RESOLUTION OF RANGE AND VELOCITY AMBIGUITY FOR A MEDIUM PULSE DOPPLER RADAR

Wen Lei, Teng Long, Yueqiu Han,

Beijing Institute of Technology, Beijing, I? R.China Beijing Institute of Technology, Beijing, I? R.China Beijing Institute of Technology, Beijing, I? R.China

Abstract

In medium pulse repetition frequency (MPRF) radars, ambiguities exist both in range and Doppler measurements. Some efficient techniques have been established to resolve the range and velocity ambiguity of the target using multiple PRFs.

In this paper, a simple algorithm is proposed to resolve both range and velocity ambiguity based on residue arithmetic. This algorithm makes out the unambiguous result by using a residue look-up table. The particular problem about filter bandwidth unitary in frequency is settled and assessed with another algorithm. An example for the generation of the residue look-up table is presented to resolve range ambiguity.

1. Introduction

In order to get unambiguous range and velocity value simultaneously, MPRF waveform is chosen for modem airbome radar. As MPRF waveform produces ambiguous measurements for both range and velocity, a common technique is to use multiple PRFs to settle this problem, which needs an algorithm to resolve the ambiguity of range and frequency in multiple PRFs.

For a given set of PRFs, Chinese Remainder Theorem has been established to resolve ambiguity [ 1-3 1. However, when the measurement error exists, the result error is usually very large. Clustering algorithm is also suggested with the minimum squared error criterion [4,5]. Apparently, it has good anti-error ability and expensive computational throughout. An algorithm based on the choice of particular values for the PRFs is provided for velocity ambiguity resolution, where a quasi-maximum likelihood criterion is maximized for ambiguity order estimation [6]. Considering the blind area both in time and frequency, this algorithm is so limited by particular PRFs that it is not fit for other combinations of PRFs to resolve velocity ambiguity.

In this paper, a simple algorithm is proposed which takes into account presumptive redundancy error to improve the ability against measurement errors. Its fast implementation relies on the established look-up table. The look-up table should be modified when the blind area can't be overlooked, to assure the completeness of the table.

Furthermore, the performance of the suggested method about the frequency filter bandwidth unity is compared with that of the clustering algorithm, each has its advantages.

The proposed method is so simple that it can be easily processed in real time processing.

2. Range and Frequency Ambiguity Resolution

2.1 The Principle of the Residue Look-Up Table Algorithm

This method makes use of the differences of the residues on different PRF to resolve ambiguities. First, it selects the residue of one PRF as the reference. Then it makes the differences of the residues on other PRFs and the reference PRF into a look-up table.

Take an example of m different PRFs in time. The principle of the method is the same on the frequency, as shown in figure 1 , where represents different pulse repetition time (PRT).

When the target. locates on T , its residue q on different PRF can be described as:

5 = T - N f c = mod(T,c) i = 1,2,..-,m (1) Where N i is the ambiguous order, mod(A,B)

If taking the residue of mB PRF as the reference, the represents the module of A to B.

values stored in the look-up table can be described as: ei,k = ri - rm i=1,2,...,m-l (2)

Where ei,k represents the differences between the residues of the range bin on different PRF and the reference, k represents which group table value it is. In the table, the sect value Bk , B, = NmTm, is also stored corresponding to the k" group values. Here only one Bk corresponds to some PRF residue set.

So each group in the table has m values. The first m-I values represent the differences between the measured ambiguous range and the reference. The last value is the range sect value of the reference.

The process of de-ambiguity can carry through as follows:

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Stepl: a set of measured ambiguous range values is obtained.

Step2: the measured ambiguous range value of the mth PRF is taken as the reference.

Step3: the real range sect value is derived from the matched values in the look-up table, according to the differences between the measured ambiguous range of other PRF and the reference. The range sect value is obtained as N,T, of the mth PRF.

Step4: the result will be (3)

If error exists in the measured ambiguous range values, the search will leave some room for admitting the error. The difference e,' between measured ambiguous range on other PRF and reference PRF can be described as:

Where Mi presents the measured error on different PRF, r, + Mi represents the measured ambiguous range value.

So the process of resolving the range ambiguity is to find ei,, which meet the equation ( 5 ) and (6), and some

error redundancy 6 exists between ei,, and ei' .

T = N,T, + r,,,

ei'=r, +Mi -r , -M, (4)

According to the e,+ and its corresponding range

sect value B, , the resolved ambiguous range is B, + r , +M,.

2.2 The Complete Residue Look-up Table Algorithm If the resolving power of radar decreases to some

degree, i.e., the size of range resolution AR is large enough which meets inequality (7), where t, represents the width of blind range area. And if these PRFs meet equality (8):

2AR > t, (7) k,T, =kzTz + S , SS26,- (8)

Where rb represents the width of blind range area, S,, represents the maximum measured error.

Then the former look-up table method can't find the answer in the table, and can7 resolve the ambiguity. Figure 2 shows an example of some range bin T, which will fall into the blind range area of TI and T, simultaneously. The dashed line indicates the location of T.

Because the maximum measured range error normally be f l bin, redundancy 6 can be taken as 2. If there is +1 measured error of T on TI, -1 measured error of T on T,, and the target can be seen on both PRF, and locate at the

dashed-dotted position. So T locates at the current repetition of TI, while at the former repetition of T,, Under this condition, the result can't be found in the look-up table. While modified look-up table algorithm takes into account this status, joining the difference of the residues, which has redundant error, and its corresponding range sect value into the look-up table.

2.3 Example for application Example 1: three PRFs are adopted to resolve range

ambiguity. LetT, =103,T2 =119,T3 =137,R, =1500(the

width of the range bin is taken as 1 IJ s, the maximum detection range R,,, is 225km), [hi 1 = N I 1 .

Take T, = 137 as the reference PRT, B is the range sect value of the reference PRT, A (T, - T3 ) represent the differences between the residue on the i" PRT and the residue on the third PRT.

Then the values in the look-up table are obtained as table 1.

Let the true range of the target be R = 574, then the residues on different PRT should be r, =59,r2 =98,r, =26 . If the measured ranges on different PRT are

r, =60,r, =99,r3 =25, then Sl=r, -r3 = 3 5 t

6 2 = r, -r, = 74. Let the redundant error be 2. From the table, the 14Ih group values (33,72,548) meet the requirement of inequality ( 5 ) and equation (6), and the corresponding range sect value B=548. The real range value can be got from R' = B +r3 = 573 with range error of 1.

I I I l t

I 1

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3. Algorithm Performance Analysis

As discussed in the previous section, the Chinese Remainder Theorem is easy to use, but anti-error capability is not so good.

The error performance of residue look-up table algorithm is identical to clustering algorithm by using only two PRFs. By using more than three PRFs, clustering algorithm completely utilizes the relationship between PRFs, calculating all the mean squared error of possible range. While residue look-up table algorithm only makes use of part of the relationship, i.e. only calculating the squared error between these measured ambiguous range value and the reference. But because of the number of PRF utilized increasing, i.e. the number of values which can be used to search the table are increased, the possibility of

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error is largely cut down. Let the number of PRF be m, and the sum of all the

number of the repetition on m PRF be N. By Clustering algorithm, which needs at first rank all the possible solutions, it will cost N computational steps. Then in order to get the minimum mean squared error solution, it still needs about N circular computation, each circular computation includes m-1 addition, 1 division, m subtraction, m squares. So it needs at least (3m+l)N computation. The residue look-up table algorithm also needs about N circular computation, each circulation includes m-1 compare, but it usually not needs equation (6) So the total computation is approximately (m-1)N. Then it seems that the computational throughout of residue look- up table algorithm is less than the clustering algorithm.

4. Special Issues on Frequency Bandwidth Unitary

The prerequisite in solving frequency ambiguity is to settle the fundamental problem of different filter bandwidth at different PRFs by requantizing the ambiguous Doppler measurements to a unique reference PRF. In the process of re-quantizing, it needs to round the unified frequency value.

On the base of requantization, the error performance of the residue look-up table algorithm and the clustering algorithm can be evaluated as follows:

To explain what happens during requantizing, let us consider the instance of two PRFs.

The errors of the rounded-unified frequency are shown in figure 3. Where MI and n separately represent the decimal fraction and the integer of the rounded-unified measured ambiguous frequency. Af, and NI separately represent the decimal fraction and the integer of the rounded-unified measured ambiguous order frequency. w3 represents the error of the rounded-unified real frequency, -0.5 < q3 < 0.5. Af4 represents the integer of Af3, i.e., when -0.5

there exists error 1. If the clustering algorithm is adopted, then the result will be consistent with the real result.

When 44= -1, AFl>Lv;,. The ability of de- ambiguity for residue look-up table algorithm is better than that for clustering algorithm. The principle is similar as in figure 4.

I

(2)If 4, and 42 meet: 0.5

: Ni Ni+n N?n+ 1

Figure 3c. Fraction of the Rounded-Unified Measured Frequency

V,=A(T,-T3) 0 -103 -103 34 -69 -69 68 -35 ... - 1 33 ... -72 -65 72 ... -58 v,=ACr,-Ts) 0 0 -119 18 18 -101 36 36 ...

A

Ni Ni+n Ni+n+ 1 Figure 4a. Unified Location of Real Frequency

(not rounded)

k Ni N,+1

Figure 4b. Unified Location of Ambiguous Order Frequency (not rounded)

clr n n+l

Figure 4c. Unified Location of Measured Ambiguous Frequency (not rounded)

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