Exam zone
Power Spectral Factorization Considerazero-mean,WSS,discrete-time,randomsignalwithapowerspectrum Pxx ( z) that is real and positive on the unit circle, which has a finite x average powerPave , where both Pxx ( z) and log( Pxx ( z)) are analytic in the region ½ < j zj < 1½. It can be shown that a power spectrum that satisfies these requirements can be factorized into the form: Pxx ( z) = ¾o2 Hmin ( z) H max ( z) ; ½< j zj <
1 ; 0 <½< 1 ½
where H min ( z) is the monic minimum phase part of Px ( z), H max ( z) is the monic maximum phase part of Pxx ( z) and ¾o2 is the variance of the innovations process. The c epstrumrepresentationofarandomsignalisdefinedviatheDTFT pair: C ( ej! ) = c[n ] =
log(Pxx ( ej! )) = 1 2¼
Z
X1 c[n ]exp( ¡ j!n )
(1)
n= ¼
C ( ej! ) exp(j!n ) d!;
(2)
¡ ¼
where c[n ] are referred to as the cepstral coe±cients. The parameters of the power spectral factorization of the random signal can be related to the cepstrum representation via: µ ¶ Z ¡ 1 ¼ 2 j! ¢ ¾o =exp log Pxx ( e ) d! : 2¼ ¡ ¼ Thisparameterissometimesreferredtoasthe ge ometric me an (GM)ofthe power spectrum. The minimum phase part is defined via: Ã 1 ! X H min ( z) =exp c[k ]z¡ k ; j zj >½: k =1
Notice from the ROC that this system function corresponds to a causal system, furthermore it can be shown that is a monic system function. The fact that this is a minimum-phase system function follows from the fact that both H min ( z) is minimum-phase as well as its logarithm. In a similar fashion, the maximum phase part of the power spectrum is given by: Ã ¡1 ! X 1 ¡ k H max ( z) =exp c[k ]z ; j zj < : ½ k=
In a manner similar to the minimum-phase part it can be shown that this system function corresponds to a monic maximum phase system function.
Ramifications of PSD Factorization In addition to allowing us to factor the PSD of a random signal into three parts, the PSD factorization theorem also implies the following: 1. The random signal x [n ] and its innovations process v[n ] are linearly equivalent, i.e., X1 x [n ] =
hmin [k ]v[n ¡ k ]
(3)
° [k ]x [n ¡ k ]
(4)
k= X1
v[n ] = k=
2. Theinnovationsequivalentoftherandomsignal v[n ];n 2 I constitutes an orthonormal basis for the Hilbert space of finite average power random signals that satisfy the criteria for PSD factorization. This innovationsprocessissometimesreferredtoasthe Kalman innovation process and will play a major role in optimal estimation. 3. Thepowerspectrum Pxx ( z) oftheWSSrandomsignaliscomposedof poles and zeroes that come in complex conjugate reciprocal pairs: Pxx ( z) = ¾o2
B ( z) B ¤( z1¤ ) : A ( z) A ¤( z1¤ )
4. Since we require the PSD, which is a continuous function of ! , to be positive and real on the unit-circle, the zeros of Pxx ( z) on the unit circlecomeinpairs,i.e.,themultiplicityofthezeroesof Pxx ( z) onthe UC is even.