WO2020212569A1 - Method and device for beamforming in a mimo radar system - Google Patents

Abstract

The invention proposes a method and device for detecting a target in a multiple-input multiple-output, MIMO, radar system comprising a first uniform linear array having a plurality of NT transmit antennas, and a second uniform linear array having a plurality of NR receive antennas. The proposed beamforming in accordance with embodiments of the invention allows for a low-complexity computation of the transmitted probing signals, while reducing angle- Doppler coupling in the backscattered received signals.

Description

METHOD AND DEVICE FOR BEAMFORMING IN A MIMO RADAR SYSTEM

Technical field

The invention lies in the field of multiple-input multiple -output radar systems.

Background of the invention

The recent developments in waveform design and receiver processing, access to higher frequencies, increased modulation bandwidths and advent of integrated hardware have motivated the potential use of radar for situational awareness in challenging environments comprising non-homogeneous interference sources with spatio-temporal dynamics [1].

The use of multiple antennas for transmission and reception leads to the Multiple -Input-Multiple - Output, MIMO, radar architecture. Orthogonal MIMO radar, associated with transmission of orthogonal signals from the different antennas and their processing, has been considered towards dynamic interference mitigation due to the offered waveform diversity and enhanced resolution [2]- [6] Key aspects considered therein include receive beamforming, space-time adaptive processing and waveform design [7] However, due to the isotropic nature of orthogonal signal radiation, the Signal to Noise Ratio, SNR, is rather poor, thereby impacting target detection performance.

Enhancing the SNR requires focussing the transmission in desired directions, which in turn requires the transmitted signals to be correlated. In this transmit beamforming MIMO context, several waveform designs and receiver processing schemes have been considered towards achieving higher SNR in desired directions [8]-[ 11] . A commonly used design metric is the Signal to Interference plus Noise Ratio, SINR, where the transmitted signal and the receive beamformer are jointly optimized in an attempt to maximize the SINR [12]— [17] Other design metrics are related to the auto-correlation properties of signals and include the Peak-Sidelobe-Level, PSL, and the Integrated-Sidelobe-Level, ISL, metrics [18], [19] Similar to the beamformer design [4], [8], [9], [11], the ISL and PSL metrics are quartic in the parameter of interest [ 18]— [23], which can be, depending on the waveform design constraints, NP hard. Naturally, transmission of correlated signals offers lower resolution compared to orthogonal transmission and results in a trade-off between orthogonal MIMO and transmit beamforming MIMO.

Due to their practical relevance in terms of coherency between transmitted and received signals, collocated antenna elements are extensively discussed [24], [25] A widely used waveform design restriction is the unimodular or constant modulus property and the related Peak to Average Ratio,

PAR, constraints [12], [13], [19], [22] The constant modulus constraint ensures the best power efficiency for a transmitted signal [23] The PAR constraint is a generalized version of the unimodular design constraint and also accounts for the power efficient waveform design. Due to the low hardware complexity, arising from fewer radio-frequency chains, time multiplexed signals are also relevant design constraints [26]- [30] A combination of unimodular waveform with discrete phase stages and time multiplexing is investigated in [31]

In this context, use of multiple antennas for transmission and reception coupled with advanced signal processing at either end is seen as a promising way towards dynamic interference mitigation and target scene enhancement. The known architectures and processing schemes require significant hardware complexity and flexibility with regards to waveform design and implementation, a fact that can limit the use of advanced radar systems to niche markets, e.g., luxury cars.

Technical problem to be solved

It is an objective to present method and device, which overcome at least some of the disadvantages of the prior art.

Summary of the invention

In accordance with an aspect of the invention, a method for detecting a target in a multiple-input multiple-output, MIMO, radar system is provided. The system comprises a first uniform linear array having a plurality of transmit antennas, a second uniform linear array having a plurality of receive antennas, and a multiplexer for generating multiplexed and discrete phase modulated probing signals, comprising the steps of:

i) providing a block-circulant signal matrix S defining a probing beam pattern to be generated by said transmit antennas, wherein each column of said signal matrix describes a plurality of probing signals to be generated by a plurality of said transmit antennas in a given time transmission slot;

ii) generating , using a data processor, a random permutation of the columns of said signal

matrix, and transmitting, using said transmit antennas, a probing beam in accordance with said generated matrix permutation;

iii) receiving, using said receive antennas, reflection signals of said probing signals, said

reflection signals being backscattered from at least one target;

iv) processing the reflection signals to determine the presence range and angular position of a target within a field of view of the transmit antennas.

Preferably, at least three transmit antennas may preferably be transmitting a probing signal in at least one time transmission slot. Preferably, said step i) may further comprise the following steps:

a) providing a desired beam pattern representation;

b) using a data processor, approximating said desired beam pattern by a weighted sum of basis vectors from a signal dictionary;

c) generating said block-circulant signal matrix S by circular shifting of each basis vector, such that the resulting number of circulant blocks in the signal matrix S is equal to the weight associated with said basis vector in said approximation.

Preferably, the weights in said approximation may be integer values.

Said signal dictionary may preferably comprise all feasible possible phase modulations given the number of active transmit antennas in a given time transmission slot.

Preferably, said signal dictionary may comprise discretized Fourier basis functions, and the transmitted signals may preferably comprise unimodular signals.

The signal dictionary may preferably be pre-provided in a memory element to which said data processor has a read access.

In accordance with a further aspect oft he invention, a computer program is provided. It comprises computer readable code means, which, when run on a computer, causes the computer to carry out the method according to an aspect of the invention.

In accordance with yet another aspect of the invention, a computer program product comprising a computer readable medium is provided, on which the computer program according to an aspect of the invention is stored.

In accordance with a further aspect of the invention, a detection system for sensing a target is provided. The system comprises a multiplexer coupled to a plurality NT of transmit antennas forming a sparse transmit uniform linear array, ULA, a plurality NR of receive antennas forming a dense receive ULA. The multiplexer is configured to generate multiplexed transmit signals based on signals from a local oscillator. The system further comprises processing circuitry coupled to the multiplexer, wherein the processing circuitry, the transmit antennas and the receive antennas are configured to perform the method according to aspects of the invention.

In accordance with a final aspect of the invention, a vehicle comprising a detection system according to an aspect of the invention is provided. The vehicle may preferably comprise an automotive vehicle. The invention provides a method which allows for achieving/surpassing the performance of state-of- art solutions in multiple antenna radar systems, while relaxing the requirements of the corresponding hardware platform. A signal matrix having a block circulant structure is used for inter-pulse modulation of a multiplexed signal applied to a sparse transmit antenna array. This allows for a significant reduction in the complexity involved with forming a desired beam pattern, while at the same time making sure that the desired cross-correlation values are well met. Before applying the signal matrix, its columns are randomly rearranged. This allows for mitigating the angle-Doppler ambiguity by randomizing the perceived center motion, PCM, of the beam pattern’s source, which is formed by multiple antennas transmitting within a same time slot of the multiplexing scheme. By using a dictionary approach to approximate a desired probing beam pattern, any combination of multiplexing and multiple phase shift keying, MPSK, is covered, which allows the proposed method to be implemented at low complexity on heterogeneous hardware setups.

Brief description of the drawings

Several embodiments of the present invention are illustrated by way of figures, which do not limit the scope of the invention, wherein:

Figure 1 illustrates the workflow of the principal method steps in accordance with a preferred embodiment of the invention;

Figure 2 illustrates a detection system in accordance with a preferred embodiment of the invention;

Figure 3 shows examples of beam pattern synthesis with a rectangular desired pattern in accordance with a preferred embodiment of the invention, wherein the offset in the desired beam pattern is typical for multiplexed signal designs;

Figure 4 shows examples of beam pattern synthesis with two rectangular main lobes as desired pattern in accordance with a preferred embodiment of the invention, wherein difference between patterns is justified by the multiplexing design constraint;

Figure 5 shows the quantization error between an optimization output v and its discretized version as a function of the number of blocks for the desired beam pattern in figure 3;

Figure 6 illustrates an angle-Doppler ambiguity function for a block circulant signal matrix, wherein due to the linear phase center motion, high sidelobes appear within the angle-Doppler plane;

Figure 7 illustrates an angle-Doppler ambiguity function for a randomize signal matric in accordance with a preferred embodiment of the invention, wherein the sidelobes transform to a flat sidelobe floor; Figure 8 shows a signal quantization error as a function of circulant blocks, in accordance with a preferred embodiment of the invention;

Figure 9 provides a radiation pattern design example of one main lobe in accordance with a preferred embodiment of the invention;

Figure 10 provides an illustration of a radiation pattern design example of two main lobes in accordance with a preferred embodiment of the invention.

Detailed description of the invention

This section describes aspects of the invention in further detail based on preferred embodiments and on the figures. The figures do not limit the scope of the invention.

Figure 1 illustrates the main method steps in accordance with a preferred embodiment of the invention, while figure 2 illustrates a detection system in accordance with a preferred embodiment of the invention, which is configured for implementing the method in accordance with embodiment of the invention.

In accordance with embodiments of the invention, a detection system 100 involving a Multiple-Input Multiple-Output, MIMO, radar system is provided. The system comprises a first uniform linear array having a plurality of NT transmit antennas 110, preferably evenly arranged along a line. The system further comprises a second uniform linear antenna array comprising NR receive antennas 120. Each receive antenna is preferably coupled to an analog-to-digital converter unit , ADC, and a received signal processor 150 is used to interpret the received signal- Both arrays may be physical collocated in a single antenna device. The system further comprises a multiplexer 130 for generating time- multiplexed and discrete phase modulated probing signals. All signals may preferably be generated using a single common oscillation source 160. A data processor 140, or a plurality of such processors, may preferably be used to implement the method in accordance with embodiments the invention. The data processor may for example comprise a Central Processing Unit, CPU of a computing device, having access to at least one memory element through a data connection bus. The memory element may comprise a hard disk drive, HDD, solid state drive, SDD, a volatile memory element such as a Random Access Memory, RAM, element, or other storage devices known to the skilled person.

The probing signals of the transmit antennas 110 within each multiplexing time slot are designed so as to generate a probing beam pattern. Consequently, the beam is steerable. A signal matrix .S' defines the probing beam pattern to be generated by said transmit antennas 110. Each column of the signal matrix defines a plurality of probing signals that are to be generated by an equal plurality of said transmit antennas within a given multiplexing time slot. The matrix S is a block-circulant matrix, which will be further described in the following, preferred, embodiment of the invention. The signal matrix .S' is preferably provided in a memory element to which the data processor has access. This corresponds to method step i).

As multiple antennas 110 transmit within each multiplexing slot, and as the antennas that are transmitting probing signals may change from one slot to the next, the angular information of a target 10 that moves relative to the transmit antennas will be coupled to the Doppler shift that is present in the reflection signals that are backscattered from the target. In order to avoid this, in accordance with a preferred embodiment of the invention, a random permutation of the columns of said signal matrix is performed using the data processor 140. The resulting randomly permuted signal matrix is used to generate a corresponding probing beam. This corresponds to method step ii). As will be shown in what follows, this approach effectively decouples the angular and Doppler information when multiple transmit antennas are active within the same multiplexing time slot. At the same time, the proposed solution to achieve this effect provides a low computational complexity.

As the receive antennas 120 obtain reflected signal versions of said probing signals at method step iii), wherein the reflections are formed by backscattering from a target 10, the correspondingly obtained angular information retrieved from the reflected signals using the processing unit 150 is available at a largely higher resolution as compared to known systems, which present strong angle-Doppler coupling. The processing of the reflection signals and detection of a target 10 based on the received signals corresponds to method step iv).

In accordance with an embodiment of the invention, the block-circulant signal matrix is designed so as to approximate a pre-determined desired beam pattern. The desired beam pattern may for example comprise a previously used beam pattern, which gave rise to a detection of a target in a given direction. Typically, the desired beam pattern representation may be provided as a continuous signal that will be discretized through sampling, or the beam pattern will be represented by a discrete vector. Using an accordingly programmed data processor, which may preferably be configured by appropriately formulated software code instructions, the shape of the desired beam pattern is approximated by a weighted sum of basis vectors. This may be achieved for example by well-known matching pursuit algorithms or by similar algorithms known as such in the art, which will not be explained in detail in the context of the present description for the sake of clarity. All available basis functions form a signal dictionary, which preferably comprises the available discrete phase modulations for any given number of active transmit antennas for a given multiplexing slot. Such a dictionary may be pre-computed offline and pre-stored in a memory element, further reducing the runtime complexity of the method. The block-circulant signal matrix S is then generated by adding circularly shifted copies of each basis vector that is represented in said approximation. The resulting number of circulant blocks in the signal matrix .S' is equal to the weight associated with said basis vector in said approximation. As will be shown in the context of the following, preferred, embodiment of the invention, this sub-optimal yet low-complexity algorithm allows for an efficient implementation of the beam-forming algorithm, while remaining close to optimal performance bounds.

In accordance with aspects of the invention, it is thus proposed to use a plurality of multiplexing channels for transmitting radar probing signals, with a subsequent discrete phase modulation, where the number of discrete phase stages may comprise any arbitrary number of phase stages.

In what follows, a preferred embodiment in accordance with the invention will be described.

In accordance with a preferred embodiment of the invention, an implementation amenable transmit beamforming approach for MIMO radar systems is proposed, to improve target detection and resolution. The inter-pulse signal design is based on block circulant signal matrix structure which implicitly ensures good signal cross-correlation properties between transmitted signals for enhanced target discrimination on the receiver while exploiting the virtual MIMO concept. It uses fewer radio frequency chains due to the incorporation of multiplexing while ensuring efficient uniform power transmission. A dictionary based approach transforms the signal design into a convex optimization problem, even when the transmitted signals are restricted to time multiplexing with multiple phase shift keying. Closed-form expressions for the signal design are also provided for special system configurations. The issue of angle-Doppler coupling in such multiplexing architectures is discussed, and a methodology to suppress it without influencing the desired radiation pattern is proposed.

Simulation results corroborate the flexibility and enhanced performance of the proposed design which is motivated by practical constraints.

I. INTRODUCTION

The proposed approach is motivated by the aforementioned trade-off between orthogonal MIMO and transmit beamforming MIMO. It is desired to have the resolution capabilities of orthogonal MIMO systems, whereas the enhanced SINR as in the case for transmit beamforming MIMO is desired. Further, the proposed waveform design accounts for practical hardware constraints like discrete phase modulation with a subsequent multiplexing as implemented in current chipsets including the Texas Instruments chipset AWR1243. Therefore, the proposed approach focuses on discrete phase modulation with a subsequent multiplexing scheme, while optimizing the desired beampattem for enhanced SINR. In this context, the authors in [21] propose a semidefmite quadratic program in order to match a desired beampattem while minimizing the cross-correlation between the transmitted signals; however, the authors do not consider power efficiency or hardware constraints. In [13], the beampattem design is addressed by a sequence of convex optimization problems under the constant modulus and similarity constraints. The authors, however, do not account for the cross-correlation properties of the transmitted signal which impacts target resolution. Unlike the earlier works, a closed form solution for the waveform based on Discrete Fourier Transform, DFT, is proposed in [32], which extends the earlier works on Fourier-based direct waveform design [33], [34] The key contribution of [32] is the closed form solution for the waveform under a discrete unimodular constraint and a uniform element power constraint. While the design is simple, the cross-correlation properties of the transmitted signal are not investigated further; this impacts the resolution performance in a virtual MIMO configuration. The authors in [35] deal with a unimodular constraint by reformulating the problem to a biconvex optimization, but do not account for cross-correlation properties, which impacts the target

resolvability. Other related literature on transmit beampattem design include [11], [36]— [38] .

A block circulant decomposition based waveform design employing discrete phases is introduced in [30] for two modulation channels. The present work extends the block circulant approach to an arbitrary amount of channels, phase stages and transmit antennas, which describes the most general case of multiple phase shift keying with a subsequent time multiplex. The presented approach has the following contributions:

A novel Block circulant stmcture for the signal matrix is proposed in order to achieve good signal cross-correlation properties and the desired beampattem. This stmcture implicitly imposes the desired cross-correlation unlike their omission, e.g., [32], [35] or explicit inclusion in the objective function, e.g., [21] This reduces the problem dimensions and simplifies the related optimization problem as well.

_• The proposed solution also addresses relevant hardware constraints including the uniform power, inter-pulse modulation and the possibility of multiplexing few RF chains over multiple antennas. The time multiplexing constraint is not included in many related works [21], [32], [35] and makes the already complex quartic optimization problem even harder to solve.

Formulation of the convex optimization problem for beampattem design by introducing a signal dictionary matrix in order to meet the requirement of discrete phase modulation (any number of discrete phase stages and modulation channels) with a subsequent time multiplexing to an arbitrary number of transmit antennas. The dictionary approach enables a low complexity convex solution instead of bi-convex [35] or a sequence of convex optimization problems [13] The choice of the dictionary and its impact on the design and performance is investigated.

An explicit stmcture for the dictionary matrix is provided for special cases of two channels and for the no multiplexing case. Exploiting these, closed form solutions for the signal matrix are provided building on works [30], [39] A Fourier transform based approach is considered for the special case of two active antennas and four phase stages [30] . A DFT based discrete phase modulation (discrete unimodular) is derived when the number of channels is equal to the number of transmitters and phases [39] This solution is similar to the DFT approach presented in [32] and [33], but with the additional properties of arbitrarily long waveform sequences and inclusion of desired crosscorrelation implicitly in the signal design. Moreover, in [32], [30], [39] there is no investigation of coupling issues on range nor Doppler (depending on the choice of interor intra pulse modulation).

_• Multiplexing schemes naturally lead to angle-Doppler coupling. This work considers a simple manipulation of the signal design process - random permutation of waveform sequence - to suppress the angle-Doppler coupling. In this context, it generalizes the investigations from randomized phase centre motion in single channel multiplexing techniques [29] to multiple channel schemes for achieving a desired beampattem in MIMO radar. This methodology translates the peakier side-lobes into a nearly flat floor, enhancing unambiguous Doppler.

The remainder of the description for the present preferred embodiment of the invention is structured as follows. A detailed model of the considered system including the architecture, constraints and the transfer function is described in Section II. The proposed block circulant signal design for beampattem synthesis is presented in Section III where its properties are highlighted and the resulting optimization is formulated. Dictionary based approach to the solution of the beampattem optimization is presented in Section IV and special design cases offering closed-form solutions highlighted. The angle-Doppler issue is introduced in Section V along with the mechanism to suppress it within the proposed framework. Section VI corroborates the findings with numerical examples and Section VII concludes the paper.

In this work, the operator ^ defines the l_p- norm. A matrix entry is defined by | -

where h denotes the row index and y the column index. A vector entry | - [, is defined by one index h. The E { } denotes the expectation operator. The set of complex numbers is defined as C, while 3 = -ϊ represents the complex number. The Kronecker and the Hadamard product is defined as ® and _°, respectively. Further, ^T,^H denote the transpose and Hermitian operations.

II. SYSTEM MODEL

The underlying radar system comprises a local oscillator, generating a train of I_c consecutive pulses in each Coherent Processing Interval, CPI. This pulse train has a center frequency f, and a modulation bandwidth B. Each pulse of duration T, is further modulated in the transmit modulation unit on a pulse- by-pulse manner. Further, the transmit modulation unit comprises N_c phase modulation channels; each channel has the capability to modulate the pulse using A), distinct phase stages that are distributed uniformly around the unit circle. The Ay phase modulated channels are multiplexed among N_T transmit antennas, where N £N_T, to reflect the capabilities of evolving hardware systems. The N_R receive antennas are arranged in a Uniform Linear Array, ULA, configuration along x-axis with an inter-element spacing of

⁼ f . The transmit antennas are also uniformly mounted along the x-axis with an inter-element spacing of ^{rl / =} - ^/·' 2. This leads to a sparse transmit ULA. The transmitted signal propagates towards T targets which are assumed to be located in the plane-wave far field region. The back-scattered signal is captured by Ay receive antennas, each having its own digital processing chain; this leads to the use of N_R Analog to Digital Converters, ADCs. The Ay received signals are processed by a matched filter bank in the digital domain. The information gathered at the output of the matched filter will be used later in the proposed signal design.

A. Transmitted Signal

Since the receiver is operating in the digital domain, the signal model of the transmitted waveform is described in that domain as well. Each pulse consists of \_s samples, where i_s denotes the sample index. The pulse vector, p e C^l represents the sampled version of a pulse p(t). i.e., | p |_/v = /J( / _v). where T_s denotes the sampling time. As indicated before, the transmitted waveform is composed of a train of pulses which are further modulated on a pulse-by-pulse basis by the transmit modulation thereby leading to an inter-pulse modulation. In particular, the transmitted signal during the /_c-th pulse radiated from the n- th transmit antenna takes the form, ¾„(«)p, where ^sy(ⁿ) corresponds to the inter-pulse modulation symbol. Thus, each pulse vector p is modulated in phase and in amplitude before the signal is radiated by N_T antennas. Further, ^sA (^rt)e W₀, where the set W₀ = {0} U W includes zero to c

model the multiplexing, and the set

comprising N_p distinct uniformly distributed phase stages to enable low complexity realizable platforms for radar.

Let ^sv A ^o ^{1 x 1} denote the stacking of the N_T transmit antenna weightings, {·¾ (ⁿ) }n=i, for the /_c-th pulse. The signal modulation vector, s ^c --o^{' X l} ' further contains the antenna weighting vectors

The entire transmit signal vector x e C^I*^{Io ¥ x l}can be denoted as the Kronecker product of inter pulse signal modulation vector, s and pulse vector,

x = s ® p. (2)

Design of p has been extensively considered in literature; however, the focus of this paper is on the design of s towards better angle-Doppler resolution. B. System Transfer Function and Received Signal

It is reasonable to assume the pulse duration to be much greater than the propagation delay for the target in far-field, i.e..

^ T , where c₀ is the speed of light and r_max is the maximum target range.

Further, if the maximum target velocity is small such that the target position can be seen as constant during the pulse duration T_c. the Doppler and range information become separable. In fact, the range and Doppler information can be obtained from the fast-time (within a pulse), and slow -time domains (across pulses) respectively. The radar environment comprises T targets whose Radar Cross Sections (RCS) follow the Swerling-I statistics [40] Particularly, a_K e C, the RCS for /c-th target, is a statistical parameter and the RCS of distinct targets are independent of each other. The angular information for the c-th target is modelled into the receive steering vector ^a-R,«^€ C^{A fl X 1} _anc[ transmit steering vector

[a _R,K]_m = exp (jk₀ sin (p_K)d_R(m - 1)) (3)

I a_{? K}\_n = exp (jk₀ sin(f_K)d {n - 1 )), (4)

with^^{0 ~} T (free-space wavenumber) and c/y being the angle of arrival of the /c-th target.

The received signal experiences a Doppler shift co_Kfor the /c-th target, whose effect is modelled using the diagonal Doppler matrix

^e C¹

D_w„ = diag (exp(ju_KT_c), · · · , exp(jw_KT_cl_c}) . (5)

On the other hand, the range information for the /c-th target appears in the time shift of the pulse vector p. The time shift and, therefore, the /c-th target range information is modelled by a shift matrix J_K€ {0; l}¹' ^1* [41] The received signal vector corresponding to /_cth pulse at the rth receive antenna, denoted by ^Zr _>*= ^e C^{1* 1}, taL_es the form,

Let ^{z e} C^{A' RI}„^I, ^{X I} ^_ehoί6 t e received signal vector obtained by stacking ^Zr _>k of (6) over received antennas and I_c pulses. To model z, it is convenient to exploit the interpulse modulation and define the inter-pulse transfer function ¾ e ^{CI / H XX A} for the K- th target as [30],

H_K = a (ajt,_Ka^_K) ¾ D_Us . )

Using (6) and (7), it follows that,

It follows from (8) that the slow-time information H_Ks is separable from the fast-time information J_Kp due to the Kronecker structure. Further, since this paper focusses on inter-pulse modulation, it suffices to only investigate the slow-time information. Referring to (8), the term of interest is the slow-time receive signal vector V ^{e x 1} d_ef_med as, Due to the definition of H_Kin (7), the signal design only affects the angle-Doppler response of each target. In the sequel, only the inter-pulse receive vector in (9) is considered.

C. Matched Filter

The information in the received signal is extracted by a matched filter with the coefficients y = H s. Since the RCS is not known a priori, the matched filter is parametrized by OJ_K and ^ similar to the receive signal model as,

HM = (^aAM^a _T,M) 8 D.M, (10)

where a_7: . a_{/; ¾/}· D arc obtained from (3), (4), (5) by using design parameters {f_M,w_M} instead of {f_k,w_k} respectively. Since the target RCS follow a Swerling-I model [40], the parameter of interest is the ensemble average of the squared matched filter output over the random {a_K}. Denoting m as the average matched filter output, it follows that,

Exploiting the block diagonal structure of H and H_K, (kindly refer to (7), (10), respectively), the expression z_^K= i ^{s h}«^H¾ above can written as a sum over I_c blocks. Further, the uncorrelated RCS fluctuations of distinct targets results in the expected matched filter output to be a superposition of K target responses,

ttenuation factor and recall that ^sx --n ' is the modulation vector for the /_c-th pulse.

The Doppler shift is omitted explicitly in the initial investigation to ease waveform design. The waveforms thus designed will then be adapted in Section V to cater to Doppler. Thus, without

2 considering the Doppler shift, (12) reduces to J^j- =

^aT«

Recalling that a_/ and a_j-Kare functions of the corresponding angles of arrival i/i and <p_k respectively.

( M^Ic s ; s¹¹ ) a _T

the term a Vw*^{c=1 c} / is a function of y_M,y_k respectively. To highlight this dependence, literature [21], [22], [30], [35] defines the cross-correlation beam pattern as, In fact, if the actual target angle coincides with the matched filter design, i. e. f_M= f_k, R{y_k,y_M) specializes to the radiated beam pattern along f_M. In other settings, R(f_k,f_M) indicates the resolution capability of the matched filter output highlighting the utility of R(f_k,f_M)·

D. Signal Design Objective

Let the signal matrix S^e - -o ^{, /å'} contain all the modulation vectors s in its columns,

S = ( si · · · s_{ic sZc} ) . (14)

Then, the covariance matrix of transmitted signal, e€^{A rXiVr} defined as,

is central to the design of the auto- and cross-correlation beam pattern (kindly refer (13)). In fact, for transmit signals that are orthogonal over a CPI, R becomes diagonal; the virtual MIMO concept is then satisfied resulting in the maximum angular resolution [3] But if the signals are orthogonal, the transmit radiation pattern is inherently isotropic, which leads to a loss in Signal to Noise Ratio (SNR) [1]. On the other hand, traditional beamforming with coherent transmissions, improve SNR, but have a poor angular resolution.

In view of the discussion above, the objective is to design a transmit signal such that a desired radiation pattern is achieved, while also enabling the virtual MIMO paradigm for maximum angular resolution. Towards this, the design of the signal dependent cross-correlation beam pattern, R(f_k,f_M) is investigated. It is well known that orthogonal transmit signals achieve best resolution performance for the virtual MIMO configuration [3] and R(f_k,f_M) also reflects this property. Further, the cross correlation beam R(f_k,f_M) is shown to be related to the transmit signal covariance matrix R 121 1. Therefore, the cross-correlation beam pattern incorporates the desired properties and provides a suitable design objective for enhanced angular resolution and beam pattern design.

III. BEAM PATTERN FORMULATION BASED ON BLOCK CIRCULANT SIGNAL DESIGN

Based on the derived model, the inter-pulse modulation matrix S or the related inter-pulse modulation vector s chosen from W₀, has to be designed, such that a desired cross-correlation beam pattern is achieved. Towards this, the cross-correlation beam pattern is investigated in more detail and a suitable signal design framework is formulated. A. Cross-Correlation Beam Pattern and Signal Covariance

This section builds on the choice of design metric and provides a formulation for the design of the signal covariance matrix to achieve a desired beam-pattern. A first step towards this is the sampling of P( ) in (13), which is described now.

1) Sampled Cross-Correlation Beampattern: Let the 2-D desired cross-correlation beam pattern, R(f_k,y_M), be sampled along />_Kand <9 _\/ w ith Ayand Ay_/samplcs respectively. These samples can be compactly arranged in a matrix,

p _€ C^{JVM x A}' _with i^p] «being P(·, ·) evaluated at the p,q sample of f_M& nd ^respectively. Let

the transmit array and matched fdter steering vectors obtained by using the p,q sample of f_M · nd /y in (3) and (4) respectively. Then, letting ^Ar _^ ^{K =} (^ar-«n ; - ^aT,_KN ^<E £^{NT XN} _ancj number of

samples for the matched filter steering matrix, ^AT,M and the transmit array steering matrix Άt,k, are set equally to N₀, i.e., N₀ = N_K = NM, then

and the cross-correlation beam pattern can be written as,

P = A R_sA_r> (16)

where ^Ar is the ULA steering matrix. It now remains to describe the samples {/¾}; inf act, the angular domain is sampled non-uniformly in yi, such that,

l No N₀

sin (fi) = 2 < l <

JV 2 2 (18)

The non-uniform sampling is reasonable, because the field of view is between ^;:t 2 and the sine function is monotonic within this domain. Using ( 18).^^{7 ~}

from Section II in

(17), yields a steering vector matrix having a Discrete Fourier Transform (DFT) like structure,

Since the maximum possible angular resolution and therefore the best target discrimination is achieved for orthogonal signals [3], the cross-correlation beam pattern for orthogonal transmit signals, denoted by P_j_, is defined as,

where each transmit signal is assumed to have unit power. The elements of P_± have the structure of a sinus-cardinal function,

The structure of P_± shows the limit on physical resolution imposed by the antenna array size. When N₀ =N_TN_r, the ^NT x ft RNT matrix A T in (19) reduces to,

where, ^^JVH is a

^{x 1} column of ones and A_T is the N_T x N_T scaled DFT matrix. It can then be shown that r₁ = ^{Lί I¾ 1}¾¹ϊ„. This clearly illustrates the repetitive structure of the transmit radiation pattern for the sparse transmit array configuration; its impact is studied in Section III-A2.

Due to the structure of ^At· ^N° ^>

does not provide additional information. Further, ft o < L'AAT does not capture the periodicity well. Hence, it suffices to consider Ay =N_TN_R. Furthermore, that the radiation pattern (k n) for the orthogonal transmit signal case is always constant, leading to an isotropic radiation pattern discussed earlier.

2) Beampattern with sparse transmit array: The sparse transmit ULA imposes a periodic structure on the beampattern. The replication of N_R scaled DFT matrices in (22) is justified by the sparse nature of the transmit ULA. This structure simplifies P in (17), as ^ ^— (A_r R_sAr) ®

Due to the induced periodicity, it suffices to consider only one period of the transmit radiation pattern as it is the one that can be actually designed. This enables the reasonable simplification where only the principle N_T x/V matrix of P in (17) is investigated henceforth (alternatively only the first N_T columns of ^AT are investigated). Noting that the first N_T columns of ' is A₇ (refer to (22)) and letting P to be the relevant N_T ^c N_T submatrix of P , it follows that,

P = A", R ,. (24)

Equations (23) and (24) will be used in the sequel as beamsteering matrix and transmit radiation pattern.

3) Desired Beampattern and Signal Design: In order to relate the insights of the orthogonal transmit signal investigation to an arbitrary beam pattern design, the desired beam has to be defined. Since a maximum resolution is desired and that rank (P) < N_T, the desired cross-correlation matrix P e k^{Lt cL¾} has to be diagonal. Further, the diagonal elements contain the desired radiation pattern vector Pd e - - comprising non-negative elements,

P_d= diag(p_d). (25)

With the desired pattern defined, it now remains to formulate a signal design (or inter-pulse modulation matrix) criteria. In order to ensure that the objective function satisfies the multiplexing paradigm, a zero norm constraint is necessary. Further, a uniform power transmission from all antennas is desired. Therefore, the optimization problem can be formulated as: The optimization represents a quartic problem [21], [35] under finite alphabet, uniform power and zero norm constraints, which is in general hard to solve. When N₀ =N_TN_R, noting

= L«l/2 i follows from (23) that ^ίAt^ ^L

deduced that ^{Ar =}

where F in the N_{T c}

N_T unitary DFT matrix. It then follows from (24) that,

P = JV_TF¾F. (27)

Since P_r/ is diagonal, so is Py infact, P_d= diag(p_r/). The design problem then becomes,

Incorporation of additional structure to the transmitted signal leads to tractable, albeit sub-optimal, solutions of (28); these are further discussed in the sequel. Further, P_d,P would be continued to be referred to as desired and generated cross-correlation beampattems respectively in the sequel.

B. Block Circulant Decomposition of Signal Covariance

The goal of the Block Circulant Decomposition (BCD) approach is to construct the inter-pulse modulation matrix S such that the off-diagonal elements of the generated cross-correlation beampattem matrix, P, become zero. From (27), P becomes diagonal if the eigenvalue decomposition of R contains the DFT matrix as its eigenvector matrix. In fact,

P = A ^H _T R A, = N_TF^HR_SF = N_TA (29)

if R = FAF^is the eigenvalue decomposition of R . where A e I¾^{LGT CAG G}·_{8 tilc} eigenvalue matrix with non-negative eigenvalues. From the definition of P_d, these non-negative eigenvalues, or equivalently, the diagonal elements of P contain information about the sampled radiation pattern. Therefore, ensuring the construction of S such that its eigenvector matrix is a DFT matrix for any set of associated eigenvalues is particularly interesting as it simplifies the design. Circulant matrices exhibit this property [42], a fact exploited henceforth.

Let ^{A i:}— Av be an integer and B' ^e .f^{T XNT ,} b = 1 2 Ay be a circulant matrix with entries drawn from W₀. Further, let the signal matrix, S, be constructed using B_¾ as,

where, Y* is the eigenvalue matrix of B_¾ [42] . The structure of S, comprising blocks of circulant matrices, leads to the terminology BCD. It can be shown that the S generated from (30) leads to a circulant Hermitian R .

From (31) and earlier discussions, it follows that the BCD based design in (30) provides an attractive framework for the sampled cross-correlation optimization, as the off-diagonal elements are zero by construction. Further, the resulting circulant R implicitly leads to a uniform power transmission from the transmit antenna elements; this ensures the constraint of uniform element power is satisfied from construction as well.

C. Beampattern design with BCD

Since each B_¾ is circulant, it is completely defined by its first column, denoted here as, c'- ^c: - -o '

[42] Hence, S in (30) is parametrized by N_B column vectors, {*¾} ^e - -u ^{, z} , where the b denotes the index of the 6-th block. Since the eigenvalues of the 6-th block circulant matrix B_¾ are given by the the DFT of C/,. i. e., Y* = diag(Fc_¾), the eigenvalues of R arc given by actual radiation pattern vector p e lf ^{x l}

Since only the diagonal elements of the optimization problem

(26) contribute to the cost and the uniform power constraint is inherently satisfied by a BCD, the optimization problem under the BCD construction simplifies to,

The BCD provides a suitable method for obtaining good cross-correlation beam pattern properties for ULA under the uniform power constraint. It can be seen as a framework for enabling the ULA virtual MIMO paradigm. The solution to the optimization in (33) is discussed in the sequel.

IV. DICTIONARY BASED OPTIMIZATION FRAMEWORK

The finite alphabet constraint problem in (33) needs additional operations towards easing the optimization process. This section provides a framework based on a dictionary matrix approach, in order to transform (33) into a convex problem. Further, special cases of this framework are presented and shown to be solvable in closed form. Therefore, the proposed framework extends the works [30] and [39] to the general case of having an arbitrary number of channels and phase stages. A. Optimization Framework for Signal Design

The solution considered for the optimization in (33) is based on a dictionary approach. Accordingly, the designed radiation pattern p can be written in terms of N_D distinct atoms, where the z-th atom and the corresponding coefficient are denoted by Z+, respectively, thereby leading to,

Here, Q^G ° denotes the dictionary matrix and v e

denotes the atom coefficient vector.

While there is no restriction on the number of atoms, to ensure that (34) represents (32), the following constraints are imposed,

1) Each atom is generated using an arbitrary (as yet) transmit array excitation c ^e --u ⁷ ' l!wlo = A as,

qi = (F^H<¾) o (F^ffc_¾) \ / e [l, A¾] ₍₃₅₎

c

contains the possible phase modulations and zero (for multiplexing). This enables a structural similarity of (34) to (32) and renders q, to be non-negative.

2) Note that (32) requires no more than N_B vectors, {c, }. while (35) considers N,, vectors. To ensure consistency, scaling factors v, have to be non-negative integers and satisfy the /_|-norm constraint, l|v|i = NB These two constraints ensure no more than N_B terms in the summation of (34) including repetitions of q„ even when N_D > N_B. In particular, this ensures

where f(ί) e [1,L¾] is the mapping set and c,_Ml) is used to construct v, circulant blocks.

3) The optimization problem (33) can be reformulated as,

The dictionary Q can be designed offline using arbitrary c _*

I Io = V, f_rom (35)· this implicitly satisfies the multiplexing and discrete phase modulation constraints. The problem (36) is non-convex due to the non-negative integer value constraint of ^{v e}

It can be simplified if v is relaxed to K^{¥lJ x 1} , a vector of non-negative reals . However, in order to relate the coefficient vector to the column vectors { C_/, } . a subsequent rounding to the nearest integer is necessary. The problem in (36) can be reformulated as,

where e > 0 is a given design constraint and the relationship between the real v and the integer v is defined by the rounding function X(c) = max { : e Z\k < x + 0.5}, as,

The aforementioned construction of v closely approximates the requirement IMIi = ^LB in (36) and the slight deviation, if any, arises due to the rounding. In this context, it is clear that a good rounding property, and hence closer approximation of l _\ norm constraint, is achieved for a sparse solution of the v e m^{Nn X l}. This motivates the /₀ objective in (37). In case H^vlli ¹ N_B, it is adjusted to N_B by increasing/ decreasing randomly chosen entries of v. While this is indeed sub-optimal, it is easy to implement with, possibly, limited performance degradation.

Robust Design Criteria: Since the desired beam pattern can be only approximated due to physical limitations, a suitable real non-negative error ^{e e} R+ has to be considered to ensure robust design. Equation (37) can be relaxed to a convex problem by the well-known /_rnorm relaxation [43],

The optimization tuning parameter h E R₊ takes into account the trade-off between sparsity of the coefficient vector and the beam pattern error. In other words, this trade-off deals with the trade-off between rounding error and beam pattern error. The optimization problem (39) is convex and can be easily solved in polynomial time using well-known solvers. Together with the BCD approach, the optimization problem (39) provides a cross-correlation beam pattern design framework for any waveform alphabet. Table I summarizes the proposed BCD framework for beam pattern design.

The error in the signal design framework is justified by the rounding error as well as by the design of a suitable dictionary matrix Q. The design of the dictionary Q is tackled for particular modulation sets in the sequel, whereas the rounding error is investigated through simulations.

TABLE I

ALGORITHM FOR GENERAL PHASE MODULATION AND MULTIPLEX

SEQUENCE DESIGN

B. Discrete Phase Unimodular Design

The unimodular design for S provides the best power efficiency [23] and is therefore of great interest. Further, the discrete phase stages are an attractive practical design constraint. For the particular case with identical number of transmit antennas, channels and phase stages N_T= N_P = N_c, the problem in (39) can be solved in closed form, due to an appropriate choice of dictionary matrix [39]

In particular, the z-th vector c, is chosen from the Fourier bases as,

Using (35), the resulting dictionary simplifies to,

Q = N_TI_nt. (41)

This choice is only satisfied for the case of N_T= N_P= Ay.

1) Design Algorithm: Since the dictionary matrix reduces to a scaled identity, it is not necessary to solve (39). Step. 1 in the signal design algorithm in Table I is replaced by a closed form,

(scaled desired radiation pattern) due to (41). Therefore Step 2 of the signal design algorithm directly considers the desired radiation pattern to yield,

The proposed signal design algorithm for discrete phase unimodular sequences is a closed form solution without any iterative steps. Performance in this scenario depends on N_B due to their ability to reduce the quantization error. Performance of such designs is illustrated in the numerical results section.

C. Two Channel QPSK Based Design

The two-channel multiplexing is of interest as it reflects the minimum number of channels needed to design a beam pattern from isotropic radiators. Further, this design considers the 2D Quadrature Phase Shift Keying (QPSK) modulation, i.e., W = { 1,- 1,-/,/} . This approach lends to a simple solution benefitting from a structured dictionary formulation. 1) Dictionary Formulation: Since two channels are switched on, only two elements of any arbitrary c„, say /i and i₂ with /) 6= i₂ are non-zero. Hence only two column vectors of FT denoted by f,i and f_i are oft i interest f tor when evaluating (F” ^cn ) ° (F^H c_n ) . With q„ being the atom generated from this c„, it then follows that,

where 1 _\T in a N_T dimensional column vector of ones. Due to the two-channel assumption, the m-th element of q„, denoted as | q„ takes the form,

where, <¾,i = |c„|,i and f^a = |c„| ₂ reflect the phase information of the / ₁ -th and /₂-th vector entry of c„. It can be seen from (44) that each q„ is characterized by two parameters

1) The relative distance between excited antenna elements, i_{\ - /2}, determines the frequency of cosine function. Due to the limited transmit antenna array size, there can only be N_{T -} 1 different frequencies, thereby leading to Ay - 1 linearly independent functions.

2) Due to the QPSK modulation, there are 4 possible phase difference per frequency given by f,,,

{0,p/2,p,3p/2} irrespective of n,i_bi2 for each frequency. The relative phase of the excited antenna elements y_{/ /2 -} q>_nj\ determines if a positive/ negative sine or cosine function appears in the atom.

Hence, the number of distinct atoms is 4(N_T ^~ 1). Using

the atoms are positive and take the form,

2) Reformulation of Optimization: With the dictionary matrix defined, the steps in Table I can be pursued towards signal design. However, the closed-form expression for q„ can be exploited to avoid Step 1 as in Section IV-B. Towards this, at first, a reformulation q„ in (44) using cos(x + y) = cos(x)cos(y) - sin(x)sin(v) is undertaken. Letting , n =4(k -1) + l, l e [1,4],L e 1 1 , Ay - 1], this leads to

With the 4(N_T- 1) atoms defined, the n- th element of the generated radiation pattern in (34) takes the following form,

coefficients of the sine/ cosine terms. Since these sine/ cosine terms are not the atoms per-se (given in (44)), a, a arc different from v of (34).

The result in (47) is an expression similar to the Fourier series. Therefore, the elements of a_c,a_s can be obtained

analogously to Fourier coefficients as,

k-ih Fourier coefficient

TABLE II

Signal Matrix Generation Principle for QPSK

ALGORITHM FOR TWO CHANNEL TDM QPSK MODULATION

The Fourier coefficients have to be mapped to the actual transmit signal matrix S. The first step towards this is to round these coefficients to a positive integer as,

The function X denotes the rounding function. Further, the following relationship has to be satisfied,

L^Gb = (l»cli + I¾ li ) (52)

The rounding in (50) and (51) meets the requirement in (52) closely. As in the previous discussions, a randomly chosen coefficient is increased/ decreased to satisfy the requirement.

3) Signal Matrix Generation: Table II illustrates how to generate the BCD blocks from the Fourier coefficients a, a . As an example, if [aj* > 0, it follows from (46), (47) that the frequency of the sine wave, and hence the distance between non-zero elements, [c„],i , [c„]_i2 of c„ is k. Due to the BCD, the actual choice / 1, 7₂ does not matter as long as |/i— /₂| = k Further, the phase difference between the excitations need to be p/2, kindly refer to (46), (47). Further, the determined c„, is used in

blocks. New excitation vectors are similarly determined for all other cases.

It can be seen from (44) that only the phase difference Accounts, leading to a degree of freedom in the design procedure. A way is to set [c„],i equal to one and determine [c„]_i2 accordingly. Another way is to select [c„],i randomly out of the given set and determine [c„]_i2 accordingly. Table III shows the signal design algorithm for the two channel QPSK modulation. This algorithm provides a closed form, possibly sub-optimal, solution of the optimization problem (33) under the aforementioned constraints.

In contrast to the closed form solution in [32], [34] and [8] the proposed approach takes the virtual MIMO configuration into account and further considers finite alphabet design constraints. The presented two channel QPSK approach can be seen as a special case of the proposed framework where (39) is solved for the dictionary in (44). Further, in contrast to the discrete phase unimodular approach the dictionary size is larger than the number of transmit antennas N_D > N_T, but the interpretation using Fourier series enables the solution to the problem in closed-form.

V. DOPPLER COUPLING ANALYSIS The aforementioned signal design for generating the desired cross-correlation radiation pattern omitted Doppler, which limits its application only to stationary targets. In addition, it is known from TDM MIMO that the pulse-by-pulse antenna switching leads to angle-Doppler coupling. These merit the consideration of Doppler and the potential coupling issues. A first step towards this is the investigation of the matched filter output in the presence of Doppler.

A. Angle-Doppler coupling in Generalized TDM MIMO

The angle-Doppler matched filter output from (12) becomes,

The second term of (53) captures the angle-Doppler response; denoting this for the target k as m_AO, using \_C = N_BN_T , \o , = w_M w_{k 3}aά the construction in (30), it follows that,

Note that the sum over I_cin (53) is replaced by a sum over b and /. The first sum represents the sum over blocks, denoted by the block index b, and the inner sum over the columns within each block

(denoted by p). Since the columns within a block i.e, Sv_{Th p}.p = 1. Ay. are circular shifted versions of

C_/,₊1, using the shift theorem of the Fourier transform, it can be shown that,

a“_Ms_NTb+k = e-J^k»®*>^pa“_'Mc_b+l'

r

where \k_K = ₀(sin(¾ _f) - sin(^_K)) denotes the relative angle response, between the matched filter output ¾ _f and the actual target position f_k. Further, the term e ^{1 A7}' illustrates a linear phase center motion (PCM), which leads to a strong angle-Doppler coupling [29], as illustrated by the first term in (56). Since this linear PCM is justified by the block circulant construction, modifications to this construction in (30) are warranted. In this context, it is known from [29] that the angle-Doppler coupling due to linear PCM can be overcome by signal randomization. This motivates the subsequent investigation on signal randomization. The second term in (56) represents a DFT operation as well, but the spectrum is inherently periodic by N_T. This periodicity is also justified by the block circulant construction of (30). Further, it is desired that the argument of the second DFT i. e., &^H _T fi._bc^H _{b &TK}, coherently sums up such that a peak at Aw_k =

0 occurs. Since the vectors {c_¾} are the output of the optimization algorithm and not known a priori, a conjugate symmetric condition on { C_/, } is introduced. This simplifying condition results in &^H _TMc_bc^H _b a^being real for all b, potentially enabling coherent summation at Aw_k = 0 These modifications to signal design are discussed next.

B. Randomization of Signal Matrix

1) Motivation: The angle-Doppler coupling is described by w ,_/,_A in (56). Due to the presence of Doppler, this term is dependent on the PCM properties of the designed transmit signal matrix. In other words, the coupling is dependent on the temporal order of antenna excitation, unlike in the case of zero Doppler. Clearly, the temporal ordering adds to the complexity of design; to ease the design procedure while incorporating this dependence, a stochastic model for the signal matrix is assumed henceforth. Such an approach enables the incorporation of the low complexity BCD signal design paradigm to further mitigate angle-Doppler coupling.

2) Implementation and Modelling: A statistical design to alter the phase centre is pursued by randomly permuting the columns of S. To analyze this situation, we model the resulting s_p to be random variables which are independent for different p. This model corresponds to the view that each vector s_p is assumed as being obtained from choosing a c_b randomly and then circularly shifting by a random number of shifts. This simple stochastic model will henceforth be used to model the random column permutation of S. The impact of this modelling is highlighted in the next section.

3) Expected Matched Filter Output: Analogous to the stochastic approach in [29], the ensemble average of (53) over signal distribution is considered. Focussing on the coupling, the resulting expected angle-Doppler coupling ¾D e R+ at the output of a matched fdter in (53) takes the form,

where the expectation is over the distribution of s _C. Letting ^ZP ^ax,M^sp^sp

(57) _can be simplified to

Using the modelling assumption that s_t and _S/arc independent for

^, V p

(identically distributed), and ^ZP ^{~ a}x,M^spsp ^av (58) can be simplified as:

E (_{zh z}x)

Using this simplification in (58), it follows that,

E

Since { s_p } are identically distributed,

is independent of pulse index and (59) further simplifies to,

where, W( \w_kΊ^') is the frequency response of a rectangular window of length I_c evaluated at Aa>_KT_c. The expected matched filter output in (60) generalizes the white PCM approach in [29] and its interpretation is provided next.

C. Coupling Analysis and Design Algorithm

The expression in (60) comprises two terms, the perfect matched filter response related to the factor I_c ² and the coupling term related to the term I_c. It can be seen that in the limit of I_c tending to infinity, the coupling term vanishes. However, for a practical signal design, the sequence length is finite and the coupling term appears.

The coupling term depends strongly on the angular domain with the Doppler cuts depending mainly on the angle. The coupling is function of the transmit signal and its contribution is lower than the other term. Therefore, the coupling term shapes the sidelobe floor in accordance to cross-correlation properties. Thus, the randomization results in the transformation of the strong coupling (typically peaky side-lobes) in (56) to side-lobe floor enabling enhanced target discrimination.

The Doppler analysis completes the proposed inter-pulse signal design. Table IV shows the complete signal design framework. The algorithm is initialized by a desired radiation pattern as well as an offline designed dictionary matrix. As shown earlier, the dictionary matrix is crucial for the algorithm performance. The performance of the dictionary matrix is further discussed through simulations. The first step after the initialization is to solve the proposed convex optimization problem in (39). If the signal design constraint to unimodular or two channel QPSK modulation, the convex optimization problem can be replaced by the presented closed form solutions. The third step is the rounding of the coefficient vector. After the rounding, a block circulant matrix is constructed, where the block circulant construction accounts for the cross-correlation properties and therefore for the virtual MIMO concept. In step four the proposed randomization is exploited to address the angle- Doppler coupling issue.

TABLE IV

GENERAL SIGNAL DESIGN ALGORITHM

The devised BCD framework proposes an inter-pulse modulation signal design for any discrete phase modulation with a subsequent multiplexing. Further, the virtual MIMO concept is enabled due to good cross-correlation properties of the block circulant decomposition approach. Moreover, the proposed approach is related to phase center motion approaches, due to conjugate symmetric modulation and the angle-Doppler coupling issue is solved. The ensuing section highlights the performance advantages through numerical simulations.

VI. SIMULATION RESULTS

1) Simulation Set-up: The simulations are carried out with ten transmit antennas, N_T= 10, and four receive antennas, N_R = 4, in a MIMO configuration as described in Section II. Unless mentioned otherwise, the number of channels and phase stages are set to six N_c = 6, and the dictionary size and number of blocks are chosen to N_D = 500 and N_B = 500, respectively. This setup is called the standard simulation setup in the sequel. On the other hand, the paper also considers the special case of N_T= N_P = N_c = 10 where the dictionary is chosen as a DFT matrix, leading to N_D = N_T= 10; this is referred to as the discrete unimodular simulation setup. In the plots for the standard simulation setup, the legends, General Framework Conjugate Symmetric and General Framework refer to the design in Section IV-A with and without conjugate symmetric constraints on excitation. Further, the legends Unimodular and 2 Channel QPSK refer to the designs from Sections IV-B and IV-C respectively. It is further noted that utmost 5% additional increase/ decrease of quantized coefficients (kindly refer Step 3 in Tables I, III, IV) is needed to satisfy the norm in the ensuing simulations. 2) Enabling virtual MIMO\ the resolution performance of different dictionary matrices as been evaluated. All approaches were found to almost align with the perfectly orthogonal signal case; this shows the effectiveness of the BCD approach and its cross-correlation properties. The perfectly orthogonal sequence relates to a scaled identity matrix for the transmit signal covariance matrix. The Mean Square Error (MSE) between perfectly orthogonal signals and the algorithm output for a randomly initialized dictionary matrix averages -18 dB. The MSE for the unimodular simulation setup is practically zero, which implies that the right choice of the dictionary matrix is crucial to the design.

3) Design of arbitrary beam-pattern: Figure 3 and Figure 4 illustrate the algorithm output for two different desired radiation patterns for different dictionaries. Common to both

beam patterns is the offset (in amplitude) with respect to the desired pattern. This offset is justified by multiplexing, which leads inevitably to a non-identity dictionary matrix. The restriction to positive coefficients v, leads, in turn, to a DC value in the radiation pattern design.

4) Impact of system parameters on Beam-pattern design: Since the optimization output are positive coefficients, the discretization is necessary to satisfy the constraint. The rounding influences the beam pattern output and Figure 5 illustrates the effect on quantization. The higher the number of blocks, the lower is the error in rounding of the coefficients v. Therefore, the ratio of nonzero elements of v and the number of blocks N_B determines the fidelity of reproduction of each nonzero element of v. Since the algorithm optimizes for sparsity in v, the solution tends to have a fine resolution of v. Further, enhanced sparsity in v, in turn, affects the number of blocks which are needed for appropriate beam pattern design. In derogation from the standard simulation setup, the number of phase stages and channels can be chosen arbitrary.

The dependence of the radiation pattern error e on N_P and N_c for a randomly initialized dictionary matrix has been evaluated, where the dictionary matrix becomes identity. It was found that the error decreases if the number of channels and phase stages increase, leading to more degrees of freedom in the beam pattern design. Further, the unimodular setup demonstrates that a proper choice of the dictionary matrix improves the algorithm output significantly.

Further, if the size of the randomly initialized dictionary is increased, the probability of selecting good atoms also increases and the optimization provides better results.

5) Angle-Doppler coupling and mitigation: The block circulant property offers a good design procedure of UFA’s, but on the other hand the block circulant structure leads to a linear phase center motion. Figure 6 shows the angle-Doppler ambiguity function for a block circulant signal matrix structure. As expected and extensively discussed in literature for TDM MIMO angle-Doppler coupling appears. This can be seen from the number of peaks in the ambiguity function, which tends to restrict the unambiguous Doppler resolution.

In contrast to Figure 6, Figure 7 demonstrates that even for the multiple antenna and multiple phase configuration, the randomized PCM procedure holds without influencing the radiation pattern properties. It can be seen that the randomized angle-Doppler ambiguity function is similar to the TDM case, including the v/wc-function characteristic due to the uniform power constraint. Since the transmitted signals are correlated, the ambiguity function changes in terms of sidelobes in accordance to the desired radiation pattern as described in 60.

VII. CONCLUSION

The proposed beam pattern design framework provides an architecture and methodology for designing inter-pulse modulated transmitted signals for MIMO systems to meet the requirements of good cross correlation properties and reduced angle-Doppler coupling. The architecture based block circulant structure for signal design leads to enhanced target discrimination on the receiver. Further, the framework addresses the design of multiplexed signals with discrete phase modulation through convex optimization due to the considered dictionary based approach. Centrality of dictionary matrix on the performance of the algorithm in terms convergence and error between the desired and the designed radiation pattern is depicted and dictionary design examples presented for special cases. The crucial issue of angle-Doppler coupling inherent in multiplexed signals is analyzed and a

randomization based scheme to suppress coupling and enhance unambiguous Doppler range is highlighted. The paper, thus provides a comprehensive investigation of the block circulant signal design for transmit beamforming MIMO radar. The proposed signal structure, ease of its design and achieved performance makes it well-suited for low-cost implementation in commercial applications including automotive radar.

In what follows, a further embodiment of the invention will be described.

The present embodiment of the invention is used to the waveform design under the constraint of discrete multiphase unimodular sequences. It relies on Block Circulant decomposition of the slow time transmitted waveform. The presented closed-form solution is capable of designing orthogonal signals, such that the virtual MIMO paradigm is enabled leading to enhanced angular resolution. On the other hand, the proposed method is also capable of approximating any desired radiation pattern within the physical limits of the transmitted array size. Simulation results prove the effectiveness in terms computational complexity, orthogonal signal design and the transmit beam pattern design under constant modulus constraint.

I. INTRODUCTION

Multiple-Input-Multiple-Output, MIMO, radar exploits the waveform diversity such that a virtual array can be constructed at the receiver, where the array size with real antennas is smaller than the virtual counterpart [101]— [105]. This diversity is achieved by appropriate waveform design based on the different design metrics proposed in literature. A commonly used design metric is the Signal -to- Interference-to-Noise Ratio (SINR) [106]— [110] Another metric is the design of a desired radiation pattern as a function of the transmitted waveform [103], [104], [111]— [120] . However, in addition to the interest in radiation pattern design, the cross-correlation properties of the transmitted signals are also considered [104], [111], [118], [120] The cross-correlation properties of the transmitted signal is of significant interest in a virtual MIMO configuration, since signals with good cross-correlation properties, e.g., orthogonal signals, enable unique discrimination of the transmitted signals at the receiver and therefore facilitate the construction of a virtually filled array at the receiver [101], [102] Most of the aforementioned references reduce to the design of the transmit signal covariance matrix under certain constraints. The most common constraint is the unimodular or constant modulus signal constraint since it is considered as a power efficient modulation scheme [114] In this context, literature considers both continuous phase design constraints [114], [118] as well as discrete phase constraints [117] for unimodular sequences. Another common constraint is the uniform antenna element power constraint [111], [113], [115], [119]

Apart from the constant modulus and uniform power constraints, the algorithmic execution time or the computational complexity is of great interest. The lower the computational complexity, the faster the signal design, the better is the algorithm suitable for adaptive signal design. The most preferable solution is that of closed form as proposed in [115]— [117]

This embodiment considers the beam pattern design under discrete phase, unimodular and uniform element power constraint constraints. Further, the cross-correlation properties of the designed sequences are taken into account. Furthermore, the proposed method offers a closed form solution under the given constraints. The novelty over the state of the art lies is the design methodology considering all of the aforementioned critical properties, such as discrete phase, uniform power constraint, desirable cross-correlation properties and closed form solution. The authors in [117] propose a closed form solution under the discrete phase unimodular constraint, but the cross correlation properties are not investigated; further, the length of the transmitted symbols is limited to number of transmit antennas times number of receive antennas. The work in [115] relates closely to the present work; it proposes a closed form solution under uniform power constraint. However, [115] does not consider cross-correlation properties and unimodular signal design constraint in the design. Further, authors in [115] introduce a Toeplitz structure to the signal covariance matrix, whereas in this paper a much simpler block circulant approach similar to [120] is proposed. This paper builds on the work [120] and extends the quadrature phase two channel sequence design of [120] to the generic case of unimodular sequences with multiple phase stages. The contributions of this embodiment can be summarized as follows:

Extension of the block circulant decomposition approach [120] to multi-phase unimodular waveform design

Derivation of a closed-form solution to an optimization problem which addresses the transmit beam pattern design and minimizes the transmit signal cross-correlation under finite alphabet and uniform antenna constraints

_• Algorithm capable of real time implementation

II. SYSTEM MODEL

The system architecture comprises a local oscillator, which generates a train of Continuous Wave, CW, pulses, such as Frequency Modulated Continuous Wave, FMCW, pulses. The transmitted signal is composed of N_P pulses within one Coherent Processing Interval, CPI. Each pulse has a duration of T_P and a modulation bandwidth B with the carrier frequency f₀. The system further comprises N_R linearly mounted receive antennas with an inter-element spacing of ^{,ll! =} I, where ^ A is the wavelength and c₀ refers to the speed of light. The transmit antenna array on the other hand consists of N_T antenna elements linearly mounted with an inter-element spacing of

which leads to a sparse MIMO configuration. To enable simpler hardware implementation, each individual transmit antenna element is assumed capable of modulating the CW by Ayphasc stages, drawn from the modulation set

A. Transmitted Signal As mentioned above, the transmitted signals comprises a train of N_P CW-pulses. Since digital processing at receiver is considered, the transmitted signal is also modelled in the discrete domain. Therefore, each CW pulse is modelled as a pulse vector

where N_s is the number of discrete samples. The transmitted signal is assumed to be modulated in the slow-time domain, which further leads to an inter-pulse modulation. In particular, the transmitted signal during the >th pulse interval from «-th antenna takes the form, s_p(n)u, where s_p(n) is the corresponding modulating symbol. Let ^SP ^{e f!'Y}‘ ^{x l} denote the stacking of the A_T- transmit antenna array weightings, {^sp(ⁿ))nii, for the / th pulse. Further, define the inter-pulse modulation vector s e W ^{V7 V/}’ ¹ as,

It has to be noted that the focus of this paper lies in the interpulse modulation and therefore only the inter-pulse modulation vector s is considered as a design variable. Further, due to system constraints, s contains only symbols from the set W.

B. System Transfer Function and Received Signal

The pulse length is assumed to be much longer than the maximum propagation delay and the target velocity is assumed slow with regards to the pulse duration. The latter assumption leads to the simplification that the target appears constant for one pulse duration, which leads in turn to a Doppler shift only in the inter-pulse domain. These assumptions result in range and Doppler information being separable; the range information appears in the fast-time signal while the slow-time signal contains the Doppler information.

The target scenario comprises K distinct targets with Swerling one model statistics for their Radar Cross Section, RCS, [121] Therefore, the RCS of the k-th target, a_K e C, is a statistical parameter. The receive steering vector <º c^{A« xl} and the transmit steering vector ⁶

for the K- th target, defined below, contain the MIMO channel information under the far-field and plane wave assumptions

[120], [122],

a R_K = (exp (/¼), · · · , exp(Jk_Kd_RN_R))^T ( 102)

Ά_TK = (exp(-y r), · · · , expi-y r/V_T·))^7, . (103)

The receive and transmit steering vector contain the angular information of /c-th target in the wave number ^k> = ³™ίΆ), with ^denoting the spatial angle of the /c-th target. The Doppler frequency. /_/,,, of the /c-th target is modelled in the diagonal Doppler matrix D_K e C^NPxNP ,

D_K = diag(exp(/2^/_DK7», · · · , exp(_j2†_DKT_PN_p)). ( 104)

Therefore, the inter-pulse transfer function for /c-th target,

Towards completing the system model, the /c-th target range information is modelled with a shift matrix

[123] Due to the superposition of the target back scatters in space, the received signal vector, c e c^{Nn s r} ' over all antennas and one CPI becomes,

It can be seen, that the fast-time information ((J_Ku)) is completely separable from the slow-time information ((H_Ks)). Therefore, angular and Doppler domains are separable from any range information. Since the focus of this paper lies in the slow-time signal design, it is sufficient to only investigate the slow-time domain, such that only the slow -time received signal ^e c*^{«Ai Xl}i_{s 0}f interest,

The inter-pulse signal design only affects the angle-Doppler response. However, stationary targets are considered in the following, i.e,f_Dx= 0, V k .

C. Matched Filter

The receive signal processing is based on matched filtering, where the filter coefficients take the form similar to the received signal, y = H s. with every k replaced by a generic index M, leading to a parametric version of one target response. Since the attenuation factor is unknown, the matched filter does not contain any RCS information. Further, the RCS of each target is modelled as stochastic and therefore the matched filter output is a stochastic parameter as well. Hence, a mean matched filter output, m. is considered,

Due to the statistical independence of RCS across targets, the expected matched filter output takes the form of a superposition of K distinct target contributions,

As mentioned above, the Doppler shift is omitted in (109) leading to (110). The cross-correlation beam pattern P (f_M, y_k) is a frequently used design metric, since (i) it comprises information on both the radiation pattern information (when f = f_M= y_k) as well as the cross-correlation of transmitted signals (when y_Mf y_k) and (ii) its relevance for the underlying virtual MIMO configuration [111], [114],

[118], [120]

Let the transmit signal covariance matrix ¾ £ C ^{¥t X}* ^J ¾_e defined as, S = [S_I, S₂, ..., S_M>]·

The inter-pulse modulation matrix S e il^{,¥ x ,¥p} contains the vector s_p in the p- th column. Exploiting (110) and (111), the objective pursued in the paper involves the design of the interpulse modulation matrix, such that a desired cross-correlation beam-pattern is well-approximated.

III. CROSS-CORRELATION BEAM-PATTERN OPTIMIZATION

As mentioned above, the inter-pulse modulation matrix affects the cross-correlation beam-pattern. The cross-correlation beam pattern is a two-dimensional continuous function, but it is sampled in order to undertake signal design in the discrete domain. An uniform sampling of the wave numbers ^k'M -

A points is assumed leading to non-linear sampling of the angular domain

and f_k. The transmit steering matrix ATM = (C , »TM„ ) and

AT = (^L _TKI ^Ά _Tkn) e C ' are identical. Further, due to the Nyquist-Shannon Sampling theorem, it suffices to have number of sample points N equal to the number of transmit antennas N = N_T. In such a scenario, the steering matrices take the form of a scaled Discrete Fourier Transform (DFT) matrix ^{L = A}™ ⁼ Ar* ^{G€L cL} Due to the DFT structure, it is clear that an oversampling N > N_T does not provide additional information as the rank of A remains Ay.

With the steering vectors sampled, the cross-correlation beam-pattern can be compactly represented using a matrix,

P = A¾A (112)

where (m,n)th element of P denotes the radiation pattern in (110) evaluated at the mth and nth sample of A _/and respectively.

The virtual MIMO paradigm is satisfied, if the cross-correlation beam-pattern matrix has a diagonal structure, since it leads to the maximum array resolution [101], [102] Therefore, desired cross- correlation beam-pattern matrix

assumes a diagonal structure,

F_d= diag(pA (113)

where ^e ^^{Lt 1} contains the desired radiation pattern information. Since the cross-correlation properties are predefined, the radiation pattern P _/is the design requirement.

A. Objective Function

With the structure of desired P, _/discussed and recalling the signal modulation set W, the objective function for designing a desired cross-correlation beam-pattern is defined as, tpϊh ||P_{d -} A^H R,A|L .

sea^N-r · ^{c L} ^ (114)

The optimization problem comprises a quartic cost function in S under a finite alphabet constraint. This problem is in general hard to solve [111]. In order to achieve a closed form solution, the Degrees of Freedom, DoF, has to be reduced. One possibility to reduce the DoF is to apply a block circulant construction method as presented in [120]

B. Block Circulant Decomposition

Under the assumption of N_P being an integer multiple of N_T, the Block Circulant Decomposition (BCD) decomposes the matrix S into N_B blocks of dimension N_T ^c N_T

S = (B_1; · · · ,B_¾, · · · ,B_L®). (1 15)

The 6-th block ¾€ ίϊ^{L J x j '} is parametrized by an unique column vector ¾ e W^{L ί x l} which is circular shifted across columns to obtain B_¾ [24] This leads to a block circulant B*. Due to this property, the unitary DFT matrix, F e C^{NT XNT} serves as a eigenvector matrix for all the blocks. Thus the blocks B_¾ differ only in the eigenvalue matrix

leading to the following

simplification for the transmit signal covariance matrix,

As discussed earlier, the steering vector matrix is a scaledV version of the unitary DFT matrix A = vWrF (scaling due to column normalization). Hence, the optimization problem in

(114) simplifies as,

The BCD ensures the actual cross-correlation beam-pattern P to be diagonal by construction under the assumption of a Uniform Linear Array, ULA, configuration for the transmitter. In fact, the ULA structure guarantees a DFT structure for the steering vector matrix. Thus the BCD provides a construction based framework for ULA transmit signal cross-correlation optimization. If P becomes diagonal, the Frobenius norm of the off-diagonal elements (114) has no contribution to the cost function. Therefore, the optimization problem (117) and (114) are equivalent.

It should be noted that the DoF is lowered by a factor of A -due to BCD when simplifying (114) to (117). In Section III-C the optimization problem (117) is further simplified such that a closed form solution is achieved.

C. Fourier Basis Approach

1) Basis Expansion: Let the actual radiation pattern be denoted by p, where

p = A_T

pattern depends only on N_B vectors, s*, where each vector has to satisfy the modulation constraint, i.e., ¾ e W^{L'i C 1} i_n order to design the blocks s*, it can be observed that p lies in a N_T dimensional space and hence can be expressed as a superposition of N_T basis functions, {q,},

where the second term denotes the matrix-vector representation of _^;=i ¾¾. Recalling that

P - ^NT SL=^] (^{F hb}) ^c (F ^Sb) _{^ jt} suffices to choose, without loss of generality, the z-th basis function as ¾

Therefore, the appropriate choice of basis functions s, influences Q. Further, the matrix Q ^e ®^{L 1 cL 1} contains the vector q, in its zth column and is a function of the

gJYy x 1

vectors s, . Additionally, if the basis coefficient vector ^v £ is restricted to positive integer values, the term å =i

can be rewritten as å?=i

, where the vectors {¾)are drawn from columns of Q with repetitions such that there are N_T distinct {¾ land whose number of repetitions is governed by {v,} . In other words, the inter-pulse modulation matrix S comprises v, circulant block for each vector s,.

2) Fourier basis as {s_¾} : If each vector s, is chosen as a Ay 1 Discrete Fourier Transform, DFT, vector, the matrix ^{Q =} ^rTv, becomes a scaled identity matrix, as the DFT vectors are inserted in the definition of each q,. Further, this choice naturally renders v non-negative. In fact,

However, according to the earlier discussion, it is essential that the resulting v is an integer vector to fully utilize BCD framework. In general, this cannot be guaranteed and the modulation constraint is satisfied by rounding the real coefficient vector v -

to the next integer value. In view of this, following (18), p is approximated as Qv where,

where G(c) = max { e Z\k < x + 0.5} denotes the rounding function. With v determined, the matrix S is constructed such that it contains v (/) circulant blocks for the z-th DFT basis, z e | fAy] .

Thus the Fourier basis approach enables a closed form solution for the transmission waveform, although it reduces the DoF. The quality of the closed form solution depends only on the signal length, namely the number of blocks N_B. It is obvious from (120) that the precision of the rounding operation by G(·) improves with larger number of blocks. Table I depicts the algorithm, which provides the closed form solution of the inter-pulse signal modulation matrix S for a given desired radiation pattern

P_d

TABLE I: Algorithm for discrete phase unimodular sequence design

IV. SIMULATION

Simulations are carried out with N_T= 10 transmit antennas and N_R = 4 receive antenna elements in a virtual MIMO configuration as discussed in Section II. The number of blocks is set to N_B = 500.

As discussed in Section III, the error in the proposed approach is largely determined by the quantization error. The quantization error is a function of the number of blocks. As can be seen in Figure 8, the quantization error becomes negligible for a sufficiently large number of circulant blocks.

i

Further, the error fits the ¾ curve. This result shows the effectiveness of the presented approach. A drawback of a large number of N_B is that the CPI becomes large and at beyond certain values, the assumptions of slow and fast time separation do not hold anymore. Also, the processing time increases with the number of blocks.

A second source of error arises from the physical limitation of the array size. The finer the beam pattern resolution, the bigger the transmit antenna array size. However, the proposed algorithm is capable of designing any desired radiation pattern within the physical resolution limits, if the number of blocks is sufficiently large. Figure 9 and 10 illustrate the design capability of the proposed approach and compare it to [120] In particular, two different radiations patterns are considered

in these figures. In comparison to the method presented in [120], the proposed approach performs better in terms of beam pattern focus. This is justified by having a unimodular sequence design, instead of a multiplex approach. It can be also seen from figures 9 and 11, that the sample points perfectly match the desired beam pattern, which in turn, demonstrates that the algorithm performs optimally. It has been observed that the transmitted signals are perfectly orthogonal and the maximum virtual array resolution can be achieved, while having the capability of designing an arbitrary radiation pattern. This holds also for the approach in [120] Therefore, the BCD provides a suitable tool for achieving good cross-correlation properties for a ULA configuration.

The ability to design a wide range of signals from perfectly orthogonal ones to those yielding desired radiation pattern at certain sample points proves the flexibility and universality of the presented method.

V. CONCLUSION This embodiment relates to a closed form solution for transmit signal design under uniform element power and constant modulus constraints based on block circulant approach to signal design. The proposed block circulant decomposition enables design of transmit waveform with optimal cross correlation properties while implicitly satisfying the uniform element power constraint. The proposed approach decouples the radiation pattern design from the consideration of cross correlation properties, which further leads to an effective radiation pattern design using a discrete Fourier basis approach. While the proposed approach has lower degrees of freedom, the simulation results nonetheless corroborate the effectiveness of the proposed approach.

ACKNOWLEDGMENT

This work was supported by the

National Research Fund, Luxembourg under AFR grant for

Ph.D. project (Reference 11274469) on Enhancing Angular Resolution in Radar Through Dynamic Beam Steering and MIMO.

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It should be noted that features described for a specific embodiment described herein may be combined with the features of other embodiments unless the contrary is explicitly mentioned. Based on the description and on the figures that have been provided, a person with ordinary skills in the art will be enabled to develop a computer program for implementing the described methods without undue burden and without requiring any additional inventive skill.

It should be understood that the detailed description of specific preferred embodiments is given by way of illustration only, since various changes and modifications within the scope of the invention will be apparent to the person skilled in the art. The scope of protection is defined by the following set of claims.

Claims

Claims

1 A method for detecting a target (10) in a multiple-input multiple-output, MIMO, radar system (100) comprising a first uniform linear array having a plurality of NT transmit antennas (110), a second uniform linear array having a plurality of NR receive antennas (120), and a multiplexer (130) for generating multiplexed and discrete phase modulated probing signals, comprising the steps of:

i) providing a block-circulant signal matrix .S' defining a probing beam pattern to be generated by said transmit antennas (110), wherein each column of said signal matrix S describes a plurality of probing signals to be generated by a plurality of said NT transmit antennas in a given time transmission slot;

ii) generating, using a data processor (140), a random permutation of the columns of said signal matrix S. and transmitting, using said transmit antennas (110), said probing signals to form a probing beam in accordance with said generated matrix permutation; iii) receiving, using said NR receive antennas (120), reflection signals of said probing signals, said reflection signals being backscattered from at least one target (10);

iv) processing the reflection signals to determine the presence, range and angular position of a target (10) within a field of view of the transmit antennas.

2 The method according to claim 1, wherein at least three transmit antennas are transmitting a probing signal in at least one time transmission slot.

3. The method according to claim 1 or 2, wherein said step i) further comprises the following steps:

a) providing a desired beam pattern representation;

b) using a data processor, approximating said desired beam pattern by a weighted sum of basis vectors from a signal dictionary;

c) generating said block-circulant signal matrix .S' by circular shifting of each basis

vector, such that the resulting number of circulant blocks in the signal matrix .S' is equal to the integer weight associated with said basis vector in said approximation .

4. The method according to claim 3, wherein said signal dictionary comprises all feasible phase modulations given the number of active transmit antennas in a given time transmission slot.

5. The method according to claim 3, wherein said signal dictionary comprises discretized Fourier basis functions, and wherein the transmitted signals are unimodular.

6. The method according to any of claims 3 to 5, wherein said signal dictionary is pre-provided in a memory element to which said data processor has a read access.

7. A computer program comprising computer readable code means, which, when run on a

computer, causes the computer to carry out the method according to any of claims 1 to 6.

8 A computer program product comprising a computer readable medium on which the computer program according to claim 7 is stored.

9. A detection system (100) for sensing a target, the system comprising:

a multiplexer (130) coupled to a plurality NT of transmit antennas (110) forming a sparse transmit uniform linear array, ULA, a plurality NR of receive antennas (120) forming a dense receive ULA, the multiplexer being configured to generate multiplexed transmit signals based on signals from a local oscillator (160); and further comprising processing circuitry (140) that is coupled to the multiplexer (130), wherein the processing circuitry, the transmit antennas and the receive antennas are configured to perform the method of any of the preceding claims.

10. A vehicle comprising a detection system according to claim 9.