Exam zone

Random Signals & Multirate Systems Now that we have a basic understanding of the decimation and interpolation systemswithdeterministicinputs,wecannowlookatthesesystemswhenthey are excited by random input signals. Consider a zero mean, discrete{time wide sense stationary (WSS) random signal x [n ] that is the input to a decimation system. The power spectrum of the input signal x [n ] is denoted Pxx ( ej! ) and its average power is the area under the power spectrum function: Z ¼ 1 x Pave = Pxx ( ej! ) d!: 2¼ ¡ ¼ The output of the lowpass ¯ltering operation (LPF) is denoted z[n ] and the average of the output of the LPF is given by: Z¼ 1 z Pave = j H lp ( ej! ) j 2 Pxx ( ej! ) d!: 2¼ ¡ ¼ InourcaseLPFoperationisassumedtobeanidealLPFwithcut-o®frequency ! c = ¼=M and this relation can be reduced to: z Pave

1 = 2¼

Z

¼=M

Pxx ( ej! ) d!: ¡ ¼=M

Theoutputofthedownsamplingsystemisdenoted function (ACF) of this sequence is given by:

yd [n ]andthe autocorrelation

R yy [k ]= E f yd [n ]yd¤ [n ¡ k ]g = E f z[Mn ]z¤ [Mn ¡ Mk ]g = R zz [Mk ]: This relation is signi¯cant in that it tells us that the ACF of the sequence z[n ] also gets downsampled in the process of downsampling z[n ]. A direct consequence ofthisrelation isthat theaveragepoweroftheoutput ofthedownsampler is the same as its input: y z Pave = R yy [0]= R zz [0]= Pave

Thisresultsinascalingofthepowerspectrumof z[n ]sothattheaveragepower remains invariant in the downsampling process: M¡ 1 ³ ! 1 X Pyy ( e ) = Pzz ej ( M p=0 j!

¡ 2 p¼ M

)

´ :

Substituting the expression for Pzz ( ej! ) into this expression yields: Pyy ( ej! ) =

M¡ 1 ³ 1 X ¯ ¯ ¯H e M p=0

j(!

¡ 2 p¼ M

)

´ ¯2 ³ ! ¯ ¯ Pxx e(

¡ 2 p¼ M

)

´ ;!

2 [¡ ¼;¼]

This of course can be interpreted as a spectral zoom operation on the power spectrumoftherandomsignal x [n ]. Thecorrespondinge®ectofinterpolationon the random signal is more complex than the decimation operation. The output of the interpolator is not a wide sense stationary process but a cyclostationary process with cyclic parameter L . However, we can still use the time-averaged power spectral density representation: 1 P~o ( ej! ) = Pxx ( ej!L ) j H i ( ej! ) j 2 ; ! L

2 [¡ ¼;¼]

Theaveragepowerassociatedwiththistime-averagedpowerspectraldensityis therefore given by: Z ¼=L L o ~ Pave = Pxx ( ej!L ) d!: 2¼ ¡ ¼=L Using a substitution of variables ½= L! we obtain: Z¼ 1 o x ~ Pave = Pxx ( ej½ ) d½= Pave : 2¼ ¡ ¼ A special case of these relations occurs when the input signal is white noise obtained by sampling a bandlimited, zero mean continuous{time white noise processx c ( t ) with variance ¾x2 and sampling period Ts . In this case the output ofthedecimationsystemisagainwhitenoisebutwithareducednoisevariance: y Pave =

1 2¼

Z

¼=M

¡ ¼=M

¾x2 ¾x2 d! = : Ts TsM

Thecorrespondinginterpolatoroutput,however,isnotwhiteduetothecorrelationintroducedduringtheinterpolationprocess. Thesewillbeusefulspeci¯cally when we look at noise shaping based quantization systems.

prob theory & stochastic14
prob theory & stochastic14

Exam zone ³ e ( ! ¡ 2 p¼ Z ¼=M ³ e j ( ! ¡ 2 p¼ Z ¼ P x ave = P z ave = P z ave = Z ¼ ¯ ¯ ¯H e P xx ( e j! ) d!: P xx ( e j! ) d!: X X j H l...