5.6.1
Analysis of Continuous Array Spatial Cram´er-Rao Bound
In this section we examine the effect of varying region size, and number of sources on the spatial CRB. In calculating the bound from (5.18), the infinite matrices had to be truncated. The numerical analysis presented uses a truncation sizeN ≫ kR introducing a negligible error.
We restrict our attention to the case of reasonably high signal to noise ratios where the performance of an estimator will approach the CRB [265]. All results are presented for an effective array signal to noise ratio of 20dB (QN/σ2 = 100). This would generally be expected to be above the threshold regime.
Firstly we consider the spatial CRB for a number of equal power, equally spaced and un- correlated sources (Rs= I). By virtue of symmetry, this configuration has the lowest equal
variance for the position of each source [128] and we need only consider one term of the CRB matrix. Figure 5.6 shows the effect of increasing the number of sources. The square root of the CRB, which represents the standard deviation of the estimate, is plotted in units of degrees.
The single source case has the form (5.23). With multiple sources, the performance in es- timation approaches the single source case beyond a threshold radius. Below this critical threshold the performance diverges rapidly. This is consistent with the notion of the dimen- sionality of a spatial region [41, 84] – we would expect poor performance when the number of sources exceeds the approximate dimensionality of the spatial field. The points at which the effective dimensionality and number of sources are equal (2kR+ 1 =P) are shown on the figure. At this point the multi-source CRB is within a factor of two of the single source case. The notion of dimensionality provides an alternative to “sensor-based” identifiability constraints set out in [266] and [237].
It has been suggested that the number of sources that can be resolved is related to the co- array formed from all the unique inter element spacings of the sensor array [239, 240]. Such approaches provide a theoretical analysis based on numerical uniqueness [241, 242] and do not reflect the uncertainty introduced by noise. The result we present here encapsulates the limitations placed in practice due to the spatial extent and inherent noise of the array. Exceeding these limits becomes exceedingly difficult due to the exponential decrease in the power of the modal expansion terms beyond the critical dimensionality (see Chapter 2 for discussion).
5.6 Numerical Analysis
5.6.2
Comparison with Discrete Sensor Cram´er-Rao Bound
We provide for comparison results from a discrete sensor uniform circular array (UCA). The signal space for aQsensor UCA can be expressed as in (5.14), with the steering array
a(θ) =e−jkRτ1(θ)
, . . . , e−jkRτQ(θ)T (5.34)
whereτq(θ) = cos (θ−2π(q−1)/Q)forq = 1, . . . , Q.
The CRB is obtained from (5.18). For the single source case,AHA =Q, and forQ≥3,
DHD =k2R2 Q X q=1 sin2 2π(q−1) Q = Qk 2R2 2 (5.35) DHA= Q X q=−1 jkRsin 2π(q−1) Q = 0 (5.36)
giving the same result as that obtained for the continuous CRB (5.23). The expression for two or more sources is not easily simplified.
Figure 5.7 compares the continuous CRB to that obtained for a 15 sensor UCA. Below a threshold radius, the performance of the UCA matches the limiting case for the continuous sensor model. The threshold remains fairly constant as the number of sources is changed. This threshold is related to the essential dimensionality. At a radius of R = 1.1λ, the di- mensionality is2kR+ 1 ≈15. The performance of a 15 sensor array degrades beyond this point since insufficient sensors are present to uniquely capture the degrees of freedom of the spatial field. The continuous CRB provides a lower bound for the UCA performance. The single source CRB provides an overall bound.
It should be noted that the CRB is a measure of the localised uncertainty in an estimate and does not consider aliasing artifacts and array [230]. As the number of sensors falls below the degrees of freedom of the array, it becomes increasingly likely that the array will suffer from ambiguities. The condition for ambiguities in linear arrays has been studied [267, 268], however the case for circular arrays is more complex [269]. Generally it is accepted that the sensors should be placed no more thanλ/2apart. This corresponds toQ≥2πR2/λ= 2kR
[151]. The results presented here are consistent with this.
The numerical analysis presented demonstrates that the sensor array CRB is lower bounded by the spatial CRB. This is quite a powerful result. It shows that the performance of an array based DOA estimator will be bounded by the maximal spatial extent of the array, independent of the number of sensors. In the limit of a large number of sensors in the region, the performance converges to the spatial CRB.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 100 101 Radius (λ) CRB (Degrees) Single Source 4 Sources 8 Sources 12 Sources 16 Sources 20 Sources 2× Single Source
Figure 5.6: Impact of region size on Cram´er-Rao bound (√CRB degrees) for direction estimation given a number of equal power distributed sources. Variance of the DOA estimation withP sources approaches that for a single source when2kR+ 1> P. These points are shown on the plot.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 100 101 Radius (λ) CRB (Degrees) 4 Sources 8 Sources 12 Sources Continuous 15 Sensor UCA
Figure 5.7: Comparison of the continuous Cram´er-Rao bound with that of a 15 element uniform
circular array. The UCA achieves the limiting performance up to a threshold radius at2kR+ 1 = 15 (R≈1.1shown).