Document text
Intermodulation Noise in FM Systems
Due to Transmission Deviations and
AM/PM Conversion*
By T. G. CROSS
(Manuscript received August 2, 1966)
Two noise contributors in FM systems are: (i) intermodulation noise
due to transmission deviations; and (u) intermodulation noise due to trans-
mission deviations and AM/PM conversion, designated AM/PM inter-
modulation noise. Expressions for the second- and third-order AM/PM
intermodulation noise are derived in terms of transmission medium coeffi-
cients and a continuous pre-emphasis characteristic, with the unpre-em-
phasized baseband signal being simidated by white Gaussian noise. These
expressions have been programmed on a digital computer and representative
noise responses and properties of AM/PM intermodulation noise were
obtained. General ■properties and characteristics for the two noise contribu-
tors are documented in parallel for comparative purposes. It was found that
AM/PM intermodulation noise can be a significant noise contributor in
FM systems.
I. INTRODUCTION
Intermodulation noise is produced whenever a phase modulated signal
is passed through a linear transmission medium whose amplitude and
phase characteristics are nonlinear functions of frequency. The output
signal from this medium is both envelope and phase modulated, with the
phase modulation being a distorted replica of the input phase function.
The envelope modulation and phase modulation functions are similar in
that both consist of first (linear), second-, third-, and higher-order func-
tions of the input phase function. They differ in that the coefficients of
the terms making up the two modulating functions are related in dif-
ferent ways to the transmission medium characteristic.
The distortion terms higher than first order, in the output phase
* Portions of this paper were presented at the 1966 IEEE International Com-
munications Conference in Philadelphia, Pa., June 16, 1966.
1749
1750 THE BELL SYSTEM TECHNICAL JOURNAL, DECEMBER 1966
modulating function, produce intermodulation noise. This source of
noise has been the subject of much work over the past ten to twenty
years. The envelope distortion terms directly produce no degrading
effects in linear systems. However, when the linear transmission medium
is followed by a device that converts envelope variations at its input to
phase variations at its output then a different noise-generating mech-
anism exists. This latter source of noise will be designated as "AM/PM
intermodulation noise" to distinguish it from the intermodulation noise
produced directly by transmission deviations.* The two phenomena are
illustrated in Fig. 1 which depicts the two-step process involved in the
v(t)
Y(o»)
v,(t)
INTERMODULATION
NOISE DUE TO
TRANSMISSION
DEVIATIONS
dB
v„(t)
INTERMODULATION NOISE
DUE TO TRANSMISSION
DEVIATIONS AND AM/PM
INTERMODULATION NOISE
V(tJ = EXP L C
V 1 (t)=EXpa(t) EXP L [ tu ct+9'o( t )J
V n (t)=EXP a «W EXP L K:t-+9>o(t)+Ka(t)]
WHERE
<p{t)= PHASE MODULATING FUNCTION DUE TO MULTICHANNEL SIGNAL
<P (t) = <p(t) + PHASE DISTORTION TERMS
k =0.1516 K = PHASE MODULATION INDEX IN RADIANS DIVIDED BY
THE AMPLITUDE MODULATION INDEX
K = AM/PM CONVERSION CONSTANT MEASURED IN DE ^p EES
ASSUMING a(t)«l d °
a,(t)^ a(t) IN GENERAL
Y(ti>)= TRANSMISSION MEDIUM WITH TRANSMISSION DEVIATIONS
Fig. 1 — Model illustrating sources of intermodulation noise due to transmission
deviations and AM/PM conversion.
AM/PM intermodulation noise generation. The AM/PM converter
will be characterized by the constant K which has the dimension of
degrees/dB and can be interpreted as the peak phase change at the out-
put for a 1-dB change in envelope at the input. In reality, this K maybe
a function of a number of quantities, e.g., carrier drive power, frequency
(carrier and/or baseband), bias levels, or may even be complex. However,
many presently developed broadband radio systems use TWT amplifiers
as power output tubes which are often the major source of AM/PM
conversion within a radio repeater. These tubes, when driven at moder-
* Transmission deviations are defined as any deviation in the gain and phase
characteristics from the ideal characteristics of constant gain and linear phase for
all frequency components of the FM wave.
NOISE IN FM SYSTEMS 1751
ate, essentially constant input power level and biased from well controlled
sources, are adequately characterized for small envelope fluctuations by
a constant K degrees/dB. 1
Both noise phenomena are of prime interest in frequency modulated
systems. Intermodulation noise due to transmission deviations is of
interest because it is a recognized significant noise source. AM/PM
intermodulation noise is of interest because of the basic lack of knowledge
which has existed on this subject. Due to this deficiency, the AM/PM
phenomenon has become the underlying scapegoat for many system
problems that appear to be unexplainable using existing system knowl-
edge.
The purpose of this paper is two-fold : (i) to present the mathematical
development and ensuing solution for the problem of AM/PM inter-
modulation noise in FM systems; and (it) to provide enough general
information about the two noise contributors considered in this paper
such that one can analyze a system's performance and/or set system
requirements with some degree of confidence without having to neces-
sarily utilize the associated digital computer programs.
The analysis to follow considers a linear transmission medium, with
generalized transmission deviations, followed by an AM/PM converting
device. The baseband signal is simulated by a Gaussian distributed band
of noise with flat power density spectrum which is pre-emphasized by a
continuous pre-emphasis function before the FM process. The end result
of the treatment is the signal-to-noise ratio for second- and third-order
AM/PM intermodulation noise. The mathematical framework for this
paper is derived from a recent paper which treated the subject of inter-
modulation noise due to an imperfect transmission medium. 2 Certain
facets of that work will be included here for the sake of continuity.
II. THEORY FOR AM/PM INTERMODULATION NOISE
2.1 General Development
Consider the system model shown in Fig. 1 where an FM signal is put
into a linear transmission medium followed by an AM/PM converting
device. The transfer function of the transmission medium is
F(«) = exp [-a(w) - 0(a)] (1)
and the impulse response is
g ( x ) = _L / F(co) exp (iaz)da. (2)
SIT •'—oo
1752 THE BELL SYSTEM TECHNICAL JOURNAL, DECEMBER 1966
The FM signal input to the transmission medium is
v(t) = exp [i[ Ue t + <p(t)]} (3)
and the output signal is
«i(0 = exp [a(t)\ exp {i[u e t + <po(f)]}, (4)
where co c is the carrier frequency and <p(t) is the phase modulating signal.
Since F(a>) is a linear system, the input and output can be related by
/oo
v(t — x)g{x)dx. (5)
Substituting (3) and (4) in (5) gives
exp [a(t)] exp [i<p {t)\ = I exp [i<p(t — x) — iu c x]g(x) dx. (6)
The output function, <p„(t), was the subject of a previous paper 2 and will
not be considered further here. Our prime objective is to determine the
envelope variation in terms of its functional relationship to the phase
modulating signal, <p(t). It follows from (6) that
a(t) = Re In / exp [up(t — x) — iu c x]g{x) dx. (7)
J— 00
It can be shown that 2
a(t) = -a{f c ) + m u <p — -kt<p" + -^<p - -jr v + • • ■
■ hr I „ hr I I" X2r >2 hr „2 _i_ / \
-r-^<P<p — -q<p>p — ~2<P — -g<P -r ••• ko)
■ hi 12 „ \%i 1% _i_ _i_ ^4r H
+ 4* * --g"t» + ••• +24* "" '
where the subscripts r and i denote the real and imaginary parts of the
corresponding coefficients, and the prime notation indicates the deriva-
tive with respect to time. The argument of the phase functions in (8)
is t — td where U is an arbitrary delay. 2 The moments, m„ , in (8) are
related to the transmission medium by
ra„ =
% T^Vn Y ^° + «) exp (**,)] (9)
Me) \_d(tU) n Jo>=0
Y{o> E
and the I and X coefficients are defined as follows :
NOISE IN FM SYSTEMS 1753
It = mA — 2m 1 m 3 — mi + 2wi 2 m 2
h = m 3 — fflifflj
Z 3 = 7n 4 — miW 3
/ 5 = Mi — Wo 2
X 2 = m* — mi 2
A 3 = m 3 — 3mim 2 + 2m x 3
X 4 = m 4 — 4miWa — 3w 2 2 + 12 rai 2 m 2 — 6rai 4 .
As an example, we have
Zi» = m« — 2m 2r m 2l - — 2mum 3r — 2mi; 2 m 2l -
since m ir = (Appendix I of Ref . 2) .
For the following transmission medium
F(a) + W a ) = [1 + g\u + 02W + 03CO + 04^
-\-^Uj cos (Pj« +5/)] exp li[b 2 u + 6 3 w 3 + & 4 w 4 , . Q x
+ T) i»j sin (qju + «rj)
the moments m n given by (9) have been evaluated and expressed in
terms of the transmission deviations in Appendix I of Ref. 2.
For the analysis to follow, the transmission deviations in (10) are
limited to values typically encountered in broadband radio relay systems.
However, the ripple type transmission deviations must have ripple
periods greater than approximately twice the top baseband frequency.
These restrictions are dictated by the limited number of terms of a(t)
which are to be considered.
Referring once again to Fig. 1 , we see that when the output signal from
F(«) passes through the AM/PM converter the envelope perturbation
given by a(t) is converted into a phase perturbation, given by k a(t).
The k coefficient is related to K (degrees/dB) as follows: the envelope
distortion term expressed in dB is
20 log exp k (<)] dB = 8.686 a(t) dB
so the phase distortion, due to envelope perturbations, after AM/PM
conversion is
1754 THE BELL SYSTEM TECHNICAL JOURNAL, DECEMBER 1966
8.686X" a(t) degrees = 0.1516 K a(t) radians.
We will let
k = 0.1516 K radians.
Hence, the phase distortion function, due to envelope variations, after
the AM/PM converting device is
k a{t) radians, (11)
where a(t) is given by (8).
The analysis up to here has been perfectly general (except for the
assumption that the AM/PM process may be represented by a constant
factor). The terms in (8) consist of first- (linear), second-, third- and
higher-order functions of the input phase modulating signal, <p(t). The
linear terms produce baseband amplitude distortion which we shall not
concern ourselves with in this paper. Also, terms higher than third
order will not be considered. This is not an undue restriction because
the prime contributors of intermodulation type noise in broadband
systems are second- and third-order phase distortion terms. Therefore,
neglecting linear, fourth-, and higher-order terms in (8) gives*
1 / x\ I I *8r / * ^3r / /»/ A2r '2 far »2
k a(t) = k — <p'<p" - — <pY - -j <p - — <p" \
+ k — - if> V" jr <P radians.
Using the relationships
J^-w^ + v
(12)
d 13 /2 „
in (12) gives
k a(t) = Oi(t) + d s (t) = 6 T {t) radians, (13)
where
* It should be noted that additional second- and third-order terms exist which
are not shown in (8) nor included in (12). These additional terms are considered
to be negligible for the transmission deviation constraints previously mentioned.
NOISE IN FM SYSTEMS
1755
^4r 1 „2
with
and
^4 r = 4?3r — 3t5r
/,v , f X3i - hi d \ 13
' [^}
(14)
(15)
(16)
In the Appendix it is shown that the second-order distortion, 6 2 (t),
and the third-order distortion, 6 3 (t), are uncorrected. Hence, the total
AM/PM intermodulation noise power density spectrum, considering
only second- and third-order distortions, is the sum of the two individual
noise power density spectra.
2.2 Second-Order Noise Power Density Spectrum
In this section we will derive the equation for the second-order
AM/PM intermodulation noise power density spectrum. The time repre-
sentation for the second-order phase distortion due to AM/PM conver-
sion was derived in the previous section and is
™->[-T + ™-V9Y + h \$f- (17)
The terms in brackets are operators on their respective functions, so (17)
can be represented by the block diagram shown in Fig. 2,
where
Jaw -ft J
and
<p' 2 (t)
G.M
\2r
T
X(t)
+ i
(18)
,(t)
<r 2 (t)
G,(w)
yet)
Fig. 2 — Second-order noise block diagram.
1756 THE BELL SYSTEM TECHNICAL JOURNAL, DECEMBER 1966
!(*(«) = ±hr. (19)
It can easily be shown, using the relationship for the cross-correlation of
linearly transformed random functions, 3 that the power density spectrum
of 6 2 (t) is
<Se 2 (w) = G\( — u>) G\(u>) S^*{u>) -f- G\{ — <x>) Gi{w) <S v 'i v "j(co)
+ ft(-») &(«) S„»V»(*0 + Gi(-<*) &(«) S,"i(ft>)
where, for example, aS V 'V 2 (&0 is the cross-power density spectrum of
vs 2 (0 and p 2 (i). As in Ref. 2, (20) can be expressed as
&,(«) = 2 | Gk(«) | 2 ff[Vto] + 2 | G 2 (o>) | 2 JF[/2,»'(t)]
(21)
+ 2[ft(-«)ft(«) + <?,(-«)(&(«)] ff[^»*(r)],
where (ri(a>) and Qa(a) are given by (18) and (19), respectively, and 5
stands for the Fourier transform.
Now, redefining the transfer functions given in (18) and (19) we can
write
p&,(«) = 2\G i (u>)\ 2 Z[RAT)]+2\G,(u,)\ 2 5[R,AT)} m)
+ 2[GM-»)<W«) + GW-«)<Ww)l ffUW'V)]
where now
^)-fe« -^]+*[t w ] (23)
G*(«)=^k. (24)
Equation (22) is the second-order AM/PM intermodulation noise power
density spectrum weighted by the AM/PM conversion parameter. The
ability to pull the k out of the calculation provides great flexibility.
2.3 Third-Order Noise Power Density Spectrum
The time representation for the third-order phase distortion due to
AM/PM conversion is, from (16),
*»-*[- fc+ a s>" (25)
which can be represented by the block diagram in Fig. 3 where
NOISK IX FM .SYSTEMS
1757
<p' 3 (t)
G,M
BAt)
Fig. 3 — Third-order noise block diagram.
I
I:
fi ^-[-¥i +i \% u \
(26)
It follows that the third-order AM/PM intermodulation noise power
density spectrum is
SeM = | G 3 (a>) | 2 S^i(«).
It can he shown that 6
which can be written
SX*>) = 65F[#/(t)]
(27)
(28)
since 9 i?/(0) £,'(«) is a scaled power density spectrum of the input FM
signal and hence can be neglected since it does not contribute to the
distortion.* Therefore,
S,M = 6 I GM | 2 IFlV(f)] (29)
where G a (u>) is given by (26). Redefining the transfer function we have
i$,(«) = 6|Gi(«) | 2 5F[/e/(r)]
where now
</.(«) =
+ i
hi
12
(30)
(31)
Hence, (30) gives the third-order AM/PM intermodulation noise power
density spectrum weighted by the AM/PM conversion parameter.
A quantity of interest in engineering problems is the signal-to-noise
ratio. Thus, we now characterize the simulated multichannel baseband
signal.
2.4 Pre- Emphasized Signal Power Density Spectrum
The basic block diagram arrangement for a typical signal transmission
path is shown in Fig. 4. The unpre-emphasized baseband signal is ob-
* This term causes baseband amplitude distortion instead of intermodulation
noise.
1758 THE BELL SYSTEM TECHNICAL JOURNAL, DECEMBER 1966
BASEBAND
SIGNAL
f[W
PRE-EMPHASIS
NETWORK
P(W)
PRE-
EMPHASIZED
BASEBAND
<P'(t)
FM
MODULATOR
FM
SIGNAL
Y(«ii)
V(t) = COS [<o c t+g>(t)]
Fig. 4 — Typical signal transmission path.
tained from the frequency division multiplex terminals, directly or via a
transmission facility, and is pre-emphasized prior to being applied to a
FM modulator. The output of the FM modulator is consistent with v(t)
shown in Fig. 1 . Assume that the unpre-emphasized baseband signal has
a Gaussian distribution and a flat power density spectrum, P , between
—fb and fb , where f b is the top baseband frequency. The output power
density spectrum from the pre-emphasis network is
SA<*) = Po\P(o>)\\ |/| ^ fb (32)
where P(co) is the transfer function of the pre-emphasis network. Letting
| P(a>) | 2 = a + a 2 P + aj* + a 6 f,
where the a's are real constants, we have
SA") = Po[ao + (hf 2 + a 4 p + o,n
It can easily be shown that 2
(27T(t) 2
/I ^ fb,
(33)
l/l Sjfl. (34)
Po =
Of f . I a 2/b I a *f>>
+
7 /
, (rad/sec) /Hz
(35)
where a = rms frequency deviation, in Hz, due to the baseband signal,
and f b is in Hz. Equation (34) gives the power density spectrum of the
pre-emphasized baseband signal in terms of the coefficients of a con-
tinuous pre-emphasis characteristic, and in terms of system parameters,
a and fb .
2.5 Signal-to-Noise Ratio
We are now in a position to express the signal-to-noise ratio for second-
and third-order AM/PM intermodulation noise. The expressions given
in (22) and (30) are for PM distortions so we convert them to FM dis-
tortions by multiplying by a> 2 . Hence, the signal-to-noise ratios can be
NOISE IN FM SYSTEMS 1759
expressed as
* W - 201 °^ (36)
U log 4^1 = io log j^r: :
L *",S 9 »J 2ndorder ^ &,(«
and
)
ri0 1og4^Tl-101og/«Y &(tt)
- 20 log k, , 37 ^
where *M«) is given by (34), 1/fc 2 &,(«) is given by (22), and 1/fc 2
S«,,(a>) is given by (30). A digital computer program has been written
which will evaluate (36) and (37) for any values of the transmission
deviation coefficients, pre-emphasis coefficients, rms frequency deviation
due to the baseband signal, top baseband frequency, and AM/PM con-
version factor. The derivations of 5[R/(t)], SF[i2, V '*(r)L $IK" 2 ( T )1
and 5[R v >\t)] in a form applicable to a digital computer program are
given in Appendix II of Ref. 2.
III. NOISE PROPERTIES AND CHARACTERISTICS
The previous material provided the mathematical treatment of
AM/PM intermodulation noise. In this section we will document the
various properties of both AM/PM intermodulation noise and inter-
modulation noise due to transmission deviations.* Also, the charac-
teristics of these two noise phenomena will be explored by utilizing a
representative system model. Both noise contributors are treated in
parallel throughout the section for comparison purposes. The results
are presented in three discrete modes: (*) properties which are true in
general; (it) properties which are approximately true; and (m) charac-
teristics which are derived from a representative system model. The
theoretical treatment previously presented was for a transmission
medium given by (10). In this section we will confine our analysis to
the power series transmission deviations in (10). This is done for two
reasons: (i) the properties of the two noise phenomena can be concisely
documented for power series transmission deviations; and (ii) the gain
and phase ripple properties need more analysis as well as mathematical
treatment in order to fully characterize the effects of ripples in the
transmission medium.
* The information for this latter noise contributor was obtained from Ref. 2,
which gives it implicitly, as well as from the associated digital computer program.
1760 THE BELL SYSTEM TECHNICAL JOURNAL, DECEMBER 196(5
3.1 General Properties
Equation (8) of this paper and (23) of Ref. 2 have been expressed
in terms of the transmission deviations and tabulated as shown in Table
I. This table is an extension of Table 21-1 of Ref. 4.
Table I — -Amplitude and Phase Modulation Caused
by Transmission Deviations
Type of transmission
Resulting amplitude
Resulting phase
deviation
modulation, a(/)
modulation, ip (l) — <p{t)
Linear gain, 0i
gi<p' - toiV* + iffiV
Parabolic gain, g>
g*p'' + h 'V
-gvp" + ff»VV
Cubic gain, g 3
—gvp'" + g*p"
— 'Mhf'f"
Quartic gain, g 4
— 4g i <p'<p"' — 3g 4 <p" 2
gup"" - iSgw'W
Parabolic phase, bi
bvp" + 26 2 V V" + 6 2 V" 2
+ 262VV"
Cubic phase, 63
obvp ip
-bvp'" + b 3 >p' 3
Quartic phase, b.\
—hvp"" + (ibitp 'V"
— 4&4fp ip — Sbitp" 2
Interaction terms
-\gM + gibi]?'"
-gib-xp'" + gig*p'<p"
+ [2gib«\<p'<p" + g,g 3 <p'<p'"
— -i(gib 3 + 02&2W"
—gigvp'* + [%i&i + 4gib<
— 3<(/,6 3 + 0262V 2
— 2gri 2 6o]^>'V"
+ (301 03 - 01 2 02)<p'V
Input signal = exp \i[u r l + <p(I)]\ ; output signal = exp \a(l)] exp \i\u c t + <p o (0}} ;
transmission medium transfer function = Y(u + w d ) = [1 + giu -j- 02« 2 + 3 oj 3
+ g A co 4 } exp \i\b-iu, 2 + b 3 u 3 + b 4 u*\\.
The argument of all the amplitude and phase functions is t.
The order of the noise produced by different transmission deviations
(e.g., <7i , 6 2 ) arc given in Table II for intermodulation noise due to
transmission deviations and for AM/PM intermodulation noise. Two
rules of thumb can be stated. For intermodulation noise due to trans-
mission deviations the rule is:
Even-order gain and delay transmission deviations cause odd-order noise.
Table II — Order of Noise
Intermodulation noise
Due to transmission deviations
Due to AM/PM conversion
Linear gain (01) :
No noise
•Second and third
Parabolic gain {gi) :
Third
Second
Cubic gain (0 3 ) :
Second
Third
Quartic gain (g t ) :
Third
Second
Linear delay (62):
*Second and third
Second
Parabolic delay (63):
Third
Second
Cubic delay (64) :
Second
Third
* Indicates predominant component of the two possible.
NOISE IX FM SYSTEMS
1761
Odd-order gain and delay transmission deviations cause even-order noise.
For AM/PM intermodulation noise the rule is, for those transmission
deviations that cause significant relative noise (will become apparent
later) ,
Even- order gain and delay transmission deviations cause even-order noise.
Odd-order gain and delay transmission deviations cause odd-order noise.
The two types of intermodulation noise are related to the magnitude
of the transmission deviation coefficient by the relationships shown in
Table III. Once a noise response is obtained for a particular system and
transmission deviation coefficient value, then the system noise for any
other coefficient value typically encountered in transmission systems
can be easily predicted.
3.2 Approximate Properties
The variation in the top message channel noise, for both noise con-
tributors, with number of channels, assuming the peak frequency devia-
tion remains constant as the number of message channels increase, is
shown in Table IV for the different transmission deviations. These
approximate relationships yield results with an error of <1 dB for
smooth pre-emphasis functions typically used in broadband radio
systems.
The assumptions used were that the peak frequency deviation re-
mained constant, and that a typical frequency division multiplex plan
was used. The rms frequency deviation, due to the baseband signal,
Table III — Variation of Relative Noise with Transmission
Deviation Coefficient Value
Intermodulation noise
T m' 'on deviation
Due to transmission deviations
Due to AM/PM conversion
Linear gain (01) :
No noise
*40 log | gi'/gil, 60 log
1 0i70i 1
Parabolic gain (0») :
40 log | 2 ' /ff2 |
^20 log 1 02V02 1 (approxi-
mation error <? dB)
Cubic gain (g 3 ) :
20 log 1 g 3 '/g 3 1
20 log
03 IO\
Quartic gain (gt):
20 log | gt'/gi \
*20 log | bi'/bo | , 40 log
20 log
Q\'l S*
Linear delay (bo);
40 log
6»7&i
I b 2 '/b, |
Parabolic delay (b 3 ):
20 log I b 3 '/b 3 1
20 log 1 b 3 '/b 3 1
Cubic delay (6«):
20 log 1 h'/bt \
20 log 1 64764 1
Where the prime (') notation depicts the terminal value and the unprimed
notation indicates the initial value.
* Indicates predominant component of the two possible.
1762 THE BELL SYSTEM TECHNICAL JOURNAL, DECEMBER I960
Table IV — Variation of Top Channel Noise with Number of
Message Channels
Intermodulation noise
Transmission deviation
Due to transmission deviations
Due to AM/PM
conversion
Linear gain (gi):
Parabolic gain (gi):
Cubic gain ({73) :
Quartic gain (g A ) :
Linear delay (b 2 ):
Parabolic delay (63) :
Cubic delay (b 4 ) :
No noise
Relative lop channel noise increase
^il log N'/N
Relative top channel noise increase
S& 39 log N'/N
Relative top channel noise increase
S :41 log N'/N
Relative top channel noise increase
^211ogiVyJV
Relative lop channel noise increase
^ 23 log iV'/JV
Relative lop channel noise increase
^ 58 log N'/N
^ 20 log N'/N
^ 21 log N'/N
^ 21 log N'/N
^ 57 log N'/N
S 58 log N'/N
^ 39 log N'/N
^ 40 log N'/N
Where N' = increased number of channels; N = initial number of channels.
was allowed to change, accordingly, as the number of message channels
increased.
3.3 Noise Characteristics
3.3.1 Representative System Model
As a system model, we will use the following system parameters:
N = number of message channels = 1200
fb = top baseband frequency = 5.772 MHz
AF = peak frequency deviation = 4 MHz
a = rms frequency deviation due to the multichannel baseband
signal = 0.771 MHz.
The pre-cmphasis characteristic is shown in Fig. 5 and can be expressed
by
I P(u>) I 2 = 0.9989 + 3.5839 X 10" 1 / 2
-5.0245 X IO- 3 / 4 + 3.894 X lO" 6 / 6 ,
where / is in MHz.
3.3.2 Noise Response for the Individual Transmission Deviations
It is instructive to show the individual noise responses on a compara-
tive basis. This can be done by letting all gain transmission deviations
NOISE IN FM SYSTEMS
1763
8
/
f
1 1
l
4
1
i i
I i
i |
i '
i
i
i
i
0.312
0.5
l
1
i
:
54
4.148 5.772
l >
1
l l
I 1
0.01 0.02 0.04
0.1 0.2 0.4 0.6 1.0 2
BASEBAND FREQUENCY IN MHZ
Fig. 5 — Pre-emphasis characteristic.
have 1 -dB distortion, relative to the carrier, at 10 MHz away from the
carrier. Also, we let all delay transmission deviations have 1 nanosecond
(ns) distortion, relative to the carrier, at 10 MHz away from the. carrier.
This allows us to directly compare the noise contributions of the different
gain and phase transmission deviations, respectively, and also allows
for some sort of pseudo comparison between a 1-dB gain distortion and
a 1-ns delay distortion. The intermodulation noise response, due to
transmission deviations, for the different transmission deviations are
shown in Fig. 6. Similarly, the AM/PM intermodulation noise responses
are shown in Fig. 7. Note that the responses in Fig. 7 are for k = 1.0
radian or a 6.6 degrees/dB AM/PM conversion device. For any other
value of k, say fci , we raise or lower the responses according to 20 log
fci , as indicated by (36) and (37).
It is interesting to note that linear delay is an important contributor
to intermodulation noise, due to transmission deviations, but is not a
significant AM/PM intermodulation noise contributor. Also, we observe
that parabolic gain is a large relative contributor for AM/PM inter-
modulation noise but is a negligible relative contributor for intermodula-
tion noise due to transmission deviations. As a side point, we point out
that parabolic gain is also a significant source of baseband amplitude
distortion.
The phase transmission deviation noise responses in Figs. 6 and 7
are of particular interest because the values used in these two figures
are realistic even for an equalized system; this is not the case for the
|93l
:ldB AT 10 MHz
+ 6
|
3 2 |
3,1
: ins AT 10 MHz
: 1 dB at ioMHz-
1
— Ib-.|
: i ns AT 10 MHz
o
c
|b a |
: i ns AT 10 MHz
m
^
^
T
2
/
|g 2 |
idB at 10MHz
§-34
z
-54
-74
12 3 4 5 6 7
BASEBAND FREQUENCY IN MHZ
Fig. 6 — Intermodulation noise due to transmission deviations.
+ g 2 : 1 dB at 10 MHz*
g 4 |: 1 dB AT ioMHzJ
|g 3 |: idB at io MHz
b 3 |: i ns AT 10 MHz
g,|: 1 dB at 10MHz-
b 4 | : i ns at io MHz
♦negative value of g 2 has a
NOISE RESPONSE WITHIN 0.2 dB
b 2 : i ns AT 10 MHz
3 4 5
BASEBAND FREQUENCY IN MHZ
Fig. 7 — AM/PM intermodulation noise (k — 1.0 radian).
1764
NOISE IN PM SYSTEMS 1765
gain transmission deviation values used so one need not be unduly
alarmed at first inspection of the noise responses shown. However, the
gain deviation noise responses are of interest in order to determine
which types of gain deviations a particular noise source is most sensitive
to.
Comparison of Figs. G and 7 and Table II show that for all transmis-
sion deviations of significance, the order of the noise distortions are
different for the two noise phenomena. Coupling this with the results
of the Appendix shows that for a given transmission deviation, the two
noise responses are uncorrected.
Of great significance and importance is the parabolic delay AM/PM
intermodulation noise response. We see from Fig. 7 that this particular
delay deviation is, by far, the largest contributor of AM/PM inter-
modulation noise compared to the other delay terms. The importance
of this finding lies in the fact that TWT amplifiers, when used as output
power tubes in broadband radio systems, are separated from transmitter
modulators (used to go from IF to RF) by band pass filters which may
possess large amounts of parabolic delay. Hence, we have large parabolic
delay distortion prior to an important AM/PM conversion device.
The noise impairment due to this typical system arrangement will be
examined in a later section.
Another point of interest is the noise response for linear gain. We
see that linear gain is not a significant AM/PM intermodulation noise
contributor. This is useful knowledge because in the past system require-
ments for linear gain have been set based on speculated AM/PM inter-
modulation noise impairments, as well as on derivable baseband ampli-
tude distortion due to linear gain and AM/PM conversion.
3.3.3 Effects of Interaction Terms
Referring back to Table I we note the row marked interaction terms.
By the form of the terms involved it is obvious why they are so named.
If one were to evaluate (22) and (30) in terms of the transmission devia-
tions explicitly, he would find that over 80 percent of the terms are
interaction terms. To examine the effects of these interaction terms we
compare the response we would get if we combined the curves shown
in Fig. 6, for example, on a power basis with the response we would
obtain by using all the transmission deviations at once, i.e., by taking
into account the interaction terms. There are a large number of possi-
bilities that could be examined, but to put the problem in perspective
the analysis considered only the cases shown in Table V. The results
for the two noise phenomena are shown in Figs. 8 and 9. The responses
1766 THE BELL SYSTEM TECHNICAL JOURNAL, DECEMBER 1966
Table V — Cases Considered in the Study of
Interaction Effects
Case
Condition*
g
10
Power addition of noise responses due to individual transmission devia-
tions (all g's and 6's positive)
Noise response under the condition: g x negative and all other g's and
6's positive
Noise response under the condition: Qz negative and all other g's and
b's positive
Noise response under the condition: gz negative and all other g's and
6'a positive
Noise response under the condition: g t negative and all other ^'s and
6's positive
Noise response under the condition: b 2 negative and all other g's and
6's positive
Noise response under the condition: 63 negative and all other g's and
b'B positive
Noise response under the condition: 64 negative and all other g's and
6's positive
Noise response under the condition: all g's and 6's positive
Noise response under the condition: all g's and 6's negative
Where | gi
01
= 1 dB at 10 MHz
ffj
= 1 dB at 10 MHz
<h
= 1 dB at 10 MHz
f/4
= 1 dB at 10 MHz
6 2
= 1 ns at 10 MHz
b 3
= 1 ns at 10 MHz
b<
= 1 ns at 10 MHz
* All the conditions take into account the effects of interaction terms except
for case 1.
shown in Fig. 8 are rewarding from a systems analysis standpoint be-
cause it indicates that the interaction components for intermodulation
noise, due to transmission deviations, do not significantly perturb
the noise response obtained by adding up the individual transmission
deviation noise responses on a power basis. Hence, for this noise source,
a system analyst could set requirements based on power addition of
the individual noise responses and be fairly confident that the actual
system noise response, due to transmission deviations, will be within
a dB of that response.
We see from Fig. 9 that the above desirable property does not hold
for AM/PM intermodulation noise. The responses shown in Fig. 9
deviate significant amounts from the power addition response (case 1)
by mere shifts of signs, the greatest departures occurring for parabolic
and quartic gain distortion which are, in their own right, the largest
relative noise contributors as evident from Fig. 7. The relative tendencies
indicated in Fig. 9 also occur when typical equalized repeater trans-
3 4 5
BASEBAND FREQUENCY IN MHz
Fig. 8 — Intel-modulation noise due to transmission deviations — effects of in-
teraction terms (refer to Table V for case listing).
3 4 5
BASEBAND FREQUENCY IN MHz
Fig. 9 — AM/PM intermodulation noise — effects of interaction terms (fc =
1.0 radian) (refer to Table V for case listing).
1767
17G8 THE BELL SYSTEM TECHNICAL JOURNAL, DECEMBER 1966
mission deviation values are used. In fact, least squares approximations
on equalized repeaters, RF band pass niters, etc., yield values for para-
bolic and quartic gain which are either, or both, negative, i.e., a loss
with increasing frequency. Hence, even under practical situations one
cannot, in general, expect power addition of the individual transmission
deviation AM/PM intermodulation noise responses to yield representa-
tive AM/PM intermodulation noise performance.
3.3.4 Noise Response for a Representative Radio System Repeater
The results up to this point utilized representative system parameters,
but normalized values for the transmission deviations were used. Of
interest is the predicted noise response for a typical situation, i.e., making
use of values typically encountered in practice. We will now use the
representative gain and delay responses shown in Fig. 10 for an un-
equalized and equalized radio repeater. The predicted intermodulation
noise responses, due to transmission deviations, are shown in Fig. 11.
It is obvious that the equalization has greatly improved the system's
noise response.
To examine the AM/PM intermodulation noise we take note of the
previously mentioned fact that the TWT has a band pass filter (whose
gain and delay responses are given in Fig. 10) preceding it. The AM/PM
intermodulation noise due to the band pass filter and the TWT amplifier
(assuming 2.5 degrees/dB) is also shown in Fig. 11.
O 1.010
z
5 1.005 -
1.000 -
0.995 -
0.990 _ -
*BPF (BANDPASS FILTER PRIOR TO THE TWT)
CURVES USE THE SCALES ON THE RIGHT
- 40 m - 1.02
""5
cc
1.00 2
3
0.98 jjjj
O
0.97 p
UJ
N
0.96 -i
<
0.95 S
Z
0.94
0.93
66 68 70 72 74
FREQUENCY IN MHz
Fig. 10 — Gain and delay characteristics.
NOISE IN FM SYSTEMS
1769
INTERMODULATION
NOISE DUE TO
TRANSMISSION
DEVIATIONS -
UNEQUALIZED
REPEATER
AM/PM
INTERMODULATION
NOISE DUE TO THE ~
BPF BEFORE THE
TWT(2.5DEGREES/dB)
INTERMODULATION
NOISE DUE TO _
TRANSMISSION
DEVIATIONS-
EQUALIZED
REPEATER
2 3 4 5 6 7 8
BASEBAND FREQUENCY IN MHZ
Fig. 11 — Representative radio system noise response.
We see from Fig. 11 that the AM/PM intermodulation noise in the
top channel is much larger than the intermodulation noise due to trans-
mission deviations for an equalized repeater. The repeater equalizer
is designed to correct for gain and delay shapes obtained from measure-
ments which do not recognize the AM/PM conversion phenomenon.
Hence, repeater equalizers based on such measurements, even though
effective for reducing intermodulation noise due to transmission devia-
tions, prove ineffective for AM/PM intermodulation noise which occurs
as indicated. In other words, the system has an AM/PM intermodula-
tion noise floor which is transparent to external gain and delay measure-
ments.
The transmission deviation of the band pass filter which is the major
noise contributor is the parabolic delay term. Hence, to reduce the
AM/PM intermodulation noise one must devise some method of cor-
recting for this transmission deviation. Two means of equalizing the
band pass filter are: (i) p re-equalization at IF prior to up-converting
in the transmitter modulator; and (it) microwave equalization directly
before or after the bandpass filter. The first method may not yield
perfect correction because up-converters using varactor diodes possess
AM/PM conversion characteristics, in some cases 1 degree/dB. In
essence, it would effectively be like trading noise due to 2.5 degrees/dB
for noise due to 1.0 degrees/dB or an 8-dB improvement in the ideal
1770 THE BELL SYSTEM TECHNICAL JOURNAL, DECEMBER 1966
case, i.e., no compression in the up-converter and a perfect inverse
band pass filter characteristic. However, an improvement anywhere
near this value would greatly reduce the effects of AM/PM conversion.
IV. CONCLUSIONS
Two noise contributors in FM systems are: (*) intermodulation noise
due to transmission deviations; and (ii) intermodulation noise due to
transmission deviations and AM/PM conversion. This latter source
of noise is designated "AM/PM intermodulation noise" in this paper.
Analysis was carried out in order to predict the second- and third-order
AM/PM intermodulation noise for the transmission medium given in
(10) and a continuous pre-emphasis function. Flat Gaussian noise was
used to simulate the unpre-emphasized baseband signal so the results
are consistent with the laboratory system tests using "noise loading".
Expressions were derived which specify the signal-to-noise ratio in
terms of system parameters, transmission deviations, pre-emphasis
characteristics and AM/PM conversion parameter. The latter param-
eter, assumed to be a real constant, was separated from the bod}' of the
calculations so that the resulting noise responses could be easily altered
for any value of AM/PM conversion.
The paper presented general noise properties and characteristics for
the two noise contributors. This material was presented in parallel,
for the two noise contributors, for comparison purposes. The order of
the noise component for different transmission deviations was given
so that one would know if a given transmission deviation causes second -
or third-order noise. The variation of the relative noise with transmission
deviation coefficient value was given so that a system analyst can deter-
mine the relative detriment to a system response that would result from
a change in a given transmission deviation. Another useful result was
the variation of top channel noise with number of message channels.
This would be of use in the case where one is interested in increasing a
system's message channel capacity.
Noise responses were given using a representative radio system model.
It was found when all gain transmission deviations had the same distor-
tion and when all delay transmission deviations had the same distortion
that: (t) for intermodulation noise due to transmission deviations the
cubic and quartic gain terms created the greatest top channel noise
due to gain transmission deviations, and that linear delay created the
greatest top channel noise due to delay transmission deviations; and
(ii) for AM/PM intermodulation noise the parabolic and quartic gain
NOISE IN FM SYSTEMS 1771
terms created the greatest top channel noise due to gain transmission
deviations, and that parabolic delay created the greatest top channel
noise due to delay transmission deviations. The effects of interaction
terms were examined. It was found that interaction terms do not signifi-
cantly perturb the noise response from that of the case of power addition
of the individual noise responses for intermodulation noise due to trans-
mission deviations. However, this desirable property did not hold for
AM/PM intermodulation noise which says that power addition of the
individual noise responses may be in gross error; in other words, inter-
action terms must be considered when evaluating AM/PM intermodula-
tion noise.
The intermodulation noise due to both noise contributors was pre-
dicted for a representative radio system repeater. It was observed that
the AM/PM intermodulation noise due to the band pass filter preceding
the TWT amplifier created more top channel noise than that due solely
to the equalized transmission characteristic, i.e., intermodulation noise
due to transmission deviations. Possible correction methods were given.
A point of interest, is that the two noise contributors considered in
this paper are correlated so that combining the two spectra together
assuming random addition, i.e., power addition, is not sufficient in
general. The significance of this correlation is presently being examined
and will be reported on in a later paper.
V. ACKNOWLEDGMENT
The author wishes to express his gratitude to Miss J. D. Witkowski,
of Bell Telephone Laboratories, for programming the AM/PM inter-
modulation noise equations on a digital computer, and to M. Liou,
of Bell Telephone Laboratories, for his motivation and interest in this
paper.
APPENDIX
Uncorr elated Second- and Third-Order Distortions
We will show here that the second-order distortion, 2 (£), and the
third-order distortion, 6 3 (t), are uncorrected. Consider
9 T (t) = d 2 (t) + 6 3 (t). (38)
Now the autocorrelation function of 6 T {t) is
Rb t (t) = Re t (r) + Rg 3 (r) + B m ,(t) + Re 3 e 2 (r) (39)
1772 THE BELL SYSTEM TECHNICAL JOURNAL, DECEMBER 1966
where, e.g., Re 2 e a (r) is the cross-correlation function of 2 (O and 6 3 (t).
Taking the Fourier transform of (39) gives
Se T (u) = Se 2 (o)) + Se 3 (o>) + Se 2 $ 3 (a>) + Se 3 e 2 (t>)). (40)
From Fig. 2, we have
2 (O = x(t) + y(t) (41)
so it follows that
Rhh(T) = R x e 3 (r) + R v e 3 (r). (42)
Referring to Figs. 2 and 3 we have, using the relationship for the cross-
correlation of linearly transformed random functions, 3
&,•» = (■&(-«) (7i(») 5 /v .(«) + ft(-«) £ 3 (o>) fl,» v .(«). (43)
Now we can write
<W(«) = JF [fl/^'iCr)] = S[ave (*>V 3 )] (44)
and
S/'V«(») = * IWWl = ff t ave (^V 3 )], (45)
where 5 stands for the Fourier Transform.
The phase modulating signal, <p(t) represents the multichannel message
load and so for a large number of talkers <p(t) is Gaussian with zero
mean. 5 It follows that derivatives of <p(t) are Gaussian with zero mean.
It can be shown that 6
ave
[xi l ■ ■ ■ x n rn ] =0, £ n odd (46)
where x\ • • • x„ are Gaussian random variables with zero mean, and
ri ■ • ■ r» are any set of integers. Hence, letting
in (44), and letting
in (45) gives, using (46),
Xi = cp
X2 = <p'
Xx = <p"
Xi = <p'
Se 2 e 3 (<>)) = 0.
NOISE IN FM SYSTEMS 1773
Similarly,
*S'« 3 8 2 (o)) = 0.
Hence, 2 (O and 6z(t) are uncorrelated so
Se T (u)) = Se 2 (io) + Se 3 (o)).
REFERENCES
1. Laico, J. P., McDowell, H. L., and Moster, C. R., Minimum-Power Traveling-
Wave Tube for 6000-mc Radio Relay, B.S.T.J., 35, November, 1956, pp. 1285-
1346.
2. Liou, M. L., Noise in an FM System Due to an Imperfect Linear Transducer,
B.S.T.J., 1,5 November 1966, pp. 1537-1561.
3. Lee, Y. W., Statistical Theory of Communication, John Wiley and Sons, Inc.,
1960, pp. 348-51.
4. Members of the Technical Staff, Bell Telephone Laboratories, Transmission
Syste?ns for Communications, Revised Third Edition, 1964.
5. Holbrook, B. D. and Dixon, J. T., Load Rating Theory for Multichannel Ampli-
fiers, B.S.T.J., IS, October, 1939.
6. Laning, J. H., Jr. and Battin, R. H., Random Processes in Automatic Control,
McGraw-Hill Book Co., N. Y., 1956, pp. 82-85.
-