7.2 even and odd fourier transforms
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7.2 Even and Odd
Fourier Transforms
phase of signal frequencies
breaking a function into even and oddcomponents using calculations breaking a function into even and odd
components using graphs representation of even and odd functionsin the complex plane
spectrum of a train of rectangular pulses
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Phase of Signal Frequencies
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A signal is composed of frequencies each of which has a specificphase with respect to time zero.
By making the spectrum consist of both sines and cosines, non-zero phases can be represented. For example, the function
can be written as a cosine with a 45 phase lag.
Even temporal functions have a spectrum composed of only cosines
(0 phase lag, even function). Odd temporal functions have aspectrum composed of only sines (90 phase lag, odd function).
In order to use a graphical approach to determining the Fouriertransform it is necessary to decompose the signal into even and
odd functions.
( ) ( ) ( )0 0cos 2 sin 2F t f t f t = +
( ) 02 cos 24
F t f t
=
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Obtaining Even & Odd Functions (1)
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An even function is defined as onewhere fe(-t) = fe(+t). An oddfunction is defined as one wherefo(-t) = -fo(+t).
Any function can be written as asum of even and odd terms.
The two equations above can besolved for the even and oddfunctions.
( ) ( ) ( )
( ) ( ) ( ) ( ) ( )e
o
o e
e
oF t
F t
f t f t f t f
f t f t
t = + =
= +
( ) ( ) ( )
( ) ( ) ( )
1
2
1
2o
e
f t F t F
f t F t F t
t =
=
+
0
0.5
1
-4 -3 -2 -1 0 1 2 3 4
F(t)F(-t)
-0.5
0.5
-4 -3 -2 -1 0 1 2 3 4
even
odd
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Obtaining Even & Odd Functions (2)
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One of the goals of this section of the course is to use the basisset transform pictures to obtain a graphic solution to the Fouriertransform. Thus it would be convenient to learn how tographically decompose a picture into an even and odd parts.
Consider the triangularly-shaped temporal waveform shown at theleft below.
To obtain the even waveform, first mirror image the blue fromquadrant I to quadrant II to obtain the red. Then add the twowaveforms together.
t
1
-1
1-1 t
1
-1
1
-1+ t
1
-1
1-1=
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Obtaining Even & Odd Functions (3)
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To obtain the odd picture, mirror image the blue waveform fromquadrant I to quadrant II, and then to quadrant III. Add the blueand red waveforms together.
t
1
-1
1-1 t
1
-1
1
-1+ t
1
-1
1-1=
The two pictures need to be scaled so that their sum equals thestarting picture. This is accomplished by multiplying each by 1/2.The amplitude on the left sums to zero, while that on the rightsums to the original waveform.
+ =t
0.5
-0.5
1-1 t1-1 t
1
-1
1-1
0.5
-0.5
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Using the Complex Plane
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When an even temporal functionis transformed, it produces aneven spectrum composed ofcosines. An odd function
produces a spectrum of sines.
( ) ( ) ( )
( ) ( ) ( )
cos 2
sin 2
e e
o o
f f t ft dt
f i f t ft dt
=
=
e(f)
io(f)
R
1-1
-1
1 III
III IV
The most useful representation ofthe combined data uses thecomplex plane where theamplitude, R, and phase, , aregiven by the equations to the
right.
( ) ( ) ( )
( ) ( )
( )
2 2
oatan
e o
e
R f f f
ff
f
= +
=
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Spectrum of Rectangular Pulses
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The inter-pulse spacing is 100 s, dictatingthat all frequencies are multiples of 0.01Hz. Since the integral is non-zero, a dccomponent exists. The spectrum iscomposed of sines and cosines since thesignal is neither even nor odd.
The pulse train shown has 20 frequencies.
Their amplitude and phase are shownbelow. The fifth harmonic is 0.05 Hz, etc.
0 50 1000.5
0
0.5
1
1.5
time
0 5 10 15 20
0.2
0.2
0.4
0.6
harmonic
0 5 10 15 20
90
45
45
90
harmonic