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MIMO雷达

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**MIMO雷达OFDM Chirp波形设计** MIMO（Multiple-Input Multiple-Output）雷达是一种先进的雷达系统，它利用多个发射和接收天线来提高雷达系统的探测性能。与传统的单天线雷达相比，MIMO雷达具有更高的空间分辨率、更强的抗干扰能力和更精确的目标定位能力。OFDM（Orthogonal Frequency Division Multiplexing）是一种多载波调制技术，常用于无线通信系统，如Wi-Fi和4G/LTE。在MIMO雷达中，OFDM Chirp波形结合了OFDM的优点和Chirp信号的特性，实现了更高效的数据传输和目标检测。 **1. OFDM Chirp波形的概念** OFDM Chirp波形是OFDM与Chirp脉冲的结合。Chirp，也称为线性调频信号，是一种频率随时间线性变化的信号，具有宽频带和良好的自相关特性，适合雷达应用。OFDM则是将高速数据流分解成多个低速子载波，每个子载波进行幅度或相位调制，然后并行传输。当这两者结合，OFDM Chirp波形能同时实现宽带传输和良好的时频局部化特性。 **2. 时域表示** 在时域中，OFDM Chirp波形表现为一系列连续的Chirp脉冲，每个脉冲的频率随时间线性增加。通过改变每个Chirp的起始频率和斜率，可以形成不同的编码序列，以区分不同的数据符号。这种时域表示方式使得信号具有很好的时域局部性，有利于在雷达探测中实现高分辨率的时差估测。 **3. 频域表示** 在频域中，OFDM Chirp波形表现为多个离散的子载波，每个子载波对应一个特定的频率。这些子载波的频率间隔是正交的，避免了相互之间的干扰。Chirp的频率变化特性使得每个子载波的频率随着时间而改变，形成了一个动态的频谱分布。这种频域表示有利于在接收端通过快速傅里叶变换（FFT）进行解调和信息提取。 **4. 波形设计的关键步骤** - **信号参数选择**：确定Chirp的初始频率、结束频率、频率斜率以及每个OFDM符号的持续时间。 - **子载波分配**：根据系统带宽和信息速率，合理分配各子载波的频率和调制方式。 - **编码设计**：利用Chirp的频率变化特性，设计编码规则以实现信息的嵌入和解码。 - **信号合成**：通过合成各个子载波的信号，形成OFDM Chirp波形。 - **功率优化**：调整各子载波的功率，以平衡信号的频域和时域特性，同时满足系统功率预算。 **5. 参考文献的价值** 参考文献对于深入理解OFDM Chirp波形的设计原理、性能分析以及实际应用至关重要。它们通常包含详细的数学模型、仿真结果和实验验证，为研究者提供了理论支持和技术指导，有助于进一步研究和改进MIMO雷达系统。通过综合运用OFDM和Chirp波形，MIMO雷达能够实现更复杂的信号结构，提升系统性能。这不仅涉及基础的信号处理技术，还涵盖了信息论、通信理论和雷达系统设计等多个领域。深入理解和掌握这些知识点，对于开发更先进、更具效能的雷达系统至关重要。

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IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 53, NO. 3, MARCH 2015 1615

MIMO SAR OFDM Chirp Waveform Diversity

Design With Random Matrix Modulation

Wen-Qin Wang, Member, IEEE

Abstract—Multiple-input multiple-output (MIMO) synthetic

aperture radar (SAR) has received much attention due to its

interesting application potentials, but effective waveform diver-

sity design is still a technical challenge. In a MIMO SAR, each

antenna should transmit a unique waveform, orthogonal to the

waveforms transmitted by other antennas. The waveforms should

have a large time–bandwidth product, low cross-correlation in-

terferences, and a low peak–average ratio. To reach these aims,

this paper proposes an orthogonal frequency division multiplexing

(OFDM) chirp waveform with random matrix modulation. The

designed waveforms are time-delay and frequency-shift decor-

related. Referring to MIMO SAR high-resolution imaging, the

proposed OFDM chirp waveform parameters are optimally de-

signed, and their performances are analyzed through the am-

biguity function and range-Doppler-based MIMO SAR imaging

algorithm. Extensive and comparative simulation results show

that the waveforms have the superiorities of high range resolu-

tion, constant time domain and almost constant frequency-domain

modulus, large time–bandwidth product, low peak–average ratio,

and low time-delay and frequency-shift correlation peaks. More

importantly, this scheme can easily generate over three orthogonal

waveforms with a large time–bandwidth product.

Index Terms—MIMO SAR, multiple-input multiple-output

(MIMO), OFDM chirp waveform, orthogonal frequency division

multiplexing (OFDM), synthetic aperture radar (SAR), waveform

diversity.

I. INTRODUCTION

N RECENT years, multiple-input multiple-output (MIMO)

synthetic aperture radar (SAR) has gained popularity and

attracted much attention for its ability to enhance system perfor-

mance [1], [2]. In a MIMO SAR system, each antenna should

transmit a unique waveform, orthogonal to the waveforms

transmitted from other antennas so that their returns can be

effectively separated by a set of matched ﬁlters in the receiver.

MIMO SAR offers potentials to obtain more degrees of free-

dom and improve spatial resolution. Literature search shows

that numerous papers on MIMO radar and waveform diversity

design have been published (see [3]–[5] and the references

Manuscript received October 20, 2013; revised March 28, 2014, May 25,

2014, and July 19, 2014; accepted August 1, 2014. This work was supported

in part by the National Natural Science Foundation of China under Grant

41101317, by the Program for New Century Excellent Talents in Universities

under Grant NCET-12-0095, by Sichuan Province Science Fund for Distin-

guished Young Scholars under Grant 2013JQ0003, and by the Fundamental

Research Fund for the Central Universities under Grant ZYGX2013J008.

The author is with the School of Communication and Information Engi-

neering, University of Electronic Science and Technology of China, Chengdu

611731, China (e-mail: [email protected]).

Color versions of one or more of the ﬁgures in this paper are available online

at http://ieeexplore.ieee.org.

Digital Object Identiﬁer 10.1109/TGRS.2014.2346478

therein), but relative little work on MIMO SAR has been

reported [6]–[16]. Note that the MIMO SAR considered in this

paper is different from general MIMO radars because aperture

synthesis with moving platform is employed in MIMO SAR but

stationary platform without aperture synthesis is often assumed

in the MIMO radar literature. Even less work on MIMO SAR

waveform diversity design can be found in the open literature

[17]–[22].

Existing waveform diversity designs considered in MIMO

radar often assume some prior knowledge for the impulse

response of the target and use the knowledge to optimize the

mutual information between the received signals and impulse

response of the target [23]. However, waveforms used in SAR

should have a large bandwidth so as to obtain a high range res-

olution. Another property is that a high average transmit power

is required for the transmitted waveforms. Thus, MIMO SAR

waveforms should have a large time–bandwidth product and a

low peak–average ratio. It is also important that the waveforms

have low cross-correlation interferences and thus have a good

peak-to-sidelobe ratio (PSLR) in the impulse response. That

is, they s hould be as far as orthogonal as possible. Note that,

since perfect waveform orthogonality cannot be implemented in

real-life hardware, the orthogonal used in this paper means

pseudo-orthogonal, namely, low cross-correlation interfer-

ences. Certainly, the waveforms should also have good ambi-

guity function characteristics such as range resolution, Doppler

resolution, and matched ﬁltering sidelobe performance. For

these reasons, most of the existing waveforms are not suitable

for MIMO SAR and are difﬁcult to be implemented in the

real-life scene. In particular, the waveforms designed with the

Barker codes have a low bandwidth efﬁciency. The Costas

pulse-based waveforms [24] have good range-Doppler prop-

erties, but each chip contains only one frequency. In [25], a

cyclic transmit waveform with quadratically phase shift keyed

Hadamard codes scrambled by a pseudonoise sequence is pro-

posed, but it i s not suitable for MIMO SAR imaging due to its

limited frequency bandwidth.

Orthogonal frequency division multiplexing (OFDM) is a

popular choice for common radar and communication signal

because it offers advantages such as robustness against multi-

path fading and relative simple synchronization [26], [27]. Time

and frequency synchronization is crucial in OFDM communi-

cation to preserve subcarrier orthogonality. For radar, however,

sensitivity to synchronization is beneﬁcial since radar uses a

stored version of the transmitted signal and measures the time-

delay and frequency offsets between the transmitted signal and

received echo to determine a target’s range and closing velocity.

OFDM-like waveforms have been shown to be suitable f or

IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.

See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

1616 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 53, NO. 3, MARCH 2015

radar applications (see [28]–[33] and the references therein),

and the feasibility of integrating communication functions in

radar networks [34], [35] has also been explored. It is pointed

out in [36] that OFDM-coded radar signals are comparable with

linear frequency modulation (LFM) signals and furthermore

experience no range-Doppler coupling.

Up to now, the most practical MIMO SAR waveform is the

up- and down-chirp modulation [37]–[39], but it offers only two

pseudo-orthogonal waveforms. From a practical point of view,

we think that MIMO SAR should use chirp-based waveforms.

We have extended the conventional OFDM signal to the OFDM

chirp diverse waveform [40]. The designed OFDM chirp di-

verse waveform has a large time–bandwidth product and no

visible gaps in the covered bandwidth, but the peak–average

ratio performance should be further improved. The authors of

[16] propose a novel OFDM chirp waveform design scheme

and a whole processing technique. With the known geometric

relation between the target and radar, they are able to suppress

the signal ambiguities with digital beamforming on receive. The

scheme allows the designed two waveforms to occupy common

spectral support and thereby enables them to exploit the full

bandwidth. For each transmit signal, they need a separate

receive beam to suppress the ambiguities/grating lobes. Thus,

a complete suppression of the signal ambiguities with four

(or more) transmit signals is possible. Note that the difference

between the OFDM chirp diverse [40] and OFDM chirp [16]

waveforms lies in that the former waveforms do not need digital

beamforming and, hence, the hardware is not complex.

This paper further proposes a practical solution to generate

multiple (over three) OFDM chirp waveforms by employing

random matrix modulation. Relative to our previous OFDM

waveform discussed in [21] and [40], this paper adaptively

designs the OFDM chirp waveform with jointly random matrix

modulation and mathematical optimization, whereas in our

previous work, the waveform is manually designed and, conse-

quently, it is difﬁcult t o optimally design over three orthogonal

waveforms. The contributions of this paper can be summarized

as follows: 1) A new OFDM chirp waveform design method is

proposed by employing random matrix modulation. Compared

with [40], it is more easily implementable to design multiple

orthogonal waveforms. More importantly, multiple (over three)

pseudo-orthogonal waveforms can be easily generated by this

method, but existing methods often yield only t wo pseudo-

orthogonal waveforms. 2) The radar ambiguity function prop-

erties of the designed OFDM chirp waveform are extensively

analyzed and compared with other waveforms, and the wave-

form superiorities are validated by numerical results. 3) A

range-Doppler-based imaging algorithm is derived for the

MIMO SAR high-resolution imaging, and it is veriﬁed by

simulation results.

The rest of the sections of this paper are organized as follows.

Section II proposes the OFDM chirp waveform with random

matrix modulation. Section III derives the general MIMO SAR

signal model and analyzes the designed waveform performance

by the radar ambiguity function. Next, extensive numerical

examples and results are provided in Section IV. Finally, this

paper is concluded in Section V with a short discussion of

future work.

Fig. 1. Illustration of three chirp signals with adjacent starting frequency and

inverse chirp rate or nonoverlapping frequency b ands.

II. PROPOSED OFDM CHIRP WAV E F OR M

DIVERSITY DESIGN

The proposed OFDM chirp waveform is designed by modu-

lating the OFDM chirp basis with a random matrix. It can be

divided into two design steps: OFDM chirp basis design and

random matrix modulation. In the following, we discuss them

separately.

A. OFDM Chirp Basis Design

The OFDM chirp basis can be designed like that in [40]. We

start to present it from a general chirp (also called LFM) signal

s(t)=rect





· exp



j2π



t +



(1)

where rect[·] is a window function, f

is the starting frequency,

is the chirp rate, and T

is the chirp duration. The correla-

tion between any two chirp signals s

(t) and s

(t), where the

subscripts

and

relate the quantities to one of the two chirp

signals, can be easily derived as [41]

exp



j2πf

+ jπk

− jπ

(

−f

)

−k







− k



γ(t

)



γ(t

)

exp



√

2π



dν (2)

where t

is the time delay and

γ(t)=

+ k

τ − f



− k

−



2(k

− k

)t. (3)

An important quantity is the relative level between the au-

tocorrelation of identical chirp signals and cross-correlation of

different chirp signals. Hence, the chirp signals with high cross-

correlational suppression are desired. It can be deduced from

(2) [41] that the chirp signals with adjacent starting frequencies

and inverse chirp rate or nonoverlapping frequency bands can

provide a good cross-correlational suppression. Considering the

chirp signals shown in Figs. 1 and 2, we compare the cross-

correlation between different combinations, where the ampli-

tudes are normalized by the autocorrelation peak. Note that, if

the echo is Doppler shifted, this is also true because the Doppler

effect brings an equivalent frequency shift for the signals. More-

over, we think that practical SAR systems should use the chirp

和文献16的波形相同

why?

参数

WANG: MIMO SAR OFDM CHIRP WAVEFORM DIVERSITY DESIGN WITH RANDOM MATRIX MODULATION 1617

Fig. 2. Comparative correlations between different combinations of the three

chirp signals shown in Fig. 1 using the simulation parameters: k

= k

1 × 10

, k

= −1 × 10

,andT

=1μs.

signals with equal chirp duration and equal absolute of chirp

rate so as to reduce the hardware design complexity. Based

on these considerations, we design the OFDM chirp basis, as

shown in Fig. 3. According to the results given in Fig. 2, rectan-

gular shaped basis is used in the hybrid-chirp basis c to achieve

low cross-correlation interferences. The up-chirp or/and down-

chirp basis signals can be used by each subcarrier frequency in-

terval at the same time. However, the hybrid-chirp basis should

not be jointly used with the up-chirp basis or down-chirp basis;

otherwise, the same subchirps may be used simultaneously by

two waveforms, and consequently, the cross-correlation inter-

ferences will be signiﬁcantly increased. The OFDM chirp basis

can be represented as an N ×M matrix. The rows represent the

subcarriers, and the columns are the chips. This matrix can also

be viewed as an NM-element chirp sequence, and each element

has a duration T

, demultiplexed onto the N subcarriers s uch

that each subcarrier has M subchirp signals. The OFDM chirp

pulse duration is determined by T

= MT

The OFDM chirp basis with N subcarriers and M temporal

chips can be expressed as

s(t)=

N−1



n=0

M−1



m=0

u(t − mT

)exp{j2πf

n,m

(t − mT

)}

×exp



jπk

n,m

(t − mT

)



(4)

where f

n,m

and k

n,m

denote respectively the corresponding

subcarrier frequency and subchirp rate for the nth row and

mth column element of the matrix and u(t)=1, 0 ≤ t ≤ T

The cross-correlation between any two subcarriers during each

subchirp duration can be expressed as



[exp(j2πf

n,m

t)] [exp(j2πf



t)]

∗

=sinc[π(f

n,m

− f



] ,

m =1, 2,...,M; n =1, 2,...,N (5)

where

∗

is the conjugate operator and n



denotes an integer. To

make the two subcarriers time orthogonal, namely, sinc[π(f

n,m

−



]=0, the subcarrier frequencies f

n,m

and f



should

be spaced in frequency by

n,m

− f



,p=1, 2,... (6)

where p is an arbitrary integer.

Accordingly, the spectrum of each subcarrier over t he dura-

tion of a subchirp is



n,m

exp



sgn[k

n,m

]



exp

⎧

⎪

⎨

⎪

⎩

−jπ



f −



n,m

⎫

⎪

⎬

⎪

⎭

(7)

where sgn[·] is the signum function and f is the Fourier fre-

quency variable. Obviously, it is different from the conventional

OFDM spectra

sinc



f −





. (8)

Since sinc(fT

− p) has a ﬁrst null at the inverse of the signal

duration T

, the spectra of the subcarriers will be overlapped,

but the subcarrier signals are mutually orthogonal. This means

that the OFDM chirp basis has a lower peak–average ratio

which is desired for the radio frequency (RF) hardware trans-

mitter and an almost constant modulus like a standard chirp sig-

nal. Therefore, the OFDM chirp basis provides a good potential

to design waveform diversity for MIMO SAR imaging.

B. Random Matrix Modulation

For notation simplicity, we express the three OFDM chirp

basis signals illustrated in Fig. 3 as a general matrix function

Φ =

⎡

⎢

⎣

φ(0, 0) φ(0, 1) ... φ(0,M−1)

φ(1, 0) φ(1, 1) ... φ(1,M−1)

φ(N −1, 0) φ(N −1, 1) ... φ(N −1,M−1)

⎤

⎥

⎦

(9)

where φ(n, m), with n =0, 1,...,N − 1 and m =0, 1,...,

M −1

, denotes the subchirp signal in the nth row and mth col-

umn element of the OFDM chirp basis. φ(n, m) is expressed as

φ(n, m)=u(t −mT

)exp{j2πf

n,m

(t − mT

)}

×exp



jπk

n,m

(t − mT

)



(10)

where the chirp rate k

n,m

is determined by

n,m

> 0, up-chirp basis

< 0, down-chirp basis

> 0or < 0, hybrid-chirp basis.

(11)

Mathematically, the OFDM chirp basis of (4) can be rewritten as

s(t)=sum {Φ(:, :)} (12)

where sum[Φ(:, :)] is deﬁned as the sum of all Φ elements.

Next, to generate multiple orthogonal OFDM chirp wave-

forms, we use a random matrix to modulate the OFDM chirp

basis as follows:

X = R  Φ (13)

就是文献17 里的subchirp

对该OFDM chirp做波束形成,先解复用,然后对每个子载波做波束形成

1,2

性能分析,PAR

指频域的PAR,还有时域PAR(15)

1618 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 53, NO. 3, MARCH 2015

Fig. 3. Illustration of three OFDM chirp basis signals. (a) Up-chirp basis. (b) Down-chirp basis. (c) Hybrid-chirp basis.

Fig. 4. Three OFDM chirp waveform pulses with the hybrid-chirp basis.

where R is an N ×M random binary matrix, each of whose en-

tries independently take the value 0 or 1, and is the Hadamard

(elementwise) product. Here, 1 means that the subchirp basis

with an index corresponding to the location of that 1 in R

belongs to the modulated waveform while 0 means that the

corresponding subchirp basis is not used in the waveform. The

modulated OFDM chirp waveform can then be expressed as the

sum of all X elements

x(t)=sum {X(:, :)}. (14)

In this way, multiple orthogonal OFDM chirp waveforms can

be generated.

Furthermore, to reduce the peak-to-average ratio of the

ODFM chirp waveform in the frequency domain, we should

minimize the difference of the sum of each row elements. In

the same time, the peak-to-average ratio in the time domain can

be reduced by minimizing the difference of the sum of each

column elements. Therefore, the random modulation matrix

should be jointly optimized by

minimize |sum {R(i

, :)}−sum {R(i

, :)}|

minimize |sum {R(:, j

)}−sum {R(:, j

)}|

, i

}∈[1, 2,...,M], i

=i

, j

}∈[1, 2,...,N], j

=j

. (15)

where sum[R(i, :)] is deﬁned as the sum of the ith row elements

of R and sum[R(:, j)] is deﬁned as the sum of the jth column

elements of R.

For an illustrative example, we use the following three 8 × 8

random binary matrices:

wc1

⎡

⎢

⎣

01000000

00000001

00100000

00000010

00000100

00010000

00001000

10000000

⎤

⎥

⎦

(16a)

wc2

⎡

⎢

⎣

00000010

00100000

00000100

10000000

00001000

00000001

00010000

01000000

⎤

⎥

⎦

(16b)

wc3

⎡

⎢

⎣

00010000

00000100

10000000

00001000

00000001

01000000

00000010

00100000

⎤

⎥

⎦

(16c)

用随机矩阵调

制后的信号

跳频图案

是不是少一个up-chirp?

式(7)(8)频域PAR

不对,弄反了

在文献low peak to average

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