Digital Signal Processing Reference
In-Depth Information
0
0
Eigenanalysis
Factor analysis
Whitened eigenanalysis
p = 8, q = 1, N = 500
Nominal noise power: 0 dB
−2
−4
−5
−6
−8
−10
−10
p = 8, q = 1
Nominal noise power: 0 dB
Maximal deviation: 3 dB
−12
Eigenanalysis
Factor analysis
Whitened eigenanalysis
−14
−16
−15
0
2
4
6
8
10
12
14
10
1
10
2
10
3
Maximal deviation in noise power from nominal [dB]
N
Fig. 10
Residual interference power after projections
Here, we describe only a limited-scope simulation on synthetic data, where we
estimate a rank-1 subspace (a) using factor analysis, and for comparison (b) using
eigendecomposition assuming that
D
2
I
, or (c) using the eigendecomposition
after whitening by
D
−
1
/
2
, assuming the true
D
is known from calibration. The
correct rank is
Q
=
=
1, and we show the residual interference power after projection,
P
a
a
i.e.,
as a function of number of samples
N
, mean noise power, and deviation
in noise power. The noise powers are randomly generated at the beginning of the
simulation, uniformly in an interval. Legends in the graphs indicate the nominal
noise power and the maximal deviation. All simulations use
J
=
8sensors,anda
nominal interference to noise ratio per channel of 0 dB.
power for varying maximal deviations, the second graph shows the residual for
varying number of samples
N
, and a maximal deviation of 3 dB of the noise
powers. The figures indicate that already for small deviations of the noise powers
it is essential to take this into account, by using the FAD instead of the EVD.
Furthermore, the estimates from the factor analysis are nearly as good as can be
obtained via whitening with known noise powers.
8
Concluding Remarks and Further Reading
In this chapter, we presented a signal processing viewpoint on radio astronomy. We
showed how, with the right translations, the “measurement equations” are connected
to covariance matrix data models used in the phased array signal processing
literature. In this presentation, the resulting data models are very compact and clean,
in the sense that the most straightforward covariance data models, widely studied in
the signal processing literature as theoretical models, already seem valid. This is
because far field assumptions clearly hold, and the propagation channels are very
