INTEGRATION OF RADAR PULSES

The relationship bctwec11 the sig11al-to-110isc ratio. the probability of detection, and the prnh- ahility of false alarm as given in Fig. 2.7 applies for a single gulseQnly. However, many pulses arc usually returned from any particular target (111cach radi~r scan and can he used to improve detection. The number of pulscs₁11n)rcturncd from a point targeJ as the radar antenna scans through its

~~~\~~-~j-~!!Jt

is ' /

ts,.\..

^{G •..}

'-fJs{fp .· Os fp

11a =

7!;i ^{= ~-~ ~~;;, (2.30)}

where On

=

antenna beamwidth, deg

J~

pulse repetition frequency, Hz

(}5 antenna scanning rate, deg/s w.,, antenna scan rate, rpm

Typical parameters for a ground-based search radar might be pulse repetition frequency 300 Hz, 1.5° beamwidth, and antenna scan rate 5 rpm (30°/s). These parameters result in 15 hits from a point target on eac:h scan. The process of summing all the radar echo pulses for the purpose of improving detection is called integration. MaRy techniques might be employed for accomplishing integration, as discussed in Secs.

10.7.

All practical integration techniques employ some sort of storage device. Perhaps the most common radar integration method is the cathode-ray-tube display combined with the integrating properties of the eye and brain of the radar operator. The discussion in this section is concerned primarily with integration performed hy electronic devices in which detection is made automatically on the basis of a threshold crossing.

Integration may be accomplished in the radar receiver either before the second detector (in the IF) or after the second detector (in the video). A definite distinction must be made hctween these two cases. Integration before the detector is called predetection, or coherent, integration, while integration after the detector is called postdetection, or noncoherent, integra- tion. Predetection integration requires that the phase of the echo signal be preserved if full benefit is to be obtained from the summing process. On the other hand, phase information is destroyed by the second detector; hence postdetection integration is not concerned with preserving RF phase. For this convenience, postdetection integration is not as efficient as predetection integration.

If

n

pulses, all of the same signal-to-noise ratio, were integrated by an ideal predetection integrator, the resultant, or integrated, signal-to-noise (power) ratio would be exactly n times that of a single pulse. If the same ^ff pulses were integrated by an ideal postdetection device, the resultant signal-to-noise ratio would be less than n times that of a single pulse. This loss in integration efficiency is caused by the nonlinear action of the second detector, which converts some of the signal energy to noise energy in the rectification process.

The comparison or predetection and postdetection integration may be briefly summarized by stating that although postdetection integration is not as efficient as pn.:detection integra- tion, it is easier to implement in most applications. Postdetection integration is therefore preforred, even though the integrated signal-to-noise ratio may not be as great. As mentioned in Sec. 10.6, an alert, trained operator viewing a properly designed cathode-ray tube display is a close approximation to the theoretical postdetection integrator.

The efficiency of postdetection integration relative to ideal predetection integration has been computed by Marcum¹⁰ when all pulses are or equal amplitude. The integration efficiency may be defined as follows:

(2.31) where n ⁼ number or pulses integrated

(S/N)i = value of signal-to-noise ratio of a single pulse required to produce given probability of detection (for n = 1)

(S/

N),, = value of signal-to-noise ratio per pulse required to produce same probability or detection when n pulses are integrated

The improvement in the signal-to-noise ratio when n pulses are integrated postdetection is nE1(n) and is the integration-improvement factor. It may also be thought or as the effective number of pulses integrated by the postdetection integrator. The improvement with ideal predetection integration would be equal to n. Examples of the postdetection integration- improvement factor

/i(n) =

nEi(n) are shown in Fig. 2.8a. These curves were derived from data given by Marcum. The integration loss is shown in Fig. 2.8b, where integration loss in decibels is defined as L;(n) = 10 log [l/E;(n)]. The integration-improvement factor (or the integration loss) is not a sensitive function of either the probability of detection or the probability of false alarm.

The parameter n₁ for the curves of Fig. 2.8 is the false-alarm ,wmber, as introduced by Marcum.¹⁰ It is equal to the reciprocal of the false-alarm probability Pr,. defined previously by Eqs. (2.24) and (2.25). Some authors, like Marcum, prefer to use the false-alarm numbi..:r instead of the false-alarm probability. On the average, there will he one false decision out of _{11 1} possible decisions within the false-alarm time

Tra.

Thus the average number of possible 1.kci- sions between false alarms is defined to be n _{1 .} If r is the pulse width, Tp the pulse repetition period, and/P

=

1/TP is the pulse repetition frequency, then the number of decisions n₁ in time

Tra

is equal to the number of range intervals per pulse period r, = Tp/r

=

^1/fp r times the number of pulse periods per second/p, times the false-alarm time

Tra.

Therefore, the number of possible decisions is

n

₁

=

Trafptf =

Tra/r.

Since r:::: 1/B, where

Bis

the bandwidth, the false- alarm number is n ₁

= Tra B = 1/ Pra.

Note that

Pra = 1/Tra

Bis the probability of false alarm assuming that independent deci- sions as to the presence or absence of a target are made B times a second. As the radar scans hy a target it receives n pulses. If these n pulses are integrated before a target decision is mad<.!, then there are 8/n p~ssible decisions per second. The false-alarm probability is thus II times as great. This does not mean that there will be more false alarms, since it is the rate of detection- decisions that is reduced rather than the average time between alarms. This is another reason the average false-alarm time

Tra

is a more significant parameter than the false-alarm probabi- lity. In this text,

Pra

will be taken as the reciprocal or

Tra

^{B = 11}_{1 ,} unless stated otherwise. Some authors ¹¹ prefer to define a false-alarm number

n'.r

that takes account the number of pulses integrated, such that

n 1 ^{= n}

¹

^/n.

Therefore, caution should be exercised when using d ilTerent authors' computations for the signal-to-noise ratio as a function of probability of detection

{a)

12 - - - --·-·· ---- - - t - - - - -

en 10 · - - - - -o _</1

-

VI 0

C 8 · - - - · --··----·-

·:;:: 0 0 L O'

<l)

+-

c 6 1 - - - -

---···---···-1-/t ::: 050

f/t ==0.90

~ =099

2•--- - 1 - - - · · · - t - - - ~ - - - ,

10 100 1,000 10,000

n = number of pulses ( b)

Fi~ure 2.8 (a) Integration-improvement factor. square law detector. P, = probability or detection.

n ₁ 117;. B = false alarm number,

Tr.

average time between false alarms, B = bandwidth; (b) integra- tion loss as a function of ^11, the number of pulses integrated. P_{4 ,} and n₁^. (After Marcum.to courtesy IRE Tra11s.)

and probability of false alarm, or false-alarm number, since there is no standardiLution of definitions. They all can give the correct values for .. use . .in the radar equation provided I he assumptions used by each author are understood.

The original fa)se-alarm time of Marcum ¹⁰ is different from that used in this text. He defined

it

as the time in which the probability is l/2 that a false alarm will not occur. A comparison

of the

two definitions

is

given by Hollis.¹² Marcum's definition

of

false-alarm time is seldom used, although his definition of false-alarm number is often found.

The solid straight line plotted in Fig. 2.8a represents a perfect predetection integrator with

E,(n)

1. It is hardly ever achieved in practice. When only a few pulses are integrated postdetection (large signal-to-noise ratio per pulse), Fig. 2.8a shows that the integration- improvement factor is not much different from a perfect predetection integrator. When a large number of pulses are integrated (small signal-to-noise ratio per pulse), the difference between postdetection and predetection integration is more pronounced.

The dashed straight line applies to an integration-improvement factor proportional to

n¹¹²• As discussed in Sec. 10.6, data obtained during World War II seemed to indicate that this described the performance of an operator viewing a cathode-ray tube display. More recent experiments, however, show that the operator-integration performance when viewing a properly designed

PPI

or B-scope display is better represented by the theoretical postdetec- tion integrator as given by Fig.

2.8,

rather than by the n¹¹² law.

The radar equation with n pulses integrated can be written

4 P,GAea

Rmax =

(4n)²

kT0Bnf

₁₁

(S/N)

₁₁ ^(2.32) where the parameters are the same as that of Eq. (2.7) except that

(S/N)

11 is the signal-to-noise ratio of one of the n equal pulses that are integrated to produce the required probability of detection for a specified probability of false alarm: To use this form of the radar equation it is necessary to have a set of curves like those of Fig. 2.7 for each vaiue of 11. Such curves are available, ¹¹ but are not necessary since only Figs.

2.7

and

2.8

are needed. Substituting

Eq. (2.31)

into

(2.32}

gives

4 P,GAeanE;(n)

Rmax

=

(411:)2kT0BnFn(S/N)1

^(2.33)

The value of

(S/N)

1 is found from Fig.

2.7

as before, and nE;(n) is found from Fig. 2.8(1.

The post-detection integration loss described by Fig.

2.8

assumes a perfect integraior.

Many practical integrators, however, have

a"

loss of memory" with time. That

is,

the ampli- tude of a signal stored in such an integrator decays, so that the stored pulses are not su mmc.:d with equal weight as assumed above. Practical analog integrators such as the recirculating delay line (also called a feedback integrator), the low-pass filter, and the electronic storage tube apply what is equivalent to an exponential weig.hting to the integrated pulses; that is, if ¹¹

pulses are integrated, the voltage out of the integrator is

N

V

= L Vi

exp [ - ( i - 1

)y]

(2.34)

l= l

where

Jti

is the voltage amplitude of the ith pulse and exp ( -y) is the attenuation per pulse. In a recirculating delay-line integrator,

e-..,

is the attenuation around the loop. ln an RC low-pass filter

y

=

Tp/RC,

where

Tp

is the pulse repetition period and

RC

is the filter time constant.

In order to find the signal-to .. noise ratio for a given probability of detection and probabi- lity or false alarm, an analysis similar to that used to obtain Figs. 2. 7 and 2.8 should he

repeated for each value of)' and 11.13 This is not done here. Instead, for simplicity, an efficiency will be defined which is the ratio of the average signal-to-noise ratio for the exponential integrator to the average signal-to-noise ratio for the uniform integrator. For a dumped integrator, one which erases the contents of the integrator after ^II pulses and starts over, the efficiency is ^{1 4}

tanh

(11y/2)

p=

11

tanh (1,/2)

(2.35a)

An example of an integrator that dumps is an electrostatic storage tube that is erased whenever it is read. The efficiency of an integrator that operates continuously without dumping is

[l - exp (-ny)]²

p

= -- - --· ---

" tanh (r/2)

(2.351,)

The maximum efficiency of a dumped integrator occurs for}'= 0, but for a continuous integra- tor the maximum efficiency occurs for II}'= 1.257.