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

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

CTFT Properties (cont…)

Integration

X 1x X 0

2 2

Xx X 0

t

t

fd f

j f

jd

j

F

F