The Continuous - Time Fourier Transform (CTFT). Extending the CTFS The CTFS is a good analysis tool...
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Transcript of The Continuous - Time Fourier Transform (CTFT). Extending the CTFS The CTFS is a good analysis tool...
Extending the CTFS
• The CTFS is a good analysis tool for systems with periodic excitation but the CTFS cannot represent an aperiodic signal for all time
• The continuous-time Fourier transform (CTFT) can represent an aperiodic signal for all time
Objective
• To generalize the Fourier series to include aperiodic signals by defining the Fourier transform
• To establish which type of signals can or cannot be described by a Fourier transform
• To derive and demonstrate the properties of the Fourier transform
CTFS-to-CTFT Transition
0 0
Its CTFS harmonic function is X sincAw kw
kT T
0Consider a periodic pulse-train signal x( ) with duty cycle /t w T
0As the period is increased, holding constant, the duty
cycle is decreased. When the period becomes infinite (and
the duty cycle becomes zero) x( ) is no longer periodic.
T w
t
. . . . . .
CTFS-to-CTFT Transition
w T0
2
w T0
10
Below are plots of the magnitude of X[k] for 50% and 10% duty cycles. As the period increases the sinc function widens and its magnitude falls. As the period approaches infinity, the CTFS harmonic function becomes an infinitely-wide sinc function with zero amplitude.
CTFS-to-CTFT Transition
0 0
0
This infinity-and-zero problem can be solved by normalizing
the CTFS harmonic function. Define a new “modified” CTFS
harmonic function X sinc and graph it
versus instead of versus .
T k Aw w kf
kf k
50% duty cycle 10% duty cycle
CTFS-to-CTFT Transition
0
In the limit as the period approaches infinity, the modified
CTFS harmonic function approaches a function of continuous
frequency ( ).f kf
Definition of the CTFT (f form)
Forward
Inverse
2X x x j ftf t t e dt
F
-1 2x X X j ftt f f e df
F
x Xt fF
Commonly-used notation
X x j tj t x t e dt
F
-1 1x X X
2j tt j j e d
F
Forward
Inverse
Definition of the CTFT (ω form)
x Xt jF
Commonly-used notation
Convergence and the Generalized Fourier Transform
Let x . Then from the
definition of the CTFT,
t A
2 2X j ft j ftf Ae dt A e dt
This integral does not converge so, strictly speaking, the CTFT does not exist.
Convergence and the Generalized Fourier Transform (cont…)
x , 0tt Ae
Its CTFT integral
does converge.
2X t j ftf Ae e dt
But consider a similar function,
Convergence and the Generalized Fourier Transform (cont…)
22
2Carrying out the integral, X .
2f A
f
Now let approach zero.
220
22
2If 0 then lim 0. The area under this
2
2function is which is , independent of
2
the value of . So, in the limit as approaches zero, the
CTFT has an area of and is zer
f Af
A df Af
A
o unless 0. This exactly
defines an impulse of strength . Therefore .
f
A A A f
F
Convergence and the Generalized Fourier Transform (cont…)
By a similar process it can be shown that
0 0 0
1cos 2
2f t f f f f
F
and
0 0 0sin 22
jf t f f f f
F
These CTFT’s which involve impulses are called generalized Fourier transforms (probably becausethe impulse is a generalized function).
Negative FrequencyThis signal is obviously a sinusoid. How is it describedmathematically?
It could be described by
0 0x cos 2 / cos 2t A t T A f t
But it could also be described by
0x cos 2t A f t
Negative Frequency (cont…)
x(t) could also be described by
1 0 2 0 1 2x cos 2 cos 2 ,t A f t A f t A A A
0 02 2
x2
j f t j f te et A
and probably in a few other different-looking ways. So who isto say whether the frequency is positive or negative? For thepurposes of signal analysis, it does not matter.
or
CTFT Properties If x X or X and y Y or Y
then the following properties can be proven.
t f j t f j F F
Linearity
x y X Y
x y X Y
t t f f
t t j j
F
F
CTFT Properties (cont…)Time Shifting
0
0
20
0
x X
x X
j ft
j t
t t f e
t t j e
F
F
0
0
20
0
x X
x X
j f t
j t
t e f f
t e
F
F
Frequency Shifting
CTFT Properties (cont…)
Time Scaling
1x X
1x X
fat
a a
at ja a
F
F
Frequency Scaling
1x X
1x X
taf
a a
tja
a a
F
F
The “Uncertainty” PrincipleThe time and frequency scaling properties indicate that if a signal is expanded in one domain it is compressed in the other domain.This is called the “uncertainty principle” of Fourier analysis.
2 2/ 2 22t fe e F
2 2t fe e F
CTFT Properties (cont…)
Transform of a Conjugate
* *
* *
x X
x X
t f
t j
F
F
Multiplication-ConvolutionDuality
x y X Y
x y X Y
x y X Y
1x y X Y
2
t t f f
t t j j
t t f f
t t j j
F
F
F
F
CTFT Properties (cont…)In the frequency domain, the cascade connection multipliesthe transfer functions instead of convolving the impulseresponses.
CTFT Properties (cont…)Time
Differentiation
x 2 X
x X
dt j f f
dtd
t j jdt
F
F
Modulation
0 0 0
0 0 0
1x cos 2 X X
21
x cos X X2
t f t f f f f
t t j j
F
F
Transforms ofPeriodic Signals
20
0
x X X X
x X X 2 X
F
F
j kf t
k k
j k t
k k
t k e f k f kf
t k e j k k
F
F
CTFT Properties (cont…)
Parseval’s Theorem
Even though an energy signal and its CTFT may look quite different, they do have something in common. They have the same total signal energy.
2 2
2 2
x X
1x X
2
t dt f df
t dt j df
CTFT Properties (cont…)
Integral Definitionof an Impulse
2j xye dy x
Duality
X x and X x
X 2 x and X 2 x
t f t f
jt jt
F F
F F
CTFT Properties (cont…)
Total-AreaIntegral
2
0
2
0
0
0
X 0 x x
x 0 X X
X 0 x x
1 1x 0 X X
2 2
j ft
f
j ft
t
j t
j t
t
t e dt t dt
f e df f df
t e dt t dt
j e d j d
Total area under a time or frequency-domain signal can be found by evaluating its CTFT or inverse CTFT with an argument of zero