IEEE Geoscience and Remote Sensing Magazine - September 2021 - 20

square Toeplitz matrix Fb
representing the filters' impulse
response functions. The superscript b is used here to indicate
the BPF subband
bNsb,
12
,,
is obtained through
sF ,ns
n
b
=
wtn
b
b
(23)
which is then multiplied by the time-varying SCORE
weight ()
for each subband and channel. After applying
the weights, the subbands are combined into one data
stream for each channel, given by
Nsb
ccs wtFs
b 11 1
n // / b
b
== =
== =
n
b
n
b
wt
n
b
where ,cn
()
b
introduced earlier, represents the Nrb
element
output of channel n. Furthermore, it is emphasized that
Fb
is fixed; i.e., here, the FIR filters are independent of the
channel n and time invariant, whereas the weights ()
depend on the channel number, the subband, and time.
wtn
b
ALL-PASS CHARACTERISTIC
The frequency transfer function of the FIR-BPF filters is nonideal.
This is to say, the signal spectrum ()
Cf ctFnn ,
= "
Sf
()
at the output of the BPF bank will differ from its input
spectrum ()
nnF st= " () ,, for any channel n. Any mo -
dification of the spectrum results in the loss of range
n n
b
Nsb
Nsb
(),
(24)
! f where N bs
is the
total number of subbands or, equivalently, BPFs. The
Nrb
element signal vector of subband b and channel n
◗ Parameter values: Typically, a small number N 4sb
#
of subbands is sufficient to achieve good performance.
A suitable FIR filter length would be on the
order of N 10tap
s =
quency of f 85MHz , the number of complex multiplications
per second for one channel amounts to
NN ,
sb··f 3 400 106
Nf 340 106
tap
s .
#
for the fixed weight, plus
sb· s . # for the varying weight, respectively.
Note that an FIR filter with such a small number of taps
would not be sufficient to provide a high rejection of adjacent
subbands. This, however, is not necessary for dispersive
beamforming, as detailed in [49, Sec. 5.2.1], which also
provides an analysis of the resulting performance.
# taps. Assuming a sampling freBPF
Coefficients: Fb
Score Weights: wn(t), wn (t), ...
1
2
Sn
z-1
z-1
z-1
wn (t)
1
f1
11 11
f2
f3
f4
Cn
BPF 1
1
Sn
z-1
z-1
z-1
BPF b
wn (t)
b
f1
bb bb
f2
f3
f4
b
Sn
z-1
z-1
f1
Nsb
f2
Nsb
z-1 BPF Nsb
wn (t)
Nsb
f3
Nsb
+ ++
f4
Nsb
Sn
Nsb
Cn
Nsb
DBF Unit n
FIGURE 13. A schematic of a filter bank with Nsb BPFs, each implemented through an Ntap FIR filter. The time-varying weights are applied to
each subband, generating dispersive SCORE beams. Note that the FIR filter is shown for four taps/coefficients; in general, a larger number of
taps is used.
20
IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE SEPTEMBER 2021
b
Cn
1
Σ
Cn

IEEE Geoscience and Remote Sensing Magazine - September 2021

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