Diss / 10

means that the operation of one of those algorithms with the functions αij ≠ 0 is possible after the algorithm Ai operation. Equation 7.5 is called the transition formula for the algorithm Ai. These formulas can be designed for all elementary DSP algorithms of the complex algorithm given by the matrix flowchart. Thus, in the case of matrix form (7.4), we obtain the following system of transition formulas:

The matrix flowchart of the complex algorithm allows us to design the table reflecting both information and controlling association between the elementary DSP algorithms. We may change all the matrix elements by magnitudes

As a result, we obtain the adjacency matrix that reflects formally an information association between the elementary DSP algorithms. For instance, the algorithm given by the matrix in (7.4) has the following adjacency matrix:

We can design the graph flowchart of the complex algorithm of computational process based on the adjacency matrix.

7.2.2  Algorithm Graph Flowcharts

The algorithm graph flowchart is the finite oriented graph satisfying the following conditions:

•There are two marked nodes in the graph: the input node corresponds to the operator “Start” and the only arrow is directed from this operator; and the output node corresponds to the operator “Stop” and no arrow is directed from this node.

7.2.3  Use of Network Model for Complex Algorithm Analysis

The main problem solved by the graph flowcharts of complex DSP algorithms is a definition of rational ways to present these problems and a choice of computational software tools and microprocessor subsystems to realize the complex DSP algorithm of a radar system. In short, the problem of optimization of computational process is assigned. The solution of this problem allows us to reduce significantly a realization time and to simplify the complex DSP algorithm of the radar system. Similar problems are the problems of network planning and control [1,2].

To design the network model of computational process under DSP in CRSs, there is a need to transform the graph flowchart of complex algorithm into the network graph that satisfies the following requirements, namely, the network graph cannot consist of contours, that is, the case when the initial top is matched with the final top, and there must be a strict order of top precedence in accordance with the condition that the number of the top i is less than the number of the top j, that is, (i < j) if there is a transition from ai to aj. Henceforth, we call a network of elementary DSP algorithms organized by the corresponding way for CRS signal processing and satisfying all the requirements mentioned earlier as the network model of the complex DSP algorithm. The network tops are interpreted as an “operation” of the corresponding elementary DSP algorithm expressed by the number of computational operations. The network arcs can be interpreted as a sequence of elementary DSP algorithm operations. Network transitions can be deterministic (planned) and stochastic. In the last case, the network model is called the stochastic network model.

The deterministic network model cannot present a complex algorithm functioning in CRSs since it is impossible to predict before a set of elementary DSP algorithms and sequence of realizations for each practical situation. Therefore, the stochastic network model, in which the transitions in the network graph are defined by the corresponding probabilities of transitions given by specific conditions of CRS functioning, is more suitable to image and analyze a realization of the complex DSP algorithm by microprocessor subsystems. When the network model has been constructed, the problem of estimating the time to complete all operations, that is, the time to finish all operations by microprocessor subsystems with the given effective speed of operations, arises. This time cannot be higher than a total duration to finish a complex DSP algorithm operation defined by the most unfavorable way from the initial graph top an to the final graph top ak, that is, along such route that generates a maximal duration of operations. This route is called the extreme route. The extreme route in the stochastic network model cannot be presented in clear form as, for example, in the network model with a given structure. Because of this, the problems of defining the average time or average number of operations required to realize the complex algorithm are assigned under analysis of stochastic network models. To illustrate some principles of designing the stochastic network graph, we consider the following example.

Let the digital signal reprocessing algorithm depicted in Figure 5.13 be considered as the complex DSP algorithm. The network graph of the considered algorithm is shown in Figure 7.7. Special transformations of the algorithm logical flowchart are not needed to design the network graph. Algorithm functioning is started from selection of the immediate target pip of current scanning in the buffer memory device (the elementary DSP algorithm an). Later, the algorithm of target pip coordinate system transformation from the polar coordinate system to Cartesian one (the algorithm a1) is realized. The next stage is a comparison of new target pip coordinates with coordinates of extrapolated target pips of target tracking trajectories (the algorithm a2). If the target pip is inside the gate of selected target tracking trajectory, then it is considered as a prolongation of this trajectory. The updated target tracking trajectory parameters (the algorithm a3) are made more exact, and preparation to send the required information to the user (the algorithm a4) is produced. After that, the graph goes to the final state (the algorithm ak). If the target pip is outside of the gates of none of target tracking trajectories, then the target pip is checked whether

FIGURE 7.7  Flowchart of network graph.

it belongs to the detected target tracks (the algorithm a5). If a new target pip is able to confirm one of the detected target tracks, we carry out an accurate definition of target track parameters (the algorithm a6) and test the detection criterion (the algorithm a7). If the target pip is outside of the gates of none of detected target tracks, we assign it to beginning new target tracks (the algorithm a8). When the target pip is caught by one of the primary lock-in gates, the beginning of a new target track is carried out and initial values of the new target tracking trajectory are defined (the algorithm a9). Finally, if the target pip is outside the existing primary lock-in gates it is recorded as the initial point of the new target track (the algorithm a10).

The considered network graph is stochastic without doubt, and there is a need to define the probabilities of transitions between the network graph tops. Let the probability that the selected target pip belongs to the target tracking trajectories be denoted by P1 = Ptr and processed by the elementary DSP algorithms a1, a2, a3, a4, ak. Then,

is the probability that the target pip does not belong to the target tracking trajectories and has to be processed further. Let PD be the probability that the target pip belongs to the detected target track. Then, the probability that the target pip processing is finished by realization of the algorithms a1, a2, a5, a6, a7, ak is defined as

and the probability to proceed with a target pip processing further is determined in the following form:

where

Pbeg is the probability that the target pip belongs to beginning the new target track

P3′ is the probability that the target pip will be assigned as the beginning point of the new target track

The probabilities Ptr, P0, Pbeg depend on the number of target tracking trajectories, detected target tracks, and beginning points of target tracks under processing. For the algorithm of target pip beginning by the criterion (see Chapter 4) we have

•The average number of target pips subjected to process under each scanning is determined as

where

N‾false is the average number of false target pips

N‾true is the average number of true target pips appearing within the limits of the scanning period

•The average number of true target pips coming in the target tracking gates under each scanning is given by

where PDgate is the probability of detection of true target pips within the limits of the target tracking gates, which is the same for all target tracks

•The average number of false target pips inside the target tracking gates under each scanning is determined in the following form:

j= m+ n

where

PFscanj is the probability of the false target pip hit into the jth target tracking gate

Nscanj is the average number of jth target tracking gates for all true and false target tracking trajectories

Taking into consideration (7.16) through (7.18), the probability of an arbitrary selected target pip belonging to one of the target tracking trajectories can be expressed in the following form:

In an analogous way, we obtain the formula for the probability of target pip belonging to detected target tracks:

where

and

j= m

where

PDfalsej is the probability of the false target pip hit into the detected target track gates with the number j

PDD is the probability of the true target pip detection into the detected target track gates

ND j is the average number of gates with the number j formed by all the detected target tracking trajectories

NDtrue is the average number of the true target tracks being in the detection process

The probability of target pip belonging to the started target tracks is determined as

where

PFlockj -in is the probability of false target pip hit into the primary lock-in gates with the number j N lockj -in is the average number of the primary lock-in gates with the number j

PDlock-in is the probability of the true target pip detection into the primary lock-in gates Ntruelock-in is the average number of the primary lock-in gates of true target tracks

Thus, if the statistical characteristics and parameters of noise and target environment inside the radar coverage are known and the target pip beginning algorithm parameters are selected, in the considered case, we are able to determine the probability of transition in the network graph of the complex DSP algorithm. However, we cannot say that this possibility exists forever. In some cases, the probability of transition can be only estimated as a result of computer simulation of the complex DSP algorithm.

7.3  EVALUATION OF WORK CONTENT OF COMPLEX DIGITAL SIGNAL PROCESSING ALGORITHM REALIZATION

BY MICROPROCESSOR SUBSYSTEMS

7.3.1  Evaluation of Elementary Digital Signal Processing Algorithm Work Content

We consider the DSP algorithms realizing the main operations and control in a CRS as the elementary DSP algorithms:

•The signal preprocessing stage: DSP algorithms of matched filtering in time or frequency domains, cancellation of passive interferences by the digital moving target indicator (MTI), detection and estimation of target return signal parameters, recognition of type and estimation of interference and noise parameters and ranking samples, etc.

•The reprocessing stage: DSP algorithms of target track detection, selection of target pips and their beginning to the target tracking trajectories, target track parameters filtering, transformation coordinate system, and so on.

•The CRS control process: DSP algorithms of the scanning signal parameter determination under target searching and tracking, determination of power balance between the modes of CRS operations, etc.

Each of the aforementioned DSP algorithms is characterized by the work content expressed by the number of arithmetical operations required to realize it. Preliminary computation of the required number of arithmetical operations is possible only if there is an analytical function between the algorithm input and output. In the case of other algorithms possessing a character of logical operations, mainly, the required number of operations can be obtained only as a result of realization based on microprocessor subsystems.

Results of analytical calculation of the required number of arithmetical operations are obtained individually by the number of additions, subtractions, products, and divisions. Henceforth, there is a need to determine the number of reduced arithmetical operations. As the reduction operation, as a rule, an addition is used (short operation). The number of reduced arithmetical operations is determined for each microprocessor subsystem taking into consideration the known ratio between the time to carry out the ith long and short operations, that is, τilong /τshorti . However, the computation of the work content is not finished at this stage since we must take into consideration other nonarithmetical operations.

The DSP of target return signals has a pronounced information-logical character. Logical operations and transition operations are for about 80% of the total number of elementary DSP operations (cycles) in the process of the complex DSP algorithm realization required for CRS functioning. For example, under realization of the digital signal preprocessing algorithm of twocoordinate radar system by microprocessor subsystems with permanent memory, the following number of operations in percentage is required: forwarding or transferring—45%; reduced arithmetical operations—23%; control transfer—17%; shift—5%; logical operations—3%; information exchange—2%; other operations—5%. Consequently, under computation of the work content of the elementary DSP algorithms there is a need to take into consideration the nonarithmetical operations, too. For this purpose, we can introduce the coefficient Kna and write the following relation:

where Nai is the average number of arithmetical operations required for the ith algorithm. The number of microprocessor operations depends on a programming mode. Under the use of highlevel programming languages, a program length is two to five times greater than that of the optimal program. To take into consideration this fact, we introduce the coefficient Kprog ≈ 2. Thus, (7.24) can be written in this form:

This work content definition will be used in our next computations.

7.3.2  Definition of Complex Algorithm Work Content Using Network Model

Under analysis of the work content of complex DSP algorithms, we can use the Markov model of computational process or the stochastic network model [3–6]. The stochastic network model allows us to reduce the number of computations under the work content determination in comparison with the Markov model. Therefore, we prefer to use the stochastic network model. As we noted previously, the network graph model of complex DSP algorithm is the initial condition to evaluate the work content. There is a need that this graph would not contain the cycle paths and the graph tops must be numbered in such a way that the graph top number, to which the transition is carried out, will be higher than any graph top number, from which such transition is possible. Moreover, the graph end top must have the maximal number k. An example of such graph satisfying all the previously mentioned requirements is depicted in Figure 7.8.

rank 1 are covered by the rank 2. The number of iterations for this cycle is denoted by n(2), and so on. The graph transform is reduced to changes in cycle paths by a single operator. These changes for the graph shown in Figure 7.9a are carried out in accordance with the following formula:

The resulting graph is shown in Figure 7.9b. Thus, the network model of complex DSP algorithm allows us to define, in principle, the average work content. If we know a realization time of a single reduced operation, then we can compute the average realization time of a complex DSP algorithm. Inversely, if a limitation on the average realization time of complex DSP algorithm is given, we are able to determine the required work content of microprocessor subsystems to realize the given complex DSP algorithm. Sometimes, to solve the problems of computational resource analysis we need to know information about the work content variance. The procedure to define the work content variance is very cumbersome and we do not discuss it in this section.

7.3.3  Evaluation of Complex Digital Signal Reprocessing

Algorithm Work Content in Radar System

We continue consideration and discussion of the example on analysis of the digital signal reprocessing algorithm in CRSs started in Section 7.1. The initial conditions for analysis are given by the graph flowchart of this algorithm depicted in Figure 7.5. In our analysis, we use the following data:

•The average number of tracking targets Ntrgate = 80

•The average number of target tracks under detection NDtrue = 10

•The average number of started target tracks N‾beg = 5

•The average number of true target pips assigned as the initial points of new target tracks

Ninitialtrue = 5

Thus, the average number of target pips subjected to processing is equal to N‾Σ = 100. False target pips and miss of true target pips are not taken into consideration in this example. In accordance with the initial conditions, we can compute the following probabilities:

•Identification of new target pip among the target tracking trajectories, Ptr = 0.8.

•Identification of new target pip among the detected target track, PD = 0.1.

•Identification of new target pip among the begun target tracks, Pbeg = 0.05.

•The new target pip is assigned as the initial point of new target track, Pnew = 0.05

FIGURE 7.9  Example of graph transformation: (a) the graph of cycles different by rank, and (b) the resulting graph.