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Equation 2.34 is analogous to the convolution integral describing a process at the linear system output with impulse response h(t) if the stochastic process x(s) comes in at the linear system input:

This analogy allows us to use the linear filter to calculate the correlation integral, the impulse response of which is matched with the expected signal s(t). Matching is reduced to choice of a corresponding linear filter impulse response satisfying the following condition:

For the considered case of detection problem the impulse response of the matched linear filter must be mirrored with respect to the expected signal

where

t0 is the delay of signal peak at the matched filter output, which in the case of the pulse signal must be t0 ≥ τ0

a is the fixed scale factor

If the process x(t) = s(t, α) + w(t) comes in at the matched filter input, then according to (2.36) the process forming at the matched filter output at the instant t0 = τ0 is defined in line with the following formula:

where Rss (τ0 − t) is the autocorrelation function of expected signal s(t, α).

As it follows from (2.38) and (2.39), the signal at the matched filter output coincides with the mutual correlation function of the signal model and expected signal accurate within the fixed factor. When the white noise is absent, that is, w(t) = 0, the output signal coincides with the same accuracy with the autocorrelation function Rss (τ0 − t) of expected signal s(t, α) at the time instant (τ0 − t).

where 0.5 0 is the two-sided power spectral density of white noise. The Neyman–Pearson detector brings us the analogous results [12]. Thus, the matched filter allows us to obtain the maximal SNR at the output within the limits of classical signal detection theory. Realization of analog matched

are the coefficients determined by numerical integration using the technique of rectangles (any technique may be used):

If in (2.48) we replace n0 − l by i in the first term, after elementary mathematical transformations of the first term with the second term we obtain

Equation 2.52 represents, by analogy with (2.39), the autocorrelation function of the expected signal s(t, α). The autocorrelation function (2.52) of the signal at the DGD output is a periodic function by frequency. At low values of fs, cross-sections of the autocorrelation function (2.52) may overlap, which leads to distortions of the process forming at the DGD output. However, if fs were chosen in agreement with the sampling theorem, these distortions would be negligible.

As it follows from (2.52), when the expected signal s(iTs) comes in at the DGD input, the signal at the DGD output is matched with the autocorrelation function (2.52) accurate within the cofactor aTs. Since the autocorrelation function (2.52) is symmetric with respect to its maximum Rss (0), the data samples of the sequence {ZGDout (kTs )} at the DGD output will at first increase and after reaching the upper limit at kTs = n0Ts, that is, the maximal value of the autocorrelation function (2.52), decrease to zero within the limits of the time interval between n0Ts and 2n0Ts. The envelope of the data samples of the sequence {ZGDout (kTs )} at the DGD output coincides with an envelope of the autocorrelation function Rss (kTs ) given by (2.52). This peculiarity of data samples of the sequence {ZGDout (kTs )} at the DGD output agrees with that of analog GD.

The limiting value of the DGD output signal energy is equal to the energy of expected signal sequence s(kTs) and is reached at the finite signal bandwidth and Ts ≤ (2 fs)−1, where fs is the signal bandwidth. If the signal spectrum is infinite and we must take into account some effective signal bandwidth fseff under sampling, there are energy losses owing to superposition of unaccounted power spectral density tails under their mutual shift on k/Ts. These energy losses may be

taken into consideration introducing an additional noise within the power spectral density 0.5 ′ .

0

The receiver noise of DGD is the stationary random sequence with the power spectral density

0.5 ″ uniform within the limits of the bandwidth −(2T )−1 ≤ f < (2T )−1 and depends on T .

0 s s s

It is well known that the sampling and quantization techniques are used to digitize analog signals. For example, the sampling technique is used to discretize a signal within the limits of the time interval, and the quantization technique is used to discretize a signal by amplitude within the limits of the sampling interval. For this reason, there is a need to distinguish between the errors caused by these two digitizing techniques, which allows us to obtain a high accuracy of receiver performance.

Amplitude quantization and sampling can be considered as additional noises ζ1(kTs) and ζ2(kTs) with zero mean and finite variances σζ21 and σ2ζ 2, respectively. If the relationship between the chosen amplitude quantization step x and the mean square deviation σ′ of process at sampling, and quantization block output is determined by x < σ ′, then the absolute value of mutual correlation function between the amplitude quantization error and the signal is approximately 10−9 with respect to the values of the autocorrelation function of the signal. Therefore, it is reasonable to neglect this mutual correlation function. As a first approximation, it is reasonable to assume that the noises ζ1(kTs) and ζ2(kTs) are Gaussian.

We may suggest that the additive component ζΣ(kTs) can be presented in the form of summary uncorrelated interferences: the interference caused by quantization ζ1(kTs), normal Gaussian with zero mean and the finite variance σζ21, and the interference caused by sampling ζ2(kTs), normal Gaussian with zero mean, and the finite variance σ2ζ2 . Thus, the summary additive interference ζΣ(kTs) can be presented in the following form:

and is the normal Gaussian with zero mean and the finite variance given by

This is a direct consequence of Bussgang’s theorem [22].

Taking into consideration the aforementioned statements, the total background noise at the DGD output is defined by the receiver noise and interferences caused by quantization and sampling:

is the noise forming at the PF output of DGD input linear system consisting of the normal Gaussian noise ξPF(kTs) with zero mean and the variance σ2n ; the interference ζ1(kTs) with zero mean and the variance σζ21 , which is caused by quantization; and the interference ζ2(kTs) with

zero mean and the variance σζ22, which is caused by sampling. Noise ξPF(kTs) and interferences ζ1(kTs) and ζ2(kTs) do not correlate with each other.

is the noise forming at the AF output of DGD input linear system (additional or reference noise) [14,17–21], consisting of the normal Gaussian noise ξAF(kTs) with zero mean and the variance σ2n; the ­interference ζ1(kTs) with zero mean and the variance σ2ζ1, which is caused by quantization; and the

interference ζ2(kTs) with zero mean and the variance σζ22, which is caused by sampling. The noise ξAF(kTs) and the interferences ζ1(kTs) and ζ2(kTs) do not correlate to each other.

The probability density function of the total background noise forming at the DGD output is symmetric with respect to zero because the means of noises ξPF(kTs) and ξAF(kTs) and interferences ζ1(kTs) and ζ2(kTs) are equal to zero owing to the initial conditions. The probability density function of the total background noise forming at the DGD output is discussed in detail in Refs. [14, pp. 250–263,15].

40 Signal Processing in Radar Systems

Replacing n0 − l by i in (2.63a), we obtain

According to the analysis carried out in [15, pp. 269–282] and following the main functioning DGD condition, that is, in the considered case

the in-phase and quadrature constituents of the process at the DGD output can be presented in the following form:

where the factor 2 in the second line is caused by the presence of amplifier (see Ref. [15, pp. 269– 282]). Moreover, the corresponding terms in the first and second lines of (2.66) are compensated in the statistical sense. As a result, the quadrature constituent of the process at the DGD output is caused by the autocorrelation function of the in-phase and quadrature constituents of the narrowband noise. Thus, we can write

ξAFΣ − ξPFΣ

the DGD input linear system (PF and/or AF) bandwidth is defined by (2.43); and sinc(x) is the sinc function [1].

Further specification of digital signal processing algorithms is defined by type of convolved sig-

where 0 < t ≤ τ0 and γ = πΔF0/τ0 = const; F0 is the target return signal frequency deviation. The complex envelope can be presented in the following form: