Pd IMPROVEMENT TECHNIQUESIn the previous discussions we derived equations for detection

probability ((106), (109), (110)) and showed that the use of a matched filter will provide the maximum SNR and that can be obtained for a given set of radar parameters and a given, single transmitted pulse. We term the resultant SNRs and s single pulse SNR and . We now want to address the gains in that we can obtain by using multiple transmit pulses. We will examine three techniques

1. Coherent Integration2. Non-coherent Integration3. Cumulative Probability

COHERENT INTEGRATIONWith coherent integration we insert a coherent integrator, or signal

processor, between the matched filter and amplitude detector as shown in Figure 14. This signal processor adds returns (thus the word integrator) from

pulses. After it accumulates the pulse sum it performs the amplitude detection and threshold check. In practice the process of forming the -pulse sum is somewhat complex. In one implementation, the signal processor samples the return from each transmit pulse at a spacing equal to the range resolution of the radar. Thus, for example, if we were interested in a range window from 5 to 80 Km and had a range resolution of 150 m the signal processor would form 75,000/150 or 500 samples for each pulse. The signal processor would then accumulate (add) each of the 500 samples in 500 summers. After the signal processor has summed the first pulses it would begin dropping older pulses off of the accumulator as new pulses arrive. Thus, the signal processor will add the returns from the most recent

pulses. In analog processors the integration (summation, accumulation) is accomplished by filters. It is accomplished by FFTs or other digital signal processors in digital signal processors.

As we did in our previous studies we will separately consider the signal and noise. For the signal, we assume that the amplitude of the signal on pulse is given by (note, we assume that we are looking at the specific range cell – out of 500 for the previous example – that contains the target return)

©2005 M. C. Budge, Jr

Figure 14 – Location of the Coherent Integrator

37

. (172)

The formulation of in (172) carries several assumptions about the target. Specifically, it implies that the amplitude and phase of the signal returned from the target is constant, at least over the pulses that are to be summed. This means that we are assuming that the target is SW0/SW5, SW1 or SW3. It does not admit SW2 or SW4 targets. As we will show later, coherent integration offers no benefit for SW2 and SW4 targets.

The second assumption is that there is no Doppler on the target return. If the target is moving, and thus will have a Doppler frequency, this Doppler frequency must be removed by the signal processor before the summation takes place. In digital signal processors that use FFTs, Doppler removal is accomplished by the FFT.

If we sum over pulses the output of the summer will be

. (173)

If the signal power at the input to the summer is

(174)

the signal power at the output of the summer will be

. (175)

We can write the noise at the input to the coherent integrator as

. (176)

If we sum over pulses, the noise at the output of the summer will be

. (177)

The noise power at the output of the summer will be

. (178)

We now need to compute the two terms on the right of (178). We write

. (179)

Since is WSS and zero-mean

©2005 M. C. Budge, Jr 38

. (180)

We also assume that the noise samples are uncorrelated from pulse to pulse. This means that and are uncorrelated . Since and

are also zero-mean we get

. (181)

If we use (180) and (181) in (179) we get

(182)

where is the noise power at the output of the matched filter.By similar reasoning we have that

(183)

and, from (178)

. (184)

If we combine (175) and (184) we find that the SNR at the output of the coherent integrator is

(185)

where is the SNR at the output of the matched filter; or the SNR given by the radar range equation. Thus, we conclude that the coherent integrator provides a factor of gain in SNR, where is the number of pulses integrated.

In the above development we have made the assumption that the signal level (in voltage and power) at the input to the coherent integrator was constant from pulse to pulse. This is indicative of a SW0/SW5 target, a SW1 target or a SW3 target. The specific amplitude over the pulses integrated by the coherent integrator is governed by the probability density function for the specific target type. What this says is that we can essentially consider the output of the coherent integrator as the return from a single pulse whose SNR is times the SNR provided by the radar range equation.

The noise on each pulse is zero-mean and Gaussian. The noise samples from each pulse are uncorrelated (and thus independent since they are Gaussian) and the I and Q samples are also Gaussian. Thus the noise out of the coherent integrator has the same statistical properties of the noise out of the matched filter.

©2005 M. C. Budge, Jr 39

The consequence of the above discussions is that, when computing and for the case where the radar uses a coherent integrator (and for SW0/SW5, SW1 or SW3 targets) we use the same and equations we used before; that is (105), (106), (109) and (110). However, in place of we use where is the number of pulses that are coherently integrated and is the SNR at the output of the matched filter.

If the target is SW2 or SW4, coherent integration of no help. This stems from the fact that, for SW2 and SW4 targets, the signal is not constant from pulse to pulse but, instead, behaves like noise. This means that we must treat the target signal the same as we do noise. Thus in place of (173) we would write

. (186)

If we follow the procedure we use for the noise case we would have that

(187)

and

. (188)

This, with (184) would lead to the result

. (189)

In other words, the SNR at the coherent integrator output would be the same as the SNR at the matched filter output and the coherent integrator would offer no integration gain. For the SW4 case there is an additional complication that indicates that we may not be able to easily determine the density function of the signal-plus-noise at the output of the coherent integrator (we can determine that the density function will be Gaussian for the SW2 target). This could further complicate the computation of . The summary consequence for SW2 and SW4 type targets (targets whose RCS changes from pulse to pulse) is that coherent integration offers no benefits in terms of increasing .

In the above development we made some ideal assumptions concerning the target based on the fact that we were collecting, and summing, returns from a sequence of pulses. In particular we assumed that the target amplitude was constant from pulse to pulse. Further, we assumed that we sampled the output of the matched filter at its peak. In practice neither of these is strictly true. First, we really can’t expect to sample the matched filter output at the peak of the target return. Because of this, the SNR in (185) will not be the SNR at the matched filter output (the SNR given by the radar range equation). It will be some smaller value. We usually

©2005 M. C. Budge, Jr 40

account for this by degrading SNR by a factor we call range straddling loss. If the sample period is the pulse width, the range straddling loss is usually taken to be 3 dB.

There are other reasons that the signal into the coherent integrator will vary. One is target motion. This will create a Doppler frequency which will cause amplitude variations from pulse to pulse. It the Doppler frequency is large enough so as to cause large amplitude variations the gain of the coherent integrator will be nullified. In general, if the Doppler frequency is greater than about the coherent integration gain will be nullified. In fact, the coherent integration will most likely result an a SNR reduction. Doppler frequency offsets can be circumvented by using banks of coherent integrators that are tuned to different Doppler frequencies.

Another degradation that is related to Doppler is termed range gate walk. Because of Doppler, the target signal will move relative to the location of the various samples that are fed to the coherent integrator. This means that, over the pulses, the signal amplitude will change. As indicated above, this could result in a degradation of SNR at the output of the coherent integrator. In practical radars, designers take steps to avoid range walk by not integrating too many pulses. Unavoidable range walk is usually accounted for by including a small (less than 1 dB) SNR degradation (SNR loss).

Still another factor that causes the signal amplitude to vary is the fact that the coherent integration may take place while the radar scans its beam across the target. The scanning beam will cause the and terms in the radar range equation to vary across the pulses that are coherently integrated. As before, this will degrade the SNR and its effects are included in what is termed a beam scanning loss. This loss, or degradation, is usuall 1 to 3 dB in a well designed radar.

Phased array radars have a similar problem. For phase array radars the beam doesn’t move continuously (in most cases) but in discrete steps. This means that the phased array radar may not point the beam directly at the target. This means, in turn, that the and of the radar range equation will not be their maximum values. As with the other cases, this phenomena is accommodated through the inclusion of a loss term called, in this case, beam shape loss. Typical values of beam shape loss are 1 to 3 dB.

NON-COHERENT INTEGRATIONWe now want to discuss non-coherent, or post-detection, integration.

The name post-detection integration derives from the fact that the integrator, or summer, is placed after the amplitude or square law detector as shown in Figure 15. The name non-coherent integration derives from the fact that, since the signal has undergone amplitude or square law detection, the phase information is lost. The non-coherent integrator operates in the same fashion as the coherent integrator (see the discussion at the beginning of the previous section) in that it sums the returns from pulses before performing the threshold check.

©2005 M. C. Budge, Jr 41

A non-coherent integrator can be implemented in several ways. In older radars it was implemented via the persistence on displays plus the integrating capability of a human operator. These types of non-coherent integrators are very difficult to analyze and will not be considered in this course. The reader is referred to the text and the Radar Handbook by Skolnik.

A second implementation is termed an m-of-n detector and uses more of a logic circuit rather than a device that integrates. Simply stated, the radar examines the output of the threshold device for n pulses. If a DETECT is declared on any m of those n pulses the radar declares a target detection. This type of implementation is also termed a dual threshold detector. Again, we will not discuss this type of non-coherent integrator in this class. Its analysis is reasonably straight forward for SW0/SW5, SW1 and SW3 targets but becomes somewhat more difficult for SW2 and SW4 targets.

The third type of non-coherent detector is implemented as a summer or integrator. In older radars low-pass filters were used to implement them. In newer radars they are implemented in special purpose hardware or the radar computer as digital summers. They operate as described in the beginning of the discussion of coherent integrators.

For SW0/SW5, SW1 and SW3 targets the main advantage of a non-coherent integrator over a coherent integrator is hardware simplicity. As indicated in earlier discussions, coherent integrators must contend with the effects of target Doppler. In terms of hardware implementation this usually translates to increased complexity of the coherent integrator. Specifically, it is usually necessary to implement a bank of coherent integrators that are tuned to various ranges of Doppler frequencies. Because of this one will need a number of integrators or summers equal to the number of range cells in the search window multiplied by the number of Doppler bands needed to cover the Doppler frequency range of interest. Although not directly stated earlier, this will also require a larger number of amplitude (or square law) detectors and threshold devices.

Since the non-coherent integrator is placed after the amplitude detector, it does not need to accommodate multiple Doppler frequency bands. This lies in the fact that the amplitude detection process recovers the signal (plus noise) amplitude without regard to phase (i.e. Doppler). Because of this, number of integrators is reduced; it is equal to the number of range cells in the search window.

It will be recalled that coherent integration offers no improvement in SNR for SW2 or SW4 targets. If fact, it can degrade SNR relative to that that can be obtained from a single pulse. In contrast, non-coherent integration

©2005 M. C. Budge, Jr

Figure 15 – Location of the Non -coherent Integrator

42

can offer significant improvement in SNR relative to a single pulse. In fact, contrary to most people’s intuition, non-coherent integration can offer SNR improvement factors that are greater than the number of pulses integrated! It is interesting to note that some radar designers are using various schemes, such as frequency hopping, to force targets to exhibit SW2 or SW4 characteristics so as to exploit the significant SNR improvement offered by non-coherent integration.

Analysis of non-coherent integrators is much more complicated that analysis of coherent integrators because the integration takes place after the non-linear process of amplitude or square law detection. From our previous work we note that the density functions of the noise and signal-plus-noise are somewhat complex. More importantly, they are not Gaussian. This means that when we sum the outputs from successive pulses we cannot conclude that the density function of the sum of signals will be Gaussian (as we can if the density function of each term in the sum was Gaussian). In fact, the density functions become very complex. This has the further ramification that the computation of and become very complicated. Analysts such as DiFranco and Rubin, Marcum, Swerling, and Mayer and Meyer have devoted considerable energy to analyzing non-coherent integrators and documenting the results of these analyses. We will not attempt to duplicate the analyses here. Instead, we present the results of their labor. The equations below are those that one could use to compute , and for each of the five types of Swerling targets.

The equation for at the output of the -pulse non-coherent integrator is

. (190)

Because of arithmetic problems it is often better to use the natural log of (190) or

(191)

As with the single pulse case, one normally specifies and then use (190) or (191) to compute to use in the equations below.

The equation for the five target types we have studied areSW0/SW5:

(192)

SW1:

©2005 M. C. Budge, Jr 43

(193)

SW2:(194)

SW3:

(195)

SW4:

(196)

In the above is the single pulse, SW0/SW5 detection probability equation defined earlier (Equation (106)). is the modified Bessel function of the first kind and order r, and,

(197)

is the incomplete gamma function. The values in the above equations are the single-pulse SNR values defined by the radar range equation. Also, the above equations are based on the assumption that the amplitude detector is a square law detector. It turns out that they also apply well to the case where the amplitude detector is a magnitude detector.

In many applications it is cumbersome to implement the above and equations. As a result of this we often resort to graphical techniques and

rules-of-thumb developed from the graphical techniques.

©2005 M. C. Budge, Jr 44

An example of a figure we use for the graphical technique is shown in Figure 16. This figure was taken from an earlier edition of your text and is a

©2005 M. C. Budge, Jr

Figure 16 – Curves Used for Graphical Analysis of Non-Coherent Integration

45

plot if improvement factor, , of the non-coherent integrator versus number of pulses integrated, . (Note that this figure uses instead of the

we have been using.). To relate to our previous work, we would set

(198)

for a coherent integrator. Thus, is the effective increase in SNR, in dB, afforded by an -pulse non-coherent integrator and the type of target specified.

It will be noted that the lower right of the figure has the notation . This means that the curves apply to a of

. (199)

It turns out that the curves also apply to other values of within the range we normally encounter in practical radar applications.

To use the curves one would use the following procedure Decide on a desired , target type and number of pulses

integrated, and an estimated .

Use the appropriate curve to compute the appropriate .

Increase the single-pulse SNR of the radar range equation by .

Use the resulting increased SNR and desired in (105) and (106), (109) or (110) as appropriate to compute the actual . If the actual differs from the estimated by a significant amount it may be necessary to estimate a new and repeat the process.

ExampleAs an example of how one might compute the effects of coherent and

non-coherent integration we will consider a practical example. The radar in this example employs a continuously rotating antenna that completes a rotation in seconds. The radar operates with a fixed PRI of seconds. Finally, the antenna has an azimuth beamwidth of degrees. We will not be directly concerned with the specific elevation beamwidth. We will state that the elevation beamwidth and target elevation position are such that the antenna beam will be roughly centered on the target, in elevation, as the beam scans by it.

We want to incorporate a coherent or non-coheent integrator that will integrate target returns as the beam scans by the target. In order to

©2005 M. C. Budge, Jr 46

determind the integration gain offered by the integrators we need to determine the number of target return pulses the radar will receive as the beam scans by the target.

Since the antenna moves 360º in seconds the antenna angular rate is

. (200)

The reciprocal of the angular rate is

. (201)

We are interested in the time it takes the beam to move an azimuth beamwidht, or degrees. We denote this and write

. (202)

is often called the dwell time or the time on target and is the length of time that the antenna beam is “looking at the target”.

As the beam scans by the target the radar will receive a target return pulse every PRI, or seconds. Thus, the number of target return pulses received during will be

. (203)

is the number of pulses that can be coherently or non-coherently integrated.

As a specific example we consider a radar that has a scan period of , an azimuth beamwidth of and a PRI of .

With this we get

(204)

If we were to coherently integrate the 3 pulses we would get a coherent integration gain of 3, or about 4.8 dB (this assumes we have a SW0/SW5, SW1 or SW3 target). For the non-coherent integration gain we would use the curves of Figure 16. For a SW0/SW5, SW1 or SW3 target and a desired of 0.9, the non-coherent integration gain would be about 4 dB. If the target was SW2 (and the desired was 0.9, the integration gain wild be about 10 dB. For a SW4 target the non-coherent integration gain would be about 8 dB.

We point out that the radar will not realize all of the integration gain specified above. The reason for this is that not all of the pulses are at the

©2005 M. C. Budge, Jr 47

peak SNR predicted by the RRD. Over the beamwidth, the SNR can vary by 3 dB. To account for this we incorporate a loss term of 1.6 dB. This would reduce the effective integration gain by 1.6 dB. If we account for the tact that the target may, in fact, not be centered in the elevation beam we would need to incorporate another loss of 1.6 dB for a total loss of 3.2 dB. Thus, the coherent gain above would be, effectively, 1.6 dB. The non-coherent gain for the SW0/SW5/SW1/SW3 target would be 0.8 dB and the integration gains for the SW2 and SW4 targets would be, respectively, 6.8 and 4.8 dB.

If the radar uses a phased array antenna we would compute somewhat differently. With phased array antennas the beam (usually) doesn’t move continuously. Instead, it moves in steps, dwelling at particular beam positions for seconds. In this case the number of pulses that could be integrated would equal the number of pulses per dwell, or

. (205)

An interesting observation of the above example is that for the SW2 case the non-coherent integration gain is 10 w/w and for the SW4 case it is 6.3 w/w. Both if these non-coherent integration gains are greater than the number of pulses. This is an interesting feature of non-coherent integration gains for SW2 and SW4 targets. That is, the non-coherent integration varies with where m is a number that can be significantly larger than unity. In fact, for SW2 type targets m varies between about 0.8 and 2.5 depending upon and . For SW4 targets it varies between about 0.75 and 1.7.

Sometimes it is convenient to have a rule of thumb for computing non-coherent integration gain, instead of having to rely on figures like Figure 16. A good rule of thumb for SW0/SW5, SW1 and SW3 targets is that the integration gain is equal to .

CUMULATIVE DETECTION PROBABILITYThe third technique we examine for increasing detection probability is

through the use of multiple detection attempts. Although we group it with integration techniques, we don’t analyze it in terms of its effect on SNR. We focus more on its effects in terms of , and . The premise behind the multiple detection attempt approach is that if we attempt to detect the target several times, we will increase the overall detection probability. We can formally state the multiple detection problem as follows.

If we check for threshold crossing several on several occasions, what is the probability that the signal-plus-noise voltage for a target will cross the threshold at least once. Thus, suppose, for example, we check for a threshold crossing on 3 occasions. We want to determine the probability of a threshold crossing on any 1, 2 or 3 of the occasions.

To compute the appropriate probabilities we must use probability theory.

©2005 M. C. Budge, Jr 48

To develop the technique we start by considering the events of the occurance of a threshold crossing on two occasions. We denote these two events as

- Threshold crossing on occasion 1

- Threshold crossing on occasion 2.If we form the event

(206)where denotes the union then is the event consisting of a threshold crossing on occasion 1, or occasion 2, or occasions 1 and 2. Since is the event of interest to us we want to find the probability that it will occur. That is we want

. (207)From probability theory we can write

(208)

where represents the intersection of and and is the event consisting of a threshold crossing on occasion 1 and 2. The first two probability terms on the right side, and are computed using (106), (109), (110), (192), (193), (194), (195) or (196) depending upon the target type and whether or not non-coherent integration is used.

To compute the third term, , we need to make an assumption about the events and . Specifically, we must assume that they are independent. This, in turn places a limitation on when we can use cumulative detection concepts. Specifically, we should only use cumulative detection concepts on a scan-to-scan basis. If we do, we are assured that the detection events on successive scans are independent. If we try to use cumulative detection concepts on a pulse-to-pulse basis we must be careful. For a SW0/SW5, signal-plus-noise will be independent on a pulse-to-pulse basis because the “randomness” is due solely to noise, which is indeed independent on a pulse-to-pulse basis. Signal-plus-noise will also be independent on a pulse-to-pulse basis for SW2 and SW4 targets since both the signal and noise are independent on a pulse-to-pulse basis (by definition). Signal-plus-noise will not be independent on a pulse-to-pulse basis for SW1 and SW3 targets. This stems from the fact that the signal is random but is correlated on a pulse-to-pulse basis. Thus, the signal-plus-noise will be correlated on a pulse-to-pulse basis. Since the signal-plus-noise is correlated on a pulse-to-pulse basis it can’t, by definition, be independent on a pulse-to-pulse basis. Further, since the events and depend upon signal-plus-noise, they are also not independent on a pulse-to-pulse basis.

If and are independent we can write

(209)

©2005 M. C. Budge, Jr 49

and. (210)

As an example, suppose . Using (210) we would obtain

. (211)

While (210) is reasonably easy to implement for two events, its extension to many events is difficult. In order to set the stage for such an extension we consider a different means of determining . We begin by observing that

(212)

where is the universe and is the complement of . contains all elements that are in but not in . By the definition of we note that and are mutually exclusive. We also note that . With this we get

(213)

and

. (214)

To proceed with the derivation we let(215)

and

. (216)

From (213) we get

(217)

By DeMogan’s Law we can write

(218)

and

. (219)

Now, since and are independent so are and . If we use this along with (215) we can write

. (220)

©2005 M. C. Budge, Jr 50

Finally, making use of (214) we obtain

(221)

We can now generalize (221) to any number of events. Specifically, if(222)

where are independent then

(223)

As an example of the use of (221) or (223) we consider the previous example wherein . With this we get

(224)

We now want to restate (224) in terminology more directly related to detection probability. To that end we write

(225)

where is the cumulative detection probability over pulses and is the detection probability on the kth scan.

In addition to increasing detection probability, the use of cumulative detection techniques also increases false alarm probability. In fact, if we consider the false alarm case we can express (225) as

(226)

In the case of false alarm probability we usually have that and write

(227)

If we further recall that we can write

. (228)Equation (228) tells us that when we use cumulative detection concepts we should compute the individual ’s using where is the desired false alarm probability.

©2005 M. C. Budge, Jr 51

As a rule of thumb, you should not invoke cumulative detection concepts in a fashion that allows any to be such that the SNR per scan falls below 10 to 13 dB. If the SNR is below 10 to 13 dB the radar will likely not be able to establish track on a detection. If this is stated in terms of , you should never let any to fall below about 0.5.

RADAR LOSSESFor our last radar range equation-related topic we want to address

radar losses. In our previous discussions we have indicated various situations that cause radar losses. We now want to summarize some of these. We will not address all possible loss terms since their number can be very large. See the text and Skolnik’s Radar Handbook for a more complete discussion of losses

Transmit Losses – Typically associated with the waveguides and other components between the power amplifier and the antenna. These are typically 1 to 2 dB in a well-designed radar.

Receive Losses – Typically associated with the waveguides and other components between the antenna and RF amplifier. These are also typically 1 to 2 dB for a well-designed radar. If the noise figure is referenced to the antenna terminals, receive losses are included in the noise figure.

Atmospheric Losses – These are losses due to atmospheric absorption by the atmosphere. They are dependent upon the radar operating frequency, the range to the target and the elevation angle of the target relative to the radar. Your author has graphs depicting atmospheric losses. So does Skolnik’s Radar Handbook.

Scanning or Beamshape Loss – This loss term accounts for the fact that, as the beam scans across the target, the signal amplitudes of the pulses coherently or non-coherently integrated varies. Because of the, the full integration gain of the integrator can’t be realized. From the Skolnik Radar Handbook typical values areo 1.6 dB for a scanning, fan beam radaro 3.2 dB for a thinner beam, scanning radaro 3.2 dB for a phased array radar wherein the beams of a search

sector overlap at the 3-dB beam positions. Range-Gate Straddling Loss – If the radar samples in range at a

rate of once per range resolution cell the loss is usually taken to be 3 dB.

Doppler Straddling Loss – The loss associated with forming the Doppler dimension of a range-Doppler map. Its particular value depends upon the specific Doppler processor implementation but typical values are 1 to 2 dB.

©2005 M. C. Budge, Jr 52

Collapsing Loss – If the coherent or non-coherent integrator integrates only noise over some if its integration time (due to the fact that the beam has moved fairly far off of the target) the radar will incur a loss that is given by

where is the number of pulses containing signal-plus-noise and is the number of pulses containing only noise.

Signal Processing Loss – If the radar uses an MTI with a staggered PRF waveform, and a good MTI and PRF stagger design, it will suffer 1 to 2 dB signal processing loss.

Miscellaneous Loss – Radar designers and analysts usually include an additional 1 to 2 dB loss to account for various factors they forgot to consider.

As an example of the above losses we will consider the 13 dB loss from Homework 4.

Transmit Losses – We will be conservative an assume a transmit loss of 2 dB

Receive Losses – Since the noise figure was large at 8 dB we will assume that the receive losses are included in the noise figure.

Atmospheric Losses – We will be conservative on the atmospheric losses and consider the losses at 0º elevation. From Figure 8.18a of your text (page 523) the losses at about 75 Km and a frequency of 5 GHz are about 2.2 dB.

Scanning or Beamshape Loss – Our elevation beamwidth of 5º is on the border of what we could call a fan beam. Therefore, we will be conservative and not consider it a fan beam. In this case, we will include scanning losses of 3.2 dB.

Range-Gate Straddling Loss – We will include the standard range-gate straddling loss of 3 dB.

Doppler Straddling Loss – The radar we have specified thus far only uses single pulse processing with a narrow pulse. Therefore, we will ignore Doppler straddling loss.

Collapsing Loss – In Homework 10 we limited ourselves to non-coherently integrating only the number of pulses we expect in a beam. Thus, we can assume no collapsing loss.

Signal Processing Loss – Again, since we have assumed only single pulse processing, we will have no signal processing loss

Miscellaneous Loss – We will assume a miscellaneous loss factor of 1 dB.

The losses above are summarized in Table 1. As can be seen, the total losses sum to 11.4 dB instead of the 13 dB we assumed. We will use the reduction in losses to help us reduce the peak transmit power.

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Table 1 – Loss Budget for Homework 4

This completes our study of the radar range equation. As was indicated at the start of this class, the radar range equation is an equation that appears to be very simple but, in fact, is very complex. In closing this part of the course we will summarize some of the observations associated with the radar range equation

RADAR RANGE EQUATION THOUGHTSNow that we have completed our study of the radar range equation

(RRE) and detection we want to revisit these topics to discuss the various terms of the RRE and review some of the conditions associated with the use of the RRE and detection theory. For our purposes we want to write the RRE as

.

We want to review the properties, characteristics, etc. of the various terms in the RRE.

is termed the peak power and is really the average power of the pulse during the time the pulse is present. A related power is , which is the long-term average power. If the radar uses a constant PRI of and a constant pulse width of then the average and peak power are related by

. If the PRI and/or pulse width is not constant one would use the average ratio of to in the previous equation. In the RRE equation, as written above, must have the units of watts.

©2005 M. C. Budge, Jr 54

is termed the transmit antenna gain and is used to characterize the focusing ability of the transmit antenna. We developed several equations for computing . The two that I think are worth remembering are

and

.

In the above and are the azimuth and elevation beam widths expressed in degrees. The azimuth and elevation notation is a convenient convention that applies to most antennas. In more general applications, the two beam widths used in the computation of antenna gain are the major and minor axes lengths of the ellipse that are used to define the antenna beam. The major and minor axes don’t need to be oriented up-down and side-to-side. is termed the effective aperture and is related to the physical aperture, , of the antenna by where is the aperture efficiency of the antenna. A typical value for is 0.6. is termed the radar wavelength and is related to the carrier frequency, , of the radar by where is the speed of light. has units of watts/watts when used in the form of the RRE above.

A term related to and is called effective radiated power and is given by the equation

.

It has units of watts. is the loss between the point where is specified and the antenna feed. The two most common points where is specified is at the power amplifier output and at the antenna, where the latter refers to the antenna feed. If is specified at the antenna, is unity (0 dB). If is specified at the power amplifier output, includes the losses of all devices between the power amplifier and the antenna feed. has the units of watt/watt.

An important point is that is not a real power. In other words, the power out of the antenna is not the ; it is . is the power one would need to transmit from an isotropic antenna in order to get the same

©2005 M. C. Budge, Jr 55

power density on the target that one gets from the actual antenna. has the units of watts.

is the gain of the receive antenna and is calculated using the same equations as above. is used to characterize the ability of the receive antenna to “capture power”. If the radar uses the same antenna for transmit and receive then . In class we have associated the fact that with a monostatic radar. This is not strictly accurate because a monostatic radar simply means that the transmit and receive antennas are co-located; the radar could still use different antennas for transmit and receive.

An important point is implied about the use of and in the radar range equation. Namely, it is assumed that the transmit and receive antennas are pointed at the target. It is more accurate to note that they are the gains of the transmit and receive antennas in the (angular) direction of the target.

is defined as the radar cross-section (RCS) of the target and has the units of m2. It is a single number that is used to represent the ability of the target to capture and re-radiate power in the (angular) direction of the receive antenna. An important issue about RCS is that it is only roughly related to physical target size. In fact, it is related to target size, the material and coatings on the target, the target orientation, the target structure and a host of other factors. Even though we use a single number for RCS in the RRE, we attempt to account for the RCS variation of the target by using various target models. The most common models are the Swerling models. These models characterize RCS as a random process (a RCS that changes randomly with time). The Swerling models provide for two sizes and rates of RCS variation by using two pairs of models. The first pair, termed SW1 and SW2, is used to characterize complex targets such as aircraft, ships, tanks, or other targets with a large number of facets. This type of target is characterized by fairly large variation in RCS (because of varying target orientation). The convention used to arrive at the density function for SW1 and SW2 targets is that they consist of a large number of equal size scatterers. The SW1 target differs from the SW2 target in that the RCS of SW1 targets varies slowly with time and the RCS of SW2 targets varies rapidly with time. The standard convention is that the RCS of SW1 targets changes over periods of seconds and the RCS of SW2 targets changes over periods of microseconds to milliseconds.

The second pair of Swerling target models is termed SW3 and SW4 and is used to characterize simple targets such as missiles and streamlined vehicles. The convention used to arrive at the density function for SW3 and SW4 targets is that the target consists of a single large scatterer and a large

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number if equal size, but smaller, scatterers. The SW3 target differs from the SW4 target in that the RCS of SW3 targets varies slowly with time and the RCS of SW4 targets varies rapidly with time. The standard convention is that the RCS of SW3 targets changes over periods of seconds and the RCS of SW4 targets changes over periods of microseconds to milliseconds.

A common, fifth, target type is a constant RCS target. This is termed a SW0 target by some and a SW5 target by others. A constant RCS target doesn’t exist in practice, however it is still used by many radar analysts. The reason for this is not clear. It is probably a carryover from before Swerling developed his models.

The density function for the SW1/SW2 target model is

and the density for the SW3/SW4 target model is

.

In the above is the unit step function. is termed the average RCS of the target and is the value of RCS used in the radar range equation.

The target model is not directly used in the RRE. It comes into play when one uses the results of the RRE to predict detection range. This will be discussed later. It also comes into play in determining , the integration gain.

People have attempted to verify that “real” targets actually obey the Swerling models. To the author’s knowledge, none have been completely successful. It appears that the best fit of a density function to real targets is a Gaussian density when the RCS is expressed in dBsm (10log(m2)). This is called a log-normal target model. The log-normal model is very rarely used in detection studies. Although the Swerling models don’t perfectly match real targets they do a good job of predicting detection performance against real targets.

is the range from the radar to the target. The use of in the RRE above carries the tacit assumption that the applicable radar is monostatic. If the radar had been bi-static one would replace with where is the range from the transmit antenna to the target and is the range from the receive antenna to the target. has the units of meters. The only constraints on is that it must be much larger than the size of the antenna. More specifically one must have where is the largest dimension of the antenna(s). This constraint is imposed by the antenna and defines the “far field” of the antenna. Of course, is also constrained by the waveform

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and must satisfy the inequality, where is the uncompressed pulse width. could also have an upper bound if the radar is required to operate unambiguously in range.

is Boltzman’s constant and is equal to .

is a standard temperature of 290° K. It is the standard temperature used to compute the noise figure.

is the noise figure of the radar receiver and is a dimensionless quantity. When specifying noise figure, one must specify an input and output of the receiver. The input is called the reference point. Thus, for example, if the input is the input to the RF amplifier and the output is the matched filter, one would say that is the noise figure at the output of the matched filter referenced to the RF amplifier input. The two most common reference points are the output of the antenna feed (the antenna or the antenna terminals) and the input to the RF amplifier, if the radar contains an RF amplifier. If the radar doesn’t contain an RF amplifier, another reference point would be the first active device, which is usually a mixer. The most common output point is the matched filter. Sometimes one also specifies the output as the output of the signal processor. However, in reasonable radar designs, the noise figure at the output of the signal processor will, for all practical purposes, be the same as the noise figure at the output of the matched filter. Note: there must be no active devices between the antenna feed and the reference point for .

In some applications, one can replace the product of and with the receiver noise temperature, . Receiver noise temperature would be the preferred quantity for those cases where the noise temperature at the antenna terminals is considerably larger than , as would be the case when the antenna was looking at the sun. It would also be the preferred form for the case where a low-noise radar is looking into space (which has a temperature near absolute zero). In most applications, the use of or and is immaterial. The reader is referred to the discussions on noise figure for more further study in this area.

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is the loss term and accounts for all losses not included elsewhere. It has the units of w/w. Refer to Skolnik’s text book and handbook for a discussion of the various things that must be included in . Some rules of thumb are as follow: must include a transmit loss term, , if is not specified at the antenna feed. More specifically, must contain any losses between the point where is specified and the antenna feed. must include a receive loss term, , if the noise figure is not referenced to the antenna feed. must contain the losses of all devices between the antenna feed and the reference point used when specifying the noise figure. is one of the more difficult quantities to determine accurately. To do so, one must have intimate knowledge of the radar subsystems, radar operation and the environment. It is not unusual for the list of the components of to be very long (10 to 30 entries). Many of the entries in the list will be on the order of tenths of a dB but, when taken together, can add up to a significant loss.

is the noise bandwidth of the receiver. Throughout this course we have been careful to define as the effective noise bandwidth of the receiver. In this context, where is the uncompressed pulse width. We have taken this approach because it helps avoid problems in defining the bandwidth to use in the RRE. If the radar uses a phase or frequency modulated pulse one could use the actual waveform bandwidth, , in the RRE (assuming the one uses a matched filter). However, one would need to include the compression gain of the waveform in . If one uses an LFM (Linear Frequency Modulation) waveform, the compression gain would be

. If one uses a phase coded waveform (e.g. Barker coded waveform, Pseudo Random Noise waveform, random phase coded waveform, Frank polyphase waveform, etc.) then the compression gain is , where is the number of subpulses in the waveform (we will discuss this further in connection with the ambiguity function). One would let the waveform bandwidth be where is the subpulse width. It is usually assumed that the subpulses are adjacent to each other such that .

With some simple analyses, it is easy to show that if and the accompanying compression gain are used in the RRE then the RRE can still be reduced to the form where the effective noise bandwidth, , is used. Having said this, we need to point out an exception. If one uses a phase coding and leaves a gap between the subpulses then one should use a noise bandwidth of and include the compression gain in . As an alternative, one could use an equivalent pulse width of and define the effective noise bandwidth as .

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A final note on noise bandwidth is related to Doppler signal processors; signal processors that contain narrow-band band-pass filters. In this case, some people propose using the band-pass filter bandwidth, , as the noise bandwidth in the radar range equation. If one uses this approach one must include a factor to account for the increase in noise power spectral density caused by the sampling effect of the range gates in the receiver. Specifically, one must increase the noise power spectral density from to where is the PRI and is the pulse width. The reason for this will be discussed in EE 725. An alternate approach is to define use the standard approach of using effective noise bandwidth (i.e., ) and including the integration gain of the Doppler processor in . The integration gain would be .

is used to represent the actual or effective SNR gain associated with the signal processing that occurs after the matched filter. (As discussed above, it can also be used to account for the integration gain of the matched filter.) If the signal processor is a coherent integrator then represents the actual gain in SNR that the signal processor provides. Recall that if the coherent integrator sums the returns from n pulses (integrates n pulses), then . If one implements the coherent integrator with a FFT then n is the number of taps in the FFT. If one implements the coherent integrator as a band-pass filter then where B is the effective noise bandwidth discussed above and is the bandwidth of the band-pass filter.

If the signal processor consists of a non-coherent integrator (an integrator or a summer that operates on the target returns after they have gone through the magnitude or square-law detector) then is an effective integration gain. It does not represent the actual increase in SNR afforded by the non-coherent integrator. It is a gain that allows us to use the single-pulse false alarm and detection probability equations with the SNR that comes from the RRE when is included. The values of depends upon the target model being considered and the range of detection and false alarm probabilities being considered. Fortunately, for SW0/SW5, SW1 and SW3 targets we can approximate by the equation . This equation is valid for all detection and false alarm probabilities that are usually of interest in radar design and analysis. There is no set approximate equation for SW2 and SW4 targets. As a first approximation one can use the curves provided in class. If one desires a more accurate relation between

and n, and one should refer to the equations or curves in Meyer

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and Mayer’s book.1 If the radar being analyzed contains a coherent integrator that integrates n pulses and a non-coherent integrator that integrates m groups of n pulses then the integration gain is . It is important to note that this equation applies only to SW0/SW5, SW1 and SW3 targets. For SW2 or SW4 targets and would become

as discussed above. In either case, is the effective integration, or SNR, gain.

SNR is the signal-to-noise power ratio at a certain point in the receiver. When we first derived the RRE, without , we formulated the derivation so that SNR was the signal-to-noise measured at the output of the matched filter. Furthermore, we showed that it was the SNR measured when the signal level out of the matched filter was at its peak value. Because of this we often refer to SNR as the peak signal - to - average noise power ratio. For further discussion of this refer to your notes on the matched filter. We also termed this SNR the single-pulse SNR.

When we consider coherent integration and include in the RRE, the SNR is the signal-to-noise power ratio (peak signal - to - average noise power ratio) at the output of the coherent integrator. In terms of detection, we treat the single pulse SNR and the SNR at the coherent the same way. That is, we use the same set of equations for and . This stems from the fact that, for the SW0/SW5 through SW4 target models, coherent integration does not change the form of the noise, signal and signal-plus-noise density functions at the output of the magnitude or square law detector.

When we consider non-coherent integration and include in the RRE, the SNR is not the signal-to-noise power ratio at the output of the non-coherent integrator. It is an effective SNR that we use to account for the effects of the non-coherent integrator in computing detection probability. Essentially, it allows us to use the same set of and equations that we use when we have only a matched filter (single pulse) or a coherent integrator.

1 Meyer, Daniel P. and Mayer, Herbert A., “Radar Target Detection – Handbook of Theory and Practice”, Academic Press, San Diego, CA, 1973 ISBN 0-12-492850-1

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