on walsh differentiable dyadically stationary random processes

612 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. IT-28, NO. 4, JULY 1982

[5] -, “Truncation error for band-limited processes,” Inform. Sci., [I I] A. J. Lee, “Characterization of band-limited functions,” Inform. vol. 1, pp. 261-271, 1969. Contr., vol. 3 1, pp. 258-271, 1976.

[6] S. Cambanis and E. Masry, “Zakai’s class of band-limited functions [ 121 H. J. Piper, Jr., “Bounds for truncation error in sampling expan- and processes: Its characterization and properties,” SIAM J. Appl. sions of finite energy band-limited signals,” IEEE Trans. Inform. Math., vol. 30, pp. 10-21, 1976. Theoty, vol. IT-21, pp. 482-485, 1975.

[7] R. .I. Duffin and A. C. Schaeffer, “Some properties of functions of [ 131 Z. A. Piranshvilli, “On the problem of interpolation of random exponential type,” Bull. Amer. Math. Sot., vol. 44, pp. 236-240, processes,” Theory Prob. Appl., vol. 12. pp. 647-665, 1967. 1938. [ 141 E. M. Stein and G. Weiss, Fourier Analysis on Euclideun Spuces.

[S] I. S. Gradshteyn and I. M. Ryzhik, Tub/e of Integrals, Series, and Princeton: Princeton University, 197 1. Products. New York: Academic, 1965. [ 151 K. Yao and J. B. Thomas, “On truncation error bounds for sam-

[9] D. Jagerman, “Bounds for truncation error of the sampling expan- pling representations of band-limited signals,” IEEE Trans. Aerosp. sion,” J. SIAM, vol. 14, pp. 714-723, 1966. Electron. Syst., vol. AES-2, pp. 640-647, 1966.

[IO] T. Kawata, Fourier Annlysis in Probability Theory. New York: [16] M. Zakai, “Band-limited functions and the sampling theorem,” Academic, 1972. Inform. Contr., vol. 8, pp. 143-158, 1965.

On Walsh Differentiable Dyadically Stationary Random Processes

WOLFGANG ENGELS AND WOLFGANG SPLETTSTGSSER, MEMBER, IEEE

Ahstrucr-Some basic properties of dyadically stationary (DS) processes are introduced, including continuity and spectral representation. A sampling theorem based on the Walsh functions is investigated for random signals that are not necessarily sequency-limited. By using the concept of a dyadic derivative, the resulting aliasing error is calculated together with the speed of convergence. An example gives a glimpse into the possibilities of applying the sampling theorem as well as the dyadic derivative.

I. INTRODUCTION AND PRELIMINARY NOTIONS

I N communication theory, signals which are measured or received are mostly contaminated by noise, meaning

that unpredictable disturbances interfere with the signals. For this reason such random signals are modeled as stochastic (or random) processes. In investigating the properties of random signals it is customary to assume that the corresponding stochastic process is weakly stationary, al- though the condition of stationarity is somewhat restrictive, since time-limited signals in general do not possess this property.

In the past ten years much attention has been paid to systems that can be represented by dyadic convolution

Manuscript received January 9, 198 I ; revised September 2, 198 1. This research was supported by the DFG Grant Bu 166/36 of the “Schwer- pm&t-programm Digitale Signalverarbeitung.”

The authors are with the Lehrstuhl A fur Mathematik, Rheinisch- Westfahsche Technische Hochschule Aachen, Templergraben 55, 5100 Aachen, West Germany.

operators, the more so since the Walsh functions, which are compatible with the operation of digital computers, are fundamental to the analysis of such systems. In this matter a number of papers have appeared, including [l] dealing with the spectral representation of dyadically stationary (DS) processes and [2], where, e.g., the dyadic filtering problem (see [3] for the classical one) was solved.

The main aim of this paper is to characterize the smoothness in the mean of dyadically weakly stationary processes in terms of properties of their autocorrelation functions. In Section I, we first recall some elementary properties of the generalized Walsh functions due to Fine [4], including Walsh-Fourier analysis techniques and the Walsh-Fourier inversion theorem. In Section II we obtain a characterization of dyadic continuity in the mean of dyadically stationary processes, similar to that given in [5] for the classical (i.e., Fourier) case. We also prove the expected nonnegativity of the spectral density function. The third section is devoted to the definition of dyadically differentiable processes on the basis of the dyadic calculus developed by Gibbs [6], [7], and Butzer and Wagner [S] in a deterministic setting. It turns out that the characterization of differentiability in the mean (i.m.) is more complicated in the dyadic sense than it is in classical analysis (see [9]).

These results are applied, in Section IV and V, to the derivation of an error estimate for the Walsh sampling

0018-9448/82/0700-0612$00.75 01982 IEEE

ENGELS AND SPLETTSTikSER: WALSH RANDOM PROCESSES 613

theorem. Maqusi [lo], who established this dyadic sampling theorem for random signals with limited sequency, dealt only with the truncation error which is trivial in this context. We consider the aliasing error arising when a function that is not necessarily sequency-limited is ap- proximated by a Walsh sampling series. The methods developed in Section III enable us to investigate the rate of convergence for dyadically differentiable processes. This leads to the determination of the largest distance allowed between the sampling points if a given tolerance should not be exceeded. In a final example the aliasing error is calculated, thereby demonstrating the utility of the concept of a dyadic derivative.

Moreover, (2.6) holds uniformly in 0 E Iw + , so F be- longs to the class

c@ g {f: lR+ --) R; llfll < 03,

Ilf(*@h)-f(*)Il -O(h+O+)},

where II f II = II f( *)II denotes the usual supremum-norm. The following inversion formula is established in [ 1 I]: at each point t E Iw + of dyadic continuity off E L one has

f(t) = ~%+k(d do. (2.7)

It is well known that any 1 E IR + = [0, co) possesses a dyadic expansion

t = 5 tj2-j (2.1) j= -N(t)

II. BASICPROPERTIESOF DYADICALLY~TATIONARY PROCESSES

with tj E (0, l} and N(t) E Z k (0, ‘1, “2,. . . } being the largest integer j such that tpj # 0. The representation (2.1) is unique in case t is not dyadically rational (DR), i.e., t @ DR where DRg{xE88+; x=p/24,pEP g {0,1,2;..},

9 E q;

When dealing with functions that take random values, it is common to consider stochastic processes as a model. G iven a probability space (52, a, P), a stochastic process x= {X(t,w), t E IIt+, w E a} is an @-measurable function of o E CI for each t E Iw + . In the classical theory of random processes and its applications it is useful to assume that the process is weakly stationary. Similarly, in dyadic analysis a random process X(t, o) is said to be dyadically stationary in the weak sense (DS) if its autocorrelation function

otherwise choose the finite expansion. For each t , u E Iw, Fine’s generalized Walsh functions [4] are given by

It;(t) g eXP{~j,~~~~~~~-jtj]; (2.2)

for o = k E P (2.2) reduces to the definition of the periodic Walsh functions. Defining, as usual, the dyadic sum t@soft,sEIW+ by

Rx,&, t @ 7) g j$t, a)X(t @ 7, o) dP(w) (3.1)

is independent of t E R + ; in this case one writes R,(r) E R, ,Jt, t @ T), (see, e.g., [3]). In order to guarantee the existence of the integral in (3.1) it is necessary to suppose X E e2, i.e., X is square-integrable on Q relative to P with

IlX(t)ll,~ {E{~x(t)(2}}“2

t@s= g Itj-sj12-j (2.3) j=-K

” {L,X(t,o),2dP(w))l/2< 00.

with K = max {j E 2; ] tBj - smj ] = 1 } one derives the following fundamental properties of the generalized Walsh functions: for each t , u E Iw + one has

a> h(t) = 4(4, b) k/,(t @ 4 = d&k:(s)>

almost everywhere (a.e.) in s & [w + ,

4 42n,(t) = #“W)> for alln E Z.

For functions f E L, i.e., functions that are Lebesgue integrable on Iw + , the Walsh-Fourier transform F(v) is defined for each o E R + by

FW = &mf(t)ljq(f) dt. (2.4

The transform F is bounded by

Ilfll, k& jo”lf(f)i dt (2.5)

and is continuous in the dyadic sense, i.e., &-I~ (F(u @A) - F(v)) = 0. (2.6)

It is then easy to obtain the following properties of the autocorrelation function (see, e.g., [3]).

Lemma 3.1: If X E C2 is DS, then for each t, r E lR8+

IlX(t @ 7) - X(t)ll; = 2(R,(O) - R,(T)) (3.2)

and

P&>I 5 R*(O). (3.3) A DS process X E C2 is defined to be dyadically continuous in the mean (i.m.) at to E Iw + if

lim IIX(to @ h) - X(to)l12 = 0, h-O+

also abbreviated by

1.i.m. X( to @ h) = X( to). h-O+

Similarly one defines dyadic continuity i.m. on Iw + as well as uniform dyadic continuity. In the following lemma we give a characterization of the dyadic continuity i.m. of DS

II-” I processes in terms of their autocorrelation functions.


Lemma 3.2: Given any DS process X E C*, the following properties are equivalent

a) X is dyadically continuous i.m . at t, E R + , b) X is uniformly dyadically continuous i.m ., c) R, is (dyadically) continuous at r = 0, d) R,E C@.

Proof: It suffices to establish the implications a) * d) and c) * b). But c) * b) follows at once from (3.2). Con- cerning a) * d), one has from Schwarz’s inequality and (3.2)

= II

$(t, @ 7)(X(t0 @ II) - X(t,)) dP

5 JmllX(t, @ h) - X(t,)ll,;

this yields the assertion.

Note that in assumption c) we put “dyadic” in brackets because dyadic and (right-hand) continuity are equivalent at the origin.

Our next result concerns the spectral representation of DS processes. With S, denoting the Walsh transform (spectral density) of R,, we prove the following property, only cited in [2].

Lemma 3.3: If X E C* is a DS process with R, E L, then one has for each 2) E R ’

S,(u) 45 /mRx(t)#v(t) dt 20. (3.4) 0

The classical counterpart of this result can be found, e.g., in [3, pp. 3471 and [5, pp. 2321.

Proof: Since the Lebesgue integral over any interval [0,2”), m E P, is invariant under dyadic translation, which can be seen from the proof in [4] for the interval [0, l), one deduces with Lemma 2.1

= ~2m2-m~2mR,( u)+b;( u) du ds

= 2-mj02mi2mRx( u @ s)#“( u @ s) du ds

III. WALSH DIFFERENTIABILITY OF DS PROCESSES

In Section II the close connection between dyadic continuity i.m . of a DS process and the dyadic continuity of its autocorrelation function was investigated. Similarly (dyadic) Walsh differentiability of a DS process X leads to second-order Walsh differentiability of R,. For this purpose we start with the definition of the pointwise dyadic derivative on lfZ+ as given by Butzer and Wagner [8] and based on ideas of Gibbs [6], [7]. A real- or complex-valued function f defined on R + is called dyadically differentiable attER+ifthelimitasn-+coof

d,!(t) = d,(f; t) g $ 2j-‘[f(t) -f(t @ 2-j-‘)] j= -”

(4.1)

exists. f[‘l(t) k lim .,,d,( f; t) is then called the first dyadic derivative of f at 1 E R + . Higher order dyadic derivatives are defined successively: f [‘I( t) = ( f[r-ll)[ll( t).

If the convergence in (4.1) is taken in the norm, e.g., of L, one speaks of a strong dyadic derivative (cf. [12]). Note that for periodic functions the corresponding definition of a dyadic derivative, which is not needed here, is somewhat different (see, e.g., [13]). The obvious extension to stochastic processes of (4.1) is given by the following definition.

Definition 4.1: A random process X E C*, is said to be dyadically differentiable i.m . at t E W + , if there exists a process X 1’1 E C* with

l..?. jnzil[x(t) - x(t @ 2-j-‘)] = X[‘l(t).

(4.4 The sum in (4.2) is also abbreviated d,(X, t). Dyadic differentiability i.m . on the whole iRf as well as higher order derivatives of X are defined in an obvious manner. If, in addition, the given process X E C* is DS, dyadic differentiability i.m . has direct consequences for its, autocorrelation function.

Theorem 4.2: If the DS process X E C* is dyadically differentiable i.m . at some point to E Iw + , then its autocorrelation function R, is twice dyadically differentiable on R+ with

= 2-m/2m/2m{~X(u, o)X(s, LJ) dP(w)}$“(u)#“(r) duds R?‘(T) = Rx,+) (4.3)

0 0 for each r E lR+; moreover, XL’] is DS, too.

zr 2-“E ir/

2mX(u)$&) du * 0 II . Proof: First note that the assumption implies that X is

dyadically differentiable i.m . everywhere on Iw + , since the

As the last expression is obviously greater than or equal difference to zero, the assertion follows with m tending to infinity. Note that the interchange of the integrals in the last step of II&J(t) - d,X(t)ll; = &&L&)(O)

the proof is justified by the Tonelli-Hobson theorem, since -‘%(d,R,)(O) + 4(4&)(O) (4.4)

2-m/02m/02m{ /,I X( u, w)JI X( 0, w)I dP( ti)} du ds ( co. ;y$zp;iE;i.; ‘,:R” + . For the calculation of the first

x one deduces with Schwarz’s in-

ENGELS AND SPLETTSTijSSER: WALSH RANDOM PROCESSES 615

equality

Id&( 7) - Rx, x& t @ 7)

=

function can be reconstructed from its values at equidistant sampling points, see, e.g., the references in [ 1 I]. Recently a Walsh sampling theorem for random signals considered as DS stochastic processes was established by Maqusi [lo].

Theorem 5.1: If the DS process X E C* is continuous at somet,EIW+withR,ELandS,(u)=Oforeachv>2”, n E Z, then -kX(t)X[‘l(t 6B T) dPi

5 /~IId,X(t @ 7) - Xrll(t @ +I*.

As the last term tends to zero for n + co, one has

&I(T) = Rx,xru(t, t @T) = Rx,&);

this shows that X and XL’] are jointly DS. Similarly one calculates for the second dyadic derivative of R,

ld,R$](T) - Rxruxcu(t, t @ T)[

= L -$ 2j-‘[X(t) - x(t @32-j-‘)] i i j- n

-X[‘](t) i

X[“(t @ 7) dP

5 h&J(t) - X[‘j(t)l12/~.

For n + cc this leads to the result

showing that Xl’] is a DS process. Note that if the process X E C* has dyadic derivatives of order 2k i.m., one has the equality

Ryk1(7) = RX&), (k E N). (4.5)

If one compares (4.4) with classical results (see, e.g., [3]), it is justified to call

lim d,(d,R,)(T) 4 R$)(T) (4.6) m,n-cc

a second generalized dyadic derivative in the sense of Loeve, provided m and n tend to infinity independently of each other. The following result may be considered as a converse to Theorem 4.2. *

Corollary 4.3: If the second generalized dyadic derivative R$?) of the autocorrelation function of a DS process X E C* exists at the origin, X is dyadically differentiable everywhere on R + .

IV. THEWALSHSAMPLINGTHEOREM

Signal processing is one of the important fields where sequency theory has found several applications; in this respect see the books [ 141 and [ 151. In this section we deal with one of the theorems fundamental for digital signal processing; namely, the Shannon sampling theorem, the Walsh analog of which states that any sequency-limited

1.i.m. i X( $)J(l; 2”t 63 k) = X(t). (5.1) m-m kc0

A DS process X E C* satisfying the conditions above is said to be sequency-limited with cut-off sequency 2”. The proof of Theorem 5.1 is based on the validity of the “deterministic” sampling theorem for R, for which the inversion formula (2.7) is needed as well as Lemma 3.2 for the continuity of R,. For the nonstationary version of Theorem 5.1, in case of sequency-limited signals, see Maqusi V61.

The estimation of the round-off error in [lo] is automati- cally trivial since the restrictive condition of sequency-limi- tation leads to a step function that equals the m th partial sum of (5.1) on the interval [0, m/2”), see [ll]. It is therefore more worthwhile to treat the Walsh-sampling- series representations of functions that are not sequency- limited and to calculate the resulting so-called aliasing error, defined by

A,(X, t) s X(t) - $ X( $+(1;2”t @ k) . II k=O II 2

(5.2)

Theorem 5.2: If the DS process X E c* is continuous i.m. at some to E lR+ with R,, S, E L, then

1.i.m. g X( $)J(1;2”t @ k) = X(t) (5 -3) n-m k=O

with

A,(X; t) I2( /?S&) dv)“‘. 2"

(5.4)

Proof: For the aliasing error A,(X, t) one calculates for each n E Z

(4,(X; t))’

+ $ ; R,(~++(1;2”t~k)J(l;2Wj). k=O i=O

Using the inversion formula (2.7) and properties of the Walsh-Dirichlet kernel, one deduces by dominated conver-

616

gence that

(4(X; t>)*

= s,(v) - 2 fi f&(+h,,+)J(1; 2"t @ k)

k=O j=O

d(1; 2”t 63 k)J(l; 2”t @j) dv,

t);(v) - : ~,,,.(v)J(1;2”t @ k) *dv, 1 g ,,,R” #0,2n(k))J(l; 2”t @ k)12 dv.

k=O

Since

’ 0, elsewhere,

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. IT-28, NO. 4, JULY 1982

Lemma 5.3: If f E L is r times dyadically differentiable withft’l E L, one has for a constant M

cd( f; 6; L) I M&.7( p; 6; L) (5 -7) and, for the Walsh transform F off,

IF(v)1 S4(;)‘w(f[‘];S; L) (5.8)

for any z, > 0. The proof of (5.7) can easily be carried over from [ 111,

where the dyadic derivative in the norm sense (mentioned in Section II) is treated. For the estimate (5.8) one only needs (5.7) and the estimate

given in [ 1 l] for arbitrary 0 > 0. For the results of Theo- rems 5.2 and 5.3 concerning Walsh harmonizable stochastic processes, the reader is referred to [ 181.

We now consider the order of convergence for the approximation of, not necessarily sequency-limited, dyadi- tally differentiable DS processes by the Walsh sampling series.

Theorem 5.4: If, in addition to the assumptions of The- the k th term of the series in the last expression is nonzero only in case 2”t CB k E [0, 1). If, furthermore, e < 2”, then

orem 5.2, X E C* is r times dyadically differentiable i.m. at

definition (2.1) of the dyadic expansion yields N(v2-“) = some to E IW+ with r E 181, and Ryrl E LipK(cu; L), then

N(2”t CB k) = 1, and therefore \1;,2n(2”t @ k) = 1. Then, A,( X; t) 5 const 2-n(r+(a-1)/2). (5.9) for 0 < 2”, we have Proof: Since X is r times dyadically differentiable i.m.

g+(v) - rC;,,.(k) = rC;,y(k)(~~,y(2”t @ k) - 1) = 0, at to E F +, R[x”] exists by Theorem 4.2. From the as- sumed Ltpschitz continuity of R[x”] one deduces from (5.6)

which implies

(A,@; t>>”

- i #k,2”(v)J(1;2”t @ k) k=O I:

and Lemma 5.3 that

S,(v) 5 M( ;)lrti( Ry’]; 2; L) 2

dv, < ~~22’+av-2’-” -

The estimate (5.4) now leads to

Now the assumption S, E L completes the proof.

Note that the estimate (5.4) of the aliasing error is a discrete analogue of a result concerning the Shannon sampling series given in [ 171.

In order to obtain bounds on the speed of convergence to zero of the aliasing error A,, we need to introduce the modulus of continuity

w(f;S; L) A sup Il.&) -f(+@h)ll, (5.5) OIh4

for S > 0 (see, e.g., [12]), and the Lipschitz classes Lip,( a; L); namely, the classes of those functions f E L satisfying

o( f; 6; L) I KS* (5 -6)

for small 6, where K and (Y are positive constants. With these notations one has the following properties of dyadi- tally differentiable deterministic functions.

(A,(X, t))” 5 4MK22r+u~;v-2’-“dv

22r+2+aMK2-n~2~+a-l~ =2r+(u-1

This already implies the desired estimate (5.9) with

( MK 1 ‘/2

cOnst= 2r+a- 1 p 1 +a/*

V. ANEXAMPLE

Let us finally apply our results to a random process having a special autocorrelation function. For this purpose, consider the functions

' E Lo, l>? (6.4

72 1,

for k = 1,2,3, the periodic extensions of which play an important role in the evaluation of the dyadic calculus in

ENGELS AND SPLE’ITST6SSER: WALSH RANDOM PROCESSES 617

[12]. From the definition of the dyadic derivative (4.1) it Indeed, all the assumptions on R, = y and S, = W, of can be seen that ws is twice dyadically differentiable with Theorems 5.2 and 5.4 being satisfied, one only has to use

w3[‘1(7) = w2(7) +$(7) - 1, the estimate (5.9) with r = 1 and a = 1 to deduce (6.3).

7 E Eo, l), Corollary 6.1 shows that one is able to approximate i.m.

-2-“-2w,(7 - 29, 7 E[2”,2”+‘), n E P a DS process having an autocorrelation function that is

3 even discontinuous (in the classical sense) on an every-

and

W,(T) + y(7) - $+(7) - 2, 7 E [o, l),

wp(T) = . - +-1 + l)( w2(7 - 2”) - l),

- $(2-“-l + l)w& - 2”),

[ 7 42”,2”+‘), n E P.

Using methods of [12] it turns out that wJ2] E Lip,(l; L). The calculation of the Walsh transform of w3 leads to

2, ~[k - 1, k), k E P, k > 0.

(6.2) Hence the necessary condition (3.4) of Lemma 3.1 for w, to be the autocorrelation function of a DS process is fulfilled. As y is obviously not sequency-limited, Theorem 5.1 cannot be applied. But from Theorem 5.4 one concludes the following corollary.

Corollary 6.1: If X E C2 is a DS process with autocorrelation function R, g ws, then

A,( X; t) 5 const 2-“. (6.3)

-- -- --

where dense set of points (all dyadic rationals) by a process of step-function type given in (5.2). On the other hand, our example is so simple that we can directly estimate the integral in (5.4) in order to verify the rate of convergence given in (6.3). With (6.2) one calculates from (5.4)

A,(f; t) 5 fi(2” - 1)-l,

showing that A, is indeed 0(2-“) as n tends to infinity. So one can see, on the one hand, that the concept of

dyadic differentiation is applicable to functions having infinitely many jumps and, on the other hand, the aliasing error of the DS process X associated with it is of the order 0(2-“); the latter represents a very good rate of convergence. Let us now illustrate the functions of type (6.1) by giving a picture for the special case k = 2. Fig. 1 shows the partial sum

of the function w2(t) for n = 612 and 0 I t < 3/4. Fig. 2 is a magnified portion of s,,,(t) for t E [0,0.0625]; it shows that this function consists of infinitely many steps (at each dyadic rational number).

2.0

--

--

Fig. 1. The partial sum s,,,(t) for 0 I t < 3/4.


. . . . .

. . .

. . . . . .

.

. . .

. .

. .

.

.

. t * . .

. . .

. * .

. . .

. . . .

1 m a l-

.

.

.

.

.

.

.

h .

~EEE TRANSACTIONSON INFORMATIONTHEORY,VOL. IT-28, NO.~,JULY 1982

ACKNOWLEDGMENT

The authors would like to thank Prof. P. L. Butzer for many valuable comments. The calculations were carried out at the Rechenzentrum RWTH Aachen on the Cyber 175. The authors would like to thank Dr. G . Bleimann, Aachen, and Dipl. Ing. B. Roeckerath, Aachen, for their helpful support concerning all calculations at the Rechen- zentmm RWTH Aachen.

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[II

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False Alarm and Correct Detection Probabilities over a T ime Interval for Restricted Classes of Failure

Detection Algorithms THOMAS H. KERR, MEMBER, IEEE

Alisirad-The statistical analysis of failure detection decisions in terms of the instantaneous probabilities of false ahum and correct detection for a specified failure magnitude at each check-time have previously been per- formed for several different failure detection techniques that utilize a Kalman filter. By performing a discrete-time specialization of a result of Gallager and Helstrom on a tightened upper bound for continuous-time level-crossing probabilities, upper bounds on the probabilities of false alarm and correct detection over (I time interval have been obtained for the specific technique of CR2 failure detection (to aRow an accounting for the

Manuscript received October 4, 1979; revised February 3, 198 1. This work was supported by the Department of the Navy, through the Stra- tegic Systems Project Office, SP-24, Contract N00030-76-C-0026.

The author was with The Analytic Sciences Corporation (TASC), Reading, MA 01867. He is now with Intermetrics, Inc., 733 Concord Ave., Cambridge, MA 02138.

effect of time correlations of the filter estimates). When these upper bounds are optimized to be as tight as possible to the desired probabilities, the resulting optimization problem for discrete-time is a collection of quadratic programming (QP) problems, which may easily be solved exactly without recourse to approximate solutions as were resorted to in the continuous-time formulation. This technique for evaluating tightened upper bounds on the false alarm and correct detection probabilities may be of general interest, since it can be applied to any failure detection technique or signal detection technique that can relate an exceeding of the detemin- istic decision threshold by the test statistic directly to a deterministic level being exceeded by a scalar Gaussian random process.

I. INTRODUCTION

T HE STATISTICAL analysis of a specific failure detection technique, the two confidence regions (CR2)

0018-9448/82/0700-0619$00.75 01982 IEEE

on walsh differentiable dyadically stationary random processes

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