Notes#9, ECE594I, Fall 2009, E.R. Brown
127
Optimum PreDetection Signal Processing
(the “matched filter” concept)
Maximum SignaltoNoise Ratio (Intuitive Derivation)
•
Intuitively, detection in the presence of noise has limits imposed by physics (especially
thermodynamics).
To understand these limits, suppose we are attempting to detect the RF
pulse plotted in Fig. 1 of waveform x(t) = A
x
sin(
ω
t +
φ
), carrier frequency
ω
= 2
πν
, and
pulse duration T
P
.
•
Assume that we are trying to detect this pulse in the presence of AWGN having power
spectral density S
P
such that <(
∆
P)
2
> = S
P
∆ν
.
In this case, the RF signaltonoise ratio is
(
)
(
)
max
2
p
p
P
p
P
p
P T
U
P
SNR
S
v T
S
v T
P
⋅
=
=
≈
∆ ⋅
∆ ⋅
< ∆
>
(1)
Where U
P
is the electrical pulse energy.
If we assume that the pulse is sampled consistent
with the Nyquist condition, then the sample rate f
S
should be matched to the pulse width
f
s
= 1/T
P
,
(2)
and should be twice the
twice the RF instantaneous bandwidth
f
S
= 2
∆ν
.
(3)
Substitution of these last two into the SNR expression yields
(
)
max
0
2
2
p
p
P
U
U
SNR
S
N
≈
≡
(4)
where N
0
is another way of writing the power spectral density (following the convention in
communications theory).
This is a factor of two higher than might be expected intuitively
because it assumes implicitly that we have precise knowledge of the phase and amplitude of
a signal, not just one or the other.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '09
 O
 Digital Signal Processing, Signal Processing, LTI system theory, XIN

Click to edit the document details