Author: Danilo Korze

Mentor: Asoc. Prof. Dr. Damjan Zazula

Co mentor: Prof. Dr. Nikola Pavesic

Date: March 20, 1996

**Multichannel
System Identification using Higher-Order Statistics**

**
Keywords:** digital signal processing, multichannel system identification,
variance estimation, higher-order statistics, higher-order cumulants, ARMA
models, Cramer-Rao lower bound, impulse response, transfer function, white
noise, deconvolution, SISO systems, MISO systems, MIMO systems.

**
UDK: **681.5.015:681.32:519.233.4

**
Abstract:** In the doctoral thesis the identification of MA systems using
higher-order statistics (cumulants) is discussed for the case of basic systems
with single input, single output (SISO), then for the case of multiple input,
single output (MISO) and form the cas of multiple input, multiple output (MIMO)
systems. The overview of known methods for the identification of SISO and MIMO
systems is given.

Next we deal with the variance analysis for SISO
systems. It has been assumed that the input signal is Bernoulli distribute white
noise. Based on this assumption and knowing the procedure for computing the
third-order cumulants, we derived the probability function for the values of the
system output signal *y(n)*, respecting *h(n) *as the system impulse
response. Further on, the probability function for the cumulant value *C _{3,y}(T_{1},T_{2})*
was determined. Finally, the basic variance

*Var*was estimated, which represents the varianceof the random variable containing the values of the partial products at the third-order cumulant lag

_{1}(T_{1},T_{2})*(T*, and the covariance

_{1},T_{2})*Cor*of these variables. Based on this two estimations, the final expression for the variance

_{1}(T_{1},T_{2})*Var*which arises cvalculating the cumulant values

_{N}(T_{1},T_{2})*C*from

_{3,y}(T_{1},T_{2})*N*-samples output signals was derived. The simulations confirmed that the variance expression has been developed correctly. It was also demonstrated that the expression

*Var*is a suitable estimate of the upper bound of the variance.

_{1}(T_{1},T_{2})/NBy estimation of the cumulant from the output
signal indexes of merit *Kf _{1}, Kf_{2}* and

*Kf*were defined, based on the cumulant variances derived previously. These indexes determine the efficiency of the identification using the

_{3}*C(q, k)*method. In the derivation was supposed that the cumulant values have normal distribution. This hypotesis was tested with te statistical test

*X*and it couldn't be rejected in any of the simulation runs. The indexes have shown that the system could be well identified, when

^{2}*K f*,

_{1}= 0.5*K f*and

_{2}> 0.5*K f*Additional it was shown that this is not always feasible in the finite cases, what confirms the thesis:

_{3}> 0.2.The variance of the cumulant estimates in system identification determines the accuracy of the results irrespective of the identification method used. This variance is not depending on the input process variances only, but far more on characteristics of the system identified. Consequently, the identification of several systems using higher-order cumulants is not feasible, when the finite length output signals are applied.

The variance in the cumulants of the MISO systems
was analysed as well. It has been supposed that the input noises *w _{K}(n) *
are Bernoulli distributed and spatially uncorrelated. As well as it has been
demanded that for the Bernoulli distributed and spatially uncorrelated. As well
as it has been demanded that for the system orders the relation

*q*as with the SISO systems, the cumulant variance was estimated and its upper bounded described with the expression

_{1}< q_{2}< ... q_{K }*Var(T*. Based on the cumulant variances, indexes

_{1}, T_{2})*K f*were derived, which tested the efficiency of the identification of the input channels

_{3,i}*K f*, because they are defined the same way as with the SISO systems. For MISO systems, it is even more probable that their identification is not always feasible using the higher-order cumulants in the finite cases.

_{3}> 0.2In the sequel, the variance analysis for the case of MIMO systems was
performed. Here was again assumed that the input noises are Bernoulli
distributed and temporally and spatially independent. For the simplification of
the responses could be padded with zeros. Oposite to the SISO and MISO systems,
the variances of the cross cumulants were estimated here, which are always
computed from three output signals. The variances were designated with *Var _{ijk}(T_{1},
T_{2})*, where

*i*,

*j*and

*k*denote the considered output signals. These variances were entered into the calculation on indexes

*K f*for evaluation of the multivariate

_{3,ij}*C(q,k)*identification method giving the response

*h*. We showed as well that the identification is not always successful in the finite cases.

_{ij}In the thesis, the structure of the variance of the cumulant estimates was
observed as well. For this purpose, the quality of the estimator for the
cumulant of the Bernoulli distributed noise using the Cramer-Rao lower bound was
studied. It was shown that the variance in the cumulant of noise and the
variance in the cumulant of the output signal have similar structure which
demonstrate larger values on the *T _{1} = T_{2}* diagonal
and on the axis

*T*and

_{1}= 0*T*, while the variance at the remainings lags is usually much lower. This fact has to be considered when the identification methods are designed. Especially the division using the random variable with the mean close to the zero and larger variance have to be avoided. The identification results could be poor in such cases, exhibiting a very large variance.

_{2}= 0