Analysis and Detection of S-Shaped NLFM Signal Based on

Transcription

Journal of Communications Vol. 10, No. 12, December 2015
Analysis and Detection of S-Shaped NLFM Signal Based
on Instantaneous Frequency
Jun Song, Yue Gao, and Demin Gao
Department of information science & technology, Nanjing Forestry University, Nanjing 210037, China
Email: songjun@njfu.edu.cn
Abstract—The characteristics
typical S-shaped Nonlinear
Frequency Modulation (S-NLFM) signal are analyzed, and a
detection method for this type of signal is also proposed. First of
all, the instantaneous frequency of the S-NLFM signal based on
sinusoid modulation and tangent modulation is obtained by
phase unwrapping respectively. Then, two characteristics are
obtained from instantaneous frequency. Consequently, a
recognition method is proposed based on the two characteristics
obtained in the previous step. At last, the detection method for
the S-NLFM signal is implemented. Simulation results indicate
that the proposed method is robust and the detection rate is
more than 90% under an SNR condition of 0 dB or higher
regardless of the modulation parameters.
Index Terms—Instantaneous frequency, signal detection, Sshaped NLFM, phase unwrap
I.
INTRODUCTION
Pulse compression is used in many radar and active
sonar systems to achieve long-range performance and
fine range resolution simultaneously. Long-range
detection implies fine noise reduction, requiring long
time transmissions. To achieve fine range resolution, the
band width of the pulse must be enlarged accordingly.
These conflicting requirements can be realized by
modulating an S-shaped function (eg. sinusoid or tangent
function) with a Linear Frequency Modulation (LFM) [1],
thus constructing the typical S-shaped Nonlinear
Frequency Modulation signal (S-NLFM). In recent
decades, the S-NLFM signal processing has attracted
increasing attention in the scope of radar and
communication engineering [2]-[11].
In this work, character analysis and detection of the SNLFM signal are implemented. As all known, LFM is a
typical pulse compression method, and the side lobes are
usually high when the LFM signal is processed by a
matched filter. As a result, a weighted filter should be
utilized to obtain proper side lobe suppression. However,
a weighted filter may result in a decrease in the Signal-toNoise Ratio (SNR). A weighted filter is unnecessary for
side lobe suppression when the NLFM signal is processed
[1]. For the S-shaped NLFM signal, the chirp rate
Manuscript received June 8, 2015; revised December 7, 2015.
This work was supported by the Jiangsu Higher Education Natural
Science Found under Grant No.13KJB220003.
Corresponding author email: songjun@njfu.edu.cn.
doi:10.12720/jcm.10.12.976-982
©2015 Journal of Communications
976
increases or decreases at the end of a pulse, which is
contrary to what occurs in the middle of a pulse. The
characteristics above result in timely side lobe
suppression [2]. Therefore, the S-NLFM signal is widely
utilized in all types of wireless applications [3], [4].
Studies on the wave design and performance analysis of
S-NLFM signal in the past several years have attracted
proper attentions [2], [5]-[7]. Recently, several studies
have focused on the character analysis and detection for
this type of signal [9]-[15]. In [6] and [14], researchers
proposed a double-characters detection method for
NLFM based on fractional Fourier transform (FRFT), and
the experiments validated its robust performance.
ZHANG et al. [7] analyzed intra-pulse feature by means
of fractal dimensions and recognized 10 typical radar
signals including NLFM, LFM, BPSK and FSK etc. K. J.
You et al. [9] employed Gini’s coefficient and maximum
likelihood classifier to measure the frequency inequality
and then to divide signal modulation types into four
classes. The algorithm of [9] exhibited excellent
performance. WANG J. [10] proposed a time-frequency
tilling based detector for detection of NLFM and
Polynomial Phase Signal (PPS). The method of [10]
exploited adaptive wavelet transform and Radon-Wigner
Transform (RWT) to get proper detection performance.
Most of previous studies exploited time-frequency
analysis, such as wavelet, RWT and FRFT, which
required heavy computation. In addition, the characters of
S-NLFM were not analyzed in previous works.
In this paper, two characteristics of typical S-NLFM
signal are analyzed based on instantaneous frequency,
and then a new detection method is proposed. Unlike
existing works on S-NLFM signal detection, the
contribution of our work has two aspects. One is to
extract two characteristics of the signal’s instantaneous
frequency by phase unwrap. The other is to design a
signal detection method based on the character analysis
above and segment filter. The computation of the
proposed method is mainly concentrated on the phase
unwrap and segment filter which result in easy
implementation in engineering. In addition, the analysis
and simulations show that our method for S-NLFM signal
detection is robust and is independent of modulation style.
The remainder of this paper is organized as follows.
The S-NLFM signal model and construction are
described in the next section, specifically the construction
of the signal phase of sinusoid based and tangent based S-
7
 B 2
K (m)
2 mt  
s(t )  A exp  j 
t  BT 
cos(
)  (5)
m
T  
m 1
  T
NLFM. Then two characteristics of S-NLFM signal’s
instantaneous frequency are analyzed in section III.
Consequently, the detection method and flow are
provided in Section IV. The simulation and numerical
results are presented in Section V. And the conclusion is
drawn in the final section.
In summary, the sinusoid based NLFM signal can be
classified into two types, one is with no weighting
coefficients (as in (2)), and the other is accompanied by a
set of weightings (as in (4)). The frequency curves of the
two types S-NLFM are plotted in Fig. 1.
II. SIGNAL CONSTRUCTION OF S-NLFM
7
The frequency functions of S-NLFM signals can be
divided typically into two types, one is sinusoid based
and the other is tangent based. Firstly, we discuss the
sinusoid based S-NLFM signal, and the frequency
functions can be expressed as:
(1)
where B is the bandwidth and T is the duration of the
signal pulse. Then the signal can be given as:

 B 2
2 t  
t  BT cos(
)
 s1 (t )  A exp  j 
T 

  T

 s (t )  A exp  j   B t 2  BT cos( 2 t )  
 

 2
T 
  T

(3)
3.74
 (t ) 
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-4
x 10
(6)
 B(1   )
T
t2 
 BT  ln(1  tan(2 t T ) 2 )
4  tan( )
(7)
The wave form of a tangent based S-NLFM signal can
be expressed as follows:
s(t ) 

A exp 

t
0
(4)
  B(1   ) 2  BT ln(1  tan(2 t T ) 2 )   (8)
j
t 
 
T
4  tan( )


and the frequency curves with different α and γ are shown
in Fig. 2.
Both the sinusoid based and tangent based S-NLFM
signals mentioned above are all typical ones in
Thus, the module of the sinusoid based NLFM signal is
provided by
3.76
where B is the bandwidth, α is the balance factor, and γ is
the weight of tangent function (    2 ).
Similarly, the phase function of the tangent based SNLFM signal can be implemented by integration. That is,
 (t )  2  f  d
T
3.78
 2 1    t

f (t )   B 
   tan(2 t T ) tan( ) 
T


is the sinusoid weighting coefficients, T is the duration of
the pulse, and B is the bandwidth.
The phase function of (1) is,
K ( m)
2 mt
cos(
)
m
T
m 1
3.8
In engineering, another type of S-NLFM signal is
based on a tangent function instead of a sinusoid one.
Accordingly, the frequency modulation-time function is
based on tangent function while mixed with linear one. In
addition, the engineering realizability should be taken
into consideration just like the implementation of
sinusoid based ones. Collins and Atkins [16] presented an
extended form of tangent based S-NLFM, which employs
a set of weightings on tangent function and linear
modulation rate to construct the frequency function. The
expression is as follows,
0.0082, 0.0055,  0.004
7
3.82
Fig. 1. The frequency curves of sinusoid based S-NLFM.
(2)
K (m)   0.1145, 0.0396,  0.0202,  0.0118,
t 2  BT 
3.84
Time (s)
where

3.86
3.7
7
B
with weighting coefficients
signal-1:without weightings
signal-2:without weightings
3.72
where A is the amplitude.
The sinusoid based NLFM signal has many advantages
such as low probability interception and fine range
resolution, but every coins has two sides, it is more
sensitive to Doppler effect than LFM signal. In an effort
to reduce the side lobe, a weighting window function in
frequency domain often must be employed while
processing the S-NLFM. For the Taylor weighting
method provides an approach to Dolph-Chebyshev
function, it is recommended in engineering [1].
According to [1], the frequency function of sinusoid
based S-NLFM with Taylor −40 dB pulse compression
response can be expressed as follows:
2 mt 
t
f (t )  B    K (m)sin

T 
 T m 1
x 10
3.88
Instantaneous Frquency (Hz)

2 t 
t
 f1 (t )  B  T  sin T  ,0  t  T




 f (t )  B  t  sin 2 t  ,0  t  T


 2
T 
T
3.9
977
engineering, and they have similar frequency modulation
curves and relevant performance of low probability
interception.
symmetrical (as Fig. 3). The linear part of the modulation
rate can be constructed as follows:
B
(11)
t
T
The difference in modulation rate between nonlinear
and linear frequency modulation can be derived as
flinear (t ) 
7
3.95
x 10
 1 = -0.5,1 = 1.2
 2 = 0.6, 2 = 1.4
3.9
7
f (t )  f (t )  flinear (t )  B   K (m)sin
3.85
m 1
The curve of
3.8
f (t ) is odd symmetrical. Consequently,
the two sides/areas of curve
f (t ) demarcated by central
3.75
line t  T / 2 are equal. That is,
3.7
SL  SR 
3.65
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Time (s)
x 10
T /2
f (t )dt 
0

T
T /2
(13)
f (t )dt
7
x 10
3.9
LFM Signal
NLFM with weighting coefficients
NLFM signal-1:without weightings
NLFM signal-2:without weightings
3.88
III. CHARACTERISTICS OF S-NFLM SIGNALS
The sinusoid-based S-NLFM signal is taken as an
example in this section. The frequency modulation
function can be derived by differential as follows:
df (t )
1 7
2 m
2 mt
 (t ) 
 B[   K (m)
cos(
)]
dt
T m 1
T
T
(9)
7
B
2 mt
 [1   K (m)2 m cos(
)]
T
T
m 1
3.86
3.84
3.82
3.8
3.78
3.76
3.74
3.72
The NLFM signal will become an LFM one if all
weighting coefficients K (m) are equal to zero and the
3.7
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Time (s)
frequency modulation function is a constant B / T .
However, if K (m)  0 , the frequency modulation
function is a variable of time. That is, various modulation
rates can be calculated at three sites of the frequency
modulation function, namely, initial, end, and middle
time. That is,
1
-4
x 10
Fig. 3. The instantaneous frequency of sinusoid based S-NLFM and
LFM signal.
7
3.95
(10)
It can be found out that  (0)   (T )   (T 2) . These
results of (10) indicate that the modulation rate at the
initial moment is equal to that of the end moment but is
different from that of the middle moment. The difference
in modulation rate is an important characteristic of the SNLFM signal. Therefore, it can be utilized as a basis for
detection.
Given the sinusoid function is an odd one, the front and
back halves of the modulation rate curve are also odd

As shown in Fig. 3, the conclusion of (13) is clearly
validated.
1
-4
Fig. 2. Frequency curves with different α and γ.
7
B


(0)

[1

K (m)2 m];


T
m 1

7

B
  (T )  [1   K (m)2 m];
T
m 1

7

B
  (T 2)  [1   K (m)2 m  cos( m)].
T
m 1

2 mt
(12)
T
x 10
LFM Signal
tangent based S-NLFM signal-1#
tangent based S-NLFM signal-2#
3.9
3.85
3.8
3.75
3.7
3.65
0
0.1
0.2
0.3
0.4
0.5
Time (s)
0.6
0.7
0.8
0.9
1
-4
x 10
Fig. 4. The instantaneous frequency of tangent based S-NLFM and LFM
signal.
As the characteristics of tangent based S-NLFM are
similar to those of sinusoid-based ones (as shown in Fig.
4), we will not discuss them repeatedly.
978
According to the analysis above, the characteristics for
S-NLFM detection are summarized as follows.
1) The modulation rate of initial moment t  0 is equal
to that of the end moment t  T . The two moments are
different from middle moment t  T / 2 .
2) The instantaneous frequency curve is odd
symmetrical in the middle moment if the linear part is
eliminated.
In engineer application, the calculation of
instantaneous frequency and modulation rate may be
disturbed by noise. As a result, deviation from the
theoretical value is observed. Thus, the rules for S-NLFM
detection based on the two characteristics above should
be modified as follows.
1) The modulation rate of initial moment t  0 is
approximately equal to that of end moment t  T ;
2) The instantaneous frequency curve is almost odd
symmetrical in the middle moment if the linear part is
eliminated.
In order to weaken the disturbance of noise, the SNLFM signal should be pre-filtered to improve the SNR
level. The segment filter algorithm [17] can be
implemented to process the S-NLFM signal because there
is no sudden break or turn in the instantaneous frequency
of this type of signal.
Step 2: The considered signal is processed by segment
filtering to improve the SNR level. The signal’s phase is
unwrapped to obtain instantaneous frequency f (t ) .
Bandwidth B is estimated with (15).
Step 3: The linear frequency modulation part is
constructed as
flinear (t ) 
f (t )  f (t )  flinear (t )
should be calculated. Then, the two sides/areas of
curve f (t ) , which is demarcated by central line t  T / 2 ,
are computed. The two areas are denoted as S L and S R .
If S L and S R satisfy the statements
SL  SR
SL  SR
0.9  SL / SR  1.1
we can regard the statements above as one characteristic
for S-NLFM detection.
Step 4: The three sections of the instantaneous
frequency curve f (t ) are cut out. The three sections are
located at the initial, middle, and end moments of f (t ) .
Each section is a tenth of the total length of f (t ) . The
three sections are then subjected to regression fitting
through a linear module. The coefficients of one-degree
terms are the correct modulation rates at the three
moments above, which are denoted as  (0) ,  (T / 2) ,
and  (T ) . If the following statements are satisfied,
0.5   (T ) /  (0)  1.5
and
(14)
 (T / 2)

Bandwidth estimation is then considered. By setting
tS  0 and tE  T , the initial and terminal values of
instantaneous frequency are calculated. Then, the
estimate of bandwidth B can be obtained from the
difference of the values above, that is,
t 0
 f (t )
t T
mean  (T ),  (0)

 2 or
 (T / 2)

mean  (T ),  (0)

 0.5
they can be regarded as another characteristic for SNLFM detection.
Step 5: If the two characteristics mentioned in steps 3
and 4 are both fulfilled, the signal can be regarded as Sshaped NLFM. Otherwise, it is another type of signal.
(15)
The characteristics for S-NLFM signal detection were
studied, and the construction of the linear frequency
modulation part was also discussed. In conclusion, the SNLFM signal detection flow can be summarized as
follows.
Step 1: By using the signal arrival time estimation
algorithm in [18] and phase unwrapping, pulse width can
be estimated as (14).
 0.1
and
In the foregoing subsections, we discussed two
characteristics of S-NLFM signal for detection.
According to the analysis above, the linear frequency
modulation part of S-NLFM instantaneous frequency
should be constructed to allow for further processing.
Bandwidth B and pulse width T are essentially required
to build the linear one as shown in (11). First, the
estimation of pulse width T is considered. The initial time
(denoted as t S ) of pulse and end time (denoted as t E ) can
be estimated with the signal arrival time estimation
algorithm proposed in [18]. Pulse width is estimated by
B  f (t )
t
T
The difference,
IV. DETECTION OF S-NLFM SIGNAL
T  tE  tS
B
V. SIMULATIONS AND PERFORMANCE ANALYSIS
Computer simulations are conducted to validate the
proposed detection algorithms in this section.
Experiment 1: In the simulations, sinusoid and tangent
based S-NLFM signals are selected with pulse width T =
100 µs and bandwidth B = 2 MHz. The sampling
frequency is 200 MHz. The initial frequency is assumed
979
to be f 0  37 MHz, and the coefficients K (m) are similar
to those in (3). The coefficients of tangent-based SNLFM are   0.5 and   1.4 . Noise is added as white
complex Gaussian noise with zero mean. At each SNR
level, 1000 Monte Carlo simulations are performed to
obtain the detection results.
To improve the detection performance, the signal is
pre-processed by a segment filter in almost all the
experiments. The original signal without filter processing
is also simulated for comparison. The value of input SNR
varies from −6 dB to 16 dB at increments of 1 dB. The
detection simulation results are shown in Fig. 5.
As shown in Fig. 5, the detection performance of
sinusoid-based S-NLFM is almost equivalent to that of
the tangent-based one. However, the segment filter
influences the detection performance significantly. The
SNR level should be higher than 9 dB to achieve
detection probability that is higher than 90% for the
original signal. However, for filtered signal, only 0 dB
SNR level or higher is required to achieve the same
detection performance.
weighting coefficients. Various ones with different
parameters are chosen to compare the detection
performance. One is with pulse width T= 100 µs and
bandwidth B = 2 MHz, and the second is with pulse width
T = 150 µs and bandwidth B = 2.5 MHz, the third is with
pulse width T = 180 µs and bandwidth B = 3 MHz. In
addition, their initial frequencies are different from each
other, and are random from 30 MHz to 40 MHz. The
detection performance with different SNR levels is
plotted in Fig. 6.
Experiment 3: In these simulations, two sinusoid-based
ones are introduced, one is accompanied by weighting
coefficients as in (3) and the other is on the contrary, that
is to say there are no weighting coefficients for sinusoid
function in the second signal. The input SNR ranges also
from −6 dB to 16 dB. The numerical results are shown in
Fig. 7.
100
90
80
Detection times
70
100
90
Detection times
80
60
50
40
70
30
60
20
50
10
40
0
sinusoid-based one with weightings
sinusoid-based one without weightings
-5
0
10
15
Fig. 7. Detection results comparison of two kinds of sinusoid based SNLFM signals.
sinusoid-based S-NLFM(Orignal)
tangent-based S-NLFM(Orignal)
sinusoid-based S-NLFM(Filtered)
tangent-based S-NLFM(Filtered)
20
10
0
-5
0
5
10
Experiment 4: Three kinds of tangent-based S-NLFM
signals with different α and γ are taken into consideration.
The first signal is with α1=−0.5, γ1=1.2, the second is with
α2=0.6, γ2=1.4, and the third is with α3=0.7, γ2=1.1. The
simulation results are proposed in Fig. 8.
15
SNR (dB)
Fig. 5. Detection results of original and filtered S-NLFM in Monte
Carlo simulations.
100
100
90
90
80
80
70
70
Detection times
Detection times
5
SNR (dB)
30
60
50
40
sinusoid-based S-NLFM 1#
30
20
50
40
30
 1 = -0.5,1 = 1.2
 2 = 0.6, 2 = 1.4
 2 = 0.7, 2 = 1.1
20
10
10
0
60
0
-5
0
5
10
15
-5
0
5
10
15
SNR (dB)
SNR (dB)
Fig. 6. Detection results of three different sinusoid based S-NLFM
signals without weightings.
Fig. 8. Detection results comparison of three tangent based S-NLFM
signals.
Experiment 2: In the following experiments, we
consider sinusoid-based S-NLFM signals without
Experiment 5: In an effort to compare the detection
performance of different kinds of S-NLFM signals
980
including sinusoid based and tangent based ones, we
perform a serial of experiments at fixed 3dB and 0dB
SNR levels. There are totally five signals as mentioned in
Experiment 2-4. And the comparison results are listed in
Table I and Table II.
ACKNOWLEDGMENT
The authors would like to thank the support by the
Jiangsu Higher Education Natural Science Found under
Grant No.13KJB220003.
REFERENCES
TABLE I: DETECTION RESULTS FOR DIFFERENT SIGNALS AT 3DB SNR
Signals
sinusoid based
without weightings
sinusoid based
with weightings
[1]
tangent
based
sig.2
98
[2]
TABLE II: DETECTION RESULTS FOR DIFFERENT SIGNALS AT 0DB SNR
[3]
Detection
times
Signals
Detection
times
sig. 1
99
sig. 2
100
sinusoid based
without weightings
sig. 1
93
sig. 2
95
99
sinusoid based
with weightings
94
sig.1
99
tangent
based
sig.1
94
sig.2
93
[4]
[5]
Fig. 6 displays that the detection performance of the
proposed method is hardly affected by the modulation
parameter of sinusoid based S-NLFM signals. And if the
SNR level is higher than 0dB, the detection probability is
above 90% despite of the modulation parameter.
According to Fig. 7, we can find out that two kinds of
sinusoid based S-NLFM signals, one is with weighting
coefficients and the other is without weightings, are
conducted, and our algorithm proposes relevant detection
performance for these two kinds signals. That is to say
the proposed algorithm maintains a steady performance
regardless of the sinusoid weightings. What’s more, the
parameters of tangent based S-NLFM signals have little
impact on the detection results as indicated in Fig.8.
Table I shows that at the same 3dB SNR level, the
proposed algorithm performs almost 98%~99% detection
performance for three kinds of signals including tangent
based S-NLFM signal, sinusoid based one without
weightings and sinusoid based one with weightings.
Table II shows the detection performance is about
93%~94% at a fixed 0dB SNR level regardless of the
modulation style. Accordingly, we can draw a conclusion
that modulation style can not influence the robust of our
algorithm almost.
[6]
[7]
[8]
[9]
[10]
[11]
[12]
VI. CONCLUSIONS
[13]
S-NLFM has been attracted an increasing amount of
attention and research in recent decades. The
characteristics of two kinds S-NLFM signals are
investigated by means of phase unwrap in this study.
Furthermore, a detection algorithm based on
instantaneous frequency is developed. The numerical
simulations show that the detection probability is higher
than 90% under proper SNR levels if the signal is
preprocessed by a segment filter, and the proposed
algorithm has a robust detection performance and is
independent of modulation parameters.
[14]
[15]
[16]
981
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professor. His research interests include spectral estimation, array signal
processing, and information theory.
De-min Gao was born in Shandong Province
in 1980. He received his B.S and M.S. degree
in computer application technology from
Jingdezhen Ceramic Institute, Jingdezhen, in
2005 and 2008. During 2011-2012, he pursued
his study as a joint PhD student and joined the
research lab of Kwan-Wu Chin in School of
Electrical
Engineering,
University
of
Wollongong, Australia. His research fields
contain routing protocols for delay tolerant in wireless sensor networks.
Jun Song was born in Jiangsu Province,
China, in 1979. He received the B.S. degree
and M.S. degree from the China University of
Mining and Technology (CUMT), Xuzhou, in
2002 and in 2005 respectively. He received
the Ph.D. degree from the Nanjing University
of Aeronautics and Astronautics, Nanjing, in
2014, all in electrical engineering. He is
currently with the Department of information
science and technology, in Nanjing Forestry University as an associate
Yue Gao was born in 1995. She is a B.S. degree candidate in electronic
engineering in Nanjing Forestry University. Her current research areas
include adaptive signal processing and filter design.
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Analysis and Detection of S-Shaped NLFM Signal Based on

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