The Fourier Transform
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
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2
CTFS-to-CTFT Transition
w
Consider a periodic pulse-train signal, x(t), with duty cycle,
T0
Its CTFS harmonic function is X[ k ] =
 kw 
Aw
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.
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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
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T0
w=
10
4
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( kf0 ))
and graph it versus kf0 instead of versus k.
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CTFS-to-CTFT Transition
In the limit as the period approaches infinity, the modified
CTFS harmonic function approaches a function of continuous
frequency f ( kf0 ).
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Definition of the CTFT
Forward
X( f ) = F ( x(t )) =
∞
∫
Inverse
f form
x(t ) = F -1( X( f )) =
x(t )e − j 2πft dt
−∞
X( jω ) = F ( x(t )) =
∫
∫
X( f )e + j 2πft df
−∞
Forward
∞
∞
ω form
x(t )e − jωt dt
−∞
Inverse
∞
1
-1
+ jωt
x(t ) = F ( X( jω )) =
X
j
ω
e
dω
(
)
∫
2π −∞
Commonly-used notation:
F
x(t ) ←
→ X( f )
5/10/04
or
F
x(t ) ←
→ X( jω )
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7
Some Remarkable Implications
of the Fourier Transform
The CTFT expresses a finite-amplitude, real-valued, aperiodic
signal which can also, in general, be time-limited, as a summation
(an integral) of an infinite continuum of weighted, infinitesimalamplitude, complex sinusoids, each of which is unlimited in
time. (Time limited means “having non-zero values only for a
finite time.”)
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Frequency Content
Highpass
Lowpass
Bandpass
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Convergence and the
Generalized Fourier Transform
Let x(t ) = A . Then from the
definition of the CTFT,
X( f ) =
∞
∫
−∞
∞
Ae − j 2πft dt = A ∫ e − j 2πft dt
−∞
This integral does not converge so,
strictly speaking, the CTFT does not
exist.
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10
Convergence and the Generalized
Fourier Transform
But consider a similar function,
xσ (t ) = Ae − σ t , σ > 0
Its CTFT integral,
Xσ ( f ) =
∞
∫
Ae − σ t e − j 2πft dt
−∞
does converge.
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Convergence and the
Generalized Fourier Transform
2σ
Carrying out the integral, Xσ ( f ) = A 2
2 .
σ + ( 2πf )
Now let σ approach zero.
2σ
If f ≠ 0 then lim A 2
2 = 0 . The area under this
σ →0
σ + ( 2πf )
function is
∞
2σ
Area = A ∫ 2
2 df
−∞ σ + ( 2πf )
which is A, independent of the value of σ. So, in the limit as
σ approaches zero, the CTFT has an area of A and is zero unless
f = 0 . This exactly defines an impulse of strength, A.
Therefore
5/10/04
F
A ←
→ Aδ ( f )
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12
Convergence and the
Generalized Fourier Transform
By a similar process it can be shown that
1
cos( 2πf0t ) ←→ [δ ( f − f0 ) + δ ( f + f0 )]
2
F
and
j
sin( 2πf0t ) ←→ [δ ( f + f0 ) − δ ( f − f0 )]
2
F
These CTFT’s which involve impulses are called
generalized Fourier transforms (probably because
the impulse is a generalized function).
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13
Convergence and the Generalized
Fourier Transform
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14
Negative Frequency
This signal is obviously a sinusoid. How is it described
mathematically?
It could be described by
 2πt 
x(t ) = A cos
 = A cos( 2πf0t )
 T0 
But it could also be described by
x(t ) = A cos( 2π ( − f0 )t )
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15
Negative Frequency
x(t) could also be described by
e j 2πf 0 t + e − j 2πf 0 t
x(t ) = A
2
or
x(t ) = A1 cos( 2πf0t ) + A2 cos( 2π ( − f0 )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.
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16
CTFT Properties
If F ( x(t )) = X( f ) or X( jω ) and F ( y(t )) = Y( f ) or Y( jω )
then the following properties can be proven.
Linearity
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F
α x(t ) + β y(t ) ←
→ α X( f ) + β Y( f )
F
α x(t ) + β y(t ) ←
→ α X( jω ) + β Y( jω )
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17
CTFT Properties
Time Shifting
F
x(t − t 0 ) ←
→ X( f )e − j 2πft0
F
x(t − t 0 ) ←
→ X( jω )e − jωt 0
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18
CTFT Properties
Frequency Shifting
F
x(t )e + j 2πf 0 t ←
→ X( f − f0 )
F
x(t )e + jω 0 t ←
→ X(ω − ω 0 )
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19
CTFT Properties
1  f
x( at ) ←→ X
a  a
1  ω
F
x( at ) ←→ X j
a  a
F
Time Scaling
1 t F
x
←→ X( af )


a a
Frequency Scaling
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1 t F
x
←→ X( jaω )
a  a
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20
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.
− π2t 2
−πt
e
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− πf 2 2
−π f
←
→2ee
←
→
ee
t
−π  
 2
F
F
2
F
←→ 2 e
−π ( 2 f )2
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21
CTFT Properties
F
x* (t ) ←
→ X* ( − f )
Transform of
a Conjugate
F
x* (t ) ←
→ X* ( − jω )
F
x(t ) ∗ y(t ) ←
→ X( f ) Y( f )
MultiplicationConvolution
Duality
F
x(t ) ∗ y(t ) ←
→ X( jω ) Y( jω )
F
x(t ) y(t ) ←
→ X( f ) ∗ Y( f )
1
x(t ) y(t ) ←→
X( jω ) ∗ Y( jω )
2π
F
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CTFT Properties
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23
CTFT Properties
An important consequence of multiplication-convolution
duality is the concept of the transfer function.
In the frequency domain, the cascade connection multiplies
the transfer functions instead of convolving the impulse
responses.
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24
CTFT Properties
Time
Differentiation
d
(x(t )) ←F → j 2πf X( f )
dt
d
(x(t )) ←F → jω X( jω )
dt
F
x(t ) cos( 2πf0t ) ←
→
Modulation
Transforms of
Periodic Signals
F
x(t ) cos(ω 0t ) ←
→
x(t ) =
x(t ) =
∞
∑ X[k ]e
k =−∞
∞
∑ X[k ]e
1
X( f − f0 ) + X( f + f0 )]
[
2
[
]
1
X( j (ω − ω 0 )) + X( j (ω + ω 0 ))
2
− j 2π ( kf F ) t
− j ( kω F ) t
←→ X( f ) =
F
∑ X[k ]δ ( f − kf )
0
k =−∞
←→ X( jω ) = 2π
F
k =−∞
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∞
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∞
∑ X[k ]δ (ω − kω )
0
k =−∞
25
CTFT Properties
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26
CTFT Properties
∞
∫ x(t )
∞
2
−∞
Parseval’s
Theorem
Integral Definition
of an Impulse
dt =
∫ X( f )
2
df
−∞
∞
∞
1
2
∫−∞ x(t ) dt = 2π −∞∫ X( jω ) df
2
∞
∫
e − j 2πxy dy = δ ( x )
−∞
F
F
X(t ) ←
→ x( − f ) and X( − t ) ←
→ x( f )
Duality
5/10/04
F
F
X( jt ) ←
→ 2π x( −ω ) and X( − jt ) ←
→ 2π x(ω )
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27
CTFT Properties
Total-Area
Integral
∞

∞
= ∫ x(t )dt
X( 0 ) =  ∫ x(t )e − j 2πft dt 
 f →0 −∞
 −∞
∞

∞
x( 0 ) =  ∫ X( f )e + j 2πft df  = ∫ X( f )df
 t→0 −∞
 −∞
∞

∞
= ∫ x(t )dt
X( 0 ) =  ∫ x(t )e − jωt dt 
ω →0 −∞
 −∞
∞

1 ∞
1
+ jωt
ω
x( 0 ) = 
X
j
e
dω  =
X( jω )dω
(
)
∫
∫
 t→0 2π −∞
 2π −∞
X( f ) 1
∫−∞ x(λ )dλ ←→ j 2πf + 2 X(0)δ ( f )
t
X( jω )
F
∫ x(λ )dλ ←→ jω + π X(0)δ (ω )
−∞
t
Integration
5/10/04
F
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CTFT Properties
X( 0 ) =
∞
∫ x(t )dt
−∞
x( 0 ) =
∞
∫ X( f )df
−∞
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CTFT Properties
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Extending the DTFS
• Analogous to the CTFS, the DTFS is a good
analysis tool for systems with periodic
excitation but cannot represent an aperiodic
DT signal for all time
• The discrete-time Fourier transform (DTFT)
can represent an aperiodic DT signal for all
time
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31
DTFS-to-DTFT Transition
DT Pulse Train
This DT periodic rectangular-wave signal is analogous to the
CT periodic rectangular-wave signal used to illustrate the
transition from the CTFS to the CTFT.
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DTFS-to-DTFT Transition
DTFS of
DT Pulse Train
As the period of the
rectangular wave
increases, the period of
the DTFS increases
and the amplitude of
the DTFS decreases.
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DTFS-to-DTFT Transition
Normalized
DTFS of
DT Pulse Train
By multiplying the
DTFS by its period and
plotting versus kF0
instead of k, the
amplitude of the DTFS
stays the same as the
period increases and
the period of the
normalized DTFS
stays at one.
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34
DTFS-to-DTFT Transition
The normalized DTFS approaches this limit as the DT
period approaches infinity.
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35
Definition of the DTFT
F Form
Inverse
x[n ] = ∫ X( F )e
1
Inverse
j 2πFn
dF ←→ X( F ) =
F
Forward
∞
∑
n =−∞
Ω Form
x[n ]e − j 2πFn
Forward
∞
1
F
jΩn
− jΩn
x[n ] =
X
j
Ω
e
d
Ω
←

→
X
j
Ω
=
x
n
e
(
)
(
)
[
]
∑
2π ∫2π
n =−∞
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36
DTFT Properties
Linearity
5/10/04
F
α x[n ] + β y[n ]←
→ α X( F ) + β Y( F )
F
α x[n ] + β y[n ]←
→ α X( jΩ) + β Y( jΩ)
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37
DTFT Properties
Time
Shifting
5/10/04
F
x[ n − n0 ] ←
→ e − j 2πFn0 X( F )
F
x[ n − n0 ] ←
→ e − jΩn0 X( jΩ)
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38
DTFT Properties
Frequency
Shifting
Time
Reversal
5/10/04
F
e j 2πF0n x[n ]←
→ X( F − F0 )
F
e jΩ 0n x[n ]←
→ X( j (Ω − Ω0 ))
F
x[− n ]←
→ X( − F )
F
x[− n ]←
→ X( − jΩ)
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39
DTFT Properties
Differencing
5/10/04
F
x[n ] − x[n − 1]←
→(1 − e − j 2πF ) X( F )
F
x[n ] − x[n − 1]←
→(1 − e − jΩ ) X( jΩ)
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40
DTFT Properties
1
X( F )
∑ x[m ]←→ 1 − e− j 2πF + 2 X(0) comb( F )
m =−∞
n
Accumulation
5/10/04
F
X( jΩ) 1
 Ω
∑ x[m]←→ 1 − e− jΩ + 2 X(0) comb 2π 
m =−∞
n
F
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41
DTFT Properties
F
x[n ] ∗ y[n ]←
→ X( F ) Y( F )
MultiplicationConvolution
Duality
F
x[n ] ∗ y[n ]←
→ X( jΩ) Y( jΩ)
F
x[n ] y[n ]←
→ X( F )
Y( F )
1
x[n ] y[n ]←→
X( jΩ)
2π
F
Y( jΩ)
As is true for other transforms, convolution in the time domain is
equivalent to multiplication in the frequency domain
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42
DTFT Properties
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43
DTFT Properties
Accumulation
Definition of a
Comb Function
∞
∑
n =−∞
∞
Parseval’s
Theorem
e j 2πFn = comb( F )
∑
x[n ] = ∫ X( F ) dF
∑
x[n ] =
n =−∞
∞
n =−∞
2
2
1
2
1
2
X
j
Ω
dΩ
(
)
∫
2π 2π
The signal energy is proportional to the integral of the
squared magnitude of the DTFT of the signal over one
period.
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The Four Fourier Methods
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Relations Among Fourier Methods
Discrete Frequency
Continuous Frequency
CT
DT
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CTFT - CTFS Relationship
X( f ) =
∞
∑ X[k ]δ ( f − kf )
0
k =−∞
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CTFT - CTFS Relationship
( )
X p [ k ] = f p X kf p
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CTFT - DTFT Relationship
∞
 t
1
Let xδ (t ) = x(t ) comb  = ∑ x( nTs )δ (t − nTs )
 Ts  n =−∞
Ts
and let x[n ] = x( nTs )
There is an “information equivalence” between xδ (t )
and x[n ] . They are both completely described by
the same set of numbers.
X DTFT ( F ) = Xδ ( fs F )
X DTFT ( F ) = fs
∞
∑X
k =−∞
5/10/04
 f
Xδ ( f ) = X DTFT  
 fs 
CTFT
( fs ( F − k ))
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CTFT - DTFT Relationship
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DTFS - DTFT Relationship
X( F ) =
∞
∑ X[k ]δ ( F − kF )
0
k =−∞
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51
DTFS - DTFT Relationship
( )
1
X p [k ] =
X kFp
Np
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