Diss / 10

FIGURE 2.13  The DGD output signal amplitude ratio ZDGDout (Nb )ZDGDout (∞) as a function of the digit capacity Nb of input signal.

least two times higher than the limiting sampling rate fsmax. There is a need to take into account the fact that an increase in the sampling rate fs causes a drastic increase in DGD design and realization complexity, that is, the technical specifications to required performance and memory size of digital signal processing equipment become rigid. For this reason, in the course of designing the DGD, it is recommended to choose the sampling rate fs within the limits 2 ÷ 3 F0.

The number of bits of the sampled target return signal or digit capacity sampling width plays a very important role in the definition of the DGD performance. Computer simulation results representing a DGD output signal amplitude ratio ZDGDout (Nb )ZDGDout (∞) as a function of the digit capacity of the target return signal amplitude samples Nb are shown in Figure 2.13. Here ZDGDout (Nb ) denotes the DGD output signal at limited digit capacity Nb and ZDGDout (∞) denotes the GD output signal without sampling (Nb → ∞). As we can see from Figure 2.13, at Nb ≥ 10…12 a dependence of the DGD output signal amplitude peak on the digit capacity Nb is weak. Thus, this digit capacity Nb can be recommended under the DGD designing in digital signal processing subsystems of complex radar systems.

2.6  SUMMARY AND DISCUSSION

The discussion in this chapter allows us to draw the following conclusions.

Under the designing of analog-to-digital converters, the task of choosing a sampling interval value and the number of signal quantization levels of the sampled target return arises. Simultaneously, we should take into account the problems of designing and realization both for converters and for digital receivers processing the target return signals.

A sampling device can be considered as a make circuit with the time τ and sampling period Ts. The instantaneously sampled target return signal xδ(t) has a mathematical form similar to that of the Fourier transform of a periodic signal. Representation of the continuous function x(t) in the form of the sequence {x(nTs)} is possible only under well-known restrictions. One restriction in kind is a requirement of spectrum limitation of the sampled target return signal xδ(t). By Kotelnikov’s theorem concerning the signal space concept [4], the continuous target return signal x(t) possessing a limited spectrum is completely defined by a countable set of samples spaced Ts ≤ 1/(2fmax) seconds apart, where fmax is the cutoff frequency of target return signal spectrum.

The complex envelope of a narrowband radio signal can be presented either by the envelope and the phase, which are the functions of time, or by the in-phase and quadrature components.

Accordingly, under the narrowband radio signal sampling there is a need to use two samples: either the envelope amplitude and the phase or the in-phase and quadrature components of complex amplitude of the narrowband radio signal. Thus, we deal with two-dimensional signal sampling.

Under digital signal processing of target return signals in CRSs there is a need to carry out a quantization of sampled values of complex envelope and phase or in-phase and quadrature components of radio signals in addition to sampling. Devices that carry out this function are called quantizers. We define the main parameters of the analog-to-digital conversion: sampling and quantization errors, reliability, that is, the ability to keep accuracy at given boundaries within the limits of definite time interval under a given environment, and other parameters producing a limitation in power consumption, weight and dimensions of analog-to-digital converters, cost under serial production, time that is required for design, and so on. We would like to pay attention to the following fact that there are two types of sampling and quantization errors: the dynamical errors and statical errors. The dynamical errors are the discrete transform errors. The statical errors are the unit sample errors. The dynamical errors depend on the target return signal nature and time performance of the analog-to-digital converter. The main constituent of this error is an inaccuracy caused by variations in the target return signal parameters at the analog-to-digital converter input.

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 errors caused by these two digitizing techniques that allow us to obtain a high accuracy of receiver performance. Amplitude quantization and sampling can be considered as the additional noise ζ1(kTs) and ζ2(kTs) with zero mean and the finite variances σζ21 and σζ22, 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 equal to 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 noise ζ1(kTs) and ζ2(kTs) are Gaussian.

In the case of the phase-code-manipulated target return signal, the DGD employed by a digital signal processing subsystem in a CRS uses only four convolving blocks. Moreover, a convolution within the limits of each cycle equal to the elementary signal duration τ0 is a summation of amplitude samples of the in-phase and quadrature constituents of the target return signal at instants with signs defined by values S*[i] = ±1 in accordance with the given code of the phase-manipulated operation. Practical realization of this principle is not difficult. In the considered case, a direct realization of convolution operations by serial digital signal processing techniques is not possible. There is a need to apply specific procedures of calculations under digital signal processing. First of all, we may use a principle of parallelism that is characteristic of convolution problems. This principle allows us to calculate n0 of pairwise multiplications S*[i] × x[k − (n0 − i)], where i = 1,…, n0, simultaneously using n0 parallel multipliers with subsequent summation of partial multiplications (see Figure 2.8). In this case, each multiplier must possess the processing speed Vreq = 5 × 106 multiplications per second, that is, one multiplication operation per 40 ns. This processing speed can be easily provided by employing VLSI circuits. Another example of accelerated convolution operation is the implementation of specifically designed processor that uses the ROM blocks to store calculations of bitwise multiplications made before. Factor codes are used as result addresses of these bitwise multiplications.

Convolution of sequences in the frequency domain is reduced to a product of DFT results. For this purpose, there is a need to realize two DFTs, namely, to convolve a sequence of readings of the function {X[i]}, i = = 0, 1, 2,…, n − 1 and a sequence of readings of the impulse response of filters used by DGD. If after convolution it is necessary to make transformations into the time domain, there is a need to carry out IDFT for a sequence of spectral components {FX[k]}, k = 0, 1, 2,…, n − 1.

The principal peculiarity of the considered algorithm of the operation DFT-Convolution-IDFT is a group technology type flow procedure if a width of input data array is higher n, that is, l ≥ n. The resulting convolution width is l + n − 1. In detection problems solved by DGD, we assume that the impulse response of all filters used by DGD is not variable, at least for probing signals of the same kind. Therefore, the DFT of impulse response of all filters used by DGD is carried out in advance and stored in the memory device of a corresponding computer. In the course of convolution, there is a need to accomplish one DFT and one IDFT. It must be emphasized that under radar detection and signal processing problems, a convolved sequence width L corresponds to radar range sweep length that is greater in comparison with the width of a convolving sequence equal to a width of impulse response nim of filters used by DGD.

Evaluation of the required memory size to realize the DGD in a radar signal processing subsystem under the implementation of FFT is based on the following statements. Using the same memory storage cells for FFT and IFFT and the buffer to store the output signal (see Figure 2.11), the memory size required for functional DGD purposes is Q = M1 + M2 + M3 + l, where M1 is the number of storage cells required for input sequence cells; M2 is the number of storage cells required for FFT and IFFT; M3 is the number of storage cells required for the spectral components of impulse responses of all filters used by DGD; l is the number of storage cells required for the output sequence. Thus, the use of FFT requires higher memory size compared to the implementation of FT in the frequency domain. A frequency of convolution operations can be essentially increased using a continuous-flow FFT system. In this case, the specific processor for FFT consists of 0.5M log2M devices for arithmetic operations functioning in parallel. Each arithmetic device fulfills a transformation at a definite stage of FFT. In doing so, we can reduce the calculation time in log2 0.5M times. The continuousflow FFT system requires additional memory in the form of interstage delay registers.

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first and subjected to noncoherent accumulation before coming in at the input of decision circuit. Possible ways of packing of the in-phase and quadrature signal components are shown in Figure 3.1, the outputs 1, 2, and 3. The single-channel digital moving-target indicator can be obtained from a two-channel digital moving-target indicator version if only one of the channels of the two-channel digital moving-target indicator shown in Figure 3.1 is used. In this case, the block diagram of the digital moving-target indicator is significantly simplified, but additional losses in the SNR do occur.

In practice, the following indices are widely used for the evaluation of the efficacy of digital moving-target indicators:

•The coefficient of interference cancellation Gcan that is defined as a ratio of the passive interference power at the equalizer input to the passive interference power at the equalizer output

where

Ppiin is the power of the passive interference at the equalizer input σ2piin is the variance of the passive interference at the equalizer input Ppiout is the power of the passive interference at the equalizer output

σ2piout is the variance of the passive interference at the equalizer output

σ2n is the receiver noise

•Figure of merit for the case of linear interperiod cancellation of the passive interferences can be defined in the following manner:

where Gs is the coefficient characterizing a signal passing through the passive interferences equalizer. In general, when a nonlinear signal processing is used, the figure of merit η indicates to what extent the interference power at the equalizer input can be increased, without losses in detection performance.

•The coefficient of improvement is a characteristic of the response of the digital movingtarget indicator on passive interferences with respect to the averaged response on target return signals:

Pout /Pout

Gim = s pi , (3.4)

Psin /Ppiin Vtg

where [Psin /Ppiin]Vtg is the SNR at the equalizer input while deciding on the average values with respect to all target velocities.

The aforementioned Q-factors can be determined both for passive interferences equalizer systems using the interperiod subtraction and for equalizer systems with subsequent accumulation of residues after the passive interference cancellation.

The digital nonrecursive filters are very simple under realization, but they have very low-angle fronts of the amplitude–frequency performance, which severely affects the effectiveness of passive interference cancellation.

In addition to the nonrecursive filters, the recursive filter can be used as the rejector filter with the following signal processing algorithm:

where ai and bj are the coefficients of the digital recursive filter.

Methods of recursive filter synthesis are considered and discussed in numerous literatures [9–21]. Without going into detail, we note that a synthesis procedure based on a squared amplitude–­ frequency filter response in accordance with a technique described in Ref. [22] can be considered the most effective for the digital moving-target indicators constructed based on the digital rejector filters. As a first step, and on the basis of the given parameters of environment conditions and initial premises, an order is chosen and the amplitude–frequency characteristic of analog filter prototype is defined. After that we define an approximation of squared amplitude–frequency characteristic of digital filter to the squared amplitude–frequency characteristic of analog filter-prototype, taking into consideration the restrictions arising from structural complexity. The elliptic filter, also known as Zolotaryov–Cauer filter [23], is best suited for these requirements.

Under realization of the digital rejector filter in accordance with (3.10), it is convenient that ai and bj are simple binary numbers. In this case, there is no need to use ROM to store these coefficients, and the multiplication operations are changed by simple shift and summation operations. For example, in the case of the recursive elliptic filter of the second order, synthesized using the squared amplitude– frequency characteristic, we obtain the following values of coefficients [22]: a0 = a2 = 1; a1 = −1.9682; b1 = −0.68; and b2 = −0.4928. To simplify the realization of the digital rejector filter, the coefficients a1, b1, and b2 can be rounded in the following order: a1 = −1.875 = −21 + 2−3, b1 = −0.75 = 20 + 2−3, b2 = −0.5 = −2−1. Investigations showed that rounding off of the coefficients a1, b1, and b2 can give a very small effect on the cancellation of passive interferences; however, this effect is negligible.

The digital rejector filter ensures the best cancellation of passive interferences due to an improvement in the amplitude–frequency characteristic shape, compared with the nonrecursive rejector filter of the same order. However, the degree of correlation of the passive interference remainders at the digital recursive filter output is higher compared to the correlation degree of the passive interference remainders at the digital nonrecursive filter. Moreover, the presence of positive feedback leads to an increase in the duration of transient process and corresponding losses in efficacy if the number of target return pulses in the train is less 20, that is, N < 20.

Now, consider some results of the comparison between the efficiency levels of the digital filter and the analog filter. At first, transform (3.10) to nonrecursive form, that is, present the signal processing algorithm of the digital recursive filter in the following form:

j =1