Pulse Comperssion Using Costas in Radar Systems (PCUCR)

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Event Date/Time: Sep 04, 2009 End Date/Time: Sep 05, 2009
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Pulse Compression Using Costas in Radar Systems

Lalitha.Byram, Dhananjayulu.V, Singh .R.K, Kadernath.B

Electronics Communication Engineering Dept, Guru Nanak Engineering College
Affiliated to J.N.T.U. Hyd. Ibrahimpatnam,R.R. (Dist),A.P,India

lalithabyram@ymail.com

viceprincipalgnec@gmail.com



Abstract: - Pulse compression is a Signal processing technique mainly used in radar, sonar and echography to augment the range resolution as well as the signal to noise ratio. This is achieved by modulating the transmitted pulse and then correlating the received signal with the transmitted pulse. Pulse compression also known as pulse coding, is a signal processing technique designed to maximize the sensitivity and resolution of radar system. Radar signal designers had been continuously putting their efforts to achieve suitable codes for the optimum performance of radar. Different pulse compression codes are compared to achieve good range and Doppler Resolution. Coded signals are used for pulse compression radars, allowing improvement of resolution and target detection without the need to increase peak emitted power. Costas waveforms are a class of pulse compression waveforms, having aspects of both phase coded and stepped frequency pulse for bursts waveforms (multiple pulse). Costas codes and quadrature, congruential codes, as the two types of frequency coding techniques were studied for their application to digital pulse compression. A costas waveform is similar to a polyphase waveform in that it is a single pulse waveforms divided into N sub pulse. With proper design of the frequency sequence, the Costas waveform can be designed to have more thumbtacks like ambiguity function. Costas signals have range and Doppler resolution properties consistent with overall signal duration and bandwidth. In the proposed project we have generated costas frequency in step sequences and compared their performance for lengths 5, 6, 7, 8 and 10 respectively in both ambiguity domain and auto-correlation domain.
Keywords: - RADAR, PULSE COMPRESSION, SIGNAL ENCODING, AMBIGUITY FUNCTION, AUTO-CORRELATION, COSTAS SIGNAL, DOPPLER RESOLUTION
I. INTRODUCTION

Pulse compression allows radar to simultaneously achieve the energy of a long pulse and the resolution of a short pulse without the high peak power required of a higher-energy short-duration pulse. It is used in high power radar applications that are limited by voltage breakdown if a short pulse were to be used. Airborne radars might experience breakdown with lower voltages as compared to the ground-based radars, and might be candidates for pulse compression. Pulse compression always used in high-power radars with solid-state transmitters since solid-state devices, unlike vacuum tubes, can operate with high duty cycles, low peak power, and pulse widths much longer than normal. Pulse compression is also found in SAR and ISAR imaging systems to obtain range resolution comparable to the cross-range resolution.
Costas FM Signal is a frequency code, or time - frequency code as in the linear FM signal, but is a frequency hopping code instead of a linear stepped code.

II. PULSE COMPRESSION USING COSTAS CODES

Radar has become an important tool for the meteorologist and as an aid for safe and efficient air travel by observing and measuring precipitation, warning of dangerous wind shear and other hazardous weather conditions and for providing timely measurement of the vertical profile of wind speed & direction. Pulse Compression allowed the use of long waveforms to obtain high energy & simultaneously achieve the resolution of a short pulse by interval modulation of the long pulse.

By rapid advances in digital technology made many theoretical capabilities practical with digital signal processing & digital data processing, by using the method of Costas FM.
A constant-amplitude radar pulse of duration T consisting of N contiguous sub-pulse each of duration T1 where, in time domain T1 = T / N (1)
The N possible frequencies are available as defined by
ωi = ω 0 + i δω, i = 1, 2,……, N (2 )
where ω 0 is a constant large enough so the pulse in narrowband and
δω = 2πδf = 2π / T1 (3)
The maximum change in frequency Δ f during time T is
Δ f = Δω / 2π = N δf = N / T1 (4)
The pulse has a time-bandwidth product of
Δ f T = N2 δf T1 = N2 ( 5 )
From ( 1), (3),and( 4). The duration of the compressed pulse is 1/ Δ f = T / N2 for all Costas FM signals.
The overall pulse can be viewed as an N ×N array of chosen frequencies versus time. The use of a Costas signal in creating a radar waveform can be very desirable, because it will yield a radar ambiguity function that closely approximates the ideal “ thumbtack” form. The Costas signals frequency code, unlike linear FM signal, it is a frequency hopping code instead of a linear stepped code.

III. COSTAS FREQUENCY CODING AND COSTAS SIGNAlS

Following procedure is used to find the Costas signals for a sequence of length N. Let p be an odd, prime number. Then N = p-1. Let g be the primitive root of p. A primitive root of p is defined as the value g such that the sequence of powers g1, g2, g3,…. gp-1 modulo p generates every integer from 1 to p-1. A primitive root of p is NOT the same as a prime factor of p or N, though a primitive root may be a factor of N. Costas found ideal frequency-time sequences for N up to 12. For instance, the primitive root of 41 is 6, which is neither a factor of p = 41 or of N = 40. A compilation of all known sequences for N up to 360 is given by Golomb and Taylor.
For the Costas code of length N, create an N × N matrix, and label the columns and row respectively as:
j = 0, 1 , 2 , 3 ,……., (p-2)
i = 1, 2 , 3 , 4 ,……., (p-1)
we will place a “1” or “dot” in the matrix location ( i , j) if and only if
i = (g)j modulo p

The dot or “1” in location ( i , j) as defined above indicates that we will use the frequency fi is obtained from dividing the frequency span of the transmitted radar pulse by N.
Let’s find a costas sequence for N = 4. Note that I wrote “a” Costas sequence, not “the” costas sequence. For the value N, there are N! possible sequences one or more which may be a Costas sequence. For example, for N = 4, there are 4! = 24 possible sequences of values, but only 40 of them will yield a Costas array.

For our example, let p = 5, which yields N = 5 – 1 = 4. From the table in Abramowitz and Stegun, the primitive root of p = 5 is g = 2. Then our column and row values are:
j = 0, 1, 2, 3,
i = 1, 2, 3, 4,
We then find the (i , j) locations for our “dots” from the expression
i = (g) j modulo p
or for our example
i = (2) j modulo 5

AFTER CARRYING OUT THESE CALCULATIONS, WE CAN CONSTRUCT THE FOLLOWING TABLE I

j i = (2) j modulo 5
Element
w/ Dot
0 20 = (1)5 = 1 (1,0)
1 21 = (2)5 = 2 (2,1)
2 22 = (4)5 = 4 (4,2)
3 23 = (8)5 = 8-5 =3 (3,3)

Using the above values, we get the Costas sequence
{1, 2, 4, 3}. This will yield a Costas array or matrix that looks like the one shown in Figure 1. Note that another Costas array of order N-1 = 3 can be obtained from this one, by eliminating the first row and column (Figure2) will, then, yield the array shown in Figure 2a, with the resulting Costas sequence of {1,3,2}. Once we have the sequence, we can then construct a difference matrix, which will allow the construction of the resultant Sidelobe matrix, which will provide insight into the form of the ambiguity function.

Let’s find a Costas sequence for N=6.For the value N, there are N! possible sequences,one or more which may be a Costas sequence. For example, for N = 6, there are 6! = 720 possible sequences of values, but only 116 of them will yield a Costas array.

Other example let p = 7, which yields N = 7 – 1 = 6. From the table in Abramowitz and Stegun, the primitive root of p = 7 is g = 3. Then our column and row values are:
j = 0, 1, 2, 3, 4, 5
i = 1, 2, 3, 4, 5, 6
We, then, find the (i , j) locations for our “dots” from the expression
i = (g)j modulo p
i = (3)j modulo 7


AFTER CARRYING OUT THESE CALCULATIONS, WE CAN CONSTRUCT THE FOLLOWING TABLE II

j i = (3)j modulo 7
Element
w/ Dot
0 3 0 = (1)7 = 1 (1,0)
1 3 1 = (3)7 = 3 (3,1)
2 3 2 = (9)7 = 9 – 7 = 2 (2,2)
3 3 3= (27)7 = 27 – 21 = 6 (6,3)
4 3 4 = (81)7 = 81 – 77 = 4 (4,4)
5 3 5 = (243)7 = 243 – 238 = 5 (5,5)

Using the above values, we get the Costas sequence {1, 3, 2, 6, 4, 5}. This will yield a Costas array or matrix that looks like in Figure 3. Note that another Costas array of order N-1 = 5 can be obtained from this one, by eliminating the first row and column as in Figure 4. This will, then, yield the array shown in Figure 4a, with the resulting Costas sequence of {2, 1, 5, 3, 4}. If the primitive root of 7 had a value of 2 instead of 3, this process could be repeated one more time to yield a Costas sequence of N-2 = 4.
Once we have the sequence, we can then construct a difference matrix, which will allow the construction of the resultant Sidelobe matrix as shown in Figure5 & 6, which will provide insight into the form of the ambiguity function.

To construct the difference matrix, start with the Costas sequence {aj} = {1, 3, 2, 6, 4, 5}. This will provide the first row of the difference matrix. To obtain the elements of the remaining rows, use the expression
Di , j = ai+j – aj i + j < N
Thus, to obtain the elements in the second row, we would compute:

D11 = a2 – a1 = 3 – 1 = 2
D12 = a3 – a1 = 2 – 3 = -1
D13 = a4 – a1 = 6 – 2 = 4
D14 = a5 – a1 = 4 – 6 = -2
D15 = a6 – a1 = 5 – 4 = 1
.


I . COSTAS ARRAY FOR N = 4 AND BASED ON P = 5

0 1 2 3

1 ☻
2 ☻
3 ☻
4 ☻




Figure 1
1 ☻

2 ☻
3 ☻
4 ☻
0 1 2 3




Figure 2


0 1 2
1 ☻
2 ☻
3 ☻



Figure 2a

II . COSTAS ARRAY FOR N =6 AND BASED ON P = 7
0 1 2 3 4 5
1 ☻
2 ☻
3 ☻
4 ☻
5 ☻
6 ☻






Figure 3
0 1 2 3 4 5
1 ☻

2 ☻
3 ☻
4 ☻
5 ☻
6 ☻






Figure 4

0 1 2 3 4
1 ☻
2 ☻
3 ☻
4 ☻
5 ☻





Figure 4a
The calculations for the remaining rows and the complete difference matrix are shown in Figure7.

Finally, with the results of the difference matrix, we can construct the sidelobe matrix. The values of the first row of the difference matrix give the locations of the sidelobe contributions in the delay equals +1 column of the sidelobe matrix. For instance, the first value in the difference matrix is 2. The value two means that two shifts upward will yield a correspondence between the frequency transmitted in the


III. CONSTRUCTION OF SIDELOBE MATRIX


a1 a2 a3 a4 a5 a6

aj 1 3 2 6 4 5
i = 1 2 -1 4 -2 1
i = 2 1 3 2 -1
i = 3 5 1 3
i = 4 3 2
i = 5 4

Figure 5









D
O
P
P
L
E
R
5 1
4 1 1
3 1 1 1
2 1 1 1
1 1 1 1
0 6
-1 1 1
-2 1
-3
-4
-5
-5 -4 -3 -2 -1 0 1 2 3 4 5


Figure 6 – Construction of sidelobe matrix for positive delay




second time slot and the frequency transmitted in the first time slot. Two shifts upward represents a positive Doppler shift equal to two frequency steps. The 2 in the first row of the difference matrix will result in a “1” in the sidelobe matrix that corresponds to normalized Doppler of 2, and in the column that corresponds to a normalized delay of +1.
IV. CONSTRUCTION OF DIFFERENCE MATRIX
Figure 7

D11 = a2 – a1 = 3 – 1 2
D12 = a3 – a1 = 2 – 3 -1
D13 = a4 – a1 = 6 – 2 4
D14 = a5 – a1 = 4 – 6 -2
D15 = a6 – a1 = 5 – 4 1
D21 = a3 – a1 = 2 – 1 1
D22 = a4 – a1 = 6 – 3 3
D23 =a5 – a1 = 4 – 2 2
D24 = a6 – a1 = 5 – 6 -1
D31 = a4 – a1 = 6 – 1 5
D32 = a5 – a1 = 4 – 3 1
D33 = a6 – a1 = 5 – 2 3
D41 = a5 – a1 = 4 – 1 3
D42 = a6 – a1 = 5 – 3 2
D51 = a6 – a1 = 5 – 1 4













The values of the second row of the difference matrix give the locations of the sidelobe contributions in the delay equals +2 column of the sidelobe matrix, and so on, Figure 6 for an example. This shows the positive delay half of the sidelobe matrix. To obtain the negative delay half of the sidelobe matrix, change the signs of the values in the difference matrix as shown in Figure 8 and place the values in the sidelobe matrix, as shown in Figure 8a.





aj -1 -3 -2 -6 -4 -5
i = 1 -2 1 -4 2 -1
i = 2 -1 -3 -2 1
i = 3 -5 -1 -3
i = 4 -3 -2
i = 5 -4
V.CONSTRUCTION OF SIDELOBE MATRIX AND DIFFERENCE MATRIX
Figure 8
a1 a2 a3 a4 a5 a6













D
O
P
P
L
E
R
5 1
4 1 1
3 1 1 1
2 1 1 1 1
1 1 1 1 1 1
0 6
-1 1 1 1 1 1
-2 1 1 1 1
-3 1 1 1
-4 1 1
-5 1

-5 -4 -3 -2 -1 0 1 2 3 4 5
Delay
Figure 8a- Completion of sibelobe matrix by addition of negative delay
A long Costas Signal indicates ambiguity function approaches the thumbtack shape, the doppler resolution of the single Costas pulse remains 1/T, where T = Ntb is the pulse duration. Thus doppler resolution can be improved by using a Coherent train of Costas pulses, as shown in Figure 9, display a partial AF of a train of four Costas 20 pulses with a duty cycle of T / T4 = 1/4.
.


Figure 9- Ambiguity function of a Costas Signal (Length N= 20) at all relevant grid points

IV. CONCLUSIONS
A Costas code is, therefore, a frequency-hopped signal where there is only one coincidence between the original and the translated array. With no translation, there will be N coincidences, which is the peak at the center of the ambiguity diagram. Thus the maximum sidelobe(voltage) ratio is 1/N. This is only a approximation since discrete values are used. A more exact calculation of the ambiguity function, therefore, might have sidelobe levels that can be greater than 1/N in the vicinity of the central peak. The ambiguity function that results from coherent processing and the sidelobe matrix, which has nothing to do with phase coherence, indicates that noncoherent processing will work relatively well with costas signals. Costas’s original application was sonar, where coherency is poorly preserved.

ACKNOWLEDGMENT

This material is an abstract from several sources. The most useful ones are the following:
Costas, J. P., “A Study of a Class of Detection Waveforms Having Nearly Ideal Range-Doppler Ambiguity Properties,” Proceedings of the IEEE, Vol 72, No. 9, September 1984, pp. 996-1009. Golomb, S. W., and H. Taylor, “Construction and Properties of Costas Arrays,” Proceedings of the IEEE, Vol 72, No. 9, September 1984, pp. 1143-1163.

REFERENCES

[1] Nadav Levanon, “Radar Priniciples”, Wiley and Sons,
1988. ISBN 0-471-85881-1.See chapther 8.
[2] Peyton Z. Peebles, "Radar Principles," Wiley Interscience, 1998.
ISBN 0-471-25205-0. See chapther 7.

[3] Merrill I. Skolnik “ Introduction to Radar Systems” Tata McGraw-Hill
ISBN-13,10 0-07-044533-8. See chapter 6.

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