Page 412 - Satellite Communications, Fourth Edition

392 Chapter Twelve

2
referred to the input has a spectral density 2q/I in in A /Hz, where I in is
the transistor gate (JFET) or base (BJT) current in amperes. Using
curly brackets {.} for the spectral density values (see Jones, 1988), the
total noise current spectral density at the input is
4kT
2 2 2
tr
5i t 6 5 2qIM F 1 1 2qI A /Hz (12.69)
R
The pre-amplifier also has a noise voltage component shown as v a
resulting from the shot noise in the drain or collector current. The mean
2 2
square noise voltage spectral density is 2qI /g m V /Hz where I is the
tr
tr
transistor drain (JFET) or collector (BJT) current in amperes and g is
m
the device transconductance in siemens. The total noise voltage spec-
tral density at the input is therefore
2qI
2 2 2 tr 2
5v n 6 5 5i 6 ZZ Z 1 2 V /Hz (12.70)
L
t
g m
where Z is the impedance of R and C in parallel.
L
The average signal voltage is MI R. Increasing R should result in an
ph
increase in signal to noise ratio since the signal voltage is proportional
to R and the R component of noise current density is inversely propor-
tional to R. However, the bandwidth of the RC input network is inversely
proportional to R and therefore sets a limit to how large R can be. An
equalizing filter, which has a transfer function, given by
H (f) 1 j2 f RC can be included in the overall transfer function,
eq
which compensates for the input impedance frequency response over the
signal bandwidth. The effective spectral density for the mean square
2 2
noise voltage at the input is then 5v n 6ZH( f )Z . The mean square noise
2
voltage V at the input is obtained by integrating this expression over
n
the signal bandwidth. Only the result will be given here:
2 2 4kT 2 2qI tr (2 RCB) 2
V ca2qIM F 2qI bR 2 a1 bdB (12.71)
tr
n
R g m 3
With the average signal voltage given by V MI R the signal to noise
s
ph
ratio is
2
S V s

N V n 2
R) 2 (12.72)
(MI ph
2
2
2
ca2qIM F 4kT 2qI bR 2qI tr a1 (2 RCB) bdB
in
R g 2 m 3