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Национальный исследовательский университет «МИЭТ»

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Estimate of Stochastic Process Frequency-Time Parameters			553
If the normalized correlation function envelope takes the form
ρen (τ) = exp{−α \| τ \|},			(15.177)
the Fourier transform can be presented as
1(ω) =	2α
		.	(15.178)
	α2 + ω2

As a result, the variance of the correlation function parameter estimate takes the following form:

2	2	3	1	+ q2
Var{νm \| ν0} = α (1+	1+ q	)			(15.179)
			q4 p ,

where

q2 =	2σ2	.	(15.180)

	0α

If q2 1 and 2σ2T −01 1, the variance of correlation function parameter estimate is simplified

	8α2	(15.181)
	Var{νm \| ν0} = q4 p .	(15.181)
	Var{νm \| ν0} = q4 p .

If q2 1, then

Var{νm \| ν0} =	α2	,	(15.182)
	p

or the variance of the central frequency estimate of stochastic process spectral density is inversely proportional to the product between the correlation interval and observation time interval.

Figure 15.11 presents the root-mean-square deviation p Var{νm | ν0}α2 as a function of ratio between the variance of the investigated stochastic process and the power noise q2 within the limits of effective spectral bandwidth. In doing so, we assume that for all values of q2 the following inequality q2p 1 is satisfied.

The optimal estimate of stochastic process correlation function can be found in the form of estimations of the elements Rij of the correlation matrix R or elements Cij of the inverse matrix C. In the case of Gaussian stationary stochastic process with the multidimensional probability density

function given by (12.169), the solution of likelihood ratio equation
	∂fN (x1, x2 ,…, xN \|C)	= 0	(15.183)

	∂Cij

allows us to obtain the estimates Cij and, consequently, the estimates of elements Rij of correlation matrix R.

554				Signal Processing in Radar Systems
	pVar(νm\|ν0)1/2
5	α2
5
4
3
2
1 1					q2
1 1	2	5	10	20	50
FIGURE 15.11 Relative root-mean-square deviation			pVar{νm\|ν0}/α2		as a function of the ratio between the
variance of the investigated stochastic process and power noise.

15.4 CORRELATION FUNCTION ESTIMATION METHODS BASED ON OTHER PRINCIPLES

Under practical realization of analog correlators based on the estimate given by (15.2), a multiplication of two stochastic processes is most difficult to carry out. We discussed previously that for this purpose there is a need to use the circuits performing a multiplication in accordance with (13.201). The described flowchart consists of two quadrators. There are two methods [1] called the interference and compensation methods using a single quadrator. In doing so, it is assumed that the variance of investigated stochastic process is known very well. The interference method is based on the following relationship:

1		2	σ	2		(15.184)
R(τ) = ±	2	[ x(t − τ) ± x(t)]	σ		.	(15.184)
	2

It is natural to use the following function to estimate the correlation function given by (15.184)

			T
	1		2		2
R(τ) = ±			∫[x(t − τ) ± x(t)]	dt σ		.	(15.185)
R(τ) = ±		2T	∫[x(t − τ) ± x(t)]	dt σ		.	(15.185)
		2T	0
			0

As we can see from (15.184) and (15.185), the estimate of correlation function is not biased.

Let us define the variance of correlation function estimate assuming that the investigated process is Gaussian. Suppose that we use the sign “+” in the square brackets in (15.184) and (15.185). Then

				T					2
	1			T
	1											2
Var{R(τ)} =				∫ x(t)x(t − τ)dt							− R		(τ)
	T			0
				0
	+	1		∫T ∫T		x(t1)x(t1 − τ)[x2 (t2 ) + x2 (t2 − τ)]												dt1dt2
	+	T 2		∫T ∫T		x(t1)x(t1 − τ)[x2 (t2 ) + x2 (t2 − τ)]												dt1dt2
				0	0
								T									2
							1	T
			2				1	∫[x		2				2				4
	−2σ			R(τ) +							(t) + x				(t	− τ)]dt		− σ	.	(15.186)
	−2σ			R(τ) +							(t) + x				(t	− τ)]dt		− σ	.	(15.186)
							2T	0
								0

Estimate of Stochastic Process Frequency-Time Parameters

555

The first and second terms in the right side of (15.186) represent the variance of the correlation function estimate R*(τ) according to (15.2). Other terms on the right side of (15.186) characterize an increase in the variance of the correlation function estimate R*(τ) according to (15.185) compared to the estimate given by (15.2). Making mathematical transformations with the introduction of new variables, as it was done earlier, we can write

Var{R(τ)} = Var{R

(τ)} +

−

∫

× {2R2 (z) + R2 (z + τ) + R2 (z − τ) + 4R(z)[R(z + τ) + R(z − τ)]}dz.

(15.187)

As τ → 0

Var{R(0)} ≈

∫1 −

(z)dz,

(15.188)

and, in this case, the variance of correlation function estimate given by (15.185) exceeds by four times the variance of correlation function estimate given by (15.2).

If the condition T τcor is satisfied, the variance of correlation function estimate given by (15.185) can be presented in the following form:

∞

Var{R(τ)} =

∫[R

(z)

+ R(z

+ τ)R(z − τ)]dz

∫T {2R2 (z) + R2 (z + τ) + R2 (z − τ) + 4R(z)[R(z + τ) + R(z − τ)]}dz.

(15.189)

At T τ τcor, we have

∞

Var{R(τ)} ≈

∫R

(z)dz.

(15.190)

Under accepted initial conditions, we obtain

∫∞ R2 (z − τ)dz ≈ ∫∞ R2 (υ)dυ = 2∫∞ R2 (υ)dυ.

(15.191)

−∞

As applied to the exponential correlation function given by (12.13) and if the condition T τcor is satisfied, the variance of correlation function estimate given by (15.185) can be written in the following form:

	σ4	[3 + 4(1 + αT) exp{−αT} + (1 + 2αT) exp{−2αT}].
Var{R(τ)} =			(15.192)
	αT

556	Signal Processing in Radar Systems
Using the compensation method to measure the correlation function, the function
	µ(τ, γ ) = [ x(t − τ) − γx(t)]2	(15.193)

is formed and a selection of the factor γ ensuring a minimum of the function μ(τ, γ) is performed. In doing so, the factor γ becomes numerically equal to the normalized correlation function value. Thus, defining the minimum of the function μ(τ, γ) based on the condition

dµ(τ, γ )	= 0	if	d2µ(τ, γ )	> 0,	(15.194)
dγ			dγ 2

we obtain

γ =	x(t)x(t − τ)	= (τ).	(15.195)
	x2 (t)

Consequently, the compensation measurer of correlation function should generate the function of the following form:

µ (τ, γ ) =	1	∫T [x(t − τ) − γx(t)]2 dt.	(15.196)
	T
0

Minimizing the function μ*(τ, γ) given by (15.196) with respect to the parameter γ, we are able to obtain the estimate of normalized correlation function γ = (τ). Solving the equation

dµ (τ, γ )	= 0,	(15.197)
dγ

we can see that the procedure to define the estimate of the correlation function R*(τ) is equivalent to the estimate that can be presented in the following form:

		(1/T )∫0T x(t)x(t − τ)dt
γ	= (τ) =			.	(15.198)
γ	= (τ) =	(1/T )∫T		.	(15.198)
		(1/T )∫T	x2 (t)dt
		0

As was shown in Ref. [1], as applied to the estimate by minimum of the function μ(τ, γ) given by (15.196), the requirements of quadrator are less stringent compared to the requirements of quadrators used by the previously discussed methods of correlation function measurement.

Determine the statistical characteristics of normalized correlation function estimate of the Gaussian stochastic process. For this purpose, we present the numerator and denominator in (15.198) in the following form:

1	∫T	x(t)x(t − τ)dt = σ2 (τ) + σ2 (τ).	(15.199)
T
0

Estimate of Stochastic Process Frequency-Time Parameters

557

1		T		2			2		Var
			x		(t)dt = σ			1 +		2	.	(15.200)
	T ∫		x		(t)dt = σ			1 +	σ	2	.	(15.200)
	T ∫								σ
		0
As discussed earlier,
						(τ) = 0,						(15.201)
						Var = 0.						(15.202)

Their variances are given by (15.9) and (13.62), respectively. Henceforth, we assume that the condition T τcor is satisfied. In this case, the error of variance estimate is negligible compared to the true value of variance

( Var)2	1.	(15.203)
σ4

Because of this, we can use the following approximation of estimate given by (15.198)

(τ) + (τ)

Var

(τ) =

≈ (τ) +

(τ) 1

−

(15.204)

1 + (

)

Var/σ

Under the definition of the bias and variance of estimate, a limitation is imposed by the terms containing the moments of random variables (τ) and Var/σ2, and the order of these terms is not higher than 2. Under this approximation, the mathematical expectation of estimate of the normalized correlation function (15.204) can be presented in the following form:

	(τ) Var			Var		2
(τ) = (τ) −			+ (τ)			.	(15.205)
	σ	2		σ	2

Thus, the estimate of the normalized correlation function given by (15.198) is characterized by the bias

	(τ) Var			Var			2
b[ (τ)] = (τ) − (τ) = −			+ (τ)			.
b[ (τ)] = (τ) − (τ) = −	σ	2	+ (τ)	σ	2	.
	σ			σ

The product moment (τ) Var can be presented in the following form:

(τ) Var =

∫ x(t)x(t − τ)dt − (τ)

∫ x

(t)dt − σ

σ T

T T

∫∫ x(t1)x(t1 − τ)x2 (t2 ) dt1dt2 − σ2 (τ).

σ2T

(15.206)

(15.207)

558	Signal Processing in Radar Systems

Determining the fourth product moment and making transformations and introducing new variables under the condition T τcor, as it was done before, we can write

(τ) Var		2	∞
	=		∫ (z)[ (z + τ) + (z − τ)]dz.	(15.208)
σ2	=	T	∫ (z)[ (z + τ) + (z − τ)]dz.	(15.208)
			0

Taking into consideration (15.208) and the variance of variance estimate given by (13.63), we obtain the estimate bias in the following form:

	2	∞	4	T	2
b[ (τ)] = −		∫ (z)[ (z + τ) + (z − τ)]dz + (z)		∫		(z)dz.	(15.209)
b[ (τ)] = −	T	∫ (z)[ (z + τ) + (z − τ)]dz + (z)	T	∫		(z)dz.	(15.209)
		0		0

To define the variance of the normalized correlation function estimate

	2		2	(15.210)
Var{ (τ)} =		(τ) − [ (τ) ]

we determine 2 (τ) accurate with the terms of the moments

2		[ (τ) + (τ)]2
	(τ) =
	(τ) =	[1 + ( Var/σ2 ] 2
		[1 + ( Var/σ2 ] 2
	≈ { 2 (τ) + 2 (τ) (τ) + [
	≈ { 2 (τ) + 2 (τ) (τ) + [		(τ)]2} 1	− 2

(τ) and Var of the second order

Var			Var		2
Var		+ 3	Var

σ	2		σ	2
σ			σ

≈ 2 (τ) + 2 (τ) + 3 2 (τ)	( Var)2	− 4 (τ)	Var (τ)	.	(15.211)
	σ4
			σ2

Taking into consideration (15.205) and the earlier-given moments, we obtain

	2		∞	2	2	4		T	2
Var{ (τ)} =			∫	[	(z) + (z + τ) (z − τ)]dz +	(τ)		∫		(z)dz − (τ)
Var{ (τ)} =	T		∫	[	(z) + (z + τ) (z − τ)]dz +	(τ)	T	∫		(z)dz − (τ)
		0						0
	×		4	∫∞ (z)[ (z + τ) + (z − τ)]dz.						(15.212)
	×		T	∫∞ (z)[ (z + τ) + (z − τ)]dz.						(15.212)
				0

As we can see from (15.209) and (15.212), as τ → 0 the bias and variance of estimate given by (15.198) tend to approach zero, since at τ = 0, according to (15.198), the normalized correlation function estimate is not a random variable.

As applied to the Gaussian stochastic process with the exponential correlation function given by (12.13), the bias and variance of the normalized correlation function estimate take the following form:

	2
b[ (τ)] = −		exp{−ατ},	(15.213)
	T

Estimate of Stochastic Process Frequency-Time Parameters			559
	1	[1 − (1 + 2ατ) exp{−2ατ}].
Var{ (τ)} =			(15.214)
	Tα

The sign or polar methods of measurements allow us to simplify essentially the experimental investigation of correlation and mutual correlation functions. Delay and multiplication of stochastic processes can be realized very simply by circuitry. The sign methods of correlation function measurements are based on the existence of a functional relationship between the correlation functions of the initial stochastic process ξ(t) and the transformed stochastic process η(t) = sgn ξ(t). The stochastic process η(t) is obtained by nonlinear inertialess transformation of initial stochastic process ξ(t) by the ideal two-sided limiter with transformation characteristic given by (12.208).

As applied to the Gaussian stochastic process, its normalized correlation function (τ) is related to the correlation function ρ(τ) of the transformed stochastic process η(t) = sgn ξ(t) by the following relationship:

(τ) = sin[0.5πρ(τ)] = − cos[2πP+ (τ)],		(15.215)
where
∞ ∞
P+ (τ) = ∫∫ p2 (x1, x2; τ)dx1dx2		(15.216)
P+ (τ) = ∫∫ p2 (x1, x2; τ)dx1dx2
0	0

is the probability of coincidence between the positive signs of functions η(t) and η(t − τ). The estimate of the probability P+(τ) can be obtained as a signal averaging by time at the matching network output of positive values of the stochastic functions η(t) and η(t − τ) realizations. If the stochastic process is non-Gaussian, a relationship between the correlation functions of initial and transformed by the ideal limiter stochastic processes is very complex. For this reason, the said method of correlation function measurement is restricted. The method of correlation function measurement using additional processes by analogy with the discussed method of estimation of the mathematical expectation and variance of stochastic process is widely used.

Assume that the investigated stochastic process ξ(t) has zero mathematical expectation. Consider two sign functions

η (t) = sgn[ξ(t) − µ (t)],

1 1 (15.217)

η2 (t − τ) = sgn[ξ(t − τ) − µ2 (t − τ)],

where the mutual independent additional stationary stochastic processes μ1(t) and μ2(t) have the same uniform probability density functions given by (12.205) and the condition (12.206) is satisfied. As mentioned previously, the conditional stochastic processes η1(t|x1) and η2[(t − τ)|x2] are mutually independent at the fixed values ξ(t) = x1 and ξ(t − τ) = x2. Taking into consideration (12.209), we obtain

η1(t \| x1 )η2[(t − τ)\| x2 ] =	x1x2	.	(15.218)

	A2

The unconditional mathematical expectation of product between two stochastic functions can be presented in the following form:

	1	∞ ∞	R(τ)
η1(t)η2 (t) =	1	∫ ∫ x1x2 p2 (x1, x2 ; τ)dx1dx2 =	R(τ)	.	(15.219)
η1(t)η2 (t) =	A2	∫ ∫ x1x2 p2 (x1, x2 ; τ)dx1dx2 =	A2	.	(15.219)
		−∞ −∞

560 Signal Processing in Radar Systems

Thus, the function

	A2	N
		∑y1i y2i ,
(τ) =	N		(15.220)
		i=1

where y1i and y2i are the samples of stochastic sequences η1i and η2i, can be considered as the estimate of correlation function to be used for the investigation of stochastic process at discrete instants. In this case, the estimate will be unbiased. When additional stochastic functions are carried out to estimate the variance (see Section 13.3), the operations of product and summation in (15.220) are easily changed by operations of definition of estimate difference between the probability of polarity coincidence and noncoincidence of sampled values y1i and y2i. Delay operations of sign functions can be implemented by circuitry.

Determine the variance of correlation function estimate given by (15.220) assuming that the samples are pairwise independent, that is,

y1i y1 j = y2i y2 j = 0

(15.221)

we obtain

Var{R(τ)} =	N 2	N	N	(τ).	(15.222)
Var{R(τ)} =	N 2	∑∑ y1i y2i y1 j y2 j − R		(τ).	(15.222)
	A4		2
		i=1	j=1

The double sum can be presented in the form of two sums by analogy with (13.94). At this time, (13.55) is true. Define the conditional product moment (η1iη2iη1jη2j|x1i, x2i, x1j, x2j) if i ≠ j under the condition

ξ(ti ) = x1i ,


ξ(ti − τ) = x2i ,	(15.223)
	(15.223)
ξ(t j ) = x1 j ,

ξ(t − τ) = x .

j 2 j

Taking into consideration a statistical independence between η1 and η2, mutual independence between the conditional random values η1(ti|x1i) and η2[(ti − τ)|x2i], and (12.209), the conditional product moment can be written in the following form:

(η1i η2i η1 j η2 j \| x1i , x2i , x1 j , x2 j ) =	x1i x2i x1 j x2 j	.	(15.224)

	A4

Given that the random variables ηi and ηj are independent of each other, the unconditional product moment can be presented in the following form:

	1	∞ ∞
	1
η1i η2i η1 j η2 j =		∫ ∫
η1i η2i η1 j η2 j =	A4	∫ ∫
		−∞ −∞

	2		R	2	(τ)
x1x2 p2 (x1, x2; τ)dx1dx2		=	R		(τ)		.	(15.225)

			A			4
			A

Estimate of Stochastic Process Frequency-Time Parameters

Substituting (13.96) and (15.225) into (13.94) and then into (15.222), we obtain

561

(15.226)

According to (15.220), the correlation function estimate satisfies the condition given by (12.206), that is, σ2 A2. For this reason, the variance of correlation function estimate is defined by halfintervals of possible values of additional stochastic processes. Comparing (15.226) and (15.25), we obtain

	A	4		1 − (σ	4	4	2	(τ)
Var{R(τ)}	A			1 − (σ		/A	)	(τ)	(15.227)
Var{R (τ)} = σ4			×	1 + 2			(τ) .		(15.227)
Var{R (τ)} = σ4			×	1 + 2			(τ) .

As we can see from (15.227), since the condition σ2 A2 is satisfied, the correlation function estimate given by (15.220) is worse compared to the correlation function estimate given by (15.21).

15.5 SPECTRAL DENSITY ESTIMATE OF STATIONARY STOCHASTIC PROCESS

By definition, the spectral density of stationary stochastic process is the Fourier transform of correlation function

S(ω) = ∫∞ R(τ) exp{− jωτ}dτ.

−∞

The inverse Fourier transform takes the following form:

∞

R(τ) = 21π −∞∫ S(ω)exp{jωτ}dω .

As we can see from (15.229), at τ = 0 we obtain the variance of stochastic process:

∞

Var = R(τ = 0) = 21π −∞∫ S(ω)dω .

(15.228)

(15.229)

(15.230)

As applied to the ergodic stochastic process with zero mathematical expectation, the correlation function is defined by (15.1). Because of this, we can rewrite (15.1) in the following form:

		1	T	∞
			∫		∫
S1	(ω) = lim			x(t)		x(t − τ) exp{− jωτ}dτ dt.	(15.231)

	T →∞ T
			0	−∞

The received realization of stochastic process can be presented in the following form:

x(t) if		0 ≤ t ≤ T ,
		(15.232)
x(t) =		(15.232)
0	if	\| t \| > T.

562	Signal Processing in Radar Systems

In the case of physically realized stochastic processes, the following condition is satisfied:

∫T	x2 (t)dt = ∫∞ x2 (t)dt < ∞	(15.233)
0	−∞

For the realization of the stochastic process, the Fourier transform takes the following form:

X( jω) = ∫T	x(t) exp{− jωτ}dt = ∫∞ x(t) exp{− jωτ}dt.			(15.234)
0			−∞
Introducing a new variable z = t − τ, we can write
∫∞ x(t − τ) exp{− jωτ}dτ = X(− jω) exp{− jωτ}.				(15.235)
−∞
Substituting (15.235) into (15.231) and taking into consideration (15.234), we obtain
	S1 (ω) = lim	1	\| X( jω) \|2 .	(15.236)
	S1 (ω) = lim		\| X( jω) \|2 .	(15.236)
	T →∞ T

Formula (15.236) is not correct for the definition of spectral density as the characteristic of stochastic process averaged in time. This phenomenon is caused by the fact that the function T−1|X(jω)|2 is the stochastic function of the frequency ω. As the stochastic function x(t), this function changes randomly by its mathematical expectation and possesses the variance that does not tend to approach zero with an increase in the observation time interval. Because of this, to obtain the averaged characteristic corresponding to the definition of spectral density according to (15.228), the spectral density S1(ω) should be averaged by a set of realizations of the investigated stochastic process and we need to consider the function

		N
S(ω) = lim	1	∑\| Xi ( jω) \|2 .	(15.237)
S(ω) = lim	T	∑\| Xi ( jω) \|2 .	(15.237)
T →∞	T	i=1
N→∞		i=1

Consider the statistical characteristics of estimate of the function

S1 (ω) =	\| Xi ( jω) \|2	,	(15.238)
	T

where the random spectrum X(jω) of stochastic process realization is given by (15.234).

The mathematical expectation of spectral density estimate given by (15.238) takes the following form:

	\| Xi ( jω) \|2				1	T	T
S1 (ω) =				=		∫∫ x(t1)x(t2 ) exp{− jω(t2 − t1}dt1dt2
S1 (ω) =			T	=	T	∫∫ x(t1)x(t2 ) exp{− jω(t2 − t1}dt1dt2
						0	0
=	1	∫T ∫T R(t2 − t1)exp{− jω(t2 − t1)}dt1dt2.						(15.239)
=	T	∫T ∫T R(t2 − t1)exp{− jω(t2 − t1)}dt1dt2.						(15.239)
	0		0

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