Performance evaluation of nested costas codes

ISSN (PRINT): 2393-8374, (ONLINE): 2394-0697, VOLUME-5, ISSUE-3, 2018
17
PERFORMANCE EVALUATION OF NESTED COSTAS CODES
FOR IMPROVING RANGE RESOLUTION
K. Ravi Kumar1
, Dr. P. Rajesh Kumar2
1
Research Scholar, 2
Professor and Chairman, BoS
Dept. of ECE, AUCE (A), Andhra University, Visakhapatnam, India
Abstract
In this paper, a modified Costas code is
proposed based on the orthogonality
condition. If the time band width product is
increased beyond the ortogonality in Costas
code, the unwanted spikes known as grating
lobes are appeared. In this work, Costas code
is nested with phase codes using kronecker
product. The grating lobes are nullified and
sidelobes also decreased. The performance
evaluation of Nested Costas codes are carried
out for different time band width products to
obtain good range resolution.
Key words- Costas code, Phase codes,
Autocorrelation Function, Grating lobes,
Sidelobes and Range resolution.
I. INTRODUCTION
In multiple target environments high range
resolution is required to detect target exact
location. Indeed pulse compression is used in
radar [1][2]. In pulse compression frequency or
phase modulation is used to increase the band
width before the transmitting the signal and
reference signal is send to receiver. Received
signal is autocorrelated with the reference signal
to get the information about the target [3]. In
phase and frequency modulations different
signals are used, such as Barker codes, Poly
phase codes, m-sequence codes, Golomb codes,
Px code, Linear Frequency Modulation (LFM),
Non-Linear Frequency Modulation (NLFM),
Stepped Frequency Train of LFM(SLFM) and
Costas codes...etc[4-7].
The Costas codes are capable of
improving the Ambiguity function in range and
Doppler directions [4]. In [5, 7, 8], the authors
proposed different algorithms to construct
Costas codes. From all the approaches the
sequence of length M approaches Costas
properties by using the difference matrix method
[9]. When time band width product of Costas
code is equal to one, there are no grating lobes in
range axis [10-14]. If time bandwidth product
t ∆f > 1, the unwanted peaks (grating lobes) are
appeared in delay axis and decrease Peak Side
Lobe Ratio (PSLR). To nullify the grating lobes,
author in [15], modulate the Costas sub-pulses
with LFM ones. Grating lobes are nullified by
using search processing but side lobes are
remaining. In [16], phase codes are introduced in
Costas sub-pulses to overcome the grating lobes
and side lobes problem. In the proposed method,
phase codes such as Barker code and poly phase
codes (Frank code P3 and P4 code) are nested
with Costas code using Kronecker product to
nullify the grating lobes and thereby improve the
PSLR.
This paper is organized as, section II,
Construction of Costas code and its Ambiguity
Function (AF). Section III gives the information
about phase codes. Section IV relates the grating
lobes and sub-pulse coding. In section V, Costas
code is nested with Phase codes using Kronecker
product to nullifying the grating lobes. Section
VI, gives the conclusions.
II. COSTAS CODE
Costas waveform [4] is generated by a
long pulse width T into a series of M sub pulses,
where the frequency of each sub-pulse is selected
from M contiguous frequencies within a band. In
Costas codes the frequencies for the sub-pulses
are selected randomly. For this purpose, consider
the M×M matrix shown in Fig. 1(a), the size of
Costas code is M= 8(2, 6, 3, 8, 7, 5, 1, 4). It

INTERNATIONAL JOURNAL OF CURRENT ENGINEERING AND SCIENTIFIC RESEARCH (IJCESR)

18
allows better approximation of ideal Ambiguity
function.
(a)
(b)
Fig. 1. (a) Costas code M=8(2, 6, 3, 8, 7, 5, 1,
4). (b) Ambiguity Function of Costas code M=8
The Costas signal corresponding to the code
[a1,a2……am] is given by:
t ∑ u t m 1 t . (1)
Where 2
b
t
t
 
 
 

b
t
t
 
 
 
1 if 0 t t
0 elsewhere
For simple Costas signal
∆ /
A. Performance measure
The performance measure of Pulse
Compression is Peak side lobe ratio (PSLR). It is
the ratio of the maximum of the sidelobe level to
maximum mainlobe level.
PSLR (dB) =20 log10
| |
| |
. (2)
Where R(0) is the mainlobe level and R(k) is
maximum side lobe level among all sidelobes.
The Ambiguity function for Costas code M=8 is
shown in figure. 1 (b).
III. PHASE CODES
B. Barker code
The highest length of Barker code is 13.
The length of the Barker code is the ratio of the
mainlobe to the sidelobe level in the
autocorrelation function and the side lobes are
‘1’ or ‘-1’ [17].
Barker code 13 = [1,1,1,1,1,-1,-1,1,1,-1,1,-1,1]
C. P3 and P4 Code
P4 [18-20] is having the smallest phase
increments from sample to sample on the center
of the waveform. The P4 code is more Doppler
tolerant and better precompression bandwidth
limiting than the P3 code. It is constructed with
any length of N. P3 code 16 is given as
0
  

  
0
  

  
P4 code 16 is given as
0
  

  
0
  

  
D. Frank Code
A Frank code of N sub-pulses are
referred to as an N-phase Frank code [21]. The
first step in computing a Frank code is to divide
360 by N, and define the result as the
fundamental phase increment ∆φ.
∆φ (3)
For N-phase Frank code the phase of each sub-
pulse is computed from
0 0 0 0 ⋯ 0
0 1 2 3 … N 1
0 2 4 6 … 2 N 1
… … … … … …
… … … … … …
0 N 1 2 N 1 3 N 1 ⋯ N 1
∆ϕ
(4)
Where each row represents a group and a column
represents the sub-pulses for that group. For
example, a 4-phase Frank code has N=4, and the
fundamental phase increment is ∆φ =
90 . It follows that
frequency







∆f 
tb Time


19

0 0 0 0
0 90 180 270
0 180 0 180
0 270 180 90
⇒
1 1 1 1
1 j 1 j
1 1 1 1
1 j 1 j
(5)
Therefore, a Frank code of 16 elements is given
by
F ={1 1 1 1 1 j –1 –j 1 –1 1 –1 1 –j –1 j}
The phase of the ith
code element in the jth
row or
code group may be expressed mathematically as
Ф
,
=(2π/N)(i-1)(j-1) (6)
Where i = 1, 2, 3,..N and j = 1, 2, 3, ..., N. In
above equation the index i ranges from 1 to N for
each value of j and the number of code elements
formed is equal to N . The Ambiguity plot of
Frank Code of length 16 is shown in figure 2.
Fig. 2. Ambiguity plot for Frank Code of length
16.
IV. COSTAS SUB-PULSE CODING AND
GRATING LOBES
Costas with LFM (Linear Frequency
Modulation) is having low PSLR and large main
lobe width when it obeys the orthogonality
condition. In Costas codes, when t ∆f > 1 will
give grating lobes corresponding to the value of
t ∆f shown in figure 3, where X axis represents
the delay and Y axis represents the normalized
Autocorrelation. The Autocorrelation function
R(τ) is defined by the product of two functions,
R (τ) and R (τ) [15]. R (τ) indicates
autocorrelation function of the code and R (τ)
indicates grating lobes distribution shown in
equation number 7.
R
τ
t
R
τ
t
R
τ
t
R .
∆
∆
; τ<t
(7)
The grating lobes are appearing at maxima of R
(τ).
τ
∆
t
k=0, 1, 2, … |t ∆f| (8)
In [16], the author proposes the phase codes with
good correlation properties in the sub-pulses of
Costas code, to lower the level of grating lobes
and sidelobes. The PSLR values for sub-pulse
coding is tabulated in table 1.
(a)
(b)
(c)
Fig. 3. Autocorrelation Function plots for (a).


20
t ∆f=1, (b). t ∆f=5, (c) t ∆f=10
E. Kronecker Product
Nested codes can be obtained by using the
Kronecker product of two codes [2,22] whose
initial matched filter response is good. The
Kronecker product is denoted by ⨂, is an
operation on two matrices of arbitrary size
resulting in a block matrix. ‘C’ is an r x s matrix
and ‘D’ is a v × u matrix, then the Kronecker
product C ⨂D is the rv × su block matrix.
C⨂D
C D ⋯ C D
⋮ ⋱ ⋮
C D ⋯ C D
(9)
The Costas code of length 8 is considered
as ‘C’ and another one is Frank 16 is considered
as ‘D’.
C⨂D = [2,6,3,8,7,5,1,4][ 1,1,1,1,1,j,-1,-
j,1,-1,1,-1,1,-j,-1,j]
In the proposed approach, Costas code is
nested with Frank code 16 using Kronecker
product. The grating lobes are suppressed and
peak sidelobes also decreased shown in figures
4-6. When peak side lobes are decreased false
alarm is avoided. The range resolution is also
increased because of increasing the phase values
in Costas code by Frank 16. The PSLR values for
all cases are tabulated for Nested Costas with
Frank code 16 in table 1. When t ∆f=1 the PSLR
is obtained for Nesting of Costas with Frank code
16 is -39.3 dB. For t ∆f = 5 and t ∆f = 10, the
correspond PSLR are -41.0 dB and -44.6 dB.
The same approach is applied for Nesting
of Costas with Barker code 13, P3 code 16 and
P4 code 16 for t ∆f = 1, t ∆f = 5 and t ∆f = 10.
The figures 7-9 represents the Autocorrelation
Function plots for nesting of Costas codes with
Barker 13 code. In this case maximum PSLR -
27.7 dB is obtained for t ∆f = 5.
Fig. 4. Autocorrelation Function plot for
Costas-Frank 16 for t ∆f=1. Top: Normalized
ACF. Bottom: Normalized ACF in dB.
Costas-Frank 16 for t ∆f=5. Top:
Normalized ACF. Bottom: Normalized ACF in
dB.
Costas-Frank 16 for t ∆f=10. Top: Normalized


21
Figures 10-12 represents the Autocorrelation
Function plots for nesting of Costas code with P3
length 16 code. The corresponding PSLR values
are tabulated in table 1 and compared with Costa
sub-pulse coding. The Highest PSLR is obtained
at t ∆f=10 is -39.7 dB when compared to the
time bandwidth products t ∆f=1 and t ∆f=5.
Costas-Barker 13 for t ∆f=1. Top: Normalized
Costas-P3 16 for t ∆f=1. Top: Normalized ACF.
Bottom: Normalized ACF in dB.


22
Costas-P3 16 for t ∆f=10. Top: Normalized
Figures 13-15 represents the Autocorrelation
Function plots for nesting of Costas code with P4
length 16 code. The corresponding PSLR values
are tabulated in table 1 and compared with Costa
sub-pulse coding. The maximum PSLR is
-42.6 dB is achieved for t ∆f=1, when
compared to the time bandwidth products t ∆f=1
and t ∆f=5.
Costas-P4 16 for t ∆f=10. Top: Normalized
Table. 1. Comparison of PSLR for Costas sub
pulse coding and nested Costas coding
VI. CONCLUSIONS
In Costas code, the time band width product is
increasing beyond orthogonality condition to get
high range resolution in multiple target
environment. When time band width is increases,
grating lobes are presented. In this work, Frank
code 16 , Barker code 13, P3 code 16 and P4 code
16 are nested with Costas code using Kronecker
product to mitigate the grating lobes and
decreasing the sidelobes. When Costas code is
nested with Frank code of length 16, the PSLR =
-44.6 dB is obtained for t ∆f=10. For Costas-
Barker code of length 13, the PSLR = -27.7 dB
is obtained for t ∆f=5. With Costas-P3 16, the
PSLR = -39.7 dB is obtained for t ∆f=10. The
Costas-P4 16, the PSLR = -42.6 dB is obtained
for t ∆f=1. The performance of nesting of
Costas code with Frank code is shown better
PSLR when compared to P3, P4 and Barker
codes. When compared to Costas sub-pulse
coding improved PSLR is obtained for Nested
Costas coding.
Phase
Code
s
Costas Sub- pulse
Coding PSLR
Nested Costas
Coding PSLR
. ∆
1
. ∆
5
. ∆
10
. ∆
1
. ∆
5
. ∆
10
Frank
16
-17.2
dB
-19.2
dB
-23.3
dB
-39.3
dB
-41.0
dB
-44.6
dB
Barke
r 13
-14.1
dB
-19.8
dB
-23.6
dB
-23.6
dB
-27.7
dB
-22.9
dB
P 16 -- -- --
-39.3
dB
-36.0
dB
-39.7
dB
P 16
-16.7
dB
-18.7
dB
-20.9
dB
-42.6
dB
-36.1
dB
-39.7
dB


23
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