efficiency of the two-stage rc low-pass sano filter in comparison with

efficiency of the two-stage rc low-pass sano filter in comparison with

Revista Mexian.
de F(sica 31 No. 2 (1985) 401406
401
EFFICIENCY OF THE TWO-STAGE
LOW-PASS SANO FILTER
IN COMPARISON WITH THE
MATCHEO FILTER
RC
Francisco F. Medina E.
Departamento
Universidad
de Antioquia,
de Física
A.A. 1226, Medellín,
(recibido julio 24, 1984; aceptado
noviembre
Colombia
8, 1984)
ABsmACf
Matched Filters for detectíon the most usual sígnals (i.e., tri
angular, rectangular and trapezoidal pulses) are complex and expensive.
This work presents a general method to calculate suboptimal filters to be
constructed using simple Re circuits, which exhibit high efficiency and
low cost in comparison with those Matched Filters.
We use a triangular
pulse in basic band as input of the filters but there is no 1055 of generality because the pro cedures of calculation are completely general and
are related with detection in a carry-band.
RESlJ1.lEN
Los filtros acoplados para la detección de las señales más usua
les (¡.e •• pulsos triangulares, rectangulares y trapezoidales) son complicados y costosos.
Este trabajo presenta un método general para el
cálculo de filtros subóptimos construidos usando circuitos Re, los cua-
402
les exhiben una alta eficiencia y son poco costosos en comparación
con
los filtros acoplados.
Utilizamos
un pulso triangular en banda básica ca
mo señal para los filtros, lo que no significa pérdida de generalidad
ya
que los procedimientos
de cálculo son completamente
generales y están relacionados con la detección en una banda de transporte.
DETERMlNATICN OF '!HE SNRo
1.
The signal-to-noise
SNR
IS
(t)
o
<n2(O»
=
O
ratio
for the ~btched Filter is defined
2
1
(1)
HAX
o
2 the maximum power of the output and <n2(0»
with ~ o (t)1MX
o
average power at the output of the device (Fig. 1).
Fig.
1
as
Block
diagram
of the Matehed
the noise
Filter.
The impulse response function (IRF) of the Matched Filter for the signal
surnmergcd into white
h(t)
noisc at thc input is
(2)
S'(-t)
aoo
S(t), because it is a real and even
h(-t)
SrtTlTlctric
function.
Thcn,
15o (t)
2
1
HAX
J S(t)h(T-t)
dt
I~
IJ
S(t) S'(t-T) dtl~(3a)
403
and
15o (t) IMA2 x = r2ss (O)
(3b)
Now, the average power oí noise is
(4)
_00
2
withIN(w) 1 the noise power speetrum and lI(w) the transferenee funetion
of the Matehed Filter. In white noise we have IN(w) 12= I and the Eq. (4)
is
00
<n2 (O) >
-L
2~ J
III(w)12dw
using the Wiener-Khintehine
have
theorem.
=
o
2
<n (0»
o
=
r
ss
(S)
But, replaeing (2) into (S) we
(O)
(6)
and using (3) and (6) in (1), we obtain
SNR
o
=
For a triangular
r ss(O)
(7)
pulse, defincd as
_lU
T
( -T••t<I")
(8)
O
(otherwise)
404
summer~ed into white noise, wc have
2
3
T
SNRo =
2.
(9)
DETERMINA
TI 00 OF 1HE SNR
Now we will calculate
stage
Re Iow-pass
the signal-to-noise
band (SNR) as suboptimal
filter
case. thc IRF oí the device and its related
(Fig.
transfer
FOORIER'S TRANSFORM
t
-tit
-c
h(t) =
ratio
2
Re and
T=
SNR =
Replacing
with A
=
the SNR.
w=
2nv.
The signal-to.noise
2).
In this
funetían are
f1(w)
T
with
far the two-
I
---2.(lOa,b)
(l+iwT)
ratio is defined as
ISo(t)l~x
(11)
<n2 (O) >
o
(IOa,b)
into
T/T and x
=
(3a) and (4) and this
(t-T)/T.
All solutions
3.
result
into
(11) we obtain
For each A trere is an x which maximizes
with x>A implies that the pulse does not existo
•
DETERMINATlooOF TItE EFFICIENCY
The efficiency
E = SNR
--.100%
SNRo
of any suboptimal
filter i5
(13)
405
1
.,__,._. .__T__
e
SIH+II(.,
Fig.
2
The
two-stage
operational
w~en
Re
S u.)+n.lt)
_aT__ ._~"_
n
l
••
low-pass
band
sub-optimal
filter
Note
amplifier
with unitary gaio used as accoplator
the two steps of the circuito
the
bet-
Then, for this case the cfficicncy is given by
6
3" {IAc
A
E
-A
- (2+x)(1-e
-A
-x
)1 e
which graph is shown in Fig. 3.
+ 2(1-c
-x
)-x(l+e
-x
) + A)
2
(14)
Aftcr the maximization oí W efficiency
we obtain thc following results:
EMAX
= 88\ = -0.56 db
A
b
Fig.
3
Graph
of the efficiency
filter in comparison
curve
3.2055
l!>
10
tor the two-stage
with the Matehed
Filter.
Re low-pass
band
406
4.
i)
We can observe
twa-stage
from the results
Re low-pass
ii) In the implementatíansignal-to-noise
the high cfficiency
of the subopti.maI
with a 1055 srnaller than 1 <lb.
filter,
is usual
better gain at the output
because
to send a train of pulses
to obtain
a
SNR¡-=(N'BT)SNR¡" where SNR¡- is the
ratio fer the train, SNRp is the signal-to-noisc
far an individual
the bandwidth
C~llJSICNS
ratio
pulse oí the train, N is the number oí pulses,
and T is the duration
of any pulse.
The product
B is
BT is
the Ilumber of modes transmi ted.
üi) The forbidden
zone for the efficiency
x>aA with ex a real number.
function
In thi5 case,
is strictly
defined
as
we use a=l.
The author acknowledges the important corranents oí Dr. Pcter Barlai
fer thi5 article and the special way the colaboratíon
fieda in
a
oí Proí. Román Casta
realizatíon oí the computer programs.
BIBLIOGRAPHY
1.
2.
3.
4.
5.
6.
M. Skolnik,
IntADduction te RadaA Sy~tem6,McGraw-Hil1, New York (1962).
J. Mi11man and H. Taub,Puloe and Vigitai C<Acuith. McGraw-Hi11. ~ew
York (1965).
G.L. Turio, llAn introduction
to Digital Matched Filtersll,
PILOt.
IEEE,
64 (1976)1092.
n. Turin, "Minimax strategies for Matched Fi1ter detection", IEEE
TIW.'I.6. Como 23 (1975)1370.
1. Yamabuchi-;l.
Takagi and K. Mano. "Natched fi1ters
for rectangular,
raised cosine and sine wave signals",
IEEE, TIW.M.Com. 19 (1971)369.
G.W. Carlock. "The two-stage Re low-pass Matched Fi1ter"":" IEEE TIW.M.
Como 20 (1972) 73.