An In-Place Levinson Update Algorithm. W. SCOTT DUNBAR. Abstrrrct-An inphce Levinson update algorithm is presented which requires less storage than ...
PROCEEDINGS OF THE IEEE, VOL. 6 9 , NO. 6 , JUNE 1981
754
One of the authors incorrectly interpreted the text of his own book [ 11 as to thesignificance of Liouville’s theorem. If the generalized momentum p’ is transformedfrom M e s i a n t o spherical coordinates onehas
py=nsinesinq
p,=nsinecosq
(1)
and theinvariance of the volume W in phasespace becomes
=Jn2 dx dy
sin e cos e de dlp =
j = 1a d l ) = a M - l ( l ) + a ” )
s
n 2 dx dy cos e d n .
(2)
paper has to be re-
and therefore relation (16) fiom the above titled placed by the relation
‘ 1 - s2 --
n:
3.
(5)
The multipliers ofa d M ) are the complex conjugatesof the coefficients at the (M-1)st stage in reverse order. Consequently, an in-place algorithm may be devised by updating the coefficient a ~ - ~ (and k ) retainh thenupdatethecoefficient ing itinoneextrastoragelocation u ~ - 1 ( M- k) using a M - l ( k ) , and finally exchange h with a&). The index k varies between 1 and INT[M/2] for M > 1, where INT denotes the integer part of the quantity in brackets. A program segment for the update is given below
M > 1 THEN M2:= M / 2 FORK: = 1 STEP 1 UNTIL M 2 DO L : = M-L
That ( 5 ) is correct, is immediately seen, if a source in a medium with n l radiates at an angle e 1 through a plane boundary into a medium with n 2 . The r e h c t i o n of the light rays at the boundary gives nl
(M- 2)
IF
-which for aLambert law sourcegives i.e.
...
j = M - I u d M - l)=aM-l(M- I)+nM(M)a&-l(l).
dW dP = S ( 0 ) dx dyd n = S ( 0 ) n2 cos e
n2
a h - l ( M - 1)
j = 2 aM(2) = aM-1(2) + aM(M)
j = M - 2uM(M- 2 ) = a ~ - 1 ( M - 2 ) + a & f ) ~ h - ~ ( 2 )
Employing (2) in the conservationof power yields
S _ -- const
There presently exist published computer programs which compute of linear predictiveorspectral thecoefficients a forthepurposes analysis of data [4]-[ 101. However, these programs require 2N words of storage for the updateof (1). In the following, anin-place algorithm is described which requires N + 1 words of storage. Such an algorithm would certainly be useful for on-line applications. A similar in-place update algorithm exists in Robinson [2, pp. 44-45], but is apparently not well known. The implementation of the algorithm is easy and requires only a few program statements. Writing equation (1) out in full, one obtains
sin e , = n 2 s i n e 2
(6)
and with ( 5 ) and ( 6 ) one proves that power is conserved dP=S1dxdycossldnl=S2dxdycos82dn2.
(7)
Equation (2), by the way, is the conservationof the number of transverse modes (distinguishable light rays, distinguishable local plane waves) in a signal [2]. REFERENCES
[ 1 1 D. Marcuse, Light Transmission Optics. New Ymk: Van Nostrand Reinhold, 1972, pp. 112-124. 121 G. Grau, Quantenelektronik. Braunschweig, Germany: ViewegVerlag, 1978, p. 103, p. 287.
An In-Place Levinson Update Algorithm W. SCOTT DUNBAR
Abstrrrct-An inphce Levinson update algorithm is presented which requires less storage than update algorithm in some published pro‘Ihe algorithm may be easily implemented in a program which employs a Lerinsonrecursion.
grams.
At the Mth stage (1 < M < N) of a Levinson-type recursive algorithm, a d M ) iscomputed. thereflectionorpartialcorrelationcoefficient The remaining coefficientsaM-l 0’) are updated according to if the wherethe asterisk denotesthecomplexconjugate,required original data are complex. The limits on the indexj are 1 Q j < M - 1. Note that a(0)= 1.O by definition. See Robinson [ 11, [ 21 or Robinson and Treitel [ 31 for a completediscussion of the Levinson algorithm. Manuscript received January 31, 1981. The author is with Weidlinger Associates, Menlo Park, CA 94025.
H:= A ( K ) + A(M) X CONJG [A(L)] A(L): = A ( L ) + A(M) x CONJG [A(K)] A(K): = H END K. A segment such as the above could be directly inserted into programs published by Andersen [4], [5] and Viswanathan and Makhoul [ 101. Note that Andersen defies a(i) as a predictor or autoregressive coefficient so that the sign of A(M) in the above program segment would be negative. Programs published by Ulrych and Bishop [6] and Claerbout [7, p. 1371 start the Levinson recursion at M = 2 with a(1) = 1.0 for all M. Forthese programs, the firstfourstatementsin the aboveprogram segment should be changedto
M > 2 THEN M2: = (M+1)/2 FORK: = 2 STEP 1 UNTIL M 2 DO L : = M-K+1.
IF
Gray and Markel [8], [9] also start the Levinson recursion at M = 2. Depending on the desirability of not altering an input vector, the up date algorithm could be implemented in their program, although not as easily. REFERENCES E. A. Robinson, Statistical Communication and Detection. New York: Hafner, 1967. -, MultichannelTimeSeriesAnalysiswithDig‘talComputer Program, revised ed. San Francisco, CA: Holden-Day, 1967. E. A. Robinson and S. Treitel, Geophysical Signal Analysis. Englewood Cliffs, NJ: Prentice-Hall, 1980. N. Andersen, “On the calculation of filter coefficients for maximumentropy spectral analysis,” Geophys., vol. 39, pp. 69-72, Feb. 1974. -, “Comments on the performance of maximum entropy algorithms,”Proc. IEEE,vol.6 6 , pp. 1581-1582, Nov. 1978. T. J . Ulrych and T. N. Bishop, “Maximum entropy spectral analysis and autoregressive decomposition,” Rev. Geophys. Space Phys., vol. 13, pp. 183-200, Feb. 1975. J. F. Claerbout, Fundamentals of Geophysical Data Processing. New York: McGraw-Kill, 1976. A H. Gray and J. D. Markel,“Linear prediction analysis programs: (AUTO-COVAR),” in Programs for Digital Signal Processing. New York: IEEE Press, 1979, pp. 4.1-1-4.1-7.
0018-9219/81/060M754S00.75 0 1981 IEEE
755
PROCEEDINGS OF THE IEEE, VOL. 69, NO. 6 , JUNE 1981
-,
+l
“Linear predictorcoefficienttransformationssubroutine LPTRN,” in Programs for Digital Si@ Processing. New York: IEEE Press, 1979, pp. 4.3-1-4.3-7. [ l o ] R. Viswanathan and J. Makhoul, “Efficientlatticemethodsfor linear prediction,” in Progmm for Digital Signal Processing. New York: IEEE Press, 1979, pp. 4.2-1-4.2-14. [ 91
+l
Fig. 1. Generation scheme of a uniformly distributed random voltage x(r)by fmt-order linear fdtering of a randomtelegraph signal E ( t ) .
Correction to “Comments on ?Signal Duration and the Fourier Transform’ ” P. GALKO
AND
s. P A S W A m
In the above titled letta,’ the last sentence of the comments should have read: “These results, however, are well known and have been discussed in the literature as far back as 1960 [l,sect. 1.51, [2,p. 275 and ex. 3.131, [3, p. 571 and prob. A.1.2 and A.1.31.” Manuscript received January 22, 1981. The authors are with the Department of Electrical Engineering, University of Toronto, Toronto, Ont., Canada MSS 1A4. ‘P. Galko and S. Pasupathy, Proc. IEEE, vol. 6 8 , pp. 1452.
SURF: A Telegraph Signal Based Uniformly Distributed Analog Random Voltage Generator F. VERNIERES, N. ELLOUZE,
AND
F. CASTANIE
Absrruct-The generation of uniformly distriiuted analog random voltages is a difticultproblem, generally solved by methods using digitalteanalog conversion following a randomnumbergenerator. We describe a new and easy implement system based on linear fdtering of a random telegraph signal. I. INTRODUCTION Uniformlydistributedrandom variables-sometimescalled random references-are basic tools for many types of computers, like stochastic computers [ 11, [ 71, [9], random reference correlators [ 4 ] , etc. Suchrandom voltages are easily generatedwith digital methods; analog methods [2], [SI are farless common and giverise to many implementation problem% especially when wide bandwidths must be reached. The signal method proposed here (SURF’) aims at providing a time continuous stochastic process with a uniformly distributed (UD)amplitude. such that its outcomes are continuous functions (except a set of null probability). This method is based upon the availability of a Poisson point process, the generation of which is out of the scope of this letter; we limit ourselves to giving the conventional solution relying on level crossings of a Gaussian noise. 11. PRINCIPLE The basic idea of the generator lies in the fact that the amplitude distribution function of random process is generally altered bylinear filtering. In most cases, the knowledge of the statistical laws of the process E ( t ) and of the transfer function of a linear filter do notallow us to establish the amplitude distniution of the output process X ( t ) . Let us however consider a first-order low-pass filter operating on a telegraph signal, with a density h (mean number of occurrences per unit of time) (Fig. 1).
H(n
Manuscript received July 14, 1980;revised October 6 , 1980. F. Vernieres and F. Castanieare withthe Groupe d’Analysedes Stochastiques en Electronique(GAPSE) ENSEEIHT, 2, rue Ch.Camichel, 31071 Toulouse Cedex, France. N. Ellouze is with the B o l e NationaledesIngenieursdeTunis (ENIT). B. P. No. 37, Belvedere, Tunis, Tunisia. SURF-Source Uniformdment Rdpartie par Filtrage.
At the output X ( r ) , we have two alternated series of rising and falling exponential patterns, their duration being controlledbythe Poisson process, each pattern having a random starting point. It will be seen, in the following, that the process X ( t ) is uniformly distributed over (-1, l), if ha = 1, where h is the density of the Poisson point process, and a the time constant of the exponential pattern. 111. FIRST AND SECONDORDER PROPERTIES A complex calculation [6], far beyond the scope of this letter has shown the probability density function to be expressed by
,-2 ha
wheretheterm B ( p , q ) involved in the normalizing constant is the BebEuler function. The behavior of p x ( x ) with hor as a parameter is displayed on Fig. 2. Obviously ha = 1 leads to a uniform distribution. In order to characterize the ‘goodness’ of the spectral properties of a uniformly distributed random signal, first approach consists in analyzing its bandwidth, i.e., its correlation function. Knowing that X ( t ) is produced by a first-order linear filtering oftelegraph a signal, a straightforward application of thewell-known Weiner-Lee equation leadsto
where )f(,s
is the normalized power spectral density and
where K x ( r ) is the autocorrelation function of X ( t ) . From (6), it can be computed that the e-correlation radius (i.e., T~ such that rx(7) < E , 171 > 7 0 is given by l
70
---logh
e
2
for small values of e.
IV. IMPLEMENTATION An approximate Poisson process can easily be obtained by detecting the level crossings of a Gaussian process, with a sufficientlyhigh triggering level [3]. Fig. 3 showsone possible organization of theproposedgenerator successfully experimented by the authors. The Poisson point process is obtainedfromthe one-senselevelcrossings ofa Gaussian-Zenerbased Primary Analog Noise (PAN). It is well known that if the level p (see Fig. 3) is sufficiently high (say at least twice the rms value u of the PAN), this procedure yields agoodapproximation of a Poisson point process (see, e.g., [ 31 ) with a density h (function of p and 0 ) . The flip-flop followed by a conventional amplitude stabilizing switch delivers the random telegraph signal E @ ) . The output process X ( r ) is obtained after first-order filtering with time constant a = RC such that a = llh. If u is submitted to drift (e.g., thermal drift) a feedback loop can be added to ensure that a . h remains equal to one (dashed lines in Fig. 3). From ( 5 ) it can be shown that absolute first-order moment is a convenient ‘indicator’ of distribution uniformity. E,,d in Fig. 3 would be in this case a meanvalue estimator of the full-wave rectified version of x@). It should be noted that any post-filtering (even a linear one) will distort this amplitude distniution. For the uniformly distributed charac-
0018-9219/81/060M755$00.75 0 1981 IEEE
