The Continuous-Time Fourier Transform (CTFT)
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Transcript The Continuous-Time Fourier Transform (CTFT)
The Continuous - Time Fourier
Transform (CTFT)
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
Consider a periodic pulse-train signal x(t ) with duty cycle w / T0
. . .
. . .
kw
Aw
Its CTFS harmonic function is X k
sinc
T0
T0
As the period T0 is increased, holding w constant, the duty
cycle is decreased. When the period becomes infinite (and
the duty cycle becomes zero) x(t ) is no longer periodic.
CTFS-to-CTFT Transition
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.
T0
w
2
T0
w
10
CTFS-to-CTFT Transition
This infinity-and-zero problem can be solved by normalizing
the CTFS harmonic function. Define a new “modified” CTFS
harmonic function T0 X k Aw sinc w kf 0 and graph it
versus kf 0 instead of versus k .
50% duty cycle
10% duty cycle
CTFS-to-CTFT Transition
In the limit as the period approaches infinity, the modified
CTFS harmonic function approaches a function of continuous
frequency f (kf 0 ).
Definition of the CTFT (f form)
Forward
X f F x t
x t e j 2 ft dt
Inverse
x t F
X f X f e
-1
j 2 ft
Commonly-used notation
F
x t
X f
df
Definition of the CTFT (ω form)
Forward
X j F x t
x t e jt dt
Inverse
1
x t F -1 X j
2
jt
X
j
e
d
Commonly-used notation
F
x t
X j
Convergence and the
Generalized Fourier Transform
Let x t A. Then from the
definition of the CTFT,
X f
Ae j 2 ft dt A e j 2 ft dt
This integral does not converge so,
strictly speaking, the CTFT does not
exist.
Convergence and the Generalized
Fourier Transform (cont…)
But consider a similar function,
x t Ae
t
, 0
Its CTFT integral
X f
Ae
does converge.
t
e j 2 ft dt
Convergence and the Generalized
Fourier Transform (cont…)
Carrying out the integral, X f A
2
2 f
2
2
.
Now let approach zero.
If f 0 then lim A
0
function is A
2
2 f
2
2
2 f
2
2
2
0. The area under this
df which is A, independent of
the value of . So, in the limit as approaches zero, the
CTFT has an area of A and is zero unless f 0. This exactly
F
defines an impulse of strength A. Therefore A
A f .
Convergence and the Generalized
Fourier Transform (cont…)
By a similar process it can be shown that
1
cos 2 f 0t f f 0 f f 0
2
F
and
j
sin 2 f 0t f f 0 f f 0
2
F
These CTFT’s which involve impulses are called
generalized Fourier transforms (probably because
the impulse is a generalized function).
Negative Frequency
This signal is obviously a sinusoid. How is it described
mathematically?
It could be described by
x t A cos 2 t / T0 A cos 2 f 0t
But it could also be described by
x t A cos 2 f 0 t
Negative Frequency (cont…)
x(t) could also be described by
e j 2 f0t e j 2 f0t
x t A
2
or
x t A1 cos 2 f0t A2 cos 2 f 0 t , A1 A2 A
and probably in a few other different-looking ways. So who is
to say whether the frequency is positive or negative? For the
purposes of signal analysis, it does not matter.
CTFT Properties
If F x t X f or X j and F y t Y f or Y j
then the following properties can be proven.
F
x t y t
X f Y f
Linearity
F
x t y t
X j Y j
CTFT Properties (cont…)
Time Shifting
F
x t t0
X f e j 2 ft0
F
x t t0
X j e jt0
Frequency Shifting
F
x t e j 2 f0t
X f f0
F
x t e j0t
X 0
CTFT Properties (cont…)
Time Scaling
F
x at
1 f
X
a a
1
x at X j
a a
F
Frequency Scaling
1 t F
x X af
a a
1 t F
x X ja
a a
The “Uncertainty” Principle
The 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.
e
e
t 2
t / 2
2
e
F
f 2
2e
F
2 f
2
CTFT Properties (cont…)
Transform of
a Conjugate
F
x * t
X* f
F
x * t
X* j
F
x t y t
X f Y f
MultiplicationConvolution
Duality
F
x t y t
X j Y j
F
x t y t
X f Y f
1
x t y t
X j Y j
2
F
CTFT Properties (cont…)
In the frequency domain, the cascade connection multiplies
the transfer functions instead of convolving the impulse
responses.
CTFT Properties (cont…)
Time
Differentiation
Modulation
Transforms of
Periodic Signals
d
F
x t
j 2 f X f
dt
d
F
x t
j X j
dt
1
F
x t cos 2 f 0t
X f f 0 X f f 0
2
1
F
x t cos 0t
X j 0 X j 0
2
x t
X k e
j 2 kf F t
X f
F
k
x t
X k e
k
X k f kf
0
k
j kF t
X j 2
F
X k k
k
0
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.
x t
2
x t
dt
X f
2
df
2
1
dt
2
X j
2
df
CTFT Properties (cont…)
Integral Definition
of an Impulse
Duality
j 2 xy
e
dy x
F
F
X t
x f and X t
x f
F
F
X jt
2 x and X jt
2 x
CTFT Properties (cont…)
Total area under a time or frequency-domain signal can be found by
evaluating its CTFT or inverse CTFT with an argument of zero
X 0 x t e j 2 ft dt
x t dt
f 0
Total-Area
Integral
x 0 X f e j 2 ft df X f df
t 0
X 0 x t e jt dt
x t dt
0
1
x 0
2
X j e
jt
1
d
t 0 2
X j d
CTFT Properties (cont…)
t
Integration
X f
1
x d j 2 f 2 X 0 f
F
t
x d
F
X j
j
X 0