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In the case of the phase-code-manipulated signal with duration τΣ0 = Neτ0, where Ne is the number of elementary signals and τ0 is the duration of elementary signal, the complex amplitude envelope can be presented in the following form:

Consequently, the discrete impulse response at the MSG output matched with the phase-code- manipulated target return signal is given by

and the in-phase and quadrature constituents of process forming at the DGD output take the following form:

where the complex amplitude envelope of the DGD output process is defined as follows:

Thus, in the case of the phase-code-manipulated target return signal, the DGD employed by digital signal processing subsystem in complex radar system 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 phasemanipulated operation. Practical realization of this principle is not difficult.

Now consider the flowchart shown in Figure 2.7 and estimate a functional ability of DGD with convolution blocks in time domain to compress the chirp-modulated target return signal. As we can see from Figure 2.7, we use eight convolving blocks. Each convolving block must carry out n0 multiplications and n0 − 1 additions of Nb numbers in the course of calculation of the k-th signal value, that is, within the limits of the sampling interval Ts. Estimate a required processing speed of the convolving block, taking into account only multiplications for the widely used case of signal processing of the chirp-modulated target return signals at the

sampling rate fs = F0 and the target return signal duration τ0 = n0Ts, namely, Vreq = n0 F0. For example, when n0 = 100 and F0 = 5 × 106 Hz, the required processing speed must be about

Vreq = 5 × 108 multiplications per second.

Consequently, 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 very large-scale integration (VLSI) circuits.

Another example of accelerated convolution operation is the implementation of a specifically designed processor that uses the read-only memory (ROM) block to store calculations of bitwise multiplications made before. Factor codes are used as result addresses of these bitwise multiplications [22,23]. Consider in detail the convolution principle carried out by a specific processor with ROM. For this case, we can present the process at the DGD output as a convolution operation in the following form to simplify calculations:

we obtain a regular particular result. This operation is repeated n times, and in doing so, at the last step, the function 0 (X1(0), X2(0),…, XN(0) ) is subtracted from the accumulated sum and the final result of convolution given by (2.90) is the content of the register R3 after n cycles.

As we can see, a realization of the considered accelerated convolution is not difficult for low values of the sample size N, for example, N = 10…12. With increasing N, the required ROM size becomes very large, for example, at N = 15, QROM = 32798 words, and an access time is increased essentially. We can decrease the ROM size by partitioning a calculation process using a set of steps with later summing up results. Assume N = LM, where M is the number of functions k (X1(k) , X2(k) ,…, X ) and L is the number of binary arguments of these functions, then, in this case, (2.86) can be written in the following form:

(2.91)

Each individual sum can be determined by the earlier-described procedure. For this case, the ROM size is determined as QROM = 2LM instead of 2N = 2M × 2L without the partition of calculation process.

2.4  CONVOLUTION IN FREQUENCY DOMAIN

Now, consider the features of discrete convolution in frequency domain. In accordance with the theory of discrete presentation of continuous functions limited by time or frequency, the function X(t) that can be presented by a sequence of readings {X[i]}, i = 0, 1, 2,…, n − 1 is transformed in frequency domain by discrete Fourier transform (DFT) that for each k = 0, 1, 2,…, n − 1 takes the following form [24]:

where

and vice versa, any function presented by the limited discrete spectrum {FX[k]}, k = 0, 1, 2,…, n − 1, can be reconstructed in the time domain using the inverse discrete Fourier transform (IDFT):

Note that the number of discrete elements of the function X(t) is the same for its presentation both in the time domain and in the frequency domain.

Convolution of sequences in the frequency domain is reduced to product of DFT results. For this purpose, there is a need to realize two direct Fourier transforms, 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 sequence of spectral components {FX[k]}, k = 0, 1, 2,…, n − 1.

For complex functions (signals) an algorithm of the operation DFT-Convolution-IDFT takes the following form:

i= 0

.

where H[i] is a sequence of readings of the complex impulse response of the convolving filter:

i= 0

.

where X[i] is a sequence of complex readings of the input (convolved) function (the target return signal).

l+ n−1

4. ZoutDGD[i] = l +1 n ∑ FZDGDout [k]Wl−+ikn−1, i = 0,1,…,l + n − 1. (2.98)

k = 0

Principal peculiarity of the considered algorithm is a group technology type flow procedure if the width of an input data array is higher n, that is, l ≥ n. The resulting convolution width is l + n − 1. Under solution of detection problems 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 the 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 much more in comparison with the width of convolving sequence equal to the width of an impulse response nim of filters used by DGD. In accordance with Section 2.1, nim = n0, where n0 is the number of discrete elements of the expected target return signal. Synchronous convolution of such sequences is a very cumbersome process. Because of this, as a rule, the input sequence is divided. on blocks with the width l. Each element of the p-th block is generated from a general sequence {X[i]}, i = 0, 1, 2,…, L following the law

where In[L/l] is the integer part of the ratio in brackets.

For each input data block of the width l the (l + n − 1)-point DFT is determined. For convolving sequence of impulse response of filters used by DGD the components of the (l + n − 1)-point DFT must be determined in advance and stored in a memory device. Convolution in frequency domain for each block is obtained by multiplication between the DFT of convolved and convolving sequences at (l + n − 1) points. To determine the convolution in time domain the IDFT is accomplished. The widths of obtained sequence ZDGDout p [i] are equal to (l + n − 1) and the neighboring sequences

[i] and ZDGDout p+1 [i] are overlapped at the n − 1 points. Thus, only the l sequence values will be true. Furthermore, the sum of overlapping partial sequences is used to obtain the correct calculation

results for ZDGDout [i] in all points. In the course of design process, the problem of selecting the optimal

value l at a fixed n by criterion of minimum convolution time arises. At low values of nim ≤ 100 the condition lopt ≈ 5nim is satisfied [11,21].

Consider now the problem of work content to realize the DGD in frequency domain. DFT or IDFT requires (l + n − 1)2 operations of multiplication and (l + n − 1) operations of summation for complex values to obtain the (l + n − 1) frequency (time) samples. The total number of required operations taking into account transformations from the frequency domain to the time domain after convolution consists of 2(l + n − 1)2 + (l + n − 1) multiplications and 2(l + n − 1) summations for complex values. In doing so, we obtain the sample of the output data with the width l. To obtain the same length for output data in the time domain, l2 operations of multiplications and l − 1 operations of summations are required for complex values. Consequently, a convolution in the frequency domain is more time-consuming compared to a convolution in the time domain, approximately eight times if lopt ≈ nim. Thus, there is no purpose to implement a convolution in the discussed form for DGD.

We can essentially decrease the number of operations at convolution in the frequency domain by employing the specific DFT algorithms that are called the fast FT (FFT) [25]. Now, consider the design principles of the FFT algorithm with time decimation by modulus 2 of real sequence. Let the sequence {X[i]} that is processed by DFT have the width M corresponding to the integer power of the number 2, that is, M = 2m. This initial sequence can be divided into two parts in accordance with the following rule:

The sequence Xeven[i] consists of elements of the initial sequence with even numbers, and the sequence Xodd[i] consists of elements of the initial sequence with odd numbers. The width of each

sequence is equal to 0.5M. The sequences obtained in the issue of expansion are expanded again in two parts, while 0.5M two-point sequences are delivered. The number of expansion steps is equal

to m = log2M.

The essence of the FFT algorithm with time decimation by modulus 2 is as follows. The DFT sequences with the width l > 2 are calculated by a combination of DFT of two sequences with the width equal to 0.5 l. In accordance with this fact, at first, the 0.5M-point DFT of two sequences is carried out under processing of the M-point sequence by FFT. Then the obtained transformations are united for the purpose of creating the 0.25M four-point sequences, 0.125M eight-point sequences, and so on, while a transformation of the width M will be obtained after m steps. FFT determination is carried out by the following formulas:

i= 0

FIGURE 2.10  The directed graph for 8-point FFT.

are DFT for even and odd sequences, correspondingly; WMk = (WM )k is the k-th power of the factor WM, which is called rotating factor.

Directed graphs are used to represent graphically the FFT algorithm. The following notations are implemented under graphical representations: The point (or circle) denotes the summation– subtraction operation, the final sum appearing at the top output branch and the final subtraction appearing at the bottom output branch; the arrow on the branch denotes the multiplication by constant written over the arrow; if the arrow is absent the constant factor is equal to unit. The directed graph for 8-point FFT with time decimation by modulus 2 is shown in Figure 2.10. An order of input data assignment is obtained using a procedure of binary inversion of the numbers 0, 1, 2, 7. Such procedure makes a representation simpler and allows us to obtain the input sequence in the natural order—F[0], F[1], F[2], F[3] at the top output branches and F[4], F[5], F[6], F[7] at the bottom output branches. As we can see from Figure 2.10, the 8-point FFT is accomplished over three steps.

At the first step, the four 2-point DFTs are implemented and the condition W2 = exp{−jπ} = −1 is taken into consideration. By this reason, the multiplication operations are absent, and in accordance with (2.98), we can write

At the second step, two pairs of 2-point FFTs are combined with two 4-point FFTs according to (2.101). At the third step, two 4-point FFTs are transformed into the 8-point FFT. In general, the number of steps is defined as m = log2M. At each step, excluding the first step, one half of M multiplications and M summations for complex values are processed. For this reason, to determinate M-point FFT we need (M/2)log2M multiplication operations and Mlog2M summation operations for complex values.

Previously it was shown that to determine the DFT we need M2 multiplications and M summations for the complex values. The advantage in the number of multiplication operations under realization of FFT in comparison with the direct DFT is

For example, ν ≈ 200 at M = 1024, ν ≈ 21 at M = 128.