IEEE Aerospace and Electronic Systems Magazine - June 2020 - 25

Carotenuto et al.
The parameters involved in (13) can be reliably estimated starting from the measurements gathered during the
perception stage. The idea is to estimate s 0 and s I;k ,
k ¼ 1; . . . ; K, using the PSD spectrum portions of the
measured signal related to the detected transmission- and
stopbands, respectively. Specifically,
 s 0 is estimated as the arithmetic mean of the measured signal power levels on each transmission band;
 s I;k , k ¼ 1; . . . ; K is estimated as the power of the
signal measured over the kth stopband minus the
measured noise floor.
The overall procedure is summarized in Algorithm 1,
where Pxx ðqÞ, q ¼ 1; . . . ; Q denotes the PSD of the measured data evaluated via Welch's method in correspondence of the normalized discrete frequencies
f ðqÞ ¼

qÀ1
;
Q

q ¼ 1; . . . ; Q :

(14)

Algorithm 1. Procedure to estimate M .
Require: Q, f ðqÞ, Pxx ðqÞ, q ¼ 1; . . . ; Q
Ensure: an estimate of M
1: apply the thresholding procedure to detect
Vk ;
m ;
V

k ¼ 1; . . . ; K
m ¼ 1; . . . ; M

(stopbands)
(transmission bands)

Figure 6.
Measured PSD (based on Welch's method) compared with estimated PSD from data covariance matrix (based on Capon's
method) versus frequency.

covariance matrix predicted via Algorithm 1) or through
Welch's estimator on measured data. Figure inspection
clearly shows the ability of the covariance matrix estimate
to capture the spectral behavior of the recorded data in
terms of emitters frequency bandwidths and powers.
The last ingredient to handle the waveform design
problem in (10) concerns setting the spectral constraints
EIk , k ¼ 1; . . . ; K, which rule the amount of interference
injected on the kth stopband. Such parameters can be set
so as to guarantee an appropriate quality of service for the
different communication systems, namely according to a
desired signal-to-interference ratio (SIR) at the radar side
in each stopband, which is defined as

2: evaluate the normalized frequency bandwidths
Dfk ¼ f2k À f1k ;

SIRk ¼

k ¼ 1; . . . ; K

 m ¼ fm À fm ;
Df
2
1

m ¼ 1; . . . ; M

M
1 X
s 0;m
M m¼1

where
s 0;m

0
1
1 @ X
¼ 
Pxx ðqÞA
QDfm q : f ðqÞ2V
m

SIRk ¼ SIRk

4: estimate s I;k , k ¼ 1; . . . ; K, as
0
1
1@ X
Pxx ðqÞA À s 0 Dfk
s I;k ¼
Q q : f ðqÞ2V

(15)



4pdk 4
;


k ¼ 1; . . . ; K

(16)

where  is the wavelength. Since for the case under analysis
dk >

k

5: evaluate
k ¼ 1; . . . ; K, using (6)
6: estimate M according to (13)
R kI ,

To corroborate the effectiveness of Algorithm 1, the
PSD computed using Welch's method is compared with
that evaluated resorting to Capon's approach [35], [36].
Precisely, Figure 6 displays the estimated PSD versus frequency obtained either via Capon's method (with the data
JUNE 2020

k ¼ 1; . . . ; K

where s D;k , k ¼ 1; . . . ; K is the disturbance power level
injected by the radar in the kth stopband. Observe that
SIRk , k ¼ 1; . . . ; K represents a worst case on the achievable performance. In fact, denoted by dk the distance
between the radar and the kth licensed emitter, using oneway path-loss equation (assuming for simplicity unit gain
for transmitting and receiving antennas), the SIR value at
the communication system side can be expressed as

3: estimate s 0 as>
s0 ¼

s I;k
;
s D;k


4p

(17)

it follows that SIRk > SIRk , k ¼ 1; . . . ; K. As a consequence, the SIR defined in (15) represents a lower bound
on the achievable performance. As mentioned before, to
ensure the spectral compatibility in each stopband, it is
required that
SIRk ! SIRmin ;

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k ¼ 1; . . . ; K

(18)
25
IEEE Aerospace and Electronic Systems Magazine - June 2020

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