Ch2 rev[1]

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ELEN 4610:Analog Communications

Chapter #2: Fourier Representation of Signals and Systems

Matlab and Simulink Tutorialhttp://www.mathworks.com/academia/student_center/tutorials

Prof. Caroline González

Haykin, S., and M. Moher, Introduction to Analog & Digital Communications, 2nd ed., Wiley, 2007.

Ch 2-1


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In this chapter, we will study:

Definition of the Fourier Transform

Properties of the Fourier Transform

The Inverse Relationship between Time and Frequency

Dirac Delta Function

Fourier Transform of Periodic Signals

Power Spectral Density

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The Fourier Transform (FT)

The FT relates the frequency-domain description of a signal to its time-domain description.

− Determine the frequency content of a continuous-time signal.

− Evaluates what happens to this frequency content when the signal is passed through a linear time-invariant (LTI) system.

− A signal can only be strictly limited in the time domain or the frequency domain, but not both.

− Bandwidth is an important parameter in describing the spectral content of a signal and the frequency response of a LTI filter.

3Haykin, S., and M. Moher, Introduction to Analog & Digital

Communications, 2nd ed., Wiley, 2007.

Definition of the FT

Advantages of using frequency-domain analysis− Resolution into eternal sinusoids presents the

behavior as the superposition of steady-state effects.

− Usually the time-domain analysis involves solving differential equations, but in the frequency domain involves simple algebra equations.

− Provides the frequency content of a signal.



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Dirichlet’s Conditions

For the FT of a signal g(t) to exist, it is sufficient, but not necessary, that g(t)satisfies:

− The function g(t) is single-valued, with a finite number of maxima and minima in any finite time interval.

− The function g(t) has a finite number of discontinuities in any finite time interval.

− The function g(t) is absolutely integrableor the g(t) is an energy-like signal.

( )

( ) ∞<

∞<

∫

∫∞

∞−

∞

∞−

dttg

dttg

2



Continuous Spectrum

The FT is a complex function of frequency so that

( ) ( ) ( )

( )( )

is the continuous amplitude spectrum

is the continuous phase spectrum

j fG f G f e

where

G f

f

θ

θ

=

For a real-value function g(t) the FT has the following characteristics

( ) ( )( ) ( )( ) ( )

*G f G f

G f G f

f fθ θ

− =

− =

− = −6


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Continuous Spectrum

In conclusion

− The spectrum of a real-valued signal exhibits conjugate symmetry.

The amplitude spectrum of a signal is an even function of the frequency; the amplitude spectrum is symmetricwith respect to the origin f=0.

The phase spectrum of a signal is an odd function of the frequency; the phase spectrum is antisymmetric with respect to the origin f=0



Examples

Rectangular Pulse (Example 2.1)

−Matlab Demo

Decaying Exponential Pulse (Ex. 2)

−Matlab Demo

Rising Exponential Pulse (Ex. 2)

−Matlab Demo

Drill P2.1



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Rectangular Pulse Amplitude Spectrum

-8 -6 -4 -2 0 2 4 6 80

0.5

1

1.5

2

Time

Amplitude

Spectrum of a Rectangular Pulse

-1.5 -1 -0.5 0 0.5 1 1.5-5

0

5

10

frequency

Amplitude Spectrum



Decaying Exponent Spectrum

-1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1

Time

Amplitude

Decaying Exponential Pulse

-3 -2 -1 0 1 2 30

0.2

0.4

0.6

0.8

frequency

Magnitude

Amplitude Spectrum of Decaying Exponential Pulse

-3 -2 -1 0 1 2 3-100

-50

0

50

100

frequency

Phase in degrees

Phase Spectrum of Decaying Exponential Pulse



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Rising Exponent Spectrum

-4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 10

0.5

1

Time

Amplitude

Rising Exponential Pulse

-3 -2 -1 0 1 2 30

0.5

frequency

Magnitude

Amplitude Spectrum of Rising Exponential Pulse

-3 -2 -1 0 1 2 3-100

0

100

frequency

Phase in degrees Phase Spectrum of Rising Exponential Pulse



Properties of the FT

Linearity

Dilation

Conjugation

Duality

( ) ( ) ( ) ( )fGcfGctgctgc 22112211 +⇔+

( )

⇔a

fG

aatg

1

( ) ( )fGtg −⇔ **

( ) ( ) ( ) ( )fgtGfGtg −⇔⇔ then , If



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Time Shifting

Frequency Shifting

Differentiation

Integration

( ) ( ) 02

0

tfjefGttg

⋅⋅−⇔− π

( ) ( )ctfjffGtge c −⇔⋅⋅π2

( ) ( ) ( )fGfjtgdt

d n

n

n

⋅⋅⇔ π2

( ) ( )∫∞− ⋅

⇔t

fGfj

dgπ

ττ2

1




Area under g(t)

Area under G(f)

Modulation Theorem

Rayleigh’s Energy Theorem

( ) ( )0Gdttg =∫∞

∞−

( ) ( )∫∞

∞−

= dffGg 0

( ) ( ) ( ) ( ) λλλ dfGGtgtg −⇔ ∫∞

∞−2121

( ) ( )∫ ∫∞

∞−

∞

∞−

= dffGdttg22



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FT Theorems



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FT Properties Examples



The Inverse Relationship between Time and Frequency

The properties of the FT show that the time-domain and frequency-domain description of a signal are inversely related to each other.

− If the time-domain description of a signal is changed, the frequency-domain description of the signal is changed in an inverse manner, and vice versa.

− A signal cannot be strictly limited in both time and frequency.


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10

Bandwidth

Provides a measure of the extend of the significant spectral content of the signal for positive frequencies.

− A signal is low-pass if its significant spectral content is centered around the origin f = 0.

− A signal is band-pass if its significant spectral content is centered around ±fc , where fc is a constant frequency.


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Bandwidth

Null-to-null bandwidth

− when the spectrum of the signal is symmetric with a main lobe bounded by well-defined nulls (i.e. frequencies at which the spectrum is zero), we may use the main lobe for defining the bandwidth of the signal.

3-dB bandwidth

− the separation (along the positive frequency axis) between the two frequencies at which the amplitude spectrum of the signal drops to of the peak value.

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Time-Bandwidth Product

The product of the signal’s duration and its bandwidth is always a constant.

(duration) X (bandwidth) = constant

The time-bandwidth product is another manifestation of the inverse relationship that exists between the time-domain and frequency-domain descriptions of a signal.


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Dirac Delta Function (Unit Impulse)

The theory of the FT is applicable to only time functions that satisfy the Dirichlet conditions, but it would be helpful to extend the theory in two ways − To combine the theory of Fourier

series and FT, so that the Fourier series may be treated as a special case of the FT.

− To expand applicability of the FT to include power signals (periodic signals), signals that satisfy:

( ) ∞<

∫−

∞→

T

TT

dttgT

2

2

1lim

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Dirac Delta Function

This can be accomplished with the use of the Dirac Delta function.

( )

( )

( ) ( ) ( )

( ) 1

1

0 ,00

00

=ℑ

=−

=

≠=

∫

∫∞

∞−

∞

∞−

t

tgdttttg

dtt

t

δ

δ

δ

δ

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(Sifting Property)

Applications of the Delta Function

DC signal

Complex Exponential

Sinusoidal Functions

( )fδ⇔1

( )ctfjffe c −⇔⋅⋅ δπ2

( ) ( ) ( )[ ]ccc fffftf ++−⇔⋅⋅ δδπ2

12cos

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( ) ( ) ( )[ ]ccc ffffj

tf +−−⇔⋅⋅ δδπ2

12sin

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Applications of the Delta Function


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Dirac Delta Function Examples



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Fourier Transform of Periodic Signal

Using the Fourier series, a periodicsignal can be represented as a sum of complex exponential or into an infinite sum of sine and cosine terms.

To denotes the period of the signal.

fo denotes the fundamental frequency of the signal.


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o

oT

f1

=

FT of Periodic Signals

( ) ( ) ( ) ( ) ( )∑∑∞

−∞=

∞

−∞=

⋅−⋅=⇔−=n

o

m

fnffnGffXmTtgtx 000 δ

T0 T0t1 t1+T0

t

g(t)x(t)

( ) ( )

( ) ( ) ( ) ( )( )∑

∑∞

=

∞

−∞=

⋅⋅⋅

⋅∠+⋅⋅⋅⋅⋅⋅+⋅=

⋅⋅=

1

000

2

00

2cos20

0

n

oo

n

tfnj

fnGtfnfnGfGftx

efnGftx

π

π

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Fourier Series: Example 1

-5 -4 -3 -2 -1 0 1 2 3 4 5-0.5

0

0.5

1

1.5

time (s)

Amplitude

Periodic Waveform



Fourier Series Example 2

-5 -4 -3 -2 -1 0 1 2 3 4 5-1.5

-1

-0.5

0

0.5

1

1.5

time (s)

Amplitude

Periodic Waveform



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Power Spectral Density (PSD)

Parserval’s Theorem – relates the energy associated with a time-domain function of finite energy to the Fourier transform of the function. To calculate the PSD, it’s necessary to assume a resistor of 1Ω(normalized).

The PSD (energy) (in Watts / Hz) of a signal x(t) is

( ) 2fXSx =



Power Spectral Density (PSD)

The average power (normalized) (in Watts) is

Parseval’s Theorem(Periodic Signals)


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( )2

0∑∞

−∞=

⋅=n

ave fnXP

( )dffSP xave ∫∞

∞−

=

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Examples 1 and 2 (PSD)

-1.5 -1 -0.5 0 0.5 1 1.50

0.1

0.2

0.3

0.4Example 1 PSD

Frequency (Hz)

PSD Sx

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.50

0.1

0.2

0.3

0.4Example 2 PSD

Frequency (Hz)

PSD Sx

Pave = 0.4833 W

Pave=0.6464 W



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