Exam zone

Modulation of Random Processes Letusnowconsiderthemodulationofarandomprocess soidal signal described before :

X ( t ) withtherandom-phasesinu-

Y ( t ) = X ( t ) cos(- ct + £) ;

(51)

where £ » U ([0; 2¼]) and X ( t ) and £ are independent random variables8t 2 R . The ACF of the processY ( t ) is given by : RYY ( t 1;t 2) = E [Y ( t 1) Y ¤( t 2)] = E [X ( t 1) cos(- ct 1 + £) X ¤( t 2) cos(- ct 2 + £) ]

(52)

Using the independence ofX ( t ) & £ we have : RYY ( t 1;t 2) = RYY ( t 1;t 2) =

E £ [X ( t 1) X ¤( t 2)] E £ [cos(- (c t 1 + t 2) + 2£) cos[- (c t 1 ¡ t 2)]] Ã RXX ( t 1;t 2) ! f E £ f cos[- (c t 1 + t 2) + 2£] g + E £ f cos[- c( t 1 ¡ t 2)] gg(53) : 2

Since £ » U ([0; 2¼]) the ¯rst expectation becomes zero and then we have : RYY ( t 1;t 2) = Thisrelationimpliesthatif hence :

RXX ( t 1;t 2) cos[- c( t 1 ¡ t 2)] 2

(54)

X ( t ) isaWSSrandomprocessthen Y ( t ) isalsoaWSSprocess,

RXX ( ¿) cos(- c¿) 2 Modulationofthe X ( t ) withtherandomcosineresultsinthemodulationoftheACFofthe random process. Taking the Fourier transformation on both sides of Eq. (55) : RYY ( ¿) =

PYY (-)

=

PXX (- + - c) + PXX (- ¡ - c) 4

(55)

(56)

This result implies that modulation of a random processX ( t ) with a random cosine results in a spectral upshift and a down-shift of the power spectrumPXX (-). Let us now look at the case where the random processX ( t ) is modulated with a deterministiccosine. Inspeci¯cletuslookatthepowerspectrumofa double sideband amplitude modulated(DSB-AM) signal given by: SDSBAM ( t ) = m( t ) cos(- ct + Áo) :

(57)

Themessagesignalorbasebandsignal m( t ) (human-voice). Letusassumethatthemessage signal m( t ) isazeromeanWSSprocesswithACF R mm ( ¿). Themeanoftherandomprocess SDSBAM ( t ) is therefore: ¹ S ( t ) = E ( m( t )) cos(- ct + Áo) = 0 :

(58)

The random process SDSBAM ( t ) is therefore ¯rst-order cyclostationary. The ACF of the modulated signal denotedRSS ( t;t ¡ ¿) is given by: RSS ( t;t ¡ ¿) =

Rmm ( ¿) [cos(- c¿) + cos(2- ct ¡ - c¿ + 2Áo)] : 2 10

(59)

NotethatthisensembleACF R SS ( t;t ¡ ¿) dependsonboth t and ¿. Thisprocessistherefore notWSS.TheensembleACF,however,isperiodicinthevariable t withfundamentalperiod To = ¼=- c. Therandomprocess SDSBAM ( t ) isthereforesecond-ordercyclostationarywith ~SS ( ¿) via: parameter To. We can then evaluate the time-averaged ACF denoted R ~SS ( ¿) = R

1 Z To ¡

To 2 To 2

RSS ( t;t ¡ ¿) dt =

Rmm ( ¿) cos(- c¿) : 2

(60)

The corresponding power spectrum of DSB-AM can therefore be expressed as: P~SS (-)

=

Pmm (- + - c) + Pmm (- ¡ - c) ; 4

(61)

where Pmm (-) is the power spectrum of the baseband signalm( t ). As in the deterministic case, the baseband power spectrum has been up-shifted and down-shifted in frequency by - c to give rise to the upper sideband(USB) and the lower sideband(LSB). For a real message and WSS signalm( t ), the ACF is also real and even. It is easy to see that spectral components that are separated by 2- c= 2 ¼=To are perfectly correlated (spectral redundancy). This will be of interest in co-channel communication systems where multiple users share the same carrier frequency. Note this result is analogous to the result thatweobtainedforthepowerspectrumofarandomsignalmodulatedwitharandom-cosine.

11

prob theory & stochastic10

Exam zone Let us now look at the case where the random processX ( t ) is modulated with a deter- ministiccosine. Inspeci¯cletuslookatthepowe...

prob theory & stochastic10

Exam zone Let us now look at the case where the random processX ( t ) is modulated with a deter- ministiccosine. Inspeci¯cletuslookatthepowe...