-- 1-,-1
I
(I'
\
THE DYNAMIC RANGE OF
A PARAMETRIC AMPLIFIER
by
SUNG JAI SOHN
B. S.,
Seoul University, Korea
(1960)
SUBMITTED IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
May,
1962
7)
Signature of Author
Department of
%cti4 cal Engineering,
May 19,
1962
Certified by
Thesis Supervisor
~1
Accepted by
Chairman, Departmentalo ommittee on Graduate Students
ACKNOWLEDGEMENT
The author wishes to express his gratitude
to Professor R. P.
Rafuse, for his encourage-
ment and guidance throughout this work.
-
ii -
THE DYNAMIC RANGE OF
A PARAMETRIC AMPLIFIER
by
SUNG JAI SOHN
Submitted to the Department of Electrical Engineering on May 21, 1962
in partial fulfillment of the requirements for the degree of Master of
Science.
ABSTRACT
The application of the semiconductor capacitor diode to parametric
amplifiers has received considerable interest in the past few years.
This analysis considers the diode as a series combination of a constant
loss resistance and a current controlled nonlinear elastance, there being
good physical evidence for this choice. 7 As a direct consequence of the
series circuit, the choice of restricting the currents to certain frequen-
cies is made.
A Fourier-series approach permits the large-signal
behavior of the amplifier to be analyzed.
The impedance representation
of the varactor is used throughout. The relations among elastances are
obtained. The gain is given. The dynamic range and phase shift are also
obtained.
Thesis Supervisor:
Title:
I
Robert P.
Rafuse
Assistant Professor of Electrical Engineering
- iii -
TABLE OF CONTENTS
page
TITLE PAGE ..............................
i
ACKNOWLEDGEMENT ........................
ABSTRACT
..........
...............................
TABLE OF CONTENTS
LIST OF FIGURES
ii
iii
.iv
.......................
........
.........................
.........
vi
..
viii
LIST OF SYMBOLS ........................
CHAPTER I - INTRODUCTION
1
...................
........
1
..........................
..........
1. 1
History
1. 2
The varactor model and its implications .........
2
1. 3
Mathematical formulation .................
5
..
CHAPTER II - IMPEDANCES OF THE AMPLIFIER ..........
. . . . . . . . . . . .
12
12
2. 1
The varactor conversion matrix
2. 2
Impedances . . . . . . . . . . . . . . . . . . . . . . . . 14
2. 3
Elastances and variables . . . . . . . . . . . . . . . . . 18
2. 4
Elastance equation and approximation . . . . . . . . . . 23
CHAPTER III - THE DYNAMIC RANGE . . . . . . . . . . . . . .
29
3. 1
Gain . . . . . . . . . . . . . . . . . . . . . . - - -.
.
29
3. 2
Noise figure . . . . . . . . . . . . . . . . . . . . . . .
35
3. 3
Dynamic range . . . . . . . . . . . . . . . . . . . .. .
37
3. 4
Examples on dynamic range . . . . . . . . . . . . . . . 44
- iv -
page
CHAPTER IV - PHASE SHIFT OF THE AMPLIFIER ........
49
4.1
Signal frequency reactance ................
49
4. 2
Phase shift of the amplifier . . . . . . . . . . . . . . .
50
4. 3
Maximum phase shift . . . . . . . . . . . . . . . . . . .
53
4. 4
Examples on phase shift . . . . . . . . . . . . . . . . .
54
CHAPTER V - CONCLUSIONS . . . . . . . . . . .
. ..
..
69
...
69
5. 1
Conclusions . . . . . . . . . . . . . . . . . . . . . . . .
5. 2
Future work . . . . . . . . . . . . . . . . . . . . . . . . 72
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . .
.
74
I
LIST OF FIGURES
page
CHAPTER I
1. 1
Model of the varactor . . . . . . . . . . . . . . . . . . . . 3
CHAPTER II
2. 1
Signal-frequency impedance model . . . . . . . . . . . . 17
2. 2
Pump-source equivalent circuit . . . . . . . . . . . . . . 20
CHAPTER III
29
3. 1
Circulator imbedment of a parametric amplifier .....
3. 2
Maximum gain . . . . . . . . . . . . . . . . . . . . . . . 34
3. 3
Dynamic range on Ex. 3. 1 . . . . . . . . . . . . . . . . . 47
3. 4
Dynamic range on Ex. 3. 2 . . . . . . . . . . . . . . . .
48
CHAPTER IV
4. la Phase shift with
i = 1/4,
Gmax
b Phase shift with
i = 1/4,
Gmax
c Phase shift with
i = 1/4,
G
d Phase shift with
i = 1/4,
Gmax
- vi -
= 40db .
. . . .56
. . ...
= 30db . . . . ...
= 20db ..
. . .
= 40, 30, 20, 10db
.
...
57
o... ..
58
. . .
59
page
b Phase shift with i = 1/6,
G
= 20db .
c Phase shift with i=1/6,
Gax
= 10db
. . . . . . . .61
..
4. 3a Phase shift with i = 1/10, Gmax = 30db .
b Phase shift with i = 1/10, Gmx = 20db ..
max
c Phase shift with
4. 4
Phase shifts with G
4. 5
Phase shifts with G
4. 6
Phase shifts with G
1/
max
max
max
60
Gmax = 30db . . . . . .....
4. 2a Phase shift with i = 1/6,
max = 10db .
= 20db . . . ..
- vii -
.
..
. .
. ..
.
62
. . . . . . . .63
. ..
. . . .64
. . . . . . .65
.66
....
=30db............
= 10db
. . . ..
. . . ..
.
..
. .. ..
..
..
67
68
LIST OF SYMBOLS
-
elastance coefficient
-
diode exponent
-
diode reverse breakdown voltage
y
V
*
S
-
1 contact"
-
complex conjugate
-
Mk
max
R
S
Fourier component of elastance
normalized Fourier component of elastance
c
S
potential
-
cutoff frequency of the diode
maximum elastance of the diode
-
- diode series resistance
-
pump frequency
.
-
idler frequency
W
-
signal frequency
p
-
normalized pump frequency
i
-
normalized idler frequency
W
s Sm 0 -
normalized signal frequency
variation of normalized direct frequency component of elastance
-
6m /i
-
source resistance
R.
1
-
idler resistance
P
-
powe r
a
R
0
0
- viii -
-
diode temperature (of Rs)
-
2900 K
Ta
-
antenna temperature
T.
-
idler temperature (of R.)
E
-
Fourier component of voltage
T
T
d
n
Ik
-
s
Fourier component of current
- ix
-
CHAPTER I
}I
INTRODUCTION
1. 1
History
Amplification can be achieved by vacuum tubes, transistors,
semiconductor diodes, and many other devices.
Each method of
amplification has both advantages and disadvantages, with respect
to dynamic range, frequency range, efficiency, device complexity,
etc.
The semiconductor diode parametric amplifier has been found
to have large dynamic range, large frequency range, and low noise. 1, 2, 3,4
In 1939,
parametric devices were treated by an electric-circuit
concept in Hartley's5 paper on nonlinear reactances.
covery of the p-n junction diode,
Until the dis-
magnetic amplifiers were mainly
the subjects of discussions concerning nonlinear reactance.
world war II,
During
North6 applied the concept of variable reactance to
explain some of the characteristics of germanium diodes.
Since 1956, 1 the application of the semiconductor diode to lownoise microwave amplifiers has been investigated by many researchers.
Many of their theoretical papers dealt with the lossless case.
However,
they should not ignore the losses completely in the analysis of the
semiconductor capacitor diode.
- 1-
A fundamental approach to the problem of parametric amplifier
and many other applications of the semiconductor capacitor diode has
been evolved by Rafuse.
His theory is based upon a series model for
the varactor (variable reactor) - a constant resistance in series with
a current controlled elastance, there being good physical evidence for
this choice.
As a direct consequence of the series circuit, the choice
is made to restrict the currents to certain frequencies.
He used a
Fourier-series representation of the varactor elastance variation for
the analysis of the amplifier.
Penfield 2,3 has derived best gain and
the best excess noise figure for the small signal application of the
varactor amplifier.
From the results the optimum source resistance
and the best pump frequency can be obtained.
Rafuse
showed that the
multiple-idler parametric amplifier does not give sufficient improvement over the single-idler one to warrant the extra circuit complexity
required.
1. 2
The varactor model and its implications
A model for the varactor is comprised of a constant series resis-
tance and a capacitance varying with applied voltage as Fig. 1. 1.
This model follows from the physics of the semiconductor capacitor
diode neglecting series lead inductance, case capacitance, and shunt
conductance.
Uhlir was probably the first to recognize and use such
-2 -
a model.
And the measured behavior fits the model for the large-
area junction diodes over a substantial range of retarding region
and frequency.
e (tr)
CWitV(I)
Fig. 1. 1
Model of the varactor
With the varactor model, incremental capacitance c(t) is
expressable as
c(v)
- dq
dv
_
1
A(v + <>
a constant dependent on diode geometry and doping
b
=
v
= the applied back voltage
- 3 -
(1. 1)
=
I
the "contact" potential of the junction material at the
depletion layer edges (a slowly varying function of v)
y
an exponent dependent on doping geometry (1/3 for
linear grading and 1/2 for abrupt)
There is a minimum value of capacitance set by the breakdown
voltage of the diode, VB.
Therefore,
=1
(1. 2)
A(VB+
With the elastance expression, from Eqs. 1. 1 and 1. 2
s(v)
Smax (v+
ma B
=
4
(1.3)
Uhlir chose to specify a "cutoff" frequency
S
c
max
R
(1.4)
s
which will be independent of any operation for which the diode is to
be utilized.
- 4 -
.
-
-Vw
1. 3
-
Mathematical formulation
From Eq. 1. 1
dv
Idq =
(1.5)
A(v +
[
which gives for the case y
q+ q
=(V
4
1
(1. 6)
(1I- Y)6
+
1
1-y 1
[q5
We may rearrange Eq. 1. 6 to read
y
(q
-
y)
+ q) (
Multiplying both sides of Eq. 1. 7 by
s(t)
s(q)
=
[N(
14
-
)7
+q 4
we recognize finally that
A,
y)(q + q4 )
1
The maximum charge occurs at the breakdown voltage of the
diode.
Thus,
- 5 -
(1. 7)
(1. 8)
y
S ma(1 =
-
max
q4
qY
Y)
1
(1. 9)
Equations 1. 8 and 1. 9 yield
s(q) = Sq+q
max
(1. 10)
QB + q
The elastance is now represented as a function of the charge rather
than the voltage.
This step is necessary because the varactor must be
represented as a charge (or current) controlled elastance to be analyzed
reasonably.
The abrupt-junction diode has been chosen as the subject for this
work. And Eq. 1. 10, for y = 1/2,
q+
becomes
q
(1. 11)
max IQB + q4
Currents are allowed to flow through the varactor for some
chosen frequencies only according to the purpose of the varactor.
The Fourier-series of the current can be written as
- 6 -
N
i(t)
e
Jwkt
(1. 12)
k=-N
k/ 0
with
I_ -k
=
I*k
where the asterisk represents a complex conjugate.
Substitution of
N
jkt
Ik
q +
q(t)
0
(1. 13)
e
k=
k=- N
where
g
s(t)
to Eq. 1. 11 yields
is a constant,
=
max
N
q+mqa
sW QB + q+
5 L
+
q0
W
k
k
j
RZ=N
N
s(t)
=
S jkt
S 0 +i
(1.14)
k=-N
with
S-k
=S
- 7 -
L
s(t) has lower bound s = S min when the diode starts forward conduction.
And S
may approach zero if the varactor can be driven to the
.
point v = -
|
(q = -q4).
This is a very good approximation for diodes
characterizable by a y .
The normalization of the elastance is defined as
N
m(t)
s (t)
-Me
+
max
.~
N Mk
k=-N
kt
(1. 15)
where m(t) is now constrained to the range
if S
0 < m(t) < 1 ,
.
mmn
(1. 16)
= 0
We desire tLat the varactor be driven to both Smax and S mi
to achieve a maximum efficiency.
When the currents are sinusoidal,
time average
(1. 17)
< m(t) > = f
and
n
0
=
1/2 serves
the required d. c. bias voltage for
to specify
achieving Smax and S m.
,
if S
.
min
- 8 -
= 0 .
From Eqs.
and 1. 15,
1. 11,
1. 3,
2
q+ q
y+
m 2(t)
]
B +4
VB
for an abrupt-junction varactor,
(1. 18)
If Eq. 1. 18 is time averaged,
V 0+
where
V
0
0
max
and
1
2
is the external bias voltage and mk
thus gives the required
S
S
= 0
.
mm
(1. 19)
m 2 + 2(m 2 + M 2 +
4
VB
d. c.
(when m
= IMk I
Eq. 1. 19
bias voltage for simultaneously attaining
o
= 1/2) .
We now have a Fourier-series representation for the elastance
time variation, dependent upon the currents which are allowed to flow
Now the current-voltage relationship for the varactor
in the varactor.
can be written as
e (t)
= R i (t) + I s(t) i(t) dt
From Eqs. 1. 12 and 1. 14,
Fourier-series of the voltage becomes
- 9 -
L
(1.20)
2N
e(t) =
jo t
Ek
k
y
(1.21)
k=-2N
kf 0
N
=R
N
I leJokt
e
+
k
S
k= -N
kfO
N
x
k=-N
kf 0
Ik S I
.(w
I
j(wk + wl)t
=-N
where
N
Ek =R
I
s k
+
I
S
jCOk
k -ef f
N
S
/0
I (R + k s
jok
+
S
(1. 22)
Y4 0
I
-k
K
-K
Sk
Leenov 8 was the first to recognize that the impedance basis was
the most convenient for the series R-C diode.
- 10 -
Penfield 2 was the first
to use the impedance formulation directly from the beginning.
Fortunately,
he picked the elastance instead of the capacitance as the variable element.
This choice lead to a very much simplified analysis.
Rafuse
showed
that a complete impedance formulation is the most convenient and most
fruitful one for all devices utilizing time-varying (non-linear) semiconductor
diode capacitors.
- 11 -
L
CHAPTER II
IMPEDANCES OF THE AMPLIFIER
2. 1
The varactor conversion matrix
For three frequencies currents are allowed to flow the varactor.
os , a pump frequency
The three frequencies, a signal frequency
w ,
and an idler frequency wo , are necessary and sufficient to have ampli7
fication with the relation w
s
=w
p
- w.
showed that the
Rafuse
1
multiple-idler parametric amplifier does not give sufficient improvement over the single-idler one to warrant the extra circuit complexity
required.
The Fourier-series representations of the current and elastance
variation have the forms, as in Eqs. 1. 12 and 1. 14, with single idler,
i(t) =I s e
+ I.e
s
jo t
jW.t
t
j
1
I e
1+
p
i
+ Se
S.e
s
+ S *e
+ S.* e
p
1
-
L.
1 + Se
-jW.t
-jWo t
+ S *e
jcot
jco.t
s+
(2.1)
+ I e
p
+ I* e
jot
s(t) =S
-jo t
-jo it
-jo t
+ Ise
s
12 -
9
-jWo t
p
(2.2)
I
Fourier components of the voltage applied to the varactor are,
from E qs. 2. 1, 2. 2, and 1. 22
S I
os
jwos
E
5
I S.*
+
p
E.
1
+
S I.
SI
E
p
jw s
+
JO.1
ss
15
jop
I S*
p s
ji.
SI *
p s + R I.
1
+
(2.3)
+ R I
jwp
j p
S I.
=01
j0S
S.I
s1
+
=
S I.*
.
+ R I
s p
JO.
s 1
1
Same relations can be expressed in matrix as
V
E
-I
5
S
(] S
+
S
S
0
j s
jos
s
p
0
0
p
S
S.
E
p
cop
(R 5s
E.
1
0
S
+,.--)
jwp
p
S
-s
jio.1
- 13 -
I-
. s
0
jo P
1
1
I
S
S
(R + 0)
s jw.
1
0
0
This matrix relation is called "varactor conversion matrix" by
Penfield.
2.2
Impedance
As the varactor model is chosen with a resistance and an elastance
in series, impedances of the varactor in the three frequencies are bases
to find the characteristics of the varactor.
To have direct relation between the elastance and the current in
each frequency, from
k
k
k
q=--
k
'k Jkt
e
jk
and Eqs. 1. 2,
1. 4, and 1. 8
S
~ (
(B
max
1/2
1
k
c
R
s
1/2
-
(B+~
k
2 jwk
- 14 -
Hence,
in the three frequencies
S
M
s
-
c
2j
o
2j
1
s
co
M_1
VB + s
R
Co
M_1
p
R
Co
1
c
2jo
w
s
S
max
I
(2.5)
VB
R
ec
o.
s
B
i1
From Eqs. 2. 3 and 2. 5, the impedance of the varactor in signal
frequency is
E
I
s
SO I
S
- R
s5
= R
s
s
+
+
jo
0
S
.
jCo
jo s I
M"o
1 c
jo
+
=R s+
s
.w 1
lo
R
+
M
S
s
Co
p
M
Co
M
S
M
p
cR
jo
I
- 15 -
L,
s
S
p
jo
o R
c
s
S5
o.
1
M
Co M
s5
s
Co
p
1
jo
s5
M1M
S
S
-
+
S
s
(2. 6)
Again, in idler frequency
E.
1
I.
S
=R
s
S
jo.
1
R
+
s
S
+
p
jo..
I.
1
S
=
I
s
+
0
+
jCO.
1
1
M*M
+
.
J.)
jowl
I*
p
s
I.
1
o
s
p
e R
s
M.
jCo.
1
(2.7)
1
With the constraint in the idler frequency circuit,
E.
-R.
I.
jCO.L.
-
1
1
1
(2.8)
1
from Eq. 2. 7
M* M
s
p
M.
1
-R
s
c
R
ji
s so
-R. -jA.L. - .w
1
1
1
(2.9)
JO.
Substituting Eq. 2. 9 to Eq. 2. 6
E
s
s
S
=
R
5
+
0
jw
m
2
p
Co)
2
2
s
c
i s
R + R. - jo. L.
1 1
s
1
- 16 -
(2. 10)
jo
with m
=
p
M
being already defined.
|I
In pump frequency, the impedance is, with the same steps as in
the other frequencies,
E
S
p
R
I
s
+ -
+
Jo
R
s
+ Jo
o
jo
p
S
=
M M.
s i
M
m
+
2
o
cR
p
s
2
s
_
s
S
o.Co
P R
s
+ R. + jo.L.i +
i
i
(2.11)
0
jWi
Up to here, the impedances are expressed without special conditions
or assumptions which aim practical optimum operation.
The equivalent model of the varactor in the signal frequency
can be directly recognized from Eq. 2. 10.
t
T fo
2
Imp
2.
2
~J
KstKR
Fig. 2. 1
Signal-frequency impedance model
- 17 -
L
-LdUzLI
2. 3
Elastances and variables
The impedances obtained in the last section were functions of
the three normalized elastances.
To have further analysis of the
amplifier, it is necessary to find constraints among the variables.
Every normalized elastance is the function of the other normalized
elastances.
In this section the constraints will be derived through
the variations of the normalized elastances from the initially tuned
state when the signal is week, condition is optimum, and algebra is
simple.
Suppose that we can imagine small signal variation from null.
At the situation, tune the three frequency circuits as
S
jo
s
L
=
-
ssss
jo
S
jo
p
L
p
=
--
0
(2. 12)
j
S
jo. L.
i
1
=
and assume R.
0
-_-
-
jW.
=
0 .
To the preceding conditions, make the initial
bias and pump frequency normalized elastances
- 18 -
WMw--
-
m
-
1/2
.
o, in
(2. 13)
m
1/4
.n
p, mn
to have maximum utilization of the varactor elastance range.
stated, from Eq. 2. 11
Then, with the condition R. = 0
1
E
m
S
Ip
=R
s
+
+
0
jo
2 2
2
s
o
s c
.
S
W
1p
R
s
+
jco.L. + -~
1 1
With the pump source impedance
Z
p
=
R
s
+ jw L
p p
and the pump source voltage E 0 as in Fig. 2. 2, the pump current is
E
I
S
2R
+ jo L
s
p p
+-
0
jp
0
2 2
m 2
s
c
s
S
o. W
1'p
- 19 -
R
s
+ jo.i1iL.
+ jo
and from Eq. 2. 5
o
p
2j
R
wop
E
s
VB +
4
0
S
2R
s
0
+ jo L +
p p
j~o
+
S
o.o
R
j
R
s
+jwoL.+-0
i
1
jo.1
,L
Epo
-
Fig. 2. 2 Pump-source equivalent circuit
Since
m
. =1/4
p, n
Co
1
=
(c)
Co
p
when
m
s
= 0
from the last equation
IE 0
V B+
(2. 14)
4
- 20 -
and
(2. 15)
m
p
1/2
=
L
|2
+ jo
t
R
m
2
2
m 2o
c
s
o
o c
j p
1
L.
po.o
ip
i R
+
S
o
m
.0
jo.
C
i
Defining the variation of normalized bias elastance
6m
0
=m
0
-m
.
o, in
and
co
= 6m
a
(2. 16)
o w.i
from Eq. 2.15,
1/4
2
mp
m
[2+
2
2
(o
s
c
1a
2 m 2o
.2
2
c
s
+
o.Co
i p
i
p
a
1+2
Co
2
p
(2. 17)
- 21 -
r
From Eq. 2. 9, applying the stated initial conditions,
2
Wc
p
Ci
2
s
m.
2
1=±
2
(2. 18)
2
Now, from Eq. 1. 19
V0
m
VB
2
0
+2(m
2
2
s
i
2
(2.19)
p
Setting the bias voltage
i
V0
2
=
mo,
=
3
VB+5
in
2
+ 2m i
p, in
(2.20)
8
from Eqs. 2. 19 and 2. 20
1
8
-
6m
0
+ 6m
2
0
+ 2(m
2
s
+
m.
i
- 22 -
L
2
+
2
m )
p
(2. 21)
r
I
Hence, Eqs. 2. 21,
2. 17 and 2. 18 bring
2
8
a W
e
+
2
2
+
2m
e
s
1
m Sc
1(1+
2..
(-)
2
2
+2
1
+
I
m
2
2222
c
s
p
2+
W
i
2
m
+
CO
s c
p i
co.
a1
1
+Ua
(2.22)
This equation shows the variation of m
signal and pump frequencies.
0
Therefore,
as a function of m
S
at any
any variable can be expressed
as a function of the magnitude of signal frequency current at any frequency in operation.
2.4
Elastance equation and approximation
In the preceding section the equation between two normalized
elastances was obtained.
It is necessary to change the shape of the
equation to examine the relations among normalized elastances and
frequencies.
Define the three normalized frequencies
s
e
p
C
(2.23)
p
C
c
- 23 -
p
2
Then, the equation becomes
2 22
+ ay+ i
+ 2m
2
+
2
2 2
2 2 2.2
p i+
s
2 2 2.4
2.2 2
4
2
4
2
2
2 2 2.2
a ms - 2(1+a )a i ms+(1+a ) a i
4(1 + a ) p i +4(1+a )pim +m
1 r
+
(2. 24)
Multiplying the equation by the denominator of the last term,
0
=
22~1
2.2
4
2
22
2
m
+ (p i +p
(-1/4 + 4m )(4p i + 4pim + m
+ aI2i(4p22 + 4pim
+ a
+ m )
2
2.2
2
4
2
22
2r.2
2i (4p i + 4pim + m ) + (-1/4 + 4m )(8p i +4pim sS
S
S
Ls
2 2
2.2
+ (2p i + p ms
+ a
3r2i(8p 2 i2 + 4pims2 + m 4
L
S
-
S
- 24 -
.4
.22
2i 2M
S
+ i )
I
.22.4
2i m+
S
i4+
4
)
+ a
4r2
2.2
2i(8p i
+ 4pim
2
s
+m
4
s
2 2
4
- 2i m +i
s
2
2.2
2 2
+ (-1/4 + 4m )(4pi - 2i m
S
+
2 i( 4 p i
caL
+ a(-1/4
-
+ 4m
.4
2.27
2i m 2 + 2
s
2
s
) i4 + 2i 2 (4p 2 i 2 - 2i
2m
s
+ 2i )
2i 5
+
7
+
8[ 2i
6
j
(2. 25)
This equation is still too complicated.
An approximation to the equation is necessary which is accurate
enough in certain ranges of the normalized elastance variations and
frequencies.
Assume the possible regions,
m
1
s
1 >
> p,
< --
20
(2.26)
. 1
i
10
- 25 -
This assumption is made on the bases of practical values which are
determined from cutoff frequencies of abrupt junction diodes, optimum
or near optimum pump frequency, usual signal frequency in application
At a
with high gain, and the bias and pump currents at Eq. 2. 13.
glance, realizing that a and m
up to the terms with a
- S
+
and m
s
8
5
vary in near order, we can disregard
, then
pi + p2 ) + m4 (16pi - 1/4) + 4m 6
S s
m 2 (16p 2 i
0 =
4
2
2i(4p 2 2 + 4pim 2 + m 4)
S
2r
2.4
+ a L8pi
-i
+ a 3[2i 3 (8p
2
1 .4
S
+ m
2
(-p i + 8p i
3
2
2 .2
+ p + 32p i
+
1 .2
i
4
+ 4i)
+ 2 )]
(2.27)
To have further simplification, let us have rather rough idea about the
value of a.
Directly from Eq. 2. 24
2 2
1
8
ai + a i
O
a i+ 2m
c
2
s
1
8
+ 2m + -
2
(2.28)
s
- 26 -
-I
r
I
From this fact, the terms with a2 and a3 are negligible to the term
With this fact, recognizing
with a in Eq. 2. 27.
2
2pi >> m
The equation can be
0
2
v2 2.2
m (1 6p 1
=
- pi+p ) + m 4 (16pi - 1/4)j
s
s
L 2i(4p 2 i 2 + 4pim )
+
or
m
a=
-(
2
2.2
16p i
5
2i )2.2
- pi+p
4p i
2
+m
2
(16pi - 1/4)
5
.2s
+ 4pim
S
2
m
=(-
-pi+p 2 -1
)4
+
2i
-( )
2i
S-
1
si - -m
4pi(i
2
-s)
2
4
4 +
2
m
s7
. 2
2.2
4p i + 4pims
-si +
m
14 +
2
4
2
s
+ si+ m 2
S
.1
Si - -m
2
43
4pi
s
-
- 27 -
+
s
2
J
m
2
m
s
[ 4 _
[4
2i
16pi
2
s
.3
(2.29)
2m 2
s
or
2
m
6m
o
m
s
16pi 3
s
2
2m
2
2
s
The equation 2. 29 is good enough approximation in the range
given in Eq. 2. 26.
Eq. 2. 28 is also valuable.
Later, to analyze the
phase shift of the amplifier Eq. 2. 29 will be applied.
The author actually computed a,
through 0 < m
S
at i = 1/4 and s < 10
-3
,
< 1/16 , using Eq. 2. 27, and found that Eq. 2. 28 is
good approximation in the case.
When s >from Eq. 2. 24.
1.
10
i,
a can be obtained also as a function of m
s
But, as it will be recognized at Eq. 3. 8, at this case,
the amplifier becomes unstable for a large gain.
- 28 -
CHAPTER III
THE DYNAMIC RANGE
3. 1
Gain
We will assume that the amplifier is imbedded around signal
loop in a lossless circulator, as shown in Fig. 3. 1.
LY
(-
elgr
4rcdct
LrL
out
Fig. 3. 1
Circulator imbedment of a parametric amplifier
The exchangable gain is
R
G
0
= |L R0O
+R
-R
R
- 29 -
2
(3.1)
The resistance around signal loop is, from Eqs. 2. 10 and 2. 17,
1 Rs
4 si
R +R
0
s
L2
-
2
m
pi
1
+ 2
+
+2]
1
m
-s
(3. 2)
2
a
2
p
1+ a2
Considering the ranges of the frequencies and parameters at
Eqs. 2. 26 and 2. 29, the resistance has the following approximations
R
1 R
R
0
+R
s
4
-
pi
.
2
4
4m 2
(4+
2
(1 +
si
2
mIn
+±
2 (8+
2 .2
p 1
4m 2
.2
s
1- +
2
pi
p
2m
2
p
R
4 si(1+a
R+R
2
-
4(1+-
4
.
pi
4
+
ms
2.2
p 1
- 30 -
2
+a(8
.2
+ ---
2
p
4m 2
2m 2
.
+
pi
2
p
R
R + R
o
s
3
R s F1 - m2 1- + M4
16si L
s pi
s 422
2
-a
.2
(2+
2
- a2
1)
4p 2
2
s
(
2
2
pi
6
1
s
3 .3
2
1
2
2p
(3.3)
From Eqs. 3. 1 and 3. 3, with an approximation again,
R
R
-
L1m
1
1+
1m
16sii
S
(3.4)
o
1
2
s pi
16s i
Rs
1
Put
R
-R
= - 1 +
S
. + r
16si
(3.5)
then the equation is
r-2
G
+
2
.-
16 si
m
2 1
1
s pi
16 si
16r
- 31 -
- 32si+ 2 - m 2
16 sir
1
s p1
(3. 6)
16sir+ m 2 1_
s pi
r is defined at Eq. 3. 5.
To have large gain Eq. 3. 4 is good approxima-
tion, and this reason can be another criterion for the approximation.
Even for small maximum gain (when m
10 log 1 0 G
--
2
s
0)
to
>6db
(3.7)
2 - 32si
16sir + m
s pi
(3. 8)
2
16 si r + m
3
To check whether Eq. 2. 8 is good enough approximation, take
a case i = 1/4 and s < 10
-3
shown in the following chapter.
G
=
which is a significant one as will be
Then
1
2sr + 8m
s
-32 -
Tockegck whether Eq. 3. 8 is good enough app
a case 1 = -
1
3
and s
which is a significant o
in the following chapter.
as will be shown
Then
2sr + 8m 2
s
This is good enough approximation to almost accurate one, which is
obtained from Eqs. 3. 1 and 3. 3:
1 - 8m
2sr + 8m
2
s
2
S
-56m
4
s
Again, to check Eq. 3. 8 at ms = 0 with Eq. 3. 6 about frequencies,
take a case i
max
And,
=
1
4
Then Eq. 3. 6 becomes when m
1
2sr - 4s +
2
s
= 0
(3. 9)
2sr
from Eq. 3. 8
max
1
2sr
Figure 3. 2 is about Eq. 3. 9.
(3. 10)
The figure confirms that Eq. 3. 10 is good
approximation to Eq. 3. 9.
- 33 -
0.1X_1
-Z -01xz
--
-7(
- X
___
-v~Q1
oz
Sc
To analyze dynamic range,
it is necessary to have a simple
equation of exchangeable gain like Eq. 3. 8.
And, the equation is
derived as a reliable one and checked at an important case.
From Eqs. 3. 5 and 3. 8, smaller s brings larger gain.
This
meets a fundamental concept of the amplifier.
3. 2
Noise Figure
When there is no signal to analyze noise power, it is sufficie nt
to have same procedure as in small signal case worked by Rafuse 7
Directly following his results, assuming R. = 0 and impedances ar e
initially tuned as in Eq. 2. 12, the signal and idler impedances are
S
Zj4
=R +
s
s
--
iw
S
S
Zo
R
i
s
+
j.
o
1
2 .2
m o
p c R
W.
S51
2 2
mwo
p c
Wo.
S51
s
2
s
+ R
(3. 11)
R
R
S
0
and the squared-magnitude reverse current gain is
2
2
s
2
s
p c
Si
(R
S
+R
(3. 12)
0
)
We shall define noise sources in series with R
e
0
= 4kT R tf
00
- 35 -
0
and R
S
as
I
er
= 4kTdR s f
T
and T
(3. 13)
where
0
d
are source and diode temperatures i0
K
k is Boltzmann's constant,
1. 38 X 10-23 joules/ 0K
A f is an incremental positive frequency bandwidth in cps.
The portion of signal'tank noise current due to idler noise is
I
2
si
=
2
d s
i
I..
11
2
T R
I
'i
R
.4kAf(
W.
P)
i
2
(3. 14)
5
(R
s
+R
0
)2
The other internal noise current flowing in the signal frequency is due to
R
s
at w.
It is,
s
2 =
dds. s
-
ss
(R
0
+ R
o
4,
4kA
f
(3. 15)
)2
S
And, the source noise current flowing in the signal circuit is
I 2
so
_
0 o - . 4kAf
(R
+ R
0
(3. 16)
)2
o
S
Now, the exchangeable excess noise figure is
I
-
36 -
9
2 +I
(Fe
(3. 17)
2
so
Inserting the results from Eqs. 3. 14,
T
1)
(Fe
R
1 + (0
3. 3
3. 16, and 3. 11,
2
m W
Td
3. 15,
p c)
(3. 18)
0.
Dynamic Range
An estimate of the minimum detectable signal can be made by
considering, T
P
being antenna temperature,
. = (F - 1) . kT Af + kT Af
es, min
e
o
a
(3.19)
This power brings same strength of signal power as noise (themal)
power at the output.
The term "Dynamic range" is used here as the
input power range from the minimum input power to the power which
drops the gain of the amplifier by 3 db from the maximum gain.
This
upper bound is defined as an estimate of the maximum possible signal
power, Pes max.
From Eq. 3. 18,
when m
=
p
(F
e
-1)
1
4
T R
)
= d s (1 +
.2
T R
161
o o
- 37 -
And, with Eq. 3. 19
R
P
es,min
= kt f LT
12
+
R1
0
T
(3. 20)
16i 2
From Eq. 2. 5, the square magnitude of the signal frequency
normalized elastance is
2
s
1
4
R
c 2
w
s
2
s
vB +4
5
2
(3. 21)
Defining
(VB +
P
R
norm
4) 2
(3. 22)
5
with the output power
P
out
2
212 R
0
s
=
equation 3. 21 becomes
m
5
2
1
2
8
o
s
out
P
norm
R
R
s
0
- 38 -
(3.23)
When the gain is
2
s
3 db lower than the maximum,
P
-G
es,max
2 max
P
norm
1
max
8s
22
On the other hand, this
1
2i
m
2
s
R
s
(3.24)
0
can be obtained from Eq. 3. 8 as
max
2
.3
16si r + m2
max
d2
m
R
s
max
But
max
(3.25)
8sir
from the same equation.
1
VG
2i
max
2i
2
Hence, from the two preceding equations
2
1
+
, Gm
ax
m
s
max
and
- 39 -
oi'4v-
m
2
= 2i
2
1
G
2-
1
max
or
m
.2
2
s, max
2
,
0. 828
(3. 26)
-7--
G
max
Now, from Eq. 3. 24 and 3. 26
G
1
16s 2
max
2P
P
es, max
R
s
R.8
- 0. 828
G
o
Pnorm
.2
or
2 2
i sP
P
=
norm
13. 248
es, max
G3/2
max
R
R
o
(3.27)
s
Define,
T
n
=
2900 K
then
T kAf
n
(3. 28)
= 4 x 10-21 /f
- 40 -
L
r
From Eqs. 3. 20, 3. 2',
and 3. 28, the dynamic range is
P
DR
=
P
es, max
es, min
.2 2
13.2481 5 P
4x 10
-21
norm
T R2
3/2E d
s
L fG
m ax L
2
T R
n
R T
1
s a
+
,2
R T .
161
o n
0
3. 312 X 1021 P
R2
T
6 fG 3/2 F
max L
s
d
T
2 2
R
n
o
1+
2
16.
1
)
norm
1
R
S
2 2
i 2
R
0
Ta
7
Tn
-
(3.29)
From Eq. 3. 5 and 3. 25
R
0
1
Rs
+
169i
8,Si
G
max
or
R
s
R-
1
1
0
max
(3. 30a)
To here, the approximations were accurate enough.
R2
s
R
2
o
22
.2
64S i
-- +
4
1
;r
(3. 30b)
max
- 41 -
L
Let us examine
K
When 10 log 1 0 G
> 10, this equation is very good.
10 max
To 10 log 1 G
=6
10 max
this is still acceptable one considering the participation in the Eq. 3. 29.
R
Furthermore, in the Eq. 3. 29, usually Ta, which has coefficient
in first order,
almost determines DR comparing with T
0
partici-
d
s
pation in the other term of the denominator.
Substituting Eq. 3. 30 to Eq. 3. 29
3. 312 X 1021 P
DR =
64
A fG3/2
max
4
1
G
1+
T
1
1
max
T
,2
norm
d
n
T
8
+
a
T
1
1
2
3i(-)
n
max
0. 207 X 10 21P
norm
1
.
3fG3/2 1 6 +
Td
max L
1
d
1 +
+
T
1
Si (1+
n
T
2
)
-J
n
(3. 31)
max
max
a
Following usual expression, taking common logarithm of Eq. 3. 31
10 log 1 0 DR = 10 log
1 0 P norm -
16 +
-10 log
10
L
15 log 1 0 Gmax - 10 log 1 0 Af
1
T
T
.2
1+
4
G
max
1
+2
-.
T n+
n
T
2
+
G
)
a
n
I
max
(3.32)
+ 203
- 42 -
when
1
r
->
-i
P or iL
1.
10
'<--i
10 log
G
10i max
>6
where
P
( V B+#) 2
= VB 4
norm
R
S
Gmax = maximum gain
(when signal is negligibly small)
A f = incremental positive frequency bandwidth in cps
W.
WA
S., i
T , T
da
T
n
5
1
c
c
= diode, antenna temperature in
2900K
When Ta is comparable to Td'
at 9<<i.
K
d can be neglected in Eq. 3. 32
Following the equation, large Si brings large dynamic range for
the same maximum gains.
-
43 -
3. 4
Examples on Dynamic Range
To examine the dynamic range,
examples will follow.
Example 3. 1: When
i
T
d
=
T =2900K
a
norm
=
40W, A f=10
6
10 log 1 0 DR=153 - 15 log Gmax
10
-10
log 1 0
8
+1
4
max
1
2
max
This example is one of usual case.
Figure 3. 3 is on this example.
As a comparison, we note that very good
vacuum-tube amplifiers have dynamic ranges in the order of 80 db.
Example 3. 2:
When
.1
1 = -4
Td = 290 0 K, Ta = 30K
- 44 -
P
=40W, & f = 106
norm
10 log
10 log1 DR = 153 - 15 log 1 0 G max
3
290
8
1
0+
fr-
S(
max
2
/Q
)
max
On this example antenna temperature affects the dynamic range noticeably.
Figure 3. 4 is on this one.
Example 3. 3:
When a certain dynamic range is desired,
gain will be a function
of signal frequency at other constants determined.
I
When
10 log
10
0
=
43
-
a
15 log 1 0 Gma
-10
log 1 0
8
L
1+
I
f = 106
= 290 0 K
= T
d
40W,
=
Pnorm
T
DR = 110
1
4
+
max
- 45
+
1 +
,
2
max
I
For large gain,
0 - 43 - 15 log 1 0 Gmax + 10 log 1 0 s
- 46 -
.D
11'ix1-k-
ca
S3 o
I2-o k
I toc)K
0LEx-
[ o
3
K
701-
t/>
/0
to0
2-0
130
d:T a4AL / )VLW,7C
CctL)
\560
i40
D
130
2
3
12-0
too,
Ae
PUc-
70
2
3o40
CHAPTER IV
PHASE SHIFT OF THE AMPLIFIER
4. 1
Signal frequency reactance
From Eqs. 2. 10 and 2. 17,
reactance around signal loop is
1
4i
jX
5
= jR
- i+
ss
1
1+
2
2
2
2
1
.s
(2±
pl
+(
1
_j
pi1+
a
_
2
i 2
p
(4. 1)
The last term inside the bracket is, with a step by step algebra and
approximation,
(1 + a 2)p
2) i2 + 2ms2]2
F(1 + a 2)p
Lr-c
2 a - (1+U2 2 )U 2i 2
+ Lms2
L
1+
i
2
~2
2
1i - m1
16i s pi
+
m4
2
-
s4p2i 2s2
m 6
1
4 p2
- a 2m 2 2
s pi
- 49 -
.2
2 (2 +
)
1 )
2p2
(4. 2)
I
The criterion of approximation is the same discussed at sections 2. 4
and 3. 1.
4. 2
Phase shift of the amplifier
Through this work, impedance in three frequencies are assumed
to be tuned when the signal does not flow the varactor.
As a consequence
of this, obviously phase shift is none when the signal is small enough.
From Eqs. 3. 3 and and 4. 2, the phase shift between input aid output
signal is, with the same kind of approximation as the one discussed
at section 3. 1,
I
r
a
16si L'
ai
tan
-1
s
R
90
R
0
2
1sLI
16si Ls
+ 1
s
1
pi
(4.3)
2 1-
pi
This is practically accurate enough for 10 log 1 0 G = 6 .
From Eqs. 3. 5 and 4. 3
2
tan
-1
-m
+
90
6i L
r +
pi
2
m
s
s
pi
1
I
16 s
- 50 -
and from Eq. 2. 29
2
2
4 -
m s
tan
9
-
2
s
2s
2
s
)M
(1
16i2
pi
2
ms
pi
r +
m
1
-1+
1 6 pi 3
12
-4)
(..
4i
1
16si
1
- (3+
-)
16i
2
ms
pi
2s
2
2
ms
+
.3
16pi
4
ms
S
2 2.6
16 p i
(4.4)
1
l6si
From Eqs. 3. 23 and 3. 30a
m
P
2
i
S
s P
out
norm
11
+
2
-
(4.5)
YG
max
Defining
S=
mn
2
P
s =1
out
norm
1
1+
(4.6)
2
VG
max
I
- 51 -
r
with Eq. 3. 25, Eq. 4. 4 becomes
(1 - 16i 2) - (3 +
tan 1 0 = -2/3
1
G
+
12
16i
1 g +
2p
1
92
S2 2
16p i1
(4.7)
1
p
max
The equation 4. 7 is valid when
1/4 > i > 1/12
S <
1
i1
>6
10 log 1 0 G
where
(VB + 4)2
P
norm
G
max
R
s
= maximum gain (when the signal is small)
When i = 1/4 ,
keeping s and Gmax as constants, the phase
shift becomes minimum.
Furthermore, at the situation, the phase shift
- 52 -
L
increases as the signal increases having zero slope at zero signal.
This fact is most interesting one.
When i is larger than 1/4 the phase shift becomes positive.
The variable
s
.
#
is proportional to Pont and inverse proportional to
According to Eqs. 4. 6 and 4. 7, as G max increases the phase shift
increases in magnitude.
4. 3
Maximum phase shift
While idler frequency determines the slope of increasing phase
shift versus increasing signal, the maximum phase shift is defined as
a phase shift when the gain drops 3db from the maximum gain.
this value gives the order of the phase shift.
So,
From Eqs. 4. 6 and 3. 26
2
max
2i
max
(4. 8)
= 0. 414
G
max
and
tan
9m
= tan 1 9
|at
Pomax
with Eq. 4. 7.
- 53 -
(4.9)
4.4
Examples on phase shift
Example 1
When i=114 and
tan
s <
1
10
i ,
phase shift 9 is
16jS
-1
0
=4
max
with
1
P out /P
3s
1+
norm
2
aG
Figures 4. la, b, c, d are on phase shift at maximum gains 40, 30, 20, 10
They are brought together for comparison at 4. 1d.
decibels each.
Example 2
1
When i = 1/ 6 and
tan
I
-1
9
-
-
-41~
s<15 i1,
5
63
9
4
81
±
I
the phase shift 9 is
61A
max
- 54 -
2
Figures 4. 2 a, b, c are on phase shift at maximum gains 30, 20, 10
decibels each.
And they are drawn together at 4. 2 c.
Example 3
When i= 1/10
s<
and
1
i,
the phase shift 9
is
20
tan
-1
9 = -2g
0. 84
1
/G
46.2# + 624#
max
Figures 4. 3 a, b, c are on the phase shifts at maximum gains 30, 20, 10
decibels each.
And they are together at 4. 3 c.
The differences of slopes, convexities and magnitudes among all
the figures are interesting.
Figures 4.4, 4. 5 and 4. 6 are groupings of preceding figures in
same maximum gains and different idler frequencies.
The restrictions
on signal frequencies at Examples 2 and 3 came from algebraic con-
veniences.
And,
Eq. 4. 7 satisfies to s =i.
-
55 -
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CHAPTER V
CONCLUSIONS
5. 1
Conclusions
As a varactor to form a parametric amplifier, abrupt junction
diodes are selected.
According to the physical structure of the diodes,
a series model of the varactor with a constant resistance and a voltage
controlled capacitance is determined.
As a consequence of the series model of the varactor, a choice is
made to flow the current in specially selected frequencies only.
A
current controlled elastance has replaced the capacitance in the series
varactor model.
Fourier series analysis is used in signal, pump, and
idler frequency circuits.
Normalized Fourier components of elastance variations were
mathematical media to derive the dynamic range and phase shift from
the impedances of three frequencies.
Any one of four normalized
elastance variations can determine the other three variations.
But the
accurate relations among them are too complicated and good approximations of them are valuable.
An approximation of direct frequency component of normalized
elastance variation from the value when signal is weak,
- 69 -
2
m
6m
=
o
m
2
s
.3 )
16pi
s
2
(2. 29)
is selected to calculate the phase shift of the amplifier and used
throughout to approximate other equations.
The approximation is
accurate enough in the regions of parameters and frequencies,
m
1
<-
s
20
(2.26)
1 >1
4> i,'' p > 12
10
and this regions are wide enough for the amplifier in practical application.
The gain of the amplifier is obtained as
1
2
Gain
=
3 2i
(3. 8)
2
16si r + m
S
where
r
RR0
+
R
s
-
1
(3. 5)
16si
- 70 -
This is a good approximation, when maximum gain is larger than 6 db.
The dynamic range was given in the form, in db,
10 log 1 0 DR = 10 log 1 0 Pnorm - 15 log 1 0 G max-
16 + -- 10 log
-G
.2
TTd
dTa +
i
T
max
n
+
10 log10
T
1
si(1 +
2
)
2
n
Ga
max
+ 203
(3. 32)
For given P
norm
proportional to G
, A f, i, s, T
max
d
and T
a
, the dynamic range is inverse
(maximum gain when the signal is weak) when
maximum gain is larger than around 20 db, and extremely increases
for smaller gain than 15 db.
The range is the input power range from
the same power as noise power to the power which brings -3 db gain
from the maximum gain of the amplifier.
The phase shift of the amplifier is obtained as, 9 being the
phase shift,
- 71 -
I
2
(1 - 16i )
tan
-1
90 = -2g3
-
1
(3 +
1
16 .2
1
+
1 i +
2 p'
1
2 .2
16p
2
i
p
G
max
(4.7)
with
1
P
P
out
norm
1
1+
(4. 6)
2
Gmax
The idler frequency
shift.
is important factor to determine the phase
c
And we should not forget that
by s for given i and G
.
max
#
is proportional to Pout divided
Generally larger gain makes larger
phase shift.
The dynamic range and phase shift are good approximation when
the maximum gain is larger than
at the
6 db , and they give considerable error
6 db, but they bring valuable results even at the
6 db maximum
gain.
5. 2
Future work
The following are among some of the problems which need
future work.
- 72 -
The experimental study is needed to compare with the results
obtained here.
If the two kinds of results do not meet in some cases
or regions of frequencies or powers it will be possible to find the
factor and add them to the theory obtained here.
As a whole problem of the parametric amplifier, an analysis
of noise, when large signal is present, will be interesting.
It is necessary to study the parametric amplifier in not sinusoidal
signals.
- 73 -
BIBLIOGRAPHY
1.
A. Uhlir, Jr.,
"Possible uses of nonlinear-capacitor diodes,"
8th Interim Rept. on Task 8, Crystal Rectifiers, BTL-Sig.
Corps. Contr. No. DA 36-039 sc-5589, pp. 4-19; July 15, 1956.
2.
P. Penfield, Jr.,
"The high frequency limit of varactor
amplifiers, " Microwave Associates Internal Memo, Burlington,
Mass., August 27, 1959.
3.
P. Penfield, Jr.,
"Interpretations of some varactor amplifier
noise formulas, " Internal Memorandum, Microwave Associates,
Inc.,
4.
Burlington, Mass.,
September 1, 1959.
M. Uenohara and A. E. Bakanowski, "Low noise varactor
amplifier using germanium p-n junction diode at 6 kmc,"
11
th
Interim Report, Microwave Solid-State Devices, BTL-
USASRDL contract DA 36-039 sc-73224, pp. 18-22; November
15, 1959.
5.
R. V. L. Hartley, "Oscillations in systems with non-linear
reactance, " Bell Syst. Tech. J., vol. 15, pp. 424-440;
July 1936.
6.
H. Q. North, "Properties of welded contact germanium rectifiers, " J. Appl. Phys.,
7.
vol. 19, pp. 912-923; November 1946.
R. P. Rafuse, "Parametric Applications of Semiconductor
Capacitor Diodes, " Sc. D. Thesis, M. I. T.,
September 1960.
- 74 -
Cambridge, Mass.,
8.
D. Leenov, "Noise figure of a nonlinear capacitor up-converter,
Second Interim Report on Improved Crystal Rectifiers, BTLSCEL Contract DA 36-039 sc-73224, pp. 4-12; July 15, 1957.
9.
E. J. Baghdady, "Lectures on communication system theory, "
McGraw-Hill; 1961.
- 75 -