Array accuracy requirements

Target parameter estimation and array features

1.4 Array accuracy requirements

The Crame´r signal representation introduced in Section 1.2.2 is also well suited to analyse some error effects. This has been done in [12] and in [2 Section 16.2.3]. We recall the main results here. These results are not meant as a method of predicting errors appearing in a real system, where we have a super-position of a multitude of effects. Rather such models can demonstrate the typical consequences of some kinds of error (especially the eigenvalue leakage effect) and they are useful for rapid system simulations. For demonstration purposes we confine the presentation to analogue de-modulation (de-modulation before digitization) which is rather used for broadband signals. Digital de-modulation with a high sampling rate and the resulting typical errors are described in [1 Section 6.3].

1.4.1 IQ-de-modulation errors

An amplification error R (or H) in the I- (or Q-) channel, respectively, an orthogonality errorj(ory) and offset errorF(G), and delay errordappearing by IQ-de-modulation can be described by extending (1.6) to the form:

I tð;r;uÞ ¼RLPfs tð d;r;uÞcosð2pf₀tjÞg þF and

Q tð;r;uÞ ¼HLPfs tð d;r;uÞsinð2pf0tyÞg þG (1.33) This leads to a complex base-band signal description, [12],

sðt;r;uÞ ¼1

2e^{j2pf0ðrTu=cþdÞ}p Z B=2

B=2e^{j2pxðtþrTu=cþdÞ}dZsðxþfoÞ þ1

2e^{j2pf0ðrTu=cþdÞ}q Z _B=2

B=2e^{j2pxðtþrTu=cþdÞ}dZ_sðxþf_oÞ þg

(1.34)

Withp¼Re^jjþHe^jy; q¼Re^jjHe^jyandg¼FþjG.

Slow time

Spatial samples

t i Fast

time

Figure 1.10 Symmetric auxiliary sensor/echo processor of Klemm [3], as an example for forming space-time sub-arrays (from [4] by courtesy of W. Bu¨rger)

For an array with different errorsR_i;H_i;F_i;G_i;j_i;y_i;d_i in each channel, the covariance-correlation matrixQ_sðd0Þ ¼Efsðtþd_i;r_i;uÞsðtþd_kþd₀;r_k;uÞg_i;k¼1...N can then be written as described in [2 p. 555] by:

Q_sðd0Þ ¼1 4 bb^H

pp^H

þbb^H

qq^H

C_sðd0Þ (1.35) wheredenotes the Hadamard (element-wise) matrix product. The quantitiesb,p, q, C_s depend on the specific signal power spectrum. As an illustrative example we present here the results for the special case of a rectangular signal power density with centre frequency fs and bandwidthBs with½fsBs=2;fsþBs=2

f0B=2; f0þB=2

½ . Then one obtainsbi¼e^j2pfsrTiu=c,pi¼e^j2pfsdiðRie^jjiþHie^jyiÞ, qi¼e^j2pfsdiðRie^jjiHie^jyiÞfori¼1 . . .N, and:

C_sðd0Þ ¼Ps sinc pBs

ðrir_kÞ^Tu

c þdidkþd0

!

e^j2pDfd0

!

i;k¼1...N

(1.36) where Df ¼fsf0. For the receiver noise covariance matrix with independent white noise in each channel we obtain:

Q_nðd0Þ ¼s²sincðpB_sd₀Þe^j2pDfd0diag^N

i¼1 ðR²_i þH_i²Þ=2

A complete array output covariance matrix forMtargets then results in:

Rðd0Þ ¼X^M

i¼1

Q_s;iðd0Þ þQ_nðd0Þ þgg^H (1.37) With these formulas one can build all kinds of sub-band and space-time covariance matrices. Special cases:

● No errors (R¼H¼1,F¼G¼0,j¼y¼0), zero bandwidth (Bs¼0),d₀¼0:

Then Q_s¼bb^HPs and b contains the frequency shifted steering vector (e.g. Doppler).

● No errors (R¼H¼1,F¼G¼0,j¼y¼0) and bandwidthBsonly,d₀¼0:

ThenQ_s¼bb^HC_s.

● With errors present:

ThenQ_sof (1.35) can be written with the rules of the Hadamard product as:

Q_s¼1

4hðbpÞðbpÞ^Hþ ðbqÞðbqÞ^Hi

C_s (1.38)

Comparing this with the error-free narrow-band covariance matrix shows that a single target with steering vectorbwill result in a tilted steering vectora_errorðuÞ ¼ bpplus an ‘image’ targeta_imageðuÞ ¼bq.

Figure 1.11 shows the eigenvalues of a covariance matrix for narrowband beamforming (phase shifting only) with and without errors and also the case of zero bandwidth for reference. The scenario consists of three targets in azimuth direc- tions53,24, 0(u¼ 0.8,0.4, 0.0) with equal element signal-to-noise ratio (SNR) of 12 dB. The generic antenna of Figure 1.7 was used. In the case of no errors and zero bandwidth one can see the three dominant eigenvalues and the noise eigenvalues at constant level of 0 dB. Additional IQ-errors were modelled here by an additional complex-normal distributed component with an amplitude standard deviation (std) of 1 dB. With bandwidth and additional I- and Q-errors the typical dominant eigenvalue leakage effect can be seen. The source power, which in the narrowband error-free case is concentrated in the three dominant eigenvalues, here leaks into the smaller eigenvalues. Also the noise eigenvalues become unequal.

The bandwidth and random channel errors lead to an approximate doubling of the dominant eigenvalues. The amount of leakage depends on the signal power, the bandwidth and the amount of error. It is important to note that eigenvalue leakage is typically a strong signal effect. The leakage eigenvalues are at a certain level below

0 5 10 15 20 25 30 35

–5 0 5 10 15 20 25 30 35 40 45

nr. of EV

dB

Eigenvalues of expectation of covariance matrix/3 sources/SNR = 12 12 12 dB

No errors B = 0 No errors B = 10%

IQ errors B = 10%

Figure 1.11 Eigenvalues of covariance matrix at sub-array level for generic array with 32 sub-arrays, narrowband beamforming by phase shifting at elements. Eigenvalues are calculated from the theoretical covariance model of (1.38). Three cases: no errors and zero bandwidth, no errors and 10% relative bandwidth, IQ errors with 1 dB std amplitude fluctuation and 10% bandwidth

the dominant eigenvalues (defined by the vectors bp, and bq). For weak sources they will merge into the noise eigenvalues. This shows that the common belief that with high signal-to-noise ratio performance converges to the ideal error- free case is obviously not true for realistic types of error.

Eigenvalue leakage is also particularly strong for small number of samples for estimating the covariance matrix. Figure 1.12 shows the eigenvalues for estimated covariance matrices corresponding to the cases of Figure 1.11. The covariance matrices are estimated fromK¼64 snapshots. This is the value that would give a 3-dB loss according to a rule of thumb termed ‘Brennan’s rule’ which says that K 2Nis required for a 3 dB loss, [22,23].

The eigenvalue leakage will in particular complicate the determination of the number of dominant eigenvalues, which is important for the application of sub- space methods for jammer suppression or super-resolution methods in Sections 1.6 and 1.7.2.

1.4.2 Bandpass filter errors

For general bandpass filter functions the integrals cannot be solved analytically.

However, a special technically important case is that of a sinusoidal ripple over the

0 5 10 15 20 25 30 35

–5 0 5 10 15 20 25 30 35 40 45

nr. of EV

dB

Eigenvalues of estimated covariance matrix/3 sources/SNR = 12 12 12 dB

No errors B = 0 No errors B = 10%

IQ errors B = 10%

Figure 1.12 Eigenvalues of estimated covariance matrices at sub-array level from 64 snapshots, generic array with 32 sub-arrays, narrowband beamforming by phase shifting at elements. Three cases: no errors and zero bandwidth, no errors and 10% relative bandwidth, IQ errors with 1 dB std amplitude fluctuation and 10% bandwidth

bandwidth and this case can be solved. In order to show what effects can be observed with this error model the results are shortly presented.

Let the baseband filter functions have the form HiðfÞ ¼aiðfÞe^jfiðfÞ; i¼1. . .N, with:

aiðfÞ ¼ 1þe_icos 2pf h_i; for j jf <B=2

0; else

(

f_iðfÞ ¼2pfdi

(1.39)

ei<1 is the ripple amplitude,hi counts the number of ripples, anddiis a linear phase slope or a delay. This results in the signal model:

siðtÞ ¼e^j2pf0ti Z _B=2

B=2

e^{j2pxðtþtiÞ}aiðxÞe^jfiðxÞdZsðxþf0Þ (1.40) Again we assume a sub-bandwidth½f_sB_s=2; f_sþB_s=2 ½f₀B=2;f₀þB=2 , which may be incorporated into the definition ofdZs. As calculated in [2 p. 556]

this model leads to a signal covariance matrix:

Q_sðd0Þ ¼bb^H

p_dp^H_d

C_sðd0Þ ¼hðbp_dÞðbp_dÞ^Hi

C_sðd0Þ (1.41) with b¼e^j2pfsti

i¼1...N; ti¼r^T_iu=c; p_d¼e^j2pfsdi

i¼1...N, and C_s;ikðd0Þ ¼ Pse^j2pDfd0½sincðpBsdikÞ þe_iHikþe_kJikþe_ie_kKik , Df ¼fsf0, dik¼t_it_kþ didkþd0and

Hik ¼1

2e^j2pDfhisincpBsðdikþhiÞ þe^j2pDfhisincpBsðdikhiÞ Jik ¼1

2e^j2pDfhksincpBsðdikþhkÞ þe^j2pDfhksincpBsðdikhkÞ Hik ¼1

4½e^{j2pDfðhihkÞ}sincpBsðdikþhihkÞ þe^{j2pDfðhihkÞ}

sincpBsðdik ðhihkÞÞ þe^{j2pDfðhiþhkÞ}sincpBsðdikþhiþhkÞ þe^{j2pDfðhiþhkÞ}sincpBsðdik ðhiþhkÞÞ

From this formula, the receiver noise covariance can be obtained as a diagonal matrix Q_nðd0Þ by replacing in Q_sðd0Þ the quantities Ps¼s² and dik ¼d₀ (see [2 p. 556] for the explicit formula).

1.4.3 AD-converter limitation

A third kind of error, which is however difficult to model, arises due to limiting effects. All digital signal processing algorithms assume a linear receiver. Limiting effects are the most frequent non-linear effects appearing in real systems and the AD converters are most sensitive to limiting because radar has typically very high dynamic range: the target echo may be below receiver noise while the clutter may

be more than 60 dB above receiver noise. Suppose we have an AD-converter with bbits then the classical formula for the maximum possible SNR with respect to quantization noise is SNRmax¼6.02bþ1.76 dB. The receiver noise should be slightly higher than the quantization noise level to allow an integration effect for a signal below receiver noise. This means that for a 14 bit ADC we have a maximum SNR of 86 dB which is a value that can be easily attained in reality. Normally an automatic gain control (AGC), which is an SNR adaptive attenuation, is inserted into the digital receivers to avoid limiting. If we have strong clutter the AGC may make small targets to be sub-merged in noise, or in other words, can reduce the radar range. So, the system is not completely distorted in the case of limiting, but it will suffer a soft degradation with respect to the maximum range.

If the non-linearity is known in its functional form, it is possible to compensate the non-linear effects. This has been shown in [24] for non-linearities up to third order. This technique may be applied for analogue limiting. For hard clipping as produced by AD converter limiting there is no correction possible and only an AGC can help avoiding this.

In any case, whether produced in the analogue or digital domain, any uncompensated limiting effects will give rise to dispersive effects leading to additional eigenvalues in the covariance matrix similar to the effects described in Sections 1.4.1 and 1.4.2.