Othogonal Waveform Design for Multiple-Input Multiple-Output (MIMO) Radar

doi:10.4236/ojapps.2012.24B006

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Othogonal Waveform Design for Multiple-Input Multiple-

Output (MIMO) Radar

Hai Deng

Department of Electrical and Computer Engineering

Florida International University, Miami, Florida, 33174, USA

Hai.Deng@fiu.edu

Abstract—An effective numerical approach is developed for orthogonal waveform design for Multiple-Input Multiple-Output

(MIMO) radar. The Doppler shift tolerance is considered in the design cost function. The design results indicate that the Doppler

tolerance of the designed orthogonal waveforms is markedly improved.

Keywords-MIMO radar; waveform design; Doppler effect;

1. Introduction

Multiple-Input Multiple-Output (MIMO) radar is capable to

significantly improve radar performance over traditional

phased array radar via integrating target scattering information

from multiple diversified channels [1, 2]. The key to

implementing MIMO radar is to design, to generate, and to

process a group of orthogonal coding waveforms for each of

the antenna elements [3]. Consider an MIMO radar with L

antenna elements with each of them transmitting a distinct

waveform from a group of orthogonal coding

waveforms

Lltu

,3, ,2 ,1 ),(

at the same carrier

frequency, any two coding waveforms used by the MIMO

radar are supposed to be aperiodically orthogonal, therefore,

the cross-correlation function of two orthogonal coding

waveforms

)(tu

and

)(tu

used in MIMO radar should be

close to zero. For the coding waveforms to achieve the high

range resolution, the aperiodic autocorrelation function of any

coding waveform

)(tul

in the orthogonal waveform group

should be close to the Dirac delta function. However, the

designed waveforms are not orthogonal when a Doppler shift

exists in the radar echoes due to moving targets, leading to

significant target signal loss at matched filter outputs and

possible missing target detection. Orthogonal coding

waveforms with Doppler tolerance should be designed and

used for MIMO radar in this case. Algebraic methods for

design of orthogonal waveforms with Doppler tolerance would

be desirable, but are extremely difficult to implement and

basically ineffective. Therefore, numerical optimization is the

only feasible approach to designing orthogonal waveforms

with Doppler tolerance for MIMO radar [3]. For the

application of numerical optimization approach, a cost

function must be identified and the effective numerical

optimization algorithm must be developed to minimize the

cost function.

2. Definition of Cost Function with

Doppler Shift Constraint

Consider a group of L orthogonal phase coding MIMO radar

waveforms with code length equal to N. If the phase coding

sequence for L waveforms

is },2,1,,2,1),({ LlNnnsl , the autocorrelation

function of any waveform is required to be a thumbtack-like

function as:

mnsns

msA

lll

,...,2,1

10 ,0

0 , 1

)()(

),(



d





(1)

If the Doppler frequency exists, the autocorrelation

becomes





nfj

dlf

mnsens

fmsA

)()(

),,(

(2)

If the Doppler frequency d

fis discretized as multiple times

of a unit Doppler frequency which is based on the reciprocal of

the waveform duration T, i.e.

2 ,1 ,0,/rr ddd kTkf (3)

The function in (4) is changed to:

10 ,)()(

),,(







Nnkj

dlf

mnsens

kmsA

(4)

Open Journal of Applied Sciences

Supplement：2012 world Congress on Engineering and Technology

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The cross-correlation function of two waveforms

ss and

with Doppler shift considered is given by:

Lqpqp

mnsens

kmssC

Nnkj

dqpf

,...,2,1 , , 0

)()(

),,,(







(5)

The final cost function is:

¦¦¦¦

¦¦













,2,1,0

)1(

),,,(

),,(max

),(

dqpf

ldlf

kmC

kmA

mAC

IIO

(6)

where

and

are the weighting coefficients for

autocorrelation and cross-correlation functions with Doppler

frequency in the cost function. The design of orthogonal

waveforms

},,,{ **

sss 

for MIMO radar is carried by

minimizing the cost function in (6) and the optimum

orthogonal waveforms are given by:

)(minarg},,,{

SsssS

L:



(7)

L phase coding waveforms.

An innovative technique termed as Enhanced Simulated

Annealing (ESA) will be implemented to optimize (7) for a set

of orthogonal phase-coding waveforms with Doppler shift

tolerance for MIMO radar.

4. Statistical Approach for Orthogonal

Waveform Design

Basic Simulated Annealing (SA) algorithm has been

successfully employed for the design of orthogonal coding

waveforms with small group sizes [4, 5]. However, we have

found SA algorithm becomes much less effective when the

problem size LN is greater than 2000, where L and N are the

orthogonal waveform group size and the code length,

respectively. For conventional SA algorithms, the random

coding perturbations are used in optimization process to

“jump” out of a local minimum to reach the global optimum.

However, with large problem sizes, when the temperature is

low, SA algorithm becomes greedier and tends to be “stuck”

in the local minima. Therefore, the basic SA algorithm is not

effective for the design of more than 100 orthogonal

waveforms as required in this project. We propose an

enhanced simulated annealing (ESA) algorithm for the design

of orthogonal waveforms with large problem sizes. In ESA,

the coding perturbations are adapted to the temperature and

cost change pattern. When the temperature is high, the coding

perturbations are minor as those in the regular SA algorithms;

when temperature is lowered, the coding perturbations are

made to be strong enough for the optimization to jump out of

the local minima. The approach is motivated by the

“tunneling” method in quantum annealing [6]. The advantage

of using ESA over basic SA algorithm is illustrated in Figure 1.

As shown in Figure 1, for optimization to jump out of a local

minimum, multiple “hill-climbing” perturbations are needed in

basic SA algorithms, but adaptive perturbation can tunnel

through “the hill” directly. Therefore, by using adaptive

frequency perturbations, the ESA optimization is more rapidly

to reach equilibrium condition and more likely to find the

global minimum in the orthogonal waveform design.

3. Design Results

ESA has been implemented for Doppler shift tolerant

orthogonal waveform design based on the cost function in (5).

The number of orthogonal coding waveforms is 3, the number

of admissible phase values for each subpulse phase coding is 8

and the code lengths are chosen to be 64 and128. In addition,

the following optimization parameters are chosen: O1=1 and

O2=0.1.

Fig. 2 shows the MIMO radar matched filtering outputs of

three designed orthogonal coding waveforms with a code

length of 64 vs. different Doppler shifts in the radar echoes. It

is found that the outputs are significantly improved with

Doppler shift existing when the Doppler shift tolerant

waveform design approach is used. Fig. 3 shows the cross-

correlation functions of orthogonal waveforms when the

Doppler shift is not zero or the matched filtering outputs for

the waveforms containing Doppler shift and matched to other

matched filters. The results indicate that the cross-correlation

outputs for the designed codes with Doppler shift considered

are slightly improved compared to those for the codes

designed without considering Doppler shift. The simulation

results indicate that ESA is very effective in designing

Doppler-tolerant orthogonal waveforms for MIMO radar.

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Figure 1. New coding perturbation strategy for the optimization to

jump out of a local minimum (S0: current state) through tunneling

(a)

(b)

(c)

Figure 2. The matched filter outputs for the three designed orthogonal

waveforms with various

Doppler shifts (code length N=64).

(a)

(b)

(c)

Fig. 3: The matched filter outputs of the designed orthogonal waveforms vs.

various Doppler shifts in the radar echoes with the filter

matched to another waveform (code length N=64).

REFERENCES

[1] D. J. Rabideau and P. Parker, “Ubiquitous MIMO

multifunction digital array radar,” The Thirty-Seventh

Asilomar Conference on Signals, Systems and Computers

vo1. 1, pp 1057 – 1064, Nov. 2003.

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[2] F. C. Robey and et al, “MIMO radar theory and

experimental results,” The Thirty-Eighth Asilomar

Conference on Signals, Systems and Computers vo1. 1,

pp 300 – 304, Nov. 2004.

[3] H. Deng, “Polyphase coding signal design for orthogonal

netted radar systems,” IEEE Transactions on Signal

Processing, vol. 52, pp. 3126-3135, November 2004.

[4] S. Kirkpatrick, C. D. Gelatt and M. P. Vecchi,

“Optimization by simulated annealing,” Science, vol. 220,

pp. 671-680, May 1983.

[5] L. Ingber, Adaptive simulated annealing (ASA): Lessons

learned, Control and Cybernetics, vol. 25, pp. 33-54, no.1,

Jan. 1996.

[6] Arnab Das and Bikas K. Chakrabarti (Eds.), Quantum

Annealing and Related Optimization Methods, Lecture

Note in Physics, vol. 679, Springer, Heidelberg, Germany,

2005.

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