Signal model Strategy Key reason for non-robustness SNR wall scaling
Unknown or ‘white’ signal Energy detection Uncertain noise power O(1)
with a power constraint Optimal detection Uncertain noise moments
‘White’ signal with Matched filter Finite phase-coheren ce time
and uncertain noise power
narrowband pilot Matched filter with Finite phase-coheren ce time O
³
1
N
c
´
run-tim e noise calibration and uncertain noise color
50%-duty-cycle pulse Detector defined in (12) Finite delay-coherence time
amplitude modulated signal and uncertain noise power
(exa mple of a Detector defined in (16) Finite delay-coherence time O
³
1
√
D
c
´
cyclostationary signal) (run-time noise calibration) and uncertain noise color
TABLE I
COMPARISON OF ROBUSTNESS RESULTS FOR DIFFERENT CLASSES OF SIGNAL MODELS.
which robust detection
1
is impossible [4]. The SNR wall
expressions for the radiometer and the matched filter are
SNR
energy
wall
=
ρ
2
− 1
ρ
, (2)
and
SNR
mf
wall
=
1
N
c
· θ
µ
ρ
2
− 1
ρ
¶
, (3)
where θ is the fraction of the total signal power allocated
to the known pilot tone, and N
c
is the phase-coherence
time of the channel.
We no w give an alternate interpretation for SNR
walls using the notion of sample complexity. Consider
the robust detection problem in (1). Assume that the
signal and noise samples are independent and identically
distributed (iid), and the noise distribution lies in the
white noise uncertainty set W
ρ
. Let the detector test-
statistic be giv en by
T (Y):=
1
N
N
X
n=1
φ(Y [n])
H
1
≷
H
0
γ, (4)
where φ(·) is a known deterministic function and γ is the
detector threshold. Denote the SNR wall for this detector
by SNR
T
wall
.LetSNR > SNR
T
wall
. Then, by the
definition of an SNR wall (see [4] for a formal definition)
we know that any P
FA
<
1
2
and P
MD
<
1
2
can be
robustly achieved. That is, we can choose a detection
threshold γ such that
P
FA
=max
W∈W
ρ
P
W
(T (Y) >γ|H
0
) ,
P
MD
=max
W∈W
ρ
P
W
(T (Y) <γ|H
1
) . (5)
1
See [4] for a formal definition of robust detection.
Eliminating γ from (5) we can solve for N as a function
of the SNR, P
FA
, P
MD
and ρ. Hence, we can write
N = µ(SNR, P
FA
,P
MD
,ρ). (6)
This is called the sample complexity of the detector. For
any reasonable detector the sample complexity increases
as the SNR decreases, i.e., µ(SNR, P
FA
,P
MD
,ρ) is a
monotonically decreasing function of SNR. Under this
monotonicity assumption, it t urns out that
lim
SNR↓SNR
T
wall
µ(SNR,P
FA
,P
MD
,ρ)=∞. (7)
Equation (7) gives an alternate interpretation for an
SNR wall. It tells us that the SNR wall for a detector
is the SNR threshold at which the sample complexity
approaches ∞.
From [4] the sample complexity of the radiometer is
N ≈
2[Q
−1
(P
FA
) − Q
−1
(1 − P
MD
)]
2
h
SNR −
³
ρ
2
−1
ρ
´i
2
, (8)
where Q
−1
(·) is the inverse of the Gaussian tail prob-
ability function. Again from [4], the sample complexity
for the matched filter is
N ≈
2[Q
−1
(P
FA
) − Q
−1
(1 − P
MD
)]
2
h
θ · N
c
· SNR −
³
ρ
2
−1
ρ
´i
2
, (9)
Clearly (8) and (9) verify the assertion made in (7).
Figure 2 plots the sample complexity of the radiometer
and the matched filter . From the figureitiseasytosee
that the sample complexity curves go to infinity as the
SNR decreases to the SNR wall.