Frequency Coding Waveform with Segment LFM

Caicai Gao, Kah Chan Teh, and Aifei Liu School of Electrical and Electronic Engineering

Nanyang Technological University 50 Nanyang Avenue, 639798, Singapore {GaoCC; EKCTeh; LiuAF}@ntu.edu.sg

Abstract—In this paper, we propose a frequency coding waveform with segment linear frequency modulation (LFM) signal for the multiple-input multiple-output (MIMO) radar. This new waveform is modified from the discrete frequency coding waveform with LFM (DFC-LFM) by changing the subpulse from the LFM signal to segment LFM and extending the contiguous waveform to a pulse train. Compared to the existing frequency coding waveforms, the proposed waveform has lower autocorrelation sidelobes and lower cross correlation.

Index Terms—Cross correlation, Doppler tolerance, frequency coding waveform, MIMO radar, segment LFM.

I. INTRODUCTION For the multiple-input multiple-output (MIMO) radar [1],

[2], in order to separate the received signals at the receiving antennas, the cross correlation between signals transmitted by different transmitting antennas should be as low as possible. In addition, the designed waveforms should have low autocorrelation sidelobes simultaneously.

Several frequency diversity waveforms [3]-[7] have been designed to realize the above objectives, e.g., the discrete frequency coding waveform (DFCW) [5], the discrete frequency coding waveform with linear frequency modulation signal (DFC-LFM) [6], and the modified DFC-LFM [7]. The DFCW and DFC-LFM waveforms consist of contiguous subpulses, however, the MDFC-LFM waveform is a pulse train with discrete subpulses. The subpulses of the DFCW waveform are constant frequency signals, whereas, the subpulses of the DFC-LFM and MDFC-LFM waveforms are LFM signals. Those waveforms share the same frame, i.e, the carrier frequencies of different subpulses are different and different transmitted signals have different frequency sequences.

A code set based on a piecewise LFM has been proposed in [8]. For each transmitted signal, the LFM pulse was divided into two segments with different frequency slope polarities and same duration. The first segment is up-chirp and the second segment is down-chirp. The bandwidth of each pulse is fixed, however, the bandwidths of up-chirps of different signals are different. It was reported that this waveform has good Doppler tolerance, however, the autocorrelation sidelobes and cross correlation are not low enough. This contiguous piecewise LFM signal was digitized to subpulses with constant frequencies [9], which leads to high autocorrelation sidelobes.

In this work, a frequency coding waveform with segment LFM (FC-SLFM) is proposed. The subpulses in the DFC-LFM

Fig. 1. Time-frequency schematic diagram of the DFC-LFM waveform for

one transmitting antenna.

waveform are changed from the LFM signals to segment LFM. Each subpulse is divided into two segments with different bandwidths and opposite frequency slope polarities. The carrier frequency sequences and the bandwidth sequences are optimized by the genetic algorithm. Then, we extend the contiguous waveform to a pulse train. The autocorrelation, cross correlation and Doppler tolerance properties of the proposed waveform are presented and compared to the existing frequency coding waveforms.

II. FREQUENCY CODING WAVEFORM WITH SEGMENT LFM The time-frequency schematic diagram of the DFC-LFM

waveform for one transmitting antenna is shown in Fig. 1. Noted that and B are the duration and bandwidth of the subpulse, respectively. is the carrier frequency offset, is the total bandwidth. Assuming that the MIMO radar consists of M transmitting antennas and each transmitted signal includes N contiguous subpulses. The subpulses of the DFC-LFM waveform are LFM signals, and the nth subpulse transmitted by the mth antenna is defined by

( ) ( )22 1 /1( ) , 1

nm p pj f t n t j t B tn

m p pp

S t e e n t t ntt

π π− −= − ≤ < (1)

where m and n are the indexes of antenna and subpulse, respectively. is the carrier frequency of the subpulse and is defined by

0n n

m mf f C f= + Δ (2) where is the base frequency and is the parameter controlling the carrier frequency.

507978-1-4673-7297-8/15/$31.00 c©2015 IEEE

(a) (b) Fig. 2. (a) Time-frequency schematic diagram of the FC-SLFM contiguous

waveform. (b) Instantaneous frequency of one subpluse.

A. Proposed FC-SLFM Contiguous Waveform The frequency coding contiguous waveform with segment

LFM is proposed based on the DFC-LFM waveform by changing the subpulses from the LFM signals to segment LFM. The time-frequency schematic diagram of the FC-SLFM contiguous waveform and the instantaneous frequency of one subpulse are shown in Fig. 2(a) and (b), respectively. The subpulses are divided into two segments with the same duration. However, the bandwidths and frequency slope polarities of these two segments are different. The bandwidth of the first segment is denoted by , which is chosen from the following bandwidth set

(1 2 ) , 1,2, ,mn kB k B n N

N−= + ⋅ = (3)

where the parameter k, , is used to control the bandwidths in the above set. Hence, the bandwidth of the second segment is given by . The subpulse is defined by

( ) ( ) ( )

( ) ( ) ( )

2

2

22 1

2 2 1 2

2 , 1 2

( )2 , 2

nm

npm p

nm

n nm m p p

Bj t

tj f t n tp p

pnm B B

j tj f B B t n t t

p pp

e e n t t ntt

S t

e e nt t ntt

ππ

ππ

− −

−−

+ − − −

− ≤ <

=

≤ <

(4)

B. Autocorrelation and Cross Correlation Optimization In order to obtain lower autocorrelation sidelobes and

lower cross correlation, we use the genetic algorithm to optimize the carrier frequency sequence, the bandwidth sequence, and the bandwidth of the subpulse. The individuals in the population are defined by

1 1Ind { , , , , ; , , , , ; }i m M m MC C C B B B β= (5) where i is the index of each individual,

and are the frequency sequence and bandwidth sequence of the signal transmitted by the mth transmitting antenna, respectively. is the parameter to control the bandwidth of the subpulse, i.e.,

. The cost function is given by 1

1, 0 1 1max ( ) max ( ) , | |

M M M

m pq pm p q p

E R R Ntτ

τ γ τ τ−

= ≠ = = += + ≤ .

(6)

Fig. 3. Autocorrelation functions and cross-correlation functions of the

proposed FC-SLFM contiguous waveform.

TABLE I. ASPS AND CPS OF THE FC-SLFM CONTIGUOUS WAVEFORM

dB Signal 1 Signal 2 Signal 3 Signal 1 -28.6 -27.8 -28.2 Signal 2 -27.8 -29.3 -29.4 Signal 3 -28.2 -29.4 -27.7

The first summation term is the sum of autocorrelation

sidelobe peaks, the second summation term is the sum of cross-correlation peaks, and is a coefficient to control the weights of two summations. Here, is set to 1.

C. Numerical Results and Discussion We use the numerical results to show the performance of

the proposed waveform and compare to the existing frequency coding waveforms. The parameters are set as follows:

• Total bandwidth: Ball = 50 MHz, • Subpulse duration: tp = 5 μs, • Number of transmitting antennas: M = 3, • Number of subpulses: N = 32. • Wavelength: = 0.1 m. The parameter k in (3) is set to 0.2. With the optimized

sequences, the autocorrelation functions (ACF) and cross-correlation functions (CCF) of the FC-SLFM contiguous waveform are shown in Fig. 3. The corresponding autocorrelation sidelobe peaks (ASP) and cross-correlation peaks (CP) are listed in Table I. ASP and CP depend on the frequency and bandwidth sequences. Different signals with different and have different ASPs and CPs.

The performance comparisons among the proposed waveform, the DFCW and DFC-LFM waveforms are listed in Table II. We observe that the proposed FC-SLFM contiguous waveform has lower autocorrelation sidelobes and lower cross correlation comparing to the other two existing frequency coding waveforms. Especially, the average ASP and CP are about 5 dB lower than those of the DFC-LFM waveform.

508 2015 IEEE 5th Asia-Pacific Conference on Synthetic Aperture Radar(APSAR)

TABLE II. PERFORMANCE COMPARISONS FOR VARIOUS WAVEFORMS

Waveforms FC-SLFM DFC-LFM DFCW Average ASP (dB) -28.5 -23.9 -13.4 Average CP (dB) -28.4 -23.6 -29.5

(a)

(b)

Fig. 4. Ambiguity functions. (a) FC-SLFM contiguous waveform, (b) DFC-LFM waveform.

For one transmitted signal, the ambiguity functions of the FC-SLSM and DFC-LFM waveforms are shown in Fig. 4(a) and (b), respectively. The range sidelobes of the proposed waveform are lower than those of the DFC-LFM waveform. As the amplitude reduces quickly with the Doppler shift, the Doppler tolerance of the FC-SLFM contiguous waveform is not good. The Doppler processing with a group of match filters should be implemented to detect the moving target.

III. FREQUENCY CODING PULSE TRAIN WITH SLFM As the Doppler tolerance of the proposed frequency coding

contiguous waveform with segment LFM is not good, we extend the contiguous waveform to a pulse train, based on

Fig. 5. Autocorrelation functions and cross-correlation functions of the FC-

SLFM pulse train for .

which, the range-Doppler map can be obtained to indicate the moving target. The subpulse of the frequency coding pulse train with segment LFM is defined by

( ) ( ) ( )

( ) ( ) ( )

2

2

22 1

2 2 1 2

2 , 1 2

( )2 , 2

nm

npm r

nm

n nm m r p

Bj t

tj f t n Tp p

pnm B B

j tj f B B t n T t

p pp

e e n t t ntt

S t

e e nt t ntt

ππ

ππ

− −

−−

+ − − −

− ≤ <

=

≤ <

(7) where is the pulse repetition interval. In addition, in the genetic algorithm, different to (6), just the ASPs and CPs with the delay short than Tr are calculated. The cost function is modified to

1

1, 0 1 1max ( ) max ( ) , | |

M M M

m pq rm p q p

E R R Tτ

τ γ τ τ−

= ≠ = = += + ≤

(8) We use some numerical results to show the performance of

various waveforms. The parameters are set as follows: • Total bandwidth: Ball = 5 MHz, • Subpulse duration: tp = 50 μs, • Pulse repetition interval: Tr = 500 μs, • Number of transmitting antennas: M = 3, • Number of subpulses: N = 32. • Wavelength: = 0.1 m. The waveforms in this Section and Section II have the

same time-bandwidth product. The autocorrelation and cross-correlation functions of the FC-SLFM pulse train for are shown in Fig. 5. The corresponding ASPs and CPs are listed in Table III. The ambiguity functions of the proposed pulse train and the MDFC-LFM waveform are shown in Fig. 6(a) and (b), respectively. The performance comparisons are listed in Table IV.

2015 IEEE 5th Asia-Pacific Conference on Synthetic Aperture Radar(APSAR) 509

TABLE III. ASPS AND CPS OF THE FC-SLFM PULSE TRAIN FOR

dB Signal 1 Signal 2 Signal 3 Signal 1 -29.3 -34.1 -34.1 Signal 2 -34.1 -29.3 -34.5 Signal 3 -34.1 -34.5 -29.3

TABLE IV. PERFORMANCE COMPARISONS FOR VARIOUS WAVEFORMS

Bandwidth (MHz) FC-SLFM MDFC-LFM Average ASP

(dB) 5 -29.3 -31.7

50 -33.5 -31.3 Average CP

(dB) 5 -34.2 -31.0

50 -39.8 -31.5

(a)

(b)

Fig. 6. Ambiguity functions. (a) FC-SLFM pulse train, (b) MDFC-LFM waveform.

When the total bandwidth is equal to 5 MHz, with the delay shorter than the subpulse duration, the FC-SLFM pulse train and the MDFC-LFM waveform have the similar ambiguity function, the range sidelobes are very low and the Doppler sidelobe peaks are about -13.2 dB. It is shown in

Table IV that, while the bandwidth varies from 5 MHz to 50 MHz, the average ASP and average CP of the MDFC-LFM waveform almost remain the same, however, the average CP of the FC-SLFM pulse train reduces from -33.5 dB to -39.8 dB and the average ASP reduces from -29.3 dB to -34.2 dB. This means that unlike the MDFC-LFM waveform, the autocorrelation sidelobes and cross correlation of the proposed waveform become lower with the increase of the bandwidth. This is useful for some applications with wide bandwidth.

IV. CONCLUSIONS In this paper, a new frequency coding waveform with

segment LFM signals has been proposed. The numerical results have shown that the autocorrelation sidelobes and cross correlation of both the contiguous waveform and the pulse train are lower than the existing frequency coding waveforms. The cross-correlation peaks are improved significantly and are reduced with the increase of the bandwidth. By extending the contiguous waveform to a pulse train, the autocorrelation sidelobes and cross correlation nearby the mainlobe area of the autocorrelation function are reduced. Besides, some grating lobes with low amplitude which are required to be further nullified appear in the autocorrelation function.

ACKNOWLEDGMENT The authors would like to thank Temasek Laboratories at

Nanyang Technological University (TL@NTU) for providing the funding to carry out this research.

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510 2015 IEEE 5th Asia-Pacific Conference on Synthetic Aperture Radar(APSAR)