1

Fourier transforms.1

Fourier Transform

Reference– Chapter 2.2 – 2.5, Carlson, Communication Systems

Using the Fourier series, a signal over a finite interval can be represented in terms of a complex exponential series. If the function is periodic, this representation can be extended over the entire interval .( )∞∞− ,

∑∞

−∞=

=n

tjnn

oeFtf ω)( ∫ −=T tjn

n dtetfT

F o

0 )(1 ω

Fourier transforms.2

Fourier Transform

On the other hand, Fourier transform provides the link between the time-domain and frequency domain descriptions of a signal. Fourier transform can be used for both periodicand non-periodic signals.

( ) ∫∞

∞−

−= dtetfF tjωω )(( )∫∞

∞−= ωω ω deFtf tj)(

2

Fourier transforms.3

ExampleThe the last example of the previous lecture shows that if the period (T) of a periodic signal increases, the fundamental frequency (ωo=2π/T) becomes smaller and the frequency spectrum becomes more dense while the amplitude of each frequency component decreases. The shape of the spectrum, however, remains unchanged with varying T. Now, we will consider a signal with period approaching infinity.

Suppose we are given a non-periodic signal f(t). In order to applying Fourier series to the signal f(t), we construct a new periodic signal fT(t) with period T.

Fourier transforms.4

The original signal can be obtained back again by letting the period , that is,

)(tf

)(tfT

∞→T

)()( lim tftf TT ∞→

=

t

t

3

Fourier transforms.5

The periodic function fT(t) can be represented by an exponential Fourier series.

As the magnitude of the Fourier coefficients go to zero when the period is increased, we define

.

The Fourier series pair become

where)( ∑∞

−∞=

=n

tjnnT

oeFtf ω

∫−

− ==2/

2/ o /2 and )(1 T

T

tjnTn T dtetf

TF o πωω

( ) nnon TFFn ≡≡ ωωω and

( ) where1)( ∑∞

−∞=

=n

tjnT

neFT

tf ωω ( ) ∫−

−=2/

2/ )(

T

T

tjTn dtetfF nωω

Fourier transforms.6

The spacing between adjacent lines (∆ω) in the line spectrum of fT(t) is

Therefore, we have

Now as T becomes very large, ∆ω becomes smaller and the spectrum becomes denser. In the limit, the discrete lines in the spectrum of fT(t) merge and the frequency spectrum becomes continuous.

T/2πω =∆

( ) (1) 2

)(πωω ω ∆

= ∑∞

−∞=n

tjnT

neFtf

4

Fourier transforms.7

Mathematically, the infinite sum (1) becomes an integral

Similarly,

( )

( ) ωωπ

ωωπ

ω

ω

deFtf

eFtf

tj

n

tjnTTT

n

∫

∑∞

∞−

∞

−∞=∞→∞→

=⇒

∆=

21)(

21lim)(lim

( ) ( )

( )

( ) ∫∫

∫

∫

−

−

−

−

∞→∞→

−

−

−

−

=⇒

=⇒

≡≡=⇒

=

2/

2/

2/

2/

2/

2/

2/

2/

)(

)(limlim

and )(

)(1

T

T

tjnT

T

T

tjnTTnT

nnon

T

T

tjTn

T

T

tjnTn

dtetfF

dtetfF

TFFndtetfF

dtetfT

F

o

o

n

o

ω

ω

ω

ω

ω

ω

ωωωω Q

Fourier transform of f(t)

Inverse Fourier transform of F(ω)

Fourier transforms.8

Operators are often used to denote the transform pair.

{ } f(t)tf of ansformFourier tr )(ℑ

( ){ } ( )ωω FF of ansformFourier tr Inverse 1−ℑ

{ }[ ] )()( 1 tftf ℑℑ= −

5

Fourier transforms.9

Singularity functions

– This is a particular class of functions which are useful in signal analysis.

– They are mathematical idealization and, strictly speaking, do not occur in physical systems.

– Good approximation to certain limiting condition in physical systems. For example, a very narrow pulse.

Fourier transforms.10

Singularity functions

Impulse function– This function has the property exhibited by the following

integral:

for any f(t) continuous at t = to, to is finite. All the propertiescan be derived from this definition.

(2) elsewhere0

)()()(

<<

=−∫btatf

dttttf oob

a oδ

ot

∞

6

Fourier transforms.11

Singularity functions

Properties of the impulse functionAmplitude

All values of δ(t) for are zero. The amplitude at the point t = to is undefined.

Area (strength)If f(t) =1, (2) becomes

Therefore δ(t) has unit area. Similarly, Aδ(t) has an area of A units.

ott ≠

1)( btadttt o

b

a o <<=−∫ δ

Fourier transforms.12

Graphic representationTo display the impulse function at t = to, an arrow is used to avoid to display the amplitude. The area of the impulse is designated by a quantity in parentheses beside the arrow or by the height of the arrow. An arrow pointing down indicates negative area.

Relation to the unit step functionThe unit step function is defined by

ot

)( ottA −δ )(2 1ttA −δ

1t

<>

=−o

o

tttt

ttu01

)( 0

ot

1

7

Fourier transforms.13

Using (2) and letting

Therefore the integral of the unit impulse function is the unit step function. The converse can also be shown by differentiating both sides of the above equation.

1 and , ,- ==∞= f(t)tba

)()(01

)(

)()(

o

t

oo

ot

o

b

a o

ttudttttt

dt

dttttf

−=−⇒

<>

=−⇒

−

∫∫

∫

∞−∞−ττδττδ

δ

)()(

)()(

)()(

oo

o

t

o

o

t

o

ttudtdtt

ttudtddt

dtd

ttudt

−=−⇒

−=−⇒

−=−

∫

∫

∞−

∞−

δ

ττδ

ττδ

Fourier transforms.14

Spectral density function

represents f(t) as a continuous sum of exponential functions with frequencies lying in the interval . The relative amplitude of the components at any frequency ω is proportional to F(ω). F(ω) is called the spectral density function of f(t).

– Each point on the F(ω) curve contributes nothing to the representation of f(t); it is the area that contributes.

( ) ωωπ

ω deFtf tj∫∞

∞−=

21)(

( )ωF

),( ∞−∞

t

8

Fourier transforms.15

Spectral density function

Example– Consider a rectangular pulse train– The line spectrum of the Fourier series

of the signal is

The spectral density of the signal is

( )ωF

Impulse function

Fourier transforms.16

Existence of the Fourier transform

We may ignore the question of the existence of the Fourier transform of a time function when it is an accurately specified description of a physically realizable signal. In other words, physical realizability is a sufficient condition for the existence of a Fourier transform.

9

Fourier transforms.17

Parseval’s theorem for energy signals

Using the Parseval’s theorem, we can find the energy of a signal in either the time domain or the frequency domain.

ExampleEnergy contained in the frequency band of a

real-valued signal is

∫∫∞

∞−

∞

∞−== ωω

πdFdttfE 22 )(

21 )(

21 ωωω <<

∫

∫∫

=

+

−

−

2

1

2

1

1

2

2

22

)(1

)()(21

ω

ω

ω

ω

ω

ω

ωωπ

ωωωωπ

dF

dFdF

Fourier transforms.18

Fourier transforms of some signals

Impulse function δ(t) The Fourier transform of a unit impulse δ(t) is

It shows that an impulse function has a uniform spectral density over the entire frequency spectrum. In practice, a narrow pulse in time domain has a very wide bandwidth in frequency domain.

ExampleIf we increase the transmission rate of digital signal, a wider frequency bandwidth is needed.

{ } ∫∫∞

∞−

∞

∞−

− ====ℑ )0()()(1)()( 0 fdttftedtett jtj δδδ ω Q

10

Fourier transforms.19

Fourier transforms of some signals

Complex exponential functionThe Fourier transform of a complex exponential function is

On the other hand, the inverse Fourier transform of is

{ } ∫∞

∞−

−±± ==ℑ ?dteee tjtjtj oo ωωω

)( oωωδ ±

{ }

{ } { }{ } )(2)(21)(

21)(

1

1

ootj

tjtjoo

o

o

e

edte

ωωπδωωδπ

ωωδπ

ωωδ

ω

ωω

±=±ℑℑ=ℑ∴

=±=±ℑ

−±

∞

∞−

±− ∫

Fourier transforms.20

Sinusoidal signalsThe sinusoidal signals can be written as

The Fourier transform of these signals are

jeexeex

jxjxjxjx

2sin

2cos

−− −=

+=

{ }

{ } )()(2

cos

)()(2

cos

oo

tjtj

o

oo

tjtj

o

jjjeet

eet

oo

oo

ωωδπωωδπω

ωωπδωωπδω

ωω

ωω

+−−=

−

ℑ=ℑ

++−=

+

ℑ=ℑ

−

−

11

Fourier transforms.21

t

toωsin

oωoω− )(π

)( π−

)(ωjF

Fourier transforms.22

Periodic signalA periodic signal f(t) can be represented by its exponential Fourier series.

The Fourier transform is

Thus the spectral density of a periodic signal consists of a setof impulses located at the harmonic frequencies of the signal. The area of each impulse is 2π times the values of its corresponding coefficient in the exponential Fourier series.

TeFtf on

tjnn

o /2where )( πωω == ∑∞

−∞=

{ }

{ }

∑

∑

∑

∞

−∞=

∞

−∞=

∞

−∞=

−=

ℑ=

ℑ=ℑ

non

n

tjnn

n

tjnn

nF

eF

eFtf

o

o

)(2

)(

ωωδπ

ω

ω

12

Fourier transforms.23

Some properties of the Fourier transform

Linearity (superposition)The Fourier transform is a linear operation based on the properties of integration and therefore superposition applies.

{ } )()()()( ωω bGaFtbgtaf +=+ℑ

Fourier transforms.24

Some properties of the Fourier transform

DualityIf , then

Example

{ } )()( ωFtf =ℑ { } )(2)( ωπ −=ℑ FtF

t0.5

0.5t

ω

ω

1

1

1

π2

π2

π2

13

Fourier transforms.25

Coordinate scalingThe expansion or compression of a time waveform affects the spectral density of the waveform. For a real-valued scaling constant α and any signal f(t)

Example{ } )(1)(

αω

αα Ftf =ℑ

2te−

2)2( te−

2/2

2 ωπ −e

2/)2/( 2

2 ωπ −e

Fourier transforms.26

Time shifting (delay)

If a signal is delayed in time by to, its magnitude spectral density remains unchanged and a negative phase -ωto is added to each frequency component.

{ } otjo eFttf ωω −=−ℑ )()(

tj oe ω

14

Fourier transforms.27

Frequency shifting (Modulation)

Therefore multiplying a time function by causes its spectral density to be translated in frequency by ωo.

Example

{ } )()( otj Fetf o ωωω −=ℑ

tj oe ω

{ } [ ])()(21cos)( ooo FFttf ωωωωω −++=ℑ

)(ωF )( oF ωω −)( oF ωω +

Fourier transforms.28

Differentiation

Time differentiation enhances the high frequencycomponents of a signal.

Integration

Integration in time suppresses the high-frequency components of a signal.

)()( ωωFjtfdtd

=

ℑ

{ } ∫∫∞

∞−∞−=+=ℑ dttfFFF

jdf

t)()0( where )()0()(1)( ωδπω

ωττ