ANLOG COMMUNICATION Lecture 04

Analysis and Transmission of Signals

Lesson 04 EEE 352 Analog Communication Systems Mansoor Khan EE Dept. CIIT

Signal Energy, Parseval’s Theorem • Parseval’s theorem gives an alternative method to evaluate energy in frequency domain instead of time domain. • In other words energy is conserved in both domains • Consider an energy signal g(t), Parseval’s Theorem states that

Parseval’s Theorem – Conservation of energy in frequency domain(CFT)

Parseval’s Theorem – Conservation of energy in frequency domain(DFT)

Proof

Example

Spectral Density • The spectral density of a signal characterizes the distribution of the signal’s energy or power in the frequency domain. • This concept is particularly important when considering filtering in communication systems while evaluating the signal and noise at the filter output.

• The Energy Spectral Density (ESD) or the Power Spectral Density (PSD) is used in the evaluation of the signals

Energy Spectral Density (ESD) • The energy of the output signal y(t) is given by 1 Ey  2

Ey 



 G(w) H (w)

2

d



• Because H(w)=1 over the passband Δw and as Δw tends to zero

1 2 2 Ey  2 G( wo ) d  2 G( wo ) df 2

1 2



 G(w) H (w)



2

d

Energy Spectral Density (cont) • Energy spectral density describes the signal energy per unit bandwidth measured in joules/hertz. • The energy spectral density (ESD) ψ(t) is thus defined as

Ψ g ( )  G( ) 1 Eg  2



Ψ



g

( )d 

2



Ψ



g

( f )df

Energy of modulated signals • The AM signal is

 t   g t cos w0t

• And the fourier transform will be

1 w  Gw  w0   Gw  w0  2

• The ESD of the modulated signal will be φ(t) is [Φ(w)]2

1 2  w  Gw  w0   Gw  w0  4

Energy of modulated signals (cont)

Energy of modulated signals (cont) • If w0≥2πB, then G(w+w0) and G(w-w0) are nonoverlapping and



1 2 2  w  Gw  w0   Gw  w0  4 1  w  g w  w0   g w  w0  4







• Observe that the area under modulated signal is half the area under baseband signal

1 E  E g 2

ESD of the Input and the Output • If g(t) and y(t) are the input and the corresponding output of LTI system, then • Therefore • This shows that

• Thus, output signal ESD is |H(w)|2 the input signal ESD

Essential Bandwidth • The spectra of most of the signals extend to infinity. • For practical signals Eg(energy) must approach zero for ω→∞. • Most of the signal energy is contained within a certain bandwidth B Hz. • Energy content of components greater than B Hz is negligible. • Thus most of the energy of signal can be suppressed within a certain bandwidth B called essential bandwidth - B Hz.

Criteria for selection of B Hz • The Criteria for selecting certain bandwidth B Hz to suppress the signal energy within depends on error tolerance. • For example for a particular application B can be selected at 95% of signal bandwidth. • Essential bandwidth varies from error tolerance suited for a particular application

EXAMPLE

Signal Power and Power Spectral Density • The power Pg of a real signal g(t) is given by T 2 T 2

1 Pg  lim  T  T

2

g (t ) dt

• We take a truncated signal gT(t) • The integral on the right hand side will be the energy of the truncated signal, thus

Pg  lim t 

E gT T

Power Spectral Density (cont) • The truncated signal is an energy signal as long as T is finite. • From Parseval’s theoram 

E gT

1   g t dt  2  2 T

2



 G w dw T



• The power of the signal is given by

Pg  lim t 

E gT T

Power Spectral Density (cont) 1 Pg  2 • Where

GT w



 lim

t 



S g w  lim t 

T

2

dw

GT w

2

T

• Sg(w) is the Power Spectral Density of Power Signal, Which is actually the time average of ESD

1 Pg  2





 S wdw  2 S wdf g



g

0

Time Autocorrelation Function and PSD • For a real signal the autocorrelation function g(t) is defined as 

 g ( )   g (t ) g (t   )dt 

• Notice that

 g ( )   g ( )

• The auto correlation function is an even function

Time Autocorrelation Function and ESD • The ESD is the Fourier Transform of the autocorrelation

 g ( )  G( )

2

 g ( )  Ψ( )

Time Autocorrelation (cont) • For energy signals the ESD is the Fourier transform of the autocorrelation

 g ( )  Ψ( ) • A similar result applies to power signals

• Because

 g ( )  lim

 gT ( )

T 

T

 g ( ) G( )  g ( )  lim

T 

GT ( ) T

2

2

 S g ( )

PSD of Input and Output • We Know • then

Y ()  H ( )G() Y ( )  H ( ) G( ) 2

2

 g ( )  lim

T 

GT ( ) T

2

2

 S g ( )

S y ( )  H ( ) S g ( ) 2

PSD of Modulated Signals • The modulated signal can be represented by

 (t )  g (t ) cos 0 t • Its Fourier transform



1 S ( )  S g (  0 )  S g (   0 ) 2

1 P  Pg 2



Mathematics-H(f) = 1*exp(-j2∏fto)