Sidelobe level reduction for complex radar signals with small base

Mariusz Łuszczyk

Przemysłowy Instytut Telekomunikacji S.A.

30, Poligonowa St., 04-051,Warsaw, Poland

tel.+48 22 486 52 08 , fax: +48 22 4865 263,

email:mariusz.luszczyk@pit.edu.pl

Aleksander Łabudziński

Przemysłowy Instytut Telekomunikacji S.A.

30, Poligonowa St., 04-051,Warsaw, Poland

tel.+48 22 486 55 43 , fax: +48 22 4865 263,

email:aleksander.labudzinski@pit.edu.pl

Abstract

The mostly applicable radar signal is pulse signal with

linear frequency modulation (LFM). The main feature such

signal is time-bandwidth product (so called base) which is

calculated as product of pulse duration and signal bandwidth.

Theoretical peak to sidelobe level is achievable with time-

bandwidth product value grater than 100. For time-bandwidth

product smaller then 100 (i.e. short pulse duration or narrow

deviation of FM modulation) peak-to-sidelobe level (PSL)

and pulse compression coefficient are significantly reduced.

Non-linear frequency modulated (NLFM) radar signal for

small time-bandwidth product features better PSL and pulse

compression coefficient. The NLFM signal synthesis

algorithm and simulation results are presented in the paper.

LFM and NLFM signals with small time-bandwidth product

are compared and results are discussed in aspect of radar

resolution improvement.

Keywords: radar signal; pulse compression; waveform

design; NLFM; peak-to-sidelobe level

1. Introduction

Radar waveform design based on complex signals and

their pulse compression are important issue in modern radar

signal processing [1]. The matched filter is a filter whose

impulse response, or transfer function are determined by a

certain signal, in a way that it results in the maximum

attainable signal to noise ratio (SNR) [3], [6]. Pulse

compression involves using a matched filter to compress the

energy in a signal into a relative narrow pulse. Moreover the

pulse compression technique increases the range resolution of

transmitted pulse without having to increase the peak transmit

power [1].

Pulse compression ratio sK is defined as a relation

maximum SNR at the matched filter output to the SNR at the

matched filter input at the time 0t , when the output signal

amplitude is maximum

( ) ( )( )i

os

SNR

tSNRK 0max

= (1)

In practise pulse compression ratio is defined as relation

width of main lobe of compressed signal measured at level of

-4dB to the pulse duration (signal before compression).

The LFM, or chirp waveform, has superior performance

in pulse compression radar since they can be easily generated

and processed. Many diverse techniques and devices have

been developed to provide the required pulse compression

processing for these signals [2]. However, the LFM has large

sidelobes with respect to the mainlobe. Reducing the

sidelobes can be accomplished by linear filtering the output,

i.e., applying window functions. However, since the

cumulative filtering is no longer precisely matched to the

signal, it necessarily reduces output SNR as well, typically by

1-2 dB. A single decibel of SNR is equivalent to a 25%

transmitted power. It has a great influence on power budget in

modern power transmitters based on semiconductors power

modules.

The non-linear frequency modulation (NLFM) signal is

another continuous phase modulation waveform applicable to

pulse compression radar. The NLFM requires no time

weighting for range sidelobe suppression and that can avoid

SNR degradation associated with sidelobe filtering required

with LFM radar signals.

2. Designing non-linear frequency modulated signal

It is well-known that NLFM chirp modulation can

advantageously shape the power spectral density such that the

autocorrelation function exhibits substantially reduced

sidelobes in comparison with LFM signal [2], [3]. An

alternative method for modifying the shape of the spectrum

which avoids energy loss is to distort the frequency

modulation function rather than the amplitude window.

However precision NLFM chirps are more difficult to design,

produce, and process.

In this section the design method is explained. An

arbitrary frequency modulated signal (chirp) can be written

( ) ( ) ( )[ ]tjexptats ϕ= (2)

where: ( )ta - amplitude modulation function, ( )tϕ - phase

modulation function.

The signal is limited to finite rectangular pulse with

shape

⎪⎩⎪⎨⎧

≤=⎟⎠

⎞⎜⎝⎛

elseT

t

T

trect

02

11

(3)

where T - is pulse time duration.

The Fourier transform of the chirp can be expressed as

( ) ( ) ( )[ ] ( ) ( ){ }[ ]∫ ϕ+ω−π

=Φ=∞

∞−

dtttjexpta2

1fjexpfSfS (4)

where: ( )fS - spectrum of , ( )fS - amplitude spectrum and

( )fΦ - spectrum phase function.

A time the instaneous frequency is determined by the

time derivative of phase at

( )kk tf ϕπ

′=2

1 (5)

IRS 2012, 19th International Radar Symposium, May 23-25, Warsaw, Poland

146 978-1-4577-1837-3/12/$26.00 ©2012 IEEE

The stationary-phase concept says that the energy

spectral density at the frequency is relatively large if the rate

of change of the frequency at this time is relatively small. The

inverse relationship between spectral density and frequency

rate of changes is approximated by the expression [3].

( ) ( )( )k

k2

2

kt

ta2fU

ϕ′′π≈ (6)

In NLFM ( )kta is constant and shape the spectrum is

dependent of ( )ktϕ ′′ . ( )fU is approximate by a function

( )fV , which is independent of the amplitude function ( )tg .

According to the stationary-phase principle the second

derivative of phase spectrum is expressed [3]

( ) ( )( )ta

fVf

2

2

2π=Φ ′′ (7)

where: ( )fV is a window function which approximates

amplitude spectrum of designed NLFM signal. ( )fV is

defined over the frequency interval 22 BfB ≤≤− , the first

derivative ( )fΦ′ is obtained by integrating the second

derivative

( ) ( ) ζζ df

f

B

∫−

Φ ′′=Φ′

2

(8)

The group delay function ( )fT is given by the expression

( ) ( )ffT Φ′−=π2

1 (9)

The instantaneous frequency as a function of time is the

inverse of ( )fT

( ) ( )fTtf 1−= (10)

The phase function of the designed signal is obtained from the

frequency function

( ) ( ) ζζπϕ dft

t∫=

0

2 (11)

Phase function and amplitude function are required to

generate the complex envelope of NLFM signal.

The design method, step by step can be summarized as:

− choose pulse time duration T and signal bandwidth B ,

− choose the desired amplitude spectrum (Kaiser window

function in this case),

− choose the parameter value for window function,

− integrate a square root of window function in range over

designed signal bandwidth according to (8),

− calculate of the inverted function according to (9) and

(10),

− calculate ( )tϕ according (11),

− calculate the radar signal ( )ts according to (2).

3. Kaiser window function

To generate NLFM signal with uniform amplitude and a spectrum following a square root of one-parameter Kaiser window function the algorithm expressed in second section this paper is used. The Kaiser window function is defined by the formula

( )⎪⎪⎩

⎪⎪⎨⎧

≤≤⎟⎟⎟⎠⎞

⎜⎜⎜⎝⎛ ⎟⎠

⎞⎜⎝⎛

−−

=

otherwise

MnI

M

nI

wn

0

0

12

1

0

2

0

πα

πα (11)

where:

− 0I is the zeroth order modified Bessel function of the first

kind,

− α is an arbitrary real number that determines the shape of

the window in the frequency domain and

parameter πα=K ,

− M is an integer and the length of the sequence is N=M+1.

Obviously, it is difficult to get the explicit expressions of

( )tf , ( )tϕ to synthesize NLFM complex signal. So the

numerical method is employed to solve nonlinear equations

and perform integration in order to get their discrete values at sampling points. Kaiser window formula in Matlab

programming environment is expressed:

)(K 0))/bessel_ 1.^2-) Mn/ *(2 -(1sqrt *(K bessel_0=w (12)

Kaiser window function with arbitrary parameter value 1K =

is depicted on the Fig 1. The shape of the function is the

spectrum of designed NLFM signal. Choosing Kaiser window

parameter value ( )K , the spectrum of the designed signal can

be changed.

Fig.1. Designed spectrum of NLFM signal based on Kaiser window

with parameter K=1.

4. Simulation results for signal with small time-

bandwidth product

According to the NLFM signal synthesis algorithm

presented in the second part of this paper some simulation

results are presented. For the experiment part of this paper intrapulse modulation parameter are arbitrary chosen:

− bandwidth modulation - MHzB 75.0= ,

− pulse time duration - sT µ20= .

Time-bandwidth product is 15=BT . Kaiser window

function with arbitrary chosen parameter 1K = is used to

approximate designed NLFM spectral shape (Fig.1).

Transformation of desired spectrum to a NLFM signal is

147

depicted on group delay function (Fig.2), instantaneous

frequency modulation frequency (Fig.3).

Fig.2 The group delay function of the designed NLFM signal.

The frequency characteristic of the NLFM signal vs. pulse

time duration time is presented in Fig. 3. The frequency

modulation characteristic is obtained on numerical method

way in Matlab environment.

Fig. 3 Modulation function of NLFM signal

(pulse duration T=20µs; bandwidth B=0,75MHz)

The resulting autocorrelation function is shown in Fig.6.

Designed NLFM signal has an almost flat pedestal of less then -40dB, excluding first time sidelobe at -37dB. The time

side lobes for designed NLFM signal are much better then for

LFM signal with the same time-bandwidth product [3,5].

The effective compression ratio is higher then the one time-bandwidth product. Main lobe width at -4dB level is

about 60ns. The pulse compression ratio is about -26dB.

The ambiguity function ( )Df,τχ (AF) is the essential tool

for assessment of complex radar signal. The function is

regarded as two-dimensional autocorrelation function with

two parameters: τ - time delay and Df - doppler shift. The

autocorrelation function is particular case of AF for Doppler

shift 0=Df . Autocorrelation function of LFM signal and

NLFM signal are depicted at Fig. 4 and Fig. 5 respectively.

The main feature both signals is small time-bandwidth

product, equals 15.

Fig. 4 Autocorrelation functions of LFM complex radar signals with pulse

duration T=20µs and bandwidth B=0,75MHz.

Fig. 5 Autocorrelation functions of NLFM complex radar signals with pulse

duration T=20µs and bandwidth B=0,75MHz:

Computer simulations were taken for LFM and NLFM

signals, where pulse compression ratio sK and PSL ratio

were considered (table 1). NLFM signal improves essential characteristics of complex signal with small time-bandwidth

product in comparison with LFM signals (i.e. PSL and pulse

compression ratio). PSL ratio for LFM signals with base less

than 100 have quite different characteristics than signals with base more than 100 [4]. PSL ratio significant deterioration

can not be accepted id radar application. NLFM signals (with

the base like LFM) can significantly improve discussed

characteristics. Moreover NLFM signal features SNR improvements referring to lack of weighting processing in

matched filtering [5].

Table 1 Main characteristics of compressed signal (LFM

and NLFM)

signal parameters

LFM NLFM

Ks PSL

[dB] Ks

PSL

[dB]

B=0,75MHz

T=5µs 2,5 -13 13 -29

T=10µs 5 -20 20 -28

T=20µs 10 -26 26 -37

B=6MHz

T=1µs 4 -18 10 -41

T=5µs 20 -32 100 -56

T=10µs 40 -38 110 -61

T=20µs 87 -43 200 -63

148

4. Conclusion

In this paper the design of NLFM signal is based on the

Kaiser window function in terms of practical radar parameters. Parameter window offers possibility to control

the designed signal spectrum and the signal autocorrelation

function. Presented algorithm NLFM signal synthesis is

illustrated with essential signal characteristics in time and frequency domain. Nonlinear FM waveforms offer a radar

matched filter output with inherently low range sidelobes

(about -80dB) for signals with time-bandwidth product more

than 100. For signals with less base parameter the PSL rate and pulse compression rate are better in comparison with

LFM signal with the same base parameters.

References

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RF digital modulation“ Electrical and Electronic

Technology, 2001. TENCON. Proceedings of IEEE

Region 10 International Conference, vol. 2, 588-591, August 19-22, 2001.

[2] Kroszczyński J., „Kompresja impulsu w radiolokacji“

Prace PIT nr 58, 1967 (in Polish).

[3] Levanon N., Mozeson E., „Radar Signals“. John Wiley&Sons, Inc., Hoboken, New Jersey, 2004.

[4] Cook Ch.E., Bernfeld M., „Radar Signals. An

Introduction to Theory and Application“, Artech House

Inc., Boston 1993. [5] Leśnik C., „Nonlinear Frequency Modulated Signal

Design“. Acta Physica Polonica A, vol. 116, 2009.

[6] Endres, T. J., R. B. Hall, A. M. Lopez, „Design and

analysis methods of a DDS-based synthesizer for military spaceborne applications" Frequency Control

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149