pair:
( ) ( )x t X
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Example Let The called the unit impulse signal :
The Fourier transform of the impulse signal can be calculated as follows
So ,
)()( ttx )(t
1)()( )0(
jtj edtetX
( ) 1t
w
X(w)
t
x(t)
01
00)(
t
tt
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Basic Fourier Transform pairs Often you have tables for common Fourier transforms
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Example Consider the non-periodic rectangular pulse at zero with
duration τ seconds
Its Fourier Transform is:
2
sin)( cP
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Properties of the Fourier Transform
Linearity:
Left or Right Shift in Time:
Time Scaling:
( ) ( )x t X ( ) ( )y t Y
( ) ( ) ( ) ( )x t y t X Y
00( ) ( ) j tx t t X e
1( )x at X
a a
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properties of the Fourier Transform
Time Reversal:
Multiplication by a Complex Exponential ( Frequency Shifting) :
Multiplication by a Sinusoid (Modulation):
( ) ( )x t X
00( ) ( )j tx t e X
0 0 0( )sin( ) ( ) ( )2
jx t t X X
0 0 0
1( )cos( ) ( ) ( )
2x t t X X
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Example: Linearity
2( ) 4sinc 2sincX
The Fourier Transform of x(t) will be :
Let x(t) be : )(2
1)()( 24
tptptx
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Example: Time Shift
2( ) ( 1)x t p t
( ) 2sinc jX e
The Fourier Transform of x(t) will be :
Let x(t) be :
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Example: Time Scaling
2 ( )p t
2 (2 )p t
2sinc
sinc2
time compression frequency expansion
time expansion frequency compression
1a 0 1a
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Example: Multiplication by a Sinusoid
Let x(t) be : 0( ) ( )cos( )x t p t t
The Fourier Transform of x(t) will be :0 01 ( ) ( )
( ) sinc sinc2 2 2
X
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Examples determine the Fourier transform of the
following periodic signal
Given that
Given that x(t) has the Fourier transform X(ω), Express the Fourier transform of the following signal in terms of X(ω)
𝛿(𝑡)↔ 1
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Fourier Transform for periodic signal
We learned that the periodic signal can be represented by the Fourier series as:
We can obtain a Fourier transform of a periodic signal directly from its Fourier series
n
tjnnectx 0
T tjnn dtetx
Tc 0
01 the coefficient cn is given as
nn ncX )(2)( 0
jj ee 2
1
2
1)cos( jj e
je
j
2
1
2
1)sin(
1
sincos
j
je j
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Fourier Transform for periodic signal The resulting transform consists of a train of impulses in
the frequency domain occurring at the harmonically related frequencies, which the area of the impulse at the nth harmonic frequency nω0 is 2π times nth the Fourier series coefficient cn
So, the Fourier Transform is the general transform, it can handle periodic and non-periodic signals. For a periodic signal it can be thought of as a transformation of the Fourier Series
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Example Let The Fourier series representation of
The Fourier series coefficients The Fourier transform of
So,
tjtj eet 00
2
1
2
1)cos( 0
)cos()( 0ttf )(tf
2
1
2
111 cc
)(tf
)()()( 00 F
0 0 0cos( ) ( ) ( )t
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Example Let , find its Fourier transform ? The Fourier series representation of is
The Fourier series coefficients The Fourier transform of is
tjtj ej
ej
ttf 00
2
1
2
1)sin()( 0
)sin()( 0ttf )(tf
jc
jc
2
1
2
111
)(tf
)()()( 00 jj
F
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Example Let find its Fourier transform ? The complex Fourier series representation of is
The Fourier series coefficients The Fourier transform of is
tjtjtjtj eeeetf 6644
4
1
4
1
2
1
2
1)(
2)6cos(2
1)4cos()( 0 tttf
4
1
2
13322 cccc
)(tf
)6(2
)6(2
)4()4()( F
)(tf