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Design Principles of Digital Signal Processing Subsystems Employed

283

where

 

 

Stg =

1

(8.11)

τtg

 

 

is the information density equal to the average number of target movements within the radar cover-

–

age along the external border per unit time; τtg is the average time interval between two considered events (the target traverses). The flux with pdf of time intervals τtg given by (8.10) is called the Poisson flow. If additional requirements of stationarity, ordinariness, and absence of sequence are satisfied, then the Poisson flow is called the simplest Poisson flow [14–16]. The average number of targets in the radar coverage is given by

 

 

tg = Stg

 

 

(8.12)

N

 

 

trc ,

where – is the average time of target being within the limits of radar coverage. Furthermore, we

trc

assume that the targets are distributed uniformly within the limits of radar coverage with the same pdf per unit volume. At the input of a digital signal processing subsystem (the radar receiver) of a CRS, the spatial pattern of targets distribution within the limits of the radar coverage is transformed into the time sequence of signals adhering to process. Henceforth, this signal flux is considered as the simplest with the intensity

 

 

 

 

γ tg = N

tgTscan ,

(8.13)

where Tscan is the period of radar coverage scanning.

In many cases, the hypothesis about the simplest flux of requests at the digital signal processing subsystem does not correspond to parameters and characteristics of real incoming signal flux. Nevertheless, an implementation of this hypothesis is considered acceptable for designing radar systems by the following reasons:

•First, analysis of QS is very simple.

•Second, because the simplest incoming signal flux is a very hard problem for such QSs, the CRS designed based on the simplest flux representation ensures successful service of other possible signal fluxes with the same intensity.

Along with the target return signals, the interferences (the false signals) caused by the internal and external noise sources come in at the input digital signal processing subsystem as requests for service. This interference (the false signals) flux can also be considered as the simplest flux with definite assumptions.

The incoming flux request service is organized by the QS. The QS can be considered as both the device carrying out a direct request service and the device assigned to store a request queue adhering to service. The QS is characterized by (see Figure 8.8) the following:

•The number of devices (cells) assigned to store a request queue of the incoming signal flux (the input buffer accumulator (IBA) memory size)

•The number of devices (channels) K that can serve multiple requests simultaneously

•The number of devices (cells) that accumulate and store the digital signal processing results (the IBA memory size)

The request queue is stored by the microprocessor subsystem IBA. The waiting time and the number of queued requests (the request queue length) are the random variables depending on statistical

284

Signal Processing in Radar Systems

parameters of the incoming signal flux and service rate. Because the IBA size (memory capacity) is limited, the request queue length and, consequently, the waiting time of request queue are limited too. This QS is called the queuing system with the limited queue (the limited waiting time). Thus, the digital signal processing subsystem can be considered as the subsystem belonging to the class of QSs with limited queue.

The request queue in the digital signal processing subsystem of a CRS is realized by the microprocessors that can belong to a single or a set of parallel channels and is reduced to a single program realization of the corresponding digital signal processing algorithms. The time requested for each realization of the digital signal processing algorithm is the random variable owing to many reasons and, consequently, can be given statistically only. If we denote the time of a single request queue as

τqueue, then the sufficient characteristic of τqueue, as a random value, is the pdf p(τqueue). QSs with the exponential pdf of τqueue

p(τqueue ) = µ exp{−µτqueue}

(8.14)

play a specific role in the queuing theory, where

 

 

 

µ =

1

 

(8.15)

 

τqueue

 

is the queue intensity. The request queue approximation by the exponential pdf in the digital signal processing subsystem of a CRS allows us to carry out an analytical estimation of statistical characteristics of the request queue process during the simplest incoming signal flux using a sufficiently easy way.

If the request queue in the digital signal processing subsystem is carried out by K identical channels or microprocessors coupled with these channels and each channel or microprocessor has the exponential pdf of τqueue, then the pdf of τqueue for whole digital signal processing (microprocessor) subsystem is determined in the following form:

p(τqueue ) =

K

×

(τqueue )K −1

exp{−µK τqueue},

(8.16)

τqueue

(K − 1)!

 

 

 

 

where

τqueue =

1

(8.17)

µK

 

 

is the average time queue. Distribution law given by (8.16) is called the Erlang distribution (pdf). At K = 1, the Erlang distribution is transformed into exponential pdf. Other distribution laws that can be represented by some exponential functions with different decay rates are used too. Presentation of distribution laws based on combination principle of exponential components allows us to approximate the request queue by Markov process and to obtain an analytical solution of this problem. Unfortunately, in CRS digital signal processing subsystems, the pdf of τqueue differs from the exponential pdf and cannot be reduced to Erlang distribution law. For this rea-

son, the pdfs of τqueue must be analyzed for each specific case of designing the CRS digital signal processing subsystem.

Finally, let us discuss some observations about the flux forming at the digital signal processing subsystem output, that is, the output flux. At first, the output flux is stored by the output buffer

Design Principles of Digital Signal Processing Subsystems Employed

285

accumulator (OBA). Thereafter, the output flux with the given rate is transferred to the user. As a rule, the digital signal processing of target return signals is realized by a set of serial microprocessor subsystems. In the course of the digital signal processing, we need to make transformations of fluxes; for instance, the signal flux must be transformed into the coordinate flux or the detected target pip flux that must be transformed into the flux of target tracking trajectory parameters and so on. In doing so, in the course of the digital signal processing, the output flux of the previous stage is the input flux of the next stage.

In the considered case, the output fluxes at each stage of the digital signal processing, as well as the input fluxes, can be considered as the simplest fluxes and their intensities can be determined using the estimated parameters of target and noise situations within the limits of the radar coverage. For example, the flux density of true target pips at the digital signal preprocessing subsystem output is defined by the number of targets within the limits of the radar coverage, the probability of target detection, and the scanning rate. Analogously, the flux density of false target pips is defined by the noise environment within the limits of the radar coverage, the scanning rate, and so on. Knowledge about the output fluxes is required to establish the necessary IBA size (capacity) for designing the information buses to exchange information between the microprocessors in the microprocessor subsystem and the communication channels to transfer information to the user.

8.2.2  Functioning Analysis of Single-Microprocessor

Control Subsystem as Queuing System

Assume that the digital signal processing subsystem is realized based on a single-microprocessor subsystem (see Figure 8.9). We think that the request flux comes in at the input of the digital signal processing subsystem. The request flux is the multidimensional process consisting of M components corresponding to the given number of digital signal processing algorithms assigned to be realized

Z1 Z2

ZM

 

Interruption device

 

SR1

 

Q1

Q2

QN–1 QN

 

SR2

 

ROM1

ROMk

ROMl

 

MP and RAM

FIGURE 8.9  Single-microprocessor digital signal processing subsystem.

286

Signal Processing in Radar Systems

by a single-microprocessor subsystem. The requests z1, z2,…, zM come in at the input of interruption device included into the microprocessor structure. When the request zi comes in at the input of interruption device, this device interrupts the computation process and transfers control to the supervisory routine SR1 that organizes the receiving process and the request zi queue in accordance with assigned priority if this request zi does not belong to immediate processing. The queuing process is stored by the buffer accumulator (BA). In doing so, the groups of cells Q1, Q2,…, QN form regions where the requests will be stored with corresponding priority. Within the limits of each priority zone, the requests are written on the first come first ordering basis. After receiving and queuing, the control is transferred to the supervisory routine SR2 that organizes the digital signal processing of requests from queuing using the microprocessor.

The process of selecting a new request from a set of requests waiting in queue is the following. After the digital signal processing of immediate request, the supervisory routine SR2 investigates sequentially queuing Q1, Q2, …, QN and selects for service the request zk possessing the greatest priority. After initiation of the corresponding routine Rk, it is realized by the microprocessor. The queued request zk leaves the system and the control is transferred to the supervisory routine SR2 again. If there are no further requests, then the supervisory routine SR2 switches on the microprocessor in the idle mode. At each instance, the microprocessor can run a single program only. For this reason, the considered QS is called a single-channel or a single-microprocessor QS.

One of the important quality parameters of the microprocessor subsystem as a QS is the capacity factor that characterizes the time interval within the limits of which the QS (microprocessor) processes a request and simultaneously the probability that the QS works (no idle mode) at the present time. The input flux with the density λk can be determined in the following form:

ρk

=

λk

,

(8.18)

 

 

 

 

µk

 

where

 

 

 

 

 

µk =

 

 

1

 

(8.19)

 

τqueue k

is the intensity of the request queue of the kth flux. The total loading factor of the microprocessor subsystem by all M incoming flux is defined in the following form:

M

 

U = ∑ρk.

(8.20)

k=1

The steady stationary operation mode of QS is the mode where the probability characteristics of QS functioning are independent of time. The condition of existence of the steady mode is defined by the loading factor U < 1. The value = 1 − U is called the downtime ratio. In the steady QS functioning mode the down ratio is positive, that is, > 0. Logically, the value of facilitated by the microprocessor subsystem must be minimal. Operation quality of the microprocessor subsystem as the QS is defined by the time of request processing by the microprocessor subsystem defined from the instant of request coming in at the microprocessor subsystem until the time a service is concluded by the microprocessor

subsystem. For the kth request, this time consists of the waiting time for queuing twait k and the queuing time τqueue k:

τΣk =

 

wait k + τqueue k ,

(8.21)

t

Design Principles of Digital Signal Processing Subsystems Employed

287

–

–

–

 

where twait k and τqueue k are the average values. The waiting time twait k depends on the regulation

applied to the request queuing—the order of multidimensional flux request selection for queuing from the total number of requests. There are the following regulations for request queuing:

•Nonpriority request queuing: The queuing in the order of request coming in at the QS, that is, in other words, “who could come early, he is the first.”

•Relative priority request queuing: Priority for queuing is taken into consideration only at the instant to be served.

•Absolute priority: The incoming request with higher priority interrupts the queued request with lower priority.

During nonpriority request queuing, the average waiting time for all requests is the same and given by

 

 

M

1 + υ2k

 

 

 

 

wait = ∑

ρk τqueue k ,

(8.22)

 

t

 

2(1 − U)

 

 

k=1

 

 

 

 

 

 

 

where

 

 

 

 

 

 

σqueue k

(8.23)

 

 

υk

= τqueue k

 

 

 

is the variation factor defined by the ratio between the root mean square deviations of τqueue k to its

–

–

average value τqueue k. In accordance with (8.22), the value of twait will be minimum at the constant

value of τqueue k, that is, at υk =–0. In the case of exponential pdf of τqueue k, υk = 1 and, consequently,

there is a twofold increase in twait when υk = 0.

 

The widely used regulation in the information subsystem with the request queuing in CRSs is the regulation with the fixed relative priorities for each component of request flux. In this regulation, an appearance of the request with high priority does not interrupt the request queuing process with lower priority if it has been started. In this case, the average waiting time for the ith request inde­ pendently on distribution law of τqueue i is given by

M

twait i = ∑ k=1

where

Ui−1

1 + υ2k

ρk τqueue k ,

2(1 − Ui−1)(1 − Ui )

= ρ1 + ρ2 + + ρi−1

(8.24)

(8.25)

is the loading factor of the microprocessor subsystem created by signal fluxes with priorities higher than i; the higher priority, the lesser i;

Ui = ρ1 + ρ2 + + ρi

(8.26)

is the loading factor created by signal fluxes with priorities not lower than i. Analysis of (8.24) shows that with a decrease in priority the waiting time to start the request queuing increases monotonically. Comparing the waiting time of requests with the relative priorities with the waiting time for the nonpriority request queuing, we can see that an introduction of relative priorities leads to a

288

Signal Processing in Radar Systems

decrease in the waiting time for the request queuing with high priority owing to an increase in the waiting time for the request queuing with low priorities.

Level of total loading of the microprocessor subsystems with the QSs stimulates a waiting time for the request queuing with relative priorities. If the loading level of the microprocessor subsystems with the QSs is low, that is, U = 0.2–0.3, the presence of relative priorities for the request queuing has a very small effect on the waiting time between the requests of various signal fluxes. If the loading factor is close to unit, this effect becomes more essential and leads essentially to a reduction in the waiting time for the request queuing with high priorities owing to an increase in the waiting time for request queuing with low priority. In addition to the discussed regulations of requests queuing, in some cases, the regulations of requests queuing with absolute priorities are used for some components of the signal flux. Employing the request QSs with absolute priorities, the average waiting time is defined in the following form:

 

wait i =

Ui−1τqueue k

t

1 − Ui−1

 

 

 

 

1

 

M

 

+

 

 

∑(1 + υ2k )ρk τqueue k .

(8.27)

 

 

 

2(1

− Ui−1)(1

− Ui )

 

k=1

 

 

 

 

 

 

Evidently, an introduction of absolute priorities in the request QS leads to a reduction in the waiting time for the request queuing with high priorities, however with a simultaneous increase in the waiting time for the request queuing with low priorities.

Dependences illustrating the average waiting time of the request queuing as a function of priorities of the request queuing are shown in Figure 8.10. During the nonpriority request queuing (the curve 1, Figure 8.10), the average waiting time is constant. In the case of relative (the curve 2) and absolute (the curve 3) priorities, there takes place an increase in the waiting time for the high priority request queuing due to an increase in the waiting time for the low priority request queuing.

As a rule, we need to follow strong limitations with respect to the waiting time for request queuing of individual signal fluxes that require assigning them by absolute priorities, for example, the requests to process the target return signals. Other requests have excess waiting time and may be assigned with relative priorities. Several requests can be served by the simple queuing procedure. Thus, we see that we need to apply mixed regulations for request queuing, the investigation of which is carried out for each specific case by simulation procedures.

Referring to (8.21), we need to pay attention to the following fact. The average waiting time –

– twait k

is the constituent of the total request queuing time as well as the average request queuing time τqueue k. However, the request queuing time does not vary for different regulations of the request QS. For

this reason, the main time characteristic of the microprocessor subsystem operation is the average

twait

3

2

1

k

M = 1 M = 3

M = 6

M = 9

M = 12

FIGURE 8.10  The average request queuing waiting time versus the request queuing priorities.

Design Principles of Digital Signal Processing Subsystems Employed

289

waiting time –wait k. Other parameters and characteristics of the request QS will be introduced as t

needed.

8.2.3  Specifications for Effective Speed of Microprocessor Subsystem Operation

In the considered case, the initial data that define the specifications for effective speed of the microprocessor subsystem operation include the following:

•The digital signal processing algorithm represented in the form of flow graph for each of the signal or request flux constituents

•The work content of constituents of the complex computational process algorithm expressed by the average number and variance of number of operations carried out for a single realization of these algorithms

•The type of the request QS of the designed microprocessor subsystem

Representation of the complex computational process algorithms and determination of the work content of these algorithms are carried out by methods discussed in Chapter 7.

The request queuing type or the type of microprocessor subsystem is defined by time requirements to process the requests by request QS or, independent of the request queuing type or the type of microprocessor subsystem, by the acceptable waiting time of the request QS. Accordingly, we call the microprocessor subsystems of the first type such microprocessor subsystems in which there are no limitations on the request queuing waiting time for all constituents of incoming signal flux. These microprocessor subsystems require the effective microprocessor operation speed that satisfies the request queuing within the finite time interval limits, the limiting value of which is unlimited in the considered case.

The request queuing waiting time is finite if the microprocessor subsystem works in the stationary mode, that is, if the total loading factor U of the microprocessor subsystem is less than the unit. The condition of stationarity takes the following form:

 

 

M

 

 

M

 

 

 

 

 

∑ρk

= ∑λk τqueue k < 1.

(8.28)

 

 

k=1

 

 

k=1

 

 

 

 

–

 

 

 

 

 

 

–

and effective speed Veff of the

The average queuing duration τqueue k is related to the work content N 0k

microprocessor subsystem operation in the following form:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

τqueue k =

N

0k

,

k = 1, 2,…, M.

(8.29)

 

 

 

 

 

 

 

 

 

Veff

 

 

 

Substituting (8.29) into (8.28), we obtain

 

 

 

 

 

 

 

 

 

M

 

 

 

 

M

 

1

∑λk N

0k < 1

or Veff > ∑λk N

0k.

(8.30)

 

Veff

 

k=1

 

 

 

 

k=1

 

 

 

 

 

 

 

 

Equation 8.30 defines the minimum required speed of the microprocessor subsystem operation to establish the stationary operation mode of the microprocessor subsystem to realize the given set of digital signal processing algorithms in a CRS. If existing or assumed to be employed in the digital signal processing subsystems of CRS types of control single-microprocessor subsystems do not satisfy the main requirement given by (8.30), then the computation can be provided by

290

Signal Processing in Radar Systems

the multimicroprocessor subsystems only. In this case, the parallel problem of computational process arises and we need to design the corresponding tools and facilities of parallel digital signal processing.

If the control single-microprocessor subsystem can satisfy the specifications and requirements for the effective speed of operation of the request QSs of the first type, we should study the possibilities of organizing the request QSs of the second and third types employing this control singlemicroprocessor systems. We call the control single-microprocessor subsystem of the second class subsystem if there are limitations on the average request queuing waiting time for all or several constituents of signal flux. The limitations are given in the following form:

 

wait k ≤ tk , k = 1, 2,…, L, L ≤ M,

(8.31)

t

where tk is the limiting value of the average request-QS time for requests of the kth incoming signal flux.

Now, let us define the problem of the effective operation speed determination in the control single-microprocessor subsystem providing the request queuing with the given limitations and taking into consideration the fact that, similar to the request QSs of the first type, all requests are queued, with the so-called request QSs without loss and the average request queuing waiting time remaining limited. In the considered case, the minimal required speed of operation depends strongly on the request queuing regulation. If, for example, there is a nonpriority request QS, in which the average waiting time for all requests is the same and given by (8.22), the required effective speed of operation of the control single-microprocessor subsystem of the second type is determined by the following inequality:

 

 

 

 

 

∑M λk τqueue k < 1(1+ υ2k )

≤ t .

 

 

 

 

 

 

 

k=1

 

 

 

 

(8.32)

 

 

 

 

 

2{1 − ∑kM=1

λk τqueue k }

Solution of (8.32) can be presented in the following form:

 

 

 

 

 

 

M

 

M

 

 

2

M

 

 

1

 

 

 

 

 

 

 

 

 

 

 

 

 

Veff ≥

∑λk N0k +

0.25

 

 

 

 

(8.33)

2

∑

λk N0k + (t*)−1∑λk N0k (1 + υ2k ).

 

k=1

 

 

 

 

 

k=1

 

 

 

 

k=1

 

 

 

 

Based on (8.33), we are able to define the required speed of operation of the control single-micro- processor subsystem by the given (the same for the request queuing of all signal flux constituents) limitation on the average request QS waiting time during the nonpriority request queuing at the time of signal flux constituents coming in at the single-microprocessor subsystem input. In the limiting case, when there are no limitations in waiting time (t* → ∞), (8.33) transforms into the following form:

M

 

Veff ≥ ∑λ k N0k ,

(8.34)

k=1

that is, it coincides with (8.30). At t* < ∞, the required effective speed of operation of the second-type control single-microprocessor subsystem is greater than that of the control single-microprocessor subsystem of the first type.

> Tk

Design Principles of Digital Signal Processing Subsystems Employed

291

To organize the priority request queuing, for example, the relative priority, we need to take into consideration a difference in limitations applied to the average request QS waiting time of each priority in the second type of request QS. In principle, this problem can be solved by the following way. All requests are divided into groups with the same value of average waiting time. The loading factor and required effective speed of operation Veff of the control single-microprocessor subsystem is defined for each request group. Making summation of partial values of the effective speed of operation, we obtain

P

 

Veff = ∑Veff p,

(8.35)

p=1

where P is the number of request groups with approximately the same acceptable average waiting time of request queuing. In the considered case of the second-type request QS, as well as in the case of the first-type request QS, a decision to realize the control single-microprocessor subsystem is made as a result of comparison of requirements and specifications to effective speed of operations of the request QS with effective speed of the CMP operation. The third type of request QS is called the microprocessor subsystem that generates limitations on acceptable (absolute) time to process the request by the request QS. In these cases, the limitations are given in the following form:

P ( twait k > Tk ) ≤ P ,

(8.36)

where

Tk is the allowed waiting time for requests of the kth signal flux P* is the allowable probability to satisfy the inequality twait k

The request queuing waiting time probability distribution function of the total signal flux can be approximated with satisfactory accuracy by the following formula [17–19]:

 

Rτ

 

P(twait ≤ τ) = 1 − R exp −

 

 

 

.

(8.37)

 

 

 

 

twait

 

In this case, the probability of exceeding the allowed waiting time Tk can be determined in the following form:

P(twait > T ) = 1 − R exp

 

RT

 

−

 

 

 

.

(8.38)

 

 

 

 

 

 

 

 

 

twait

 

 

 

 

 

 

 

 

 

 

 

–

For example, in the case of the nonpriority request queuing regulation, for which twait is given by

(8.22), based on solution of the transcendent equation we can write

 

 

RT

= P

 

(8.39)

1 − R exp −

 

 

 

 

 

 

t

wait

 

 

 

 

 

 

We can obtain the required effective speed of operation of the control third-type single-micropro- cessor subsystem. Mathematical notation of (8.39) with respect to Veff is tedious, and solution can be obtained only by numerical procedures.

292

Signal Processing in Radar Systems

The required effective speed down boundary definition of the third-type control single-microproces- sor subsystem operation can be considered as the initial stage of designing the microprocessor subsystem for CRS digital signal processing. The next stage is a choice of the optimal request queuing regulation. At the system design stage, we will not have sufficient information to choose the specific­ request queuing regulation, but we should take into consideration the following circumstances:

•The request queuing regulation based on request queuing in turn to come in at the control single-microprocessor subsystem possesses the best performance among all the nonpriority request queuing regulations; in particular, the variance of the request queuing waiting time is minimal compared to other nonpriority request queuing regulations.

•Introduction of the priority request queuing regulation allows us to decrease essentially the request queuing waiting time for the most important requests of signal flux owing to increase in the delay during the queuing of signal minor fluxes.

•In general, an introduction of the priority request queuing regulations provides an equivalent performance of the control single-microprocessor subsystem compared to the request queuing one in turn. Consequently, the determined values of the required minimum speed of the control single-microprocessor subsystem operation makes it possible to choose the priority request queuing regulation without any additional increase in the effective speed of operation.

Now, let us discuss some ways of choosing the optimal speed of operation of the control singlemicroprocessor subsystem. The minimum value of the effective speed of operation ensures the boundary values of the request queuing waiting time of the earlier-considered types of the request queuing regulations for the control single-microprocessor subsystem and related to a queue length

that can be defined for the kth signal flux as lk = λktwait k. Queue existence and its storage are related to definite losses. To decrease these losses, the effective speed of control single-microprocessor

subsystem operation must be greater than the minimal one. However, in this case, the loading factor U decreases and, correspondingly, the idle time factor Q = 1 − U increases, which is not good from the point of view of hardware costs.

Evidently, we can select the optimal speed of operation taking into consideration these two contradictory factors. As the criterion of effectiveness we use the functional in the following form:

M

 

CV = βQQ(Veff ) + ∑βkλ k twait k (Veff ),

(8.40)

k=1

where

βQ is the loss caused by idle mode

βk, k = 1, 2,…, M are the losses caused by the queue length for each signal flux request queuing

The optimal value of the effective speed of control single-microprocessor subsystem operation can be obtained from solution of the following equation:

dCV = 0 at

d2CV

> 0.

(8.41)

dVeff2

dVeff

 

 

Solution of (8.41) is not difficult, but a choice of losses βQ and βk is difficult. A single acceptable procedure for this choice is the procedure of expert judgments.

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