Basic concepts and results of array antennas

Target parameter estimation and array features

1.2 Basic concepts and results of array antennas

The array principle, which consists of synthesizing a plane wave by a number of ele- mentary spherical waves in transmit or receive mode, is well known. We assume that the elements of the array are positioned in the three-dimensional space with (x,y,z)- co-ordinates, and we denote the plane wave directions by a unit direction vectoruin the (x,y,z)-co-ordinate system with kuk ¼1. Vectors are noted as column vectors and the components of the direction vector are denoted by u¼ ðu;v;wÞ^T. For a suitable definition of azimuth and elevation angles, this direction vector can also be parameterized by the corresponding angles, but the unit vector notation is a parameterization-free description and therefore more convenient.

1.2.1 Plane wave at single frequency

Suppose, we have an array with N elements at the positions r_i¼ðxi;yi;ziÞ^T; i¼1. . .N. A plane wave from directionu₀¼ðu0;v0;w0Þ^Tat frequencyfsampled at timetat these positions can then be written as:

s_iðtÞ ¼Ae^j2pfte^j2pfrTiu0=c (1.1) wherecis the velocity of light andAis a complex amplitude with an initial phase j,A¼j jeA ^jj. This is a notation viewed from a receiving antenna for an incoming wave. The signal that is actually measured at the antenna elementszi(t) is of course corrupted by some noise, which is inevitable with any real hardware:

ziðtÞ ¼siðtÞ þni; i¼1. . .N; or in vector notation zðtÞ ¼Ae^j2pfte^j2pfrTiu0=c

i¼1...Nþn (1.2)

This is the standard signal model for array antennas.

For a planar or linear array with elements at positions r_i¼ðx_i;y_iÞ^T, or r_i¼x_i; i¼1. . .N, respectively, a signal with a frequency deviation f¼af0 is indistinguishable from a signal at displaced angles ðau₀;av₀Þ at frequency f0, because f ðx_iu₀þy_iv₀Þ ¼f₀ðx_iau₀þy_iav₀Þ. Another interpretation is that the array positions seem to be stretched if a signal at higher frequency is observed with receivers at the original frequency. This may introduce bias and grating effects which we will discuss below. Clearly, such errors are zero for the boresight directionu₀ ¼ ð0;0Þ^T and the errors will increase with the angle of incidenceu0.

The frequency dependent angle shift does not appear for arrays with a true three-dimensional element distribution (volume array or crow’s nest antenna, [1 Sec- tion 4.6.1]), because the third component ofu,w, is not independent ofu,v, because kuk ¼1. Therefore, there is no way of aliasing au0with af0. It is a fundamental property of 3D volume arrays that squint and bias effects due to frequency do not appear. The 3D volume array has an inherent bandpass-filtering property.

For the detection of such an elementary plane wave, we have to sum up all array elements phase coherently in a beamforming operationSsum¼PN

i¼1a_isiwith ai¼e^{j2paf rTiu0=c}to maximize the signal power. At this point, we ignore any noise contribution. The asterisk means complex conjugate. Of course, the true signal frequency is in general unknown and usually the centre frequency or different frequencies of a frequency filter bank are used. Beamforming with the phases at the centre frequency produces for a planar array the afore mentioned angle shift by the factoraor a relative angle errorðkuau₀kÞ=ku0k ¼a1.

The relation between bandwidth and the angle error for planar arrays can be characterized by the bandwidth factor, [7 p. 13.39], which is defined as:

KB¼ bandwidthð%Þ

boresight beamwidthðdegÞ (1.3)

The bandwidth factor is used to calculate the admissible bandwidth from the maximum scan angle and the tolerable angle error in beamwidth at this scan angle,

if narrowband beamforming at centre frequency is performed. The magnitude of an acceptableKBtypically ranges from 1 to 2. For example, if one accepts a maximum squint angle of one-fourth of the beamwidth at a maximum scan angle of 60, then one obtainsK_B¼1, [7 p. 13.39]. This is the famous rule of thumb that the relative bandwidth in per cent should be equal to the beamwidth in degrees. For example, according to this rule one can use 2% bandwidth with an antenna with 2beamwidth.

1.2.2 Band-limited signals

If a broadband signal is received, we have a super-position of all frequencies in the band. Such a signal can be described by a homogeneous plane wave field, see [11 p. 5], which is a special stochastic process in time and space. It can be written in the following notation (Crame´r representation):

sðt;r;uÞ ¼

Z _f0þB=2

f0B=2 e^{j2pf tþrð Tu=cÞ}dZ_sð Þ þf

Z f0þB=2

f0B=2 e^{j2pf tþrð Tu=cÞ}dZ_sð Þf (1.4) where f0 is the centre frequency, and B is the receiving bandwidth. Zs(f) is a complex stochastic process with independent increments, i.e. with the following properties: E{Zs(f)}¼0 and EfdZsðfÞdZ_sðgÞg ¼PsðfÞdðf gÞdfdg and E dZf _sðfÞdZ_sðgÞg ¼0.Ps(f) denotes the power spectral density of the signal, the asterisk denotes complex conjugation. As the processsis assumed to be real, we havedZsðfÞ ¼dZsðfÞ, and the signal can also be written as:

sðt;r;uÞ ¼2Re

Z f0þB=2

f0B=2 e^{j2pfðtþrTu=cÞ}dZ_sð Þf

( )

(1.5) Such a stochastic integral cannot be solved as in classical analysis. This is only a representation of the stochastic process, but the moments of the process can be conveniently calculated with the rules for the stochastic differentials given above.

The complex baseband outputs (theI- andQ-components) are generated by the complex demodulation procedure which can ideally be written as:

I tð;r;uÞ ¼LPfs tð;r;uÞcosð2pf0tÞg and

Q tð;r;uÞ ¼LPfs tð;r;uÞsinð2pf0tÞg (1.6) where LP{.} denotes the ideal low-pass filtering operation. As described in [12] the complex baseband signals(t,r,u)¼I(t,r,u)þjQ(t,r,u) can then be written as:

sðt;r;uÞ ¼e^j2pf0rTu=c Z B=2

B=2e^{j2pxðtþrTu=cÞ}dZsðxþf0Þ (1.7) Such a stochastic signal is not typical for the desired radar echo. In fact, the transmit signal is a special deterministically modulated signal (e.g. linear/non- linear frequency modulated), but such a signal is included in this model by means of a suitable deterministic measuredZs. The model (1.7) is very useful in describing clutter and interference signals.

1.2.3 Narrowband and broadband beamforming

Beamforming is a linear filtering operation which extracts maximum signal energy for all possible signal spectra. This is achieved by coherently summing up the delayed signals, see [11 p. 9]. If the wave field is sampled by an array at positions r_i¼(xi,yi,zi), the sum beam output power is:

Ssum

j j²¼E X^N

i¼1

s tð t_i;r_i;uÞ

8 2

<

:

9=

; (1.8)

witht_i¼r^T_iu=c. This is the classical delay and sum beamformer.

For narrowband processing, one assumes the delays to be small,Bt_if₀t_ior B=f₀1, such one can write this summation with a steering vectorw¼ð Þw_{i i¼1...N} containing only phase shiftsw_i¼e^j2pf0rTiu=c:

Ssum

j j² ¼E X^N

i¼1

w_is tð;r_i;uÞ

8 2

<

:

9=

;

¼w^HEnðs tð;r_i;uÞsðt;r_k;uÞÞ_i;k¼1...No w

¼w^HQ_sw

(1.9)

The entries of the covariance matrixQ_sare given by:

ðQsÞ_i;k¼e^j2pf0rTiu=ce^j2pf0rTku=c Z _B=2

B=2

PsðxþfoÞe^{j2pxðrirkÞTu=c}dx (1.10) or in matrix notation:

Q_s¼aa^H

C_s (1.11)

witha¼ ðe^j2pf0rTiu=cÞ_i¼1...N,C_s¼ RB=2

B=2P_sðxþf_oÞe^{j2pxðrirkÞTu=c}dx

i;k¼1...N, and denoting the element-wise (Schur–Hadamard) matrix product. Ps is the signal spectral power density at the output of the antenna element. For example, for rectangular spectral power density we have:

C_s¼PsðsincðpBðtitkÞÞÞ_i;k¼1...N (1.12) In the narrowband case, one has B=f01 and C_sPsð Þ1_i;k¼1...N, such that with (1.11):

Q_s¼Psaa^H (1.13)

The element receiver noise is assumed to be independent in each channel and independent of the direction, i.e. the delayst¼r^Tu=cin (1.7) are zero such that nðt;r;uÞ ¼R_B=2

B=2e^j2pxtdZ_nðxþf₀Þ. The noise-alone covariance matrix is then:

Q_n¼diag

Z B=2

B=2P_n;iðxþfoÞdx

!

i¼1...N

( )

(1.14) with noise power spectral densitiesPn,i(f) fori¼1 . . .N.

In general, the received signal may be composed of a sum ofMplane waves plus receiver noise such that one obtains the array outputs:

zðt;r_iÞ ¼X^M

k¼1

skðt;r_i;u_kÞ þniðtÞ; i¼1. . .N; or in a short notation zð Þ ¼t X^M

k¼1

s_kð Þ þt nð Þt

(1.15)

with ni¼R_B=2

B=2e^j2pxtdZ_n;iðxþfoÞ and with obvious definition of the vectors s_k. According to (1.11), (1.14) one obtains for uncorrelated plane waves the array output covariance matrix R¼PM

k¼1Q_s;kþQ_n. The narrowband version of this covariance matrix is:

R¼X^M

k¼1

P_ka_ka^H_k þQ_n¼ABA^HþQ_n (1.16)

wherePkdenotes the power,a_k¼aðukÞ, andu_kindicates the direction of the kth plane wave. For uncorrelated narrowband plane waves, we haveB¼diag^M_k¼1f g;Pk

for correlated wavesBis a full matrix, but possibly of rank lower thanMif there are fully correlated sources. Broadband beamforming and narrowband beamform- ing as in (1.9) is performed with these matrices.

The delay operation for broadband beamforming may be too costly if it is applied at each element of a large array. Therefore, a hybrid solution is often pre- ferred, where sub-arrays are summed up with phase shifting at the elements (the delays are small within the sub-arrays) and where the sub-array outputs are sum- med up with delays (time delayed sub-arrays). For the application of digital multi- channel (array processing) techniques, A/D conversion of the sub-array outputs is needed (digital sub-arrays). These digital sub-arrays may be larger than the time delay sub-arrays, depending on receiver cost, bandwidth and error considerations.

However, the preferred solution is to have a sufficiently large number of time delay sub-arrays which coincide with the digital sub-arrays.

In the sequel, we will assume narrow-band beamforming. The case of broad- band beamforming and advanced broadband array signal processing methods with stepwise increasing complexity has been described in [2 Chapter 16].

1.2.4 Difference beamforming and monopulse estimation

Difference beamforming followed by monopulse estimation is mentioned in nearly all radar textbooks as a standard beamforming procedure, but in most cases, it is not mentioned why this is a good procedure and how it is related to optimum processing.

In fact, difference beams and the monopulse procedure are only needed for accurate angle estimation. In the radar search mode, the beam-pointing direction is taken as a rough direction estimate if a detection of a target has occurred. In a second step, this detection is confirmed and a more accurate angle estimate is performed.

This is a statistical parameter estimation problem and not a matter of beamforming.

The problem will be considered in more detail in Sections 1.7 and 1.9. We mention

here briefly the underlying principle and the rationale of difference beamforming, because difference beamforming is a feature that influences heavily the array design.

Radar parameter estimation can be formulated as a maximum likelihood (ML) estimation procedure, see [1 Section 11.1, 13 Section 3.2.1] or [14]. For a set of measured array data vectors (often called array snapshots)z₁;. . .z_K,k¼1. . .K; z_k2C^N, modelled as in Section 1.2.1, one can define an appropriate statistical model for the noise and the signal amplitude fluctuation. The probability density of the snapshots depending on all the desired (unknown) parametersqcan then be written aspðz1;. . .;z_KjqÞ. In general, no globally optimum procedure exists for estimating the parametersq. However, the ML procedure is a realisable way that can produce at least an asymptotically optimum estimator. Asymptotic refers here to the number of array elements or the number of time samplesKgoing to infinity. For assump- tions fulfilled in a reasonable practical system, the ML estimator is asymptotically unbiased and has asymptotically the smallest variance, i.e. it attains the Crame´r–

Rao bound. Angle estimation for a single target by the ML procedure with a given set of snapshots requires forming the sum beam for all possible directions and taking the value with maximum power as the angle estimate, as explained in [13].

This is nothing else than maximizing what we call the antenna scan pattern. The procedure of sequential lobing mentioned in standard radar textbooks [7 p. 9.16], [8] is an abbreviated way of maximizing the antenna scan pattern.

An alternative way of finding the maximum is to extrapolate the shape of the scan pattern in the vicinity of the rough direction estimate obtained from detection. One can approximate the beam shape by a parabola and determine the parabola parameters from a set of neighbouring sum beams or from the sum beam in the search direction plus its derivatives. The difference beams are just estimates of the derivatives of the sum beam.

Note that for planar or volume arrays we have derivatives for azimuth and elevation, i.e. we have multiple difference beams. The extrapolation is typically done by Taylor expansion. This Taylor expansion finally provides a direct formula for the unknown directions depending only on the sum and difference beam outputs. This is the basis of the famous monopulse formula for the estimated angle ^u¼u0mRefD=Sg.

D,Sdenote the difference and sum beam outputs, respectively, andmis a proportion- ality constant which depends on the second derivative and which is determined approximately or experimentally. The estimate for the elevation ^v is obtained analogous by an elevation difference beam. The relationship with the Taylor expansion is of importance when the monopulse procedure has to be generalized to ABF, where the beam shape is dependent on the unknown interference scenario. This will be considered in Section 1.9.

The optimum difference beam (in the ML sense) is taken as the derivative of the sum beamSsumðu;vÞ ¼aðu;vÞ^Hz, which gives:

Sdiff;az¼@S_sum

@u ¼g X^N

k¼1

xka_kðu;vÞzk

!

¼gd^H_xz¼ga^Hdiagf gzxk

fordk¼xkak; k¼1. . .N. The constant factorg¼ j2pf0=cis irrelevant and can be omitted. So, the optimum difference beam weighting is the sum beam weighting with an additional amplitude tapering according to the element positions. For a

uniform linear array (ULA), this means to form the beam by summing the differ- ences of the opposing element pairs, hence the name difference beam. If we take as a crude approximation the amplitude weighting sign(xk), then we just subtract the outputs of the left from the right half of the array which for a planar circular array is also known as four-quadrant monopulse. In any case, the beam pattern is zero in the array steering direction.