Proceedings Letters: Section Discontinued
Proceedings Letters: Section Discontinued
Proceedings Letters: Section Discontinued
Cumulants: A Powerful Tool in Signal [H(w)] stands for the impulse response (transfer function) of the
underlying NMP model. Output autocorrelation and spectrum,can
Processing be viewed as special cases of (1)and (2), respectively, when k = 2.
The input non-Gaussianity is necessary (7; # 0 for some k > 2,)
GEORGIOS B. GlANNAKlS but the i.i.d. assumption can be relaxed by a kth-order whiteness
assumption, [IO].
Based on (2), Lii and Rosenblatt[2] haveshown that using higher-
The impulse response of a linear, time-invariant system is related order periodogram techniques, the amplitude and phase of H(w)
in a simple closedform solution to the output cumulants, when can be estimated from output data only (up to a scale and time-
the input is assumed to be non-Gaussian and independent. This delay ambiguity.) The high-variance and low-resolution charac-
expression permits the use of one-dimensional processing of the teristics of the Fourier-type methods [2]-[4] suggested the use of
output cumulants for identification of non-minimum-phase sys- cumulants in conjunctionwithparametric(MA,AR,ARMA) models
tems, and opens new directions in other signal processing appli- [i7-[10]. The novelty of this letter (Section II) is to provide a very
cations. simple closed-formsolution of theimpulse response (IR) samples
in terms of the output cumulants, useful in both the parametric
I. PROBLEMSTATEMENT and nonparametric approaches. The potential of this method in
We addressthe problem of identifyingthe impulse response of various signalprocessingtasks isanalyzed in Section I l l . In Section
afinite-dimensional, Linear,Time-lnvariant(LT1) system, when out- IV, we discussproperties of the proposed solution, and comment
put (perhaps noisy) observations are provided. The input is on its computational aspects.
unknown, but is assumed to be stationary, non-Gaussian, inde-
pendent, and identically distributed (i.i.d.) When the input is II. MAINRESULT
Gaussianand/oroutputautocorrelationMmplesareused,onemay
A.FIRCase(MA Models)
obtain onlythe spectrally equivalent minimum phase (MP) part of
the system. Theunderlying reason is that second-order output sta- Considering the third-order output cumulant [c.f. (1)with k =
tistics are unaffected by all-pass factors, andas such the autocor- 31 we obtain
relation is a “phase-blind” sequence. To recover a general non- c5(ml, md = E { y ( k ) y(k + m , ) y(k + m,)}
minimum-phase (NMP) model, we need phase sensitive higher P
order output statistics. Thestatisticsthat we propose are the cumu- = y; h(i)
h(i + m,) h(i + m,) (3)
lants (in frequency domain known as polyspectra,) whose rela- ,=O
tionship with LTI systems, [I] described by
is,
wherey; I € { x 3 ( & ) } # 0,andpdenotestheorderoftheMAmodel.
OD
If we substitute m, = p , m2 = k, and m, = m2 = - p in (3), and
ct;(m,, - ,mk-1) = y;
,-0
h(i) h(i + m,) .h(i + m k - l ) (1) assume that h(0) = 1, it is easy to show that c@, k ) = y ; h ( p ) h(k),
and c<(-p, - p ) = y ; h ( p ) . Hence
or, in the frequency domain by
k-1
s~(01,. ’ ‘ t wk-1) = y;H(ol) * * H(wk-1) H(-igl w j ) (2)
Equation (4) states that the IR { h ( k ) } of an LTI system is identical
where cg( .) denotes the output kth-order cumulant, 7 ; the input (within a scale factor) to thethird-order output cumulant sequence
kth-order cumulant, St;( * ) the output kth-order spectrum, andh(i) {c~(p,k)},andconsequentlythetrueNMPsystemcanberecovered
using output cumulants only. This can also be verified using a
Manuscript received December 1,1986. graphical interpretation of (3) (see also Fig.1).Therefore, from an
The author is with the Departmentof Electrical Engineering-Systems, Sig- identification viewpoint, {cg(p, k ) } plays the role of thecross-cor-
nal and Image Processing Institute, University of Southern California, Los relation sequence { r d k ) } , because
Angeles, CA 900890781, USA.
IEEE L o g Number 8714508. 4 h(k) = rJk) = € { x ( i ) y(i + k ) } .
1 t h(i+p’
which is a closed-form solution of thesystem’s phase in terms of
the output cumulants. Furthermore, once the ARMA model has
been obtained we may use it for deconvolution (see also[2],) har-
monic retrieval, and spectral estimation. For the latter, using the
{c@, k ) } samples as constraints one may obtain a unique r e p
resentative in the family of extrapolating spectra described in [q.
IV.
CONCLUSIONS-DISCUSSION
The main contribution of this letter is avery simple, timedomain
solution of the stochastic realization problem that requires one-
dimensionalprocessing. As opposed to the methods of [2]-[4], [q,
and [8] we proved that onedimensional versions of the output
cumulantsaresufficient for NMPsystem identification.Theoutput
cumulants are computed through sample averaging, e.g.,
I
Fig. 1. Sf,, h(i) h(i + p ) h(i + k ) is always a scalar multiple of h(k).