Considered parameters of the stochastic process, namely, the mathematical expectation, the variance, the probability distribution and the density functions, do not describe statistic dependence between the values of stochastic process at different instants. We can use the following parameters of the stochastic process, such as the correlation function, spectral density, characteristics of spikes, central frequency of narrowband stochastic process, and others, to describe the statistic dependence subject to specific problem. Let us briefly consider the methods to measure these parameters and define the methodological errors associated, in general, with the finite time of observation and analysis applied to the ergodic stochastic processes with zero mathematical expectation.
As applied to the ergodic stochastic process with zero mathematical expectation, the correlation function can be presented in the following form:
R(τ) = lim
1
T
x(t)x(t − τ)dt.
(15.1)
∫
T →∞ T
0
Since in practice the observation time or integration limits (integration time) are finite, the correlation function estimate under observation of stochastic process realization within the limits of the finite time interval [0, T] can be presented in the following form:
R*(τ) =
1
∫T
x(t)x(t − τ)dt.
(15.2)
T
0
As we can see from (15.2), the main operations involved in measuring the correlation function of ergodic stationary process are the realization of fixed delay τ, process multiplication, and integration or averaging of the obtained product. Flowchart of the measurer or correlator is depicted in Figure 15.1. To obtain the correlation function estimate for all possible values of τ, the delay must be variable. The flowchart shown in Figure 15.1 provides a sequential receiving of the correlation function for various values of the delay τ. To restore the correlation function within the limits of the given interval of delay values τ, the last, as a rule, varies discretely with the step Δτ = τk+1 − τk, k = 0, 1, 2,…. If the spectral density of investigated stochastic process is limited by the maximum frequency value fmax, then according to the sampling theorem or the Kotelnikov’s theorem, in order to restore the correlation function, we need to employ the intervals equal to
Δτ =
1
2 fmax .
(15.3)
523
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Signal Processing in Radar Systems
x(t)
x(t – τ)
R*(τ)
Delay
Integrator
τ
FIGURE 15.1 Correlator.
However, in practice, it is not convenient to restore the correlation function employing the sampling theorem. As a rule, there is a need to use an interpolation or smoothing of obtained discrete values. As it was proved empirically [1], it is worthwhile to define the discrete step in the following form:
Δτ ≈
1
5 − 10 fmax .
(15.4)
In doing so, 5 – 10 discrete estimates of correlation function correspond to each period of the highest frequency fmax of stochastic process spectral density.
Approximation to select the value Δτ based on the given error of straight-line interpolation of the correlation function estimate was obtained in Ref. [2]:
τ ≈
1
0.2 | |,
(15.5)
ˆ
f
where | | is the maximum allowable value of interpolation error of the normalized correlation function (τ);
ˆ
=
∫∞ f 2 S( f )df
0
f
(15.6)
∫∞ S( f )df
0
is the mean-square frequency value of the spectral density S(f) of the stochastic process ξ(t). Sequential measurement of the correlation function at various values of τ = kΔτ, k = 0, 1, 2,…, v
is not acceptable forever owing to long-time analysis, in the course of which the measurement conditions can change. For this reason, the correlators operating in parallel can be employed. The flowchart of multichannel correlator is shown in Figure 15.2. The voltage proportional to the discrete value of the estimated correlation function is observed at the output of each channel of the multichannel correlator. This voltage is supplied to the channel commutator, and the correlation function estimate, as a discrete time function, is formed at the commutator output. For this purpose, the commutator output is connected with the low-pass filter input and the low-pass filter possesses the filter time constant adjusted with request speed and a priori behavior of the investigated correlation function.
In principle, a continuous variation of delay is possible too, for example, the linear variation. In this case, the additional errors of correlation function measurements will arise. These errors are caused by variations in delay during averaging procedure. The acceptable values of delay variation velocity are investigated in detail based on the given additional measurement error.
Methods of the correlation function estimate can be classified into three groups based on the principle of realization of delay and other elements of correlators: analog, digital, and analog-to- digital. In turn, the analog measurement procedures can be divided on the methods employing a representation of investigated stochastic process both as the continuous process and as the
Estimate of Stochastic Process Frequency-Time Parameters
525
x(t)
Delay
τ
τ
Delay
2Δτ
τ
Delay
v
τ
τ
Integrator
Integrator
Integrator
Switchboard
R*(τ)
FIGURE 15.2 Multichannel correlator.
sampled process. As a rule, physical delays are used by analog methods with continuous representation of investigated stochastic process. Under discretization of investigated stochastic process in time, the physical delay line can be changed by corresponding circuits. While using digital procedures to measure the correlation function estimate, the stochastic process is sampled in time and transformed into binary number by analog-to-digital conversion. Further operations associated with the signal delay, multiplication, and integration are carried out by the shift registers, summator, and so on.
Define the statistical characteristics of the correlation function estimate (the bias and variance) given by (15.2). Averaging the correlation function estimate by an ensemble of realizations we obtain the unbiased correlation function estimate. The variance of correlation function estimate can be presented in the following form:
1 T T
2
(15.7)
Var{R (τ)} =
T 2 ∫∫ x(t1)x(t1 − τ)x(t2 )x(t2 − τ) dt1dt2 − R (τ).
0
0
The fourth moment x(t1)x(t1 − τ)x(t2)x(t2 − τ) of Gaussian stochastic process can be determined as
Substituting (15.8) in (15.7) and transforming the double integral into a single one by introduction of the new variable t2 − t1 = τ, we obtain
Var{R (τ)} =
2
T
−
z
[R
2
(z) + R(z − τ)(z + τ)]dz.
1
(15.9)
T
∫
T
0
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Signal Processing in Radar Systems
If the observation time interval [0, T] is much more in comparison with the correlation interval of the investigated stochastic process, (15.9) can be simplified as
Var{R (τ)} =
2
∫T [R2 (z) + R(z − τ)R(z + τ)]dz.
(15.10)
T
0
Thus, we can see that maximum value of variance of the correlation function estimate corresponds to the case τ = 0 and is equal to the variance of stochastic process variance estimate given by (13.61) and (13.62). Minimum value of variance of the correlation function estimate corresponds to the case when τ τcor and is equal to one half of the variance of the stochastic process variance estimate.
In principle, we can obtain the correlation function estimate, the variance of which tends to approach to zero as τ → T. In this case, the estimate
1
T −|τ|
R(τ) =
∫
x(t)x(t − τ)dt
(15.11)
T
0
can be considered as the correlation function estimate when the mathematical expectation of stochastic process is equal to zero. This correlation function estimate is characterized by the bias
| τ |
b[R(τ)] =
R(τ)
(15.12)
T
and variance
2
T −|τ|
z + | τ |
2
Var{R(τ)} =
∫
1
−
T
[R
(z) + R(z − τ)R(z + τ)]dz.
(15.13)
T
0
In practice, a realization of the estimate given by (15.11) is more difficult to achieve compared to a realization of the estimate given by (15.2) since we need to change the value of the integration limits or observation time interval simultaneously with changing in the delay τ employing the one-channel measurer or correlator. Under real conditions of correlation function measurement, the observation time interval is much longer compared to the correlation interval of stochastic process. Because of this, the formula (15.13) can be approximated by the formula (15.10). Note that the correlation function measurement process is characterized by dispersion of estimate. If the correlation function estimate is given by (15.11), in the limiting case, the dispersion of estimate is equal to the square of estimate bias. For example, in the case of exponential correlation function given by (12.13) and under the condition T τcor, based on (15.10) we obtain
Var{R (τ)} =
σ4
[1 + (2ατ + 1)exp{−2ατ}].
(15.14)
αT
As applied to the estimate given by (15.11) and the exponential correlation function, the variance of correlation function estimate can be presented in the following form [3]:
σ4
σ4
2
2
Var{R(τ)} =
αT
[1 + (2ατ + 1)exp{−2ατ}] −
2α2T 2
[2ατ + 1(4ατ + 6α T
) exp{−2ατ}],
(15.15)
Estimate of Stochastic Process Frequency-Time Parameters
527
when the conditions T τcor and τ T are satisfied. Comparing (15.15) and (15.14), we can see that the variance of correlation function estimate given by (15.11) is lesser than the variance of correlation function estimate given by (15.2). When αT 1, we can discard this difference because it is defined as (αT)−1.
In addition to the ideal integrator, any linear system can be used as an integrator to obtain the correlation function estimate analogously as under the definition of estimates of the variance and the mathematical expectation of the stochastic process. In this case, the function
R (τ) = c∫∞ h(z)x(t − z)x(t − τ − z)dz
(15.16)
0
can be considered as the correlation function estimate, where, as before, the constant c is chosen from the condition of estimate unbiasedness
c∫∞ h(z)dz = 1.
(15.17)
0
As applied to the Gaussian stochastic process, the variance of correlation function estimate as t → ∞ can be presented in the following form:
where the function rh(z) is given by (12.131) as T − τ → ∞.
Now, consider how a sampling in time of stochastic process acts on the characteristics of correlation function estimate assuming that the investigated stochastic process is a stationary process. If a stochastic process with zero mathematical expectation is observed and investigated at discrete instants, the correlation function estimate can be presented in the following form:
N
R (τ) =
1
∑x(ti )x(ti − τ),
(15.21)
N
i=1
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Signal Processing in Radar Systems
where N is the sample size. The correlation function estimate is unbiased, and the variance of correlation function estimate takes the following form:
Var{R (τ)} = N 2
N
N
(15.22)
∑∑ x(ti )x(ti − τ)x(t j )x(t j − τ) − R2 (τ).
1
i=1
j =1
As applied to the Gaussian stochastic process, the general formula, by analogy with the formula for the variance of correlation function estimate under continuous observation and analysis of stochastic process realization, is simplified and takes the following form:
2
N
i
2
Var{R (τ)} =
∑ 1
−
[R
(iTp ) + R(iTp − τ)(iTp + τ)],
(15.23)
N
i=1
N
where we assume that the samples are taken over equal time intervals, that is, Tp = ti − ti−1.
If samples are pairwise independent, a definition of the variance of correlation function estimate given by (15.22) can be simplified; that is, we can use the following representation:
1
N
1
Var{R (τ)} =
∑ x2 (ti )x2 (ti − τ) −
R2 (τ).
(15.24)
N 2
N
i=1
As applied to the Gaussian stochastic process with pairwise independent samples, we can write
Var{R (τ)} =
σ4
[1 + 2 (τ)],
(15.25)
N
where (τ) is the normalized correlation function of the observed stochastic process. As we can see from (15.25), the variance of correlation function estimate increases with an increase in the absolute magnitude of the normalized correlation function.
The obtained results can be generalized for the estimate of mutual correlation function of two mutually stationary stochastic processes, the realizations of which are x(t) and y(t), respectively:
Rxy (τ) =
1
∫T
x(t)y(t − τ)dt.
(15.26)
T
0
At this time, we assume that the investigated stochastic processes are characterized by zero mathematical expectations. Flowcharts to measure the mutual correlation function of stochastic processes are different from the flowcharts to measure the correlation functions presented in Figures 15.1 and 15.2 by the following: The processes x(t) and y(t − τ) or x(t − τ) and y(t) come in at the input of the mixer instead of the processes x(t) and x(t − τ). The mathematical expectation of mutual correlation function estimate can be presented in the following form:
Rxy (τ) =
1
∫T
x(t)y(t − τ) dt = Rxy (τ)
(15.27)
T
0
that means the mutual correlation function estimate is unbiased.
Estimate of Stochastic Process Frequency-Time Parameters
529
As applied to the Gaussian stochastic process, the variance of mutual correlation function estimate takes the following form:
As before, we introduce the variables t2 − t1 = z and t1 = υ and reduce the double integral to a single one. Thus, we obtain
Var{Rxy (τ)} =
2
T
−
z
[Rxx (z)Ryy (z) + Rxy (τ − z)Rxy (τ + z)]dz.
(15.29)
1
T
∫
T
0
At T τcorx, T τcory, T τcorxy, where τcorxy is the interval of mutual correlation of stochastic processes and is determined analogously to (12.21), the integration limits can be expanded until [0, ∞)
and we can neglect z/T in comparison with unit. Since the mutual correlation function can reach the maximum value at τ ≠ 0, the maximum variance of mutual correlation function estimate can be obtained at τ ≠ 0.
Now, consider the mutual correlation function between stochastic processes both at the input of linear system with the impulse response h(t) and at the output of this linear system. We assume the input process is the “white” Gaussian stochastic process with zero mathematical expectation and the correlation function
Rxx (τ) =
0
δ(τ).
(15.30)
2
Denote the realization of stochastic process at the linear system input as x(t). Then, a realization of stochastic process at the linear system output in stationary mode can be presented in the following form:
∞
∞
y(t) = ∫h(υ)x(t − υ)dυ = ∫h(t − υ)x(υ)dυ.
(15.31)
0
0
The mutual correlation function between x(t − τ) and y(t) has the following form:
∞
0.5 0h(τ),
τ ≥ 0,
Ryx (τ) = y(t)x(t − τ) =
∫
(15.32)
h(υ)Rxx (υ − τ)dυ =
0,
τ < 0.
0
In (15.32) we assume that the integration process covers the point υ = 0.
As we can see from (15.32), the mutual correlation function of the stochastic process at the output of linear system in stationary mode, when the “white” Gaussian noise excites the input of linear system, coincides with the impulse characteristic of linear system accurate with the constant factor. Thus, the method to measure the impulse response is based on the following relationship:
h (τ) =
2
Ryx (τ) =
2
1
∫T
y(t)x(t − τ)dt,
(15.33)
0
T
0
0
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Signal Processing in Radar Systems
x(t)
y(t)
Noise
h(t)
Integrator
generator
Delay
Amplifier
τ
x(t – τ)
h*(t)
FIGURE 15.3 Measurement of linear system impulse response.
where y(t) is given by (15.31). The measuring process of the impulse characteristic of linear system is shown in Figure 15.3. Operation principles are clear from Figure 15.3.
Mathematical expectation of the impulse characteristic estimate
h (τ) =
2
∫T
∫∞ h(υ) x(t − υ)x(t − τ) dυdt = h(τ),
(15.34)
0T
0
0
that is, the impulse response estimate is unbiased. The variance of impulse characteristic estimate takes the following form:
4
T T
Var{h (τ)} =
∫∫ y(t1)y(t2 )x(t1 − τ)x(t2 − τ) dt1dt2 − h2 (τ).
(15.35)
20T 2
0
0
Since the stochastic process at the linear system input is Gaussian, the stochastic process at the out-
put of the linear system is Gaussian, too. If the condition T τcory is satisfied, where τcory is the correlation interval of the stochastic process at the output of linear system, then the stochastic process
at the output of linear system is stationary. For this reason, we can use the following representation for the variance of impulse characteristic estimate
Introducing new variables t2 − t1 = z, as before, the variance of impulse characteristic estimate can be presented in the following form:
Var{h (τ)} =
8
T
1
−
z
[Rxx (z)Ryy (z) + Ryx (τ − z)Ryx (τ + z)]dt1dt2
2
T
0T ∫
0
+
2
T
−
z
1
h(τ − z)h(τ + z)dz,
T
∫
T
0
where [3]
∞
Ryy (z) = 20 ∫h(υ)h(υ + | z |)dυ
0
(15.37)
(15.38)
Estimate of Stochastic Process Frequency-Time Parameters
531
is the correlation function of the stochastic process at the linear system output in stationary mode. While calculating the second integral in (15.37), we assume that the observation time interval [0, T] is much more compared to the correlation interval of the stochastic process at the output of linear system τcory. For this reason, in principle, we can neglect the term zT−1 compared to the unit. In this case, the upper integration limits can be approximated by ∞. However, taking into consideration the fact that at τ < 0 the impulse response of linear system is zero, that is, h(τ) = 0, the integration limits with respect to the variable z must satisfy the following conditions:
0 < z < ∞,
τ − z > 0,
τ + z > 0.
Based on (15.39), we can find that 0 < z < τ. As a result, we obtain
2
∞
τ
2
Var{h (τ)} =
∫h
(υ)dυ + ∫h(τ − υ)h(τ + υ)dυ .
T
0
0
As applied to the impulse responses of the form
h1 (τ) =
1
, 0 < τ < T1, T > T1
T1
and
h2 (τ) = α exp(−ατ),
the variances of impulse characteristic estimates can be presented in the following form:
1 +
τ
,
0 ≤ τ ≤ 0.5T1,
T1
Var{h1 (τ)} =
2
×
τ
TT1
2 −
,
0.5T1 ≤ τ ≤ T1,
T1
Var{h2 (τ)} =
α
[1 + 2ατ × exp{−2ατ}].
T
15.2 CORRELATION FUNCTION ESTIMATION BASED ON ITS EXPANSION IN SERIES
(15.39)
(15.40)
(15.41)
(15.42)
(15.43)
(15.44)
The correlation function of stationary stochastic process can be presented in the form of expansion in series with respect to earlier-given normalized orthogonal functions φk(t):
∞
R(τ) = ∑αkϕk (τ),
(15.45)
k=0
532 Signal Processing in Radar Systems
where the unknown coefficients αk are given by
αk = ∫∞ ϕk (τ)R(τ)dτ,
(15.46)
−∞
and (14.123) is true in the case of normalized orthogonal functions. The number of terms of the expansion in series (15.45) is limited by some magnitude v. Under approximation of correlation function by the expansion in series with the finite number of terms, the following error
v
ε(τ) = R(τ) − ∑αkϕk (τ) = R(τ) − Rv (τ)
(15.47)
k = 0
exists forever. This error can be reduced to an earlier-given negligible value by the corresponding selection of the orthogonal functions φk(t) and the number of terms of expansion in series.
Thus, the approximation accuracy will be characterized by the total square of approximated correlation function Rv(τ) deviation from the true correlation function R(τ) for all possible values of the argument τ
∞
∞
v
ε2 = ∫
ε2 (τ)dτ = ∫ R2 (τ)dτ − ∑α2k .
(15.48)
−∞
−∞
k = 0
Formula (15.48) is based on (15.45). The original method to measure the correlation function based on its representation in the form of expansion in series
v
Rv*(τ) = ∑α*k ϕk (τ)
(15.49)
k =0
by the earlier-given orthogonal functions φk(t) and measuring the weight coefficients αk was discussed in Ref. [4]. According to (15.46), the following representation
∞
T
1
∫
αk = lim
ϕk (τ)
x(t)x(t − τ)dt dτ
(15.50)
T →∞ ∫
T
−∞
0
is true in the case of ergodic stochastic process with zero mathematical expectation. In line with this fact, the estimates of unknown coefficients of expansion in series can be obtained based on the following representation:
h