EE302 Sen Lnt 002b GAM FurierTransform May10

8/13/2019 EE302 Sen Lnt 002b GAM FurierTransform May10

http://slidepdf.com/reader/full/ee302-sen-lnt-002b-gam-furiertransform-may10 1/9

Communication Theory

(EE302)

Fourier Transform

Lecturer: Dr. Ghafour Amouzad Mahdiraji

Semester: May 2010



Fourier Transform and Continuous Spectra

Consider signals whose energy is finite.This also means that the signal is concentrated to relatively short

time period (time-limited).

The Fourier transform for this kind of energy signal is defined as

V ( f ) is the spectrum of signal v(t ).

Non-periodic signals have continuous spectra => Fourier transformis used (instead of Fourier series).

Periodic signals have line spectra (discontinuous spectra); they can

be developed in Fourier series.

∫∞

∞−

−== dt et vt v F f V ft j π 2)()]([)(

∫∞

∞−= dt t v E 2

)(



Fourier Transform and Continuous Spectra

The time function v(t ) is obtained from V ( f ) by

using the inverse Fourier transform:

∫

∞

∞−

− == df e f V f V F t v ft j π 21 )()]([)(



Spectrum Properties

1. V ( f ) is a complex function. |V ( f )| is the amplitude spectrumand argV ( f ) is the phase spectrum.

2. The value of V at f = 0 equals the net area of v(t ), since

3. If v(t ) is real, then and

so again we have even amplitude symmetry and odd phasesymmetry.

)(arg)(arg |)(||)(| f V f V f V f V −=−=−

∫∞

∞−= dt t vV )()0(

)()( * f V f V =−



Example: Rectangular pulse

The notation is used for rectangular pulse. The followingrectangular pulse is defined as

Consider the signal .

Its Fourier transform derive as

∏ ⎩⎨⎧

>

<=

/2|t|

/2|t|

0)/(

τ

τ

τ

At

)(sinc)sin()(2/

2/

2τ τ τ π

τ π

τ τ

τ

π f A f f

Adt Ae f V ft j === ∫−

−

∏)/( τ t

∏= )/()( τ t At v



Rectangular Pulse Spectrum

)(sinc)( τ τ

f A f V =



Rectangular Pulse Spectrum

• It can be seen that the spectrum of therectangular pulse obtained from Fourier

transform corresponds to the envelope of thespectrum of the rectangular pulse train obtainedfrom Fourier series (the previous example).

• It can be also noted that most of the spectralcontent (signal energy) is located in thefrequency band of | f| < 1/τ . This means that the

spectrum of a narrow pulse is wide.



Rayleigh’s Energy Theorm

Rayleigh’s energy theorem is similar to Parseval’s theorem

Thus, the energy of the signal can be calculated by integrating the squareof the amplitude spectra.

Example:

The total energy of the rectangular pulse is .

The energy in the frequency band is

This is about 90% of the total energy.

τ τ τ

τ

τ

τ

τ

2/1

/1

/1

/1

222 92.0)(sinc)(|)(| Adf f Adf f V ∫ ∫− −==

∫∫∫ ∞

∞−

∞

∞−

∞

∞−=== df f V df f V f V dt t v E 2*2 |)(|)()(|)(|

∏ )/( τ t A τ 2 A E =

)/1(|| τ < f



Rectangular Pulse – Time and Frequency Domains

Squared Fourier transform(signal spectrum) and theenergy distribution per

frequency bands.

EE302 Sen Lnt 002b GAM FurierTransform May10

Documents

Transcript of EE302 Sen Lnt 002b GAM FurierTransform May10