PROBLEMS 66514.3.4(a) The transfer function of a notch filter is given
byH(jω)=( 100 +jω−ω^2 )/( 10 +jω)^2.
SketchHdBversusω.
(b) A notch filter centered atf 0 =ω 0 / 2 πcan be
formed by using a resonant circuit arranged
such thatH(jω)=
j Q(ω/ω 0 −ω 0 /ω)
1 +j Q(ω/ω 0 −ω 0 /ω)
Investigate|H(f)|atf=0 andf=∞;
also atfl=f 0 ( 1 − 1 /( 2 Q), andfu=f 0 (1+
1 /( 2 Q), forQ>>1.
(c) Withω 0 = 1 /√
LCandQ=( 1 /R)√
L/C,
show that a seriesRLCcircuit can perform
as a notch filter when the output voltage is
taken acrossLandC.
(d) For the purpose of rejecting 1-kHz interfer-
ence, in order to getfl=980 Hz andfu=
1020 Hz, find the values ofLandCof the
seriesRLCcircuit withR= 50 .
(e) Now consider a tuned circuit in whichRis
connected in series with a parallel combi-
nation ofLandC, and the output voltage
is taken acrossR. Withω 0 = 1 /√
LCand
Q= R√
C/L, show that the circuit can
perform as a notch filter.
(f) For the purpose of rejecting 60-Hz interfer-
ence, in order to getfl=58 Hz andfu= 62
Hz, find the values ofLandCof the circuit
in part (e) withR=1k.
14.3.5A noisy amplifier hasNout= 600 μW whenT=
T 0 ; butNoutdrops to 480μW when the source is
immersed in liquid nitrogen atT=80 K. Find
the amplifier’s noise temperatureTa.
14.3.6Two noisy amplifiers (with noise temperatures
Ta 1 andTa 2 ) having the same bandwidth are con-
nected in cascade so that the overall gain isG
=G 1 G 2. If the input noise to the first amplifier
isN=kTB, determine the total output noise
power and the effective noise temperatureTaof
the cascade, so thatNout=Gk(T+Ta)B.*14.3.7In order to measure the amplifier noise tempera-
tureTa, a thermal source at temperatureT 0 is con-
nected and the correspondingNoutis observed;
then the source temperature is increased toTR
when the correspondingNoutis doubled. FindTa
in terms ofTRandT 0.
14.3.8A signal (withP= 1 μW andB=250 kHz)
contaminated by white noise at noise temperature
T= 2 T 0 is applied to an amplifier. Obtain the
condition onTasuch thatPout/Nout≥80.8 dB.
14.3.9The noise figure of an amplifier is given byF=
1 +Ta/T 0. Express the output signal-to-noise
ratio in terms ofF, input noise powerN, and
input noise temperatureT. See how the result is
simplified whenT=T 0.
14.3.10A system for measuring the constant signal value
xhasPout/Nout=40 dB andB=8 Hz. In order
to obtain an accuracy of±0.2%, how long must
the output be observed?
14.3.11An amplifier has a noise equivalent bandwidthB
=25 kHz and a maximum available power gain
ofG=30 dB. If its output noise power is 10^8 kT 0 ,
determine the effective noise temperature and the
noise figure (given byF= 1 +Ta/T 0 ). Assume
the input noise source temperature to beT 0.
*14.3.12The overall noise figure of a cascade ofKam-
plifiers with gainsGkand corresponding noise
figuresFk,1≤k≤K,isF=F 1 +
F 2 − 1
G 1
+
F 3 − 1
G 1 G 2
+...+
FK− 1
G 1 G 2 ...GK− 1If an amplifier is designed having three identical
states, each of which has a gain ofGi=5 and a
noise figure ofFi=6,i=1, 2, 3, determine
the overall noise figure of the cascade of the
three stages. Looking at the result, justify the
statement that the front end of a receiver should
have a low noise figure and a high gain. (Note
that the noise figure of an amplifier isF =
1 +Ta/T 0 .)
14.3.13A radio antenna with a noise temperature of 60
K is pointed in the direction of the sky. The
antenna feeds the received signal to the preampli-
fier, which has a gain of 30 dB over a bandwidth
of 10 MHz and a noise figure (F= 1 +Ta/T 0 )
of 2 dB.
(a) Determine the effective noise temperature at
the input to the preamplifier.
(b) Determine the noise power at the output of
the preamplifier.