IEEE Aerospace and Electronic Systems Magazine - November 2020 - 13

Daum
engineers use much more detailed industrial strength cost
models with many design variables for important practical
applications.
The Nelder-Mead algorithm is a very popular
gradient-free optimization method, but it suffers from several problems, including getting stuck in false local minima, and hence, we use a randomized version that is
guaranteed to converge to the global minimum under certain conditions (see [30]-[32] for details). Our cost function is highly multimodal, and hence, a simple gradient
search will fail to converge to the global minimum. Moreover, the constraints for our QR application are not
smooth, and hence, we cannot use a gradient method.
Details on the cost models for modern radars are given in
[55]-[58]; such cost models can be used in our randomized Nelder-Mead algorithm.
Our cost analysis has assumed that the optimal QR
has 6 dB better effective SNR than the corresponding CR
with the same transmit power and signal bandwidth for
low photon flux per mode (see [2] and [16] for details).
The precise meaning of the 6 dB for QR versus CR is
improvement in the ROC curve for detecting targets
rather than an actual improvement in SNR itself. But
other papers [1] and books [7, p. 85] claim that the
effective SNR is improved by a factor of d (the timebandwidth product of the transmitted waveform) rather
than only 6 dB. However, this seems too good to be true,
and it is in fact not practical to implement, as explained
in detail in [20]. In particular, the analysis in [1] compares two QRs, one with entanglement and the other
without entanglement, rather than comparing a QR with
a CR (see [20] for details). Moreover, the factor of d in
[1] seems to arise from confusing the increase in SNR as
a result of matched filtering, which is indeed a factor of
d for both QR and CR. Furthermore, the analysis in [1]
implicitly assumes a constant target RCS, with no change
in polarization due to scattering, which is rarely the case
in practice. For a fluctuating target with polarization
change, the increase in SNR with QR is much less than
6 dB. Also, the analysis in [1] implicitly assumes that the
target is known a priori to be in a single given resolution
cell in range, Doppler, and angle, for the duration of the
transmit waveform, whereas this is very unlikely in the
real world for radars with large time-bandwidth products,
which is essentially the only case of interest for QR.
Large unknown Doppler shift is a severe problem for random waveforms, which is what a QR transmits. For CR
using linear FM waveforms (or similar smooth and deterministic frequency modulation), a Doppler shift is equivalent to a time shift after matched filtering, resulting in
no loss in SNR despite large unknown Doppler shifts,
but this is not the case for random waveforms used in
QR. All of these limiting assumptions are also made for
the 6 dB QR advantage over CR derived in [16] as well
as in the experiments reported in [2] and [3]-[5].
NOVEMBER 2020

APPLICATIONS FOR QRS
In this section, we shall consider various potential applications of QR that have been suggested in papers and books.
As shown in Fig. 1, QR are many orders of magnitude
more expensive than CR for any range and any target
RCS. This rules out almost all traditional applications for
QR, because nobody would be willing to pay such a high
premium for a QR with no apparent benefits. Hence, we
need to look for nontraditional applications.
First, it has been suggested that QR would be much
safer than CR for medical imaging because the required
transmit power for QR is 6 dB lower than for the corresponding CR [2]. Moreover, it has also been implied that
CR would not be safe for medical imaging, owing to 6 dB
higher transmit power for CR compared with QR [2]. But
this last implication in [2] and elsewhere is incorrect. In
fact, CR can also operate at extremely low power levels,
resulting in safe medical imaging with excellent quality
images (see [41] and [42] for details). The fact that the
QR might have 6 dB less transmit power than the corresponding CR does not improve safety for humans, because
30 dB of safety margin for the CR is ample and another
6 dB of margin with the QR would not help. In particular,
microwaves are deemed to be safe for humans at 5 mW
per square centimeter (the US FDA and DoD standard). A
typical cellphone radiates about 1 W, and humans routinely put such cellphones next to their skulls with no
reported ill effects. Likewise, microwave ovens typically
leak about 1 W without any reported ill effects. The FCC
safety standard for communication systems is 1.6 W per
kg (of human skull), and there has been no serious debate
that this standard needs to be tightened. CR can create
excellent medical images with transmit power that is
less than the most stringent international standards (e.g.,
Austrian). Moreover, the QR would be many orders of
magnitude more expensive than the corresponding CR,
without any apparent benefit to safety or image quality.
Second, many papers have asserted that QR would be
superior to CR for detecting and tracking stealthy targets
(i.e., low RCS targets) [17]. But detection of stealthy targets requires high transmit power levels, which QR lacks.
Moreover, Figure 1 makes it obvious that QR are many
orders of magnitude more expensive than CR at any range
and for any target RCS. Hence, the idea that QRs are better than CR for detecting stealthy targets is erroneous.
This notion seems to be based on the fact that QRs have
6 dB better SNR than CR with the same transmit power
and bandwidth at low photon flux per mode, but ignoring
the extremely high cost of building such a QR. The detection of stealthy targets is often foiled by clutter, jamming,
RFI, or chaff rather than radar front-end noise. There is no
reason to believe that QR would detect stealthy targets in
clutter, jamming, RFI, or chaff any better than CR. A QR
might be slightly better than a CR against a barrage

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IEEE Aerospace and Electronic Systems Magazine - November 2020

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