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# AMBIGUITY FUNCTION GENERATION USING THE FFT

By Gloria Boyd,2014-11-26 11:24
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AMBIGUITY FUNCTION GENERATION USING THE FFT

PULSE COMPRESSION WAVEFORMS

There is a class of waveforms termed pulse compression waveforms. These types of waveforms, and their associated signal processors, are useful because the overall signal duration is long, which helps SNR, but the output of the signal processor has the response of a much shorter signal, which provides good range resolution. Thus, it allows the radar to achieve good SNR, at reasonable power levels, and good range resolution.

Almost all pulse compression waveforms are constant amplitude, phase (or frequency) modulated waveforms. That is, they are of the form

；?tjt;；rect ute. ;；~?(~?p??

Usually the signal processor (or matched filter) is matched to so that, ut;；

using the terminology of the ambiguity function, . vtut;；;；

The pulse compression waveform that we have already considered is the

2LFM waveform. With this waveform . That is, is a quadratic ，，tbtt;；;；

phase function. The term linear frequency modulation comes from the form of the derivative of . Specifically, t;；

dt;；tftbt？？？22 ；，，;；;；dt

or

, ftbt;；

which says that the frequency of changes linearly with time. As an aside, ut;；

Bb(the total frequency excursion is . The means that contains ut;；p

Bb(frequencies that extend over a band of . Thus, we say that the LFM p

Bb(bandwidth is . p

The figure below contains a plot of the matched-Doppler range cut (normalized to a peak of one) of an LFM pulse with a bandwidth of 15 MHz and a duration of 1 µs (the solid curve). It also contains similar plots for a 1/15-µs (dotted curve) and a 1-µs (dashed curve) unmodulated pulse. It will be noted that the central peak of the range cut for the LFM pulse has close to the same shape as that of the 1-µs, and is much narrower than the 15-µs unmodulated pulse. The implications here are that the (15-MHz) LFM waveform and the 1/15-µs pulse have about the same range resolution. However, since the LFM waveform duration is 1 µs it has the same energy as the unmodulated, 1-µs pulse.

Some further notes on the figure concerns the two regions of the plot indicated on the figure. The central region is termed the mainlobe and the outlying areas are termed the sidelobes. Ideally, we would like the mainlobe to ?2005 M. C. Budge, Jr 1

be very narrow and the sidelobes to be zero. In practice, this is not possible. Having said this, for a waveform to qualify as a pulse compression waveform, its range cut must have a fairly narrow mainlobe and fairly low sidelobes.

matched-Doppler range cut1

1 us LFM w/15 MHz BW1 us unmodulated pulse0.91/15 us unmodulated pulse

0.8

0.7

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0.5

amplitude (v/v)Main Lobe 0.4

0.3Sidelobes

0.2

0.1

0-1-0.8-0.6-0.4-0.200.20.40.60.81range offset (us)

Other examples of pulse compression waveforms are

Barker Codes

Pseudo Random Noise Codes

Random Phase Codes

Frank Polyphase Codes

Frequency coding

In the first four examples, the waveforms consists of a concatenation of short pulses (termed subpulses) where the phase, , is constant over the duration t;；

of each subpulse but varies from subpulse-to-subpulse. This is depicted in the figure below. A characteristic of these phase coded waveforms is that the width

of the mainlobe of the range cut is twice the width of the subpules. The total extent of the sidelobes is twice the total length of the overall pulse, which equals to the width of the subpulses times the number of subpulses.

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subpulse 1

1234Nsubpulse 2

An equation for the waveform can be written as subpulse 3

subpulse 4N1；?tk(spjkuterect ;；~?(~?spk0??

(where the are the phases of the individual subpulses, is the width of the spksubpulse Nindividual subpulses and N is the number of subpulses that make up the

N(((overall pulse. The overall pulse width, is . ppsp

In many applications the phase shifts for Barker Codes, Pseudo Random

Noise Codes and Random Phase Codes are either 0 or . This is due primarily

to the fact that these two phase shifts are easy to generate on transmit and in the signal processor. Some modern radars are using other than these two phase shifts since modern hardware can support more complicated phase coding. With Frank Polyphase Codes the phase changes in a quadratic fashion from subpulse to subpulse. Frank Polyphase Codes are an approximation to LFM. To the authors knowledge, they are not used very much.

With frequency coded waveforms, the frequency is changed on a

subpulse-to-subpulse basis. Radar analysts have studied frequency coded waveforms for many years but they were very seldom used because available hardware could not support their generation and processing. With the introduction of direct digital synthesizers (DDSs) it is expected that such coding could find wider application in newer radars.

There are only a few Barker codes as noted on the in the table below. Barker codes are characterized by the fact that the sidelobes of their range cut have a peak value that is 1/N where N is the number of subpulses in the waveform. A seldom realized problem with Barker coded waveforms is that their ambiguity function has ridges in range-Doppler space. These ridges are only about 6 dB below the peak value and could thus cause interference problems. A range cut and a plot of the full ambiguity function of a 13-subpulse Barker coded waveform is shown in the two figures following the table of Barker codes.

There are many Pseudo Random Noise (PRN) waveforms. These

waveforms are characterized by the fact that the phase changes in an apparently random fashion from subpulse to subpulse. However, the changes are not random but are very predictable and follow a very strict set of rules. This is why they are called Pseudo Random coded waveforms. The sidelobes of

the range cut of the ambiguity function of PRN waveforms are not as low as for Barker coded waveforms or for LFM waveforms. However, their ambiguity function also has no discernable ridges in range-Doppler space. The ambiguity function of PRN coded waveforms approaches what is called the thumb tack

ambiguity function. This ambiguity function is characterized by a large central ?2005 M. C. Budge, Jr 3

peak and uniform sidelobes in the rest of range-Doppler space. Plots of a range cut and the full ambiguity function of a 15-pulse PRN coded waveform is shown in the last two figures below.

Known Barker Codes

Code Length Subpulse phases, t;；

2 0,

3 0,0,

4 0,0,,0

4 0,0,0,

5 0,0,0,,0

7 0,0,0,,,0,，，，

11 0,0,0,,,,0,,,0,，，，，，，

13 0,0,0,0,0,,,0,0,,0,,0，，，，

1

0.8

0.6

0.4Amplitude

0.2

0-10-50510

Range Mismatch (s)

Range Cut for a 13-Subpulse Barker-Coded Waveform

With Random Phase coded waveforms, the phase code on each subpulse is determined by drawing random numbers. As an example, on each subpulse one would draw a number from a random distribution that extends from -1 to 1. If the random number is positive, the subpulse would have a phase of 0 and if

the random number is negative, the subpulse would have a phase of .

Unfortunately, one cannot determine in advance whether the resulting waveform will have sidelobes (in range-Doppler space) that are uniformly low. ?2005 M. C. Budge, Jr 4

The only way to find out is to create a waveform and form the ambiguity function.

Range-Doppler Map of a 13-Subpulse Barker-Coded Waveform

It should be noted that LFM waveforms have good sidelobe levels over large range-Doppler regions but their ambiguity function has a ridge in range-Doppler space. As with Barker coded waveforms, this can cause problems from an interference perspective.

As a note, a common feature of the ambiguity function is that the volume of the ambiguity function is the same for all pulses of the same length (assuming constant amplitude pulses). Its peak value is also the same. What this means is that if one decreases sidelobes in one area of range-Doppler space, the sidelobes must increase in other regions. With LFM pulses, we get low range-Doppler sidelobes by “pushing” the volume along the ridge. With Barker codes, we get low range sidelobes (sidelobes along the range cut) at the expense of ridges elsewhere in range-Doppler space. For PRN and random phase coded waveforms we get generally mediocre, but fairly constant, sidelobe levels in range-Doppler space. Finally, for unmodulated pulses, we get low sidelobes in the Doppler direction at the expense of a large ridge at matched Doppler.

It should be noted that in some applications the ridge of the LFM pulse is an advantage and in other cases the thumb tack ambiguity function of PRN or random phase coded waveforms are an advantage. An example of the former ?2005 M. C. Budge, Jr 5

would be in a search application. Because of the LFM ridge, the LFM waveform is fairly Doppler insensitive. Therefore, even if the signal processor is not matched to the target Doppler its output will have a discernable peak. This is good in terms of simplifying signal processor and detection hardware. If one were to use a PRN waveform in the same application one would need to have a bank of signal processors matched to a range of expected target Doppler frequencies. This is because of the fact that the ambiguity function of a PRN waveform has a peak only at the range and Doppler of the target. For track purposes, the PRN waveform may be better because it has a narrow response in both range and Doppler. This means that it can provide accurate range and Doppler measurement. It also means that it could be less likely to experience interference from other targets that are nearby. The ridge associated with LFM waveforms can cause errors in range and/or Doppler measurement because the two are coupled: an error in Doppler tuning of the signal processor can cause range errors and vice versa. This means that the tracking algorithms will need logic to cope with the ambiguity caused by the range-Doppler ridge.

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Amplitude0.4

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0-15-10-5051015

Range Mismatch (s)

Range Cut for a 15-Subpulse PRN-Coded Waveform

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Range-Doppler Map of a 15-Subpulse PRN-Coded Waveform

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