Document text
Transient Response of an FM Receiver
By MANVEL K. ZINN
Introduction
•* I ""HIS paper develops various formulas for the response of an FM
■^ receiver to signal or noise input voltages of arbitrary form. The prin-
cipal object in view is to obtain a more complete understanding of how an
FM receiver responds to transient voltages, such as those arising from
ignition interference, but the more general aspects of the theory have other
applications as well. In particular, general formulas are given for the re-
sponse of a linear circuit to an applied voltage, or current, of variable fre-
quency. The Fourier transforms, or frequency spectra, of the response, and
the envelope thereof, are determined.
Two examples are given: (1) the audio response of an FM receiver to a
very large impulse and (2) the response, including harmonic distortion, to
a sinusoidal signal wave.
The element of an FM receiver that demands most discussion is the bal-
anced frequency detector. The greater part of the paper accordingly deals
with that important clement. The general problem can be stated as follows:
A limiter and frequency detector are transmitting a steady unmodulated
carrier wave to an audio output circuit. At time, / = 0, frequency modula-
tion of arbitrary form is applied to the carrier (either by signal modulation or
a superposed noise transient) . What is the audio output voltage that re-
sults?
Frequency Detector
Except for the greater bandwidth, the amplifiers and selective circuits
between the antenna and the limiter of an FM receiver are similar to those
of an AM receiver in their transmission features. If the selective circuits
have a bandwidth ample to accommodate the maximum frequency swing of
the FM transmitter, and if the transmission over the band is substantially
"flat" and the phase shift nearly linear with frequency, the amplifiers will
introduce little distortion. The limiter and frequency detector are therefore
regarded as the distinctive elements of an FM receiver meriting theoretical
discussion.
The literature contains descriptions of frequency detectors of several
types together with adequate analyses of the action of the circuits based on
the variable impedance concept.^ The more generally used circuits can be
^ See Items 3 to 6 in list of references attached.
714
TRANSIENT RESPONSE OF AN FM RECEIVER 715
reduced to the circuit shown schematically in Fig. 1, which can be taken to
illustrate a generic form of frequency detector. Zi and Z2 are two resonant
impedances tuned to different frequencies, one above, the other below, the
carrier frequency.^ For example, the simplest version of Zi and Z2 could be,
for each, a parallel combination of R, L and C. Across each of these
impedances is connected a rectifier with load circuit so proportioned that
the rectification is substantially linear. The rectifiers are poled so that their
low-frequency outputs are opposed, thereby obtaining cancellation of even-
order demodulation products. With this arrangement, the low-frequency
output voltage Vo, which is applied to the audio amplifier, is substantially
proportional to the difference between the envelopes of the voltage drops
across Zi and Zj.
The resistance elements of the impedances, Zi and Z2, each include a shunt-
ing resistance equal to half the load resistance of the associated rectifier,
which therefore determines, to some extent, the Q of the tuned circuit. The
output diode load, Ro Co, has negUgible impedance at the carrier frequency.
Under these conditions, the low-frequency output voltage across the two-
rectifier load impedances is
Vo= V ([Vi] - [V2])
where rj = detection efficiency (nearly unity)
y^^V2 ^ high-frequency voltages across Zi , Z2 , respectively (Fig. 1)
[V] = envelope of V.
All this is in accord with the accepted understanding of the operation of a
properly designed hnear rectifier working at an efficiency approaching 100
per cent.
The amplitudes of the voltages, Vi and V2 , across the resonant imped-
ances, Zi and Zi , of Fig. 1 are shown in Fig. 2. In the practical engineering
analysis of this frequency detector circuit, employing the idea of impedance
that varies in step with the instantaneous frequency, the two voltages of
Fig. 2 are subtracted (owing to the opposed polarities of the rectifiers) to
obtain the over-all voltage-frequency characteristic shown in Fig. 3. Then
it is inferred, by physical intuition, that if the instantaneous frequency of
the carrier is varied at the input, the output voltage wave will vary as in-
dicated by the curve of Fig. 3. Strictly speaking, this is a false assumption,
but where the rate of variation of the instantaneous frequency is at an audio
signal frequency far below the carrier frequency, the error in the assump-
tion is of no importance, whereas the simplification in thinking accomplished
^ The term carrier frequency will be used to designate the value of the unmodulated
received frequency after all heterodyne conversions. {This frequency is equal to the mid-
band frequency of the last intermediate frequency amplifier ahead of the limiter, if tuning
is perfect.)
716
BELL SYSTEM TECHNICAL JOURNAL
by it is considerable. It is only where the rate of variation of the instan-
taneous frequency is high, as it can be in the case of a large noise transient
caused by impulse excitation, that the error in the assumption in question
TO
AUDIO
° >=.[{v,}-{v.}]
I 1-
T} - DETECTION
EFFICIENCY
- ENVELOPE OF
FUNCTION OF
TIME ,V
Fig. 1 — Circuit of a balanced frequency detector.
1
l<-^_A >+«-
--A— ->t
yA^ 1
.'-r^v.
^ 1 ^^ ^
/ ' N
<
/ ' \ '
/ ' ^
1-
7
1 1 >
\ '
/ 1
lU
|v,|/
' 3^'
1 \
()
^^^v
a
1 _.,-
\^^^^ 1
I'-'^i-
_,!,-'''
! 1
CO, OJo iJJa C[j-2;rf^>
I'ig. 2 — Voltages across tuned circuits of frequency detector.
Fig. 3^0utput voltage of frequency detector.
can become serious. A particularly subtle error that can arise from the
assumption is to fall into the habit of regarding the characteristic curve of
Fig. 3 as a frequency transmission curve of the sort obtained by measuring
the ratio of output to input of a hnear network over a range of frequencies.
TRANSIENT RESPONSE OF AN FM RECEIVER
717
The curve of Fig. 3 is not such a transmission curve, because the principle of
superposition does not apply and a frequency conversion is involved.
Owing to the considerations discussed above, the analysis to follow avoids
the assumption of variable impedance associated with the varying
instantaneous frequency. This does not imply that the assumption, as
employed by various writers, is considered seriously erroneous, but, rather,
that it seems preferable to develop the theory without invoking the as-
sumption, provided that this can be done without falling into unmanageable
comphcations. Briefly, the procedure in the work to follow is to determine
directly the envelopes of the voltages W and Fo as functions of time, one
envelope then being subtracted from the other to obtain the output wave.
7i ffh
^ 4
t-
Z
Mi
a.
% n-h
O 4
TIME.t-
^--f
-J
Fig. 4 — Current wave out of limiter.
General Theory
When a carrier is being received, the limiter can be regarded as substanti-
ally a constant current source having an internal shunt admittance small
compared to the admittance of the tuned impedance elements, Zi and Zi ,
of the frequency detector. If the limiting is severe, as it should be for good
operation, the current delivered by the hmiter is a rectangular wave as
illustrated in Fig. 4. When this current is driven through the impedances,
Zi and Za , the voltage drops, Vi and V^ , that arise across these elements
are substantially sinusoidal in form, owing to the selectivity, which practi-
cally extinguishes all harmonics of the carrier frequency. We therefore take
the current input to be sinusoidal in the first place, namely
/(/) = //COS 27r/o/
This is the unmodulated current, irh/^ being the current cutoff point of the
hmiter and fa the frequency of the carrier. When the carrier is modulated
in frequency, we write
/(/) = //COs[27r/J+ e{l)] ■ (O
where 6{i) is the phase angle varying with time. The instantaneous fre-
quency then is
/f'> ^ T- T, tMo/ + Gil)] =Io + ^ e'U).
l-K at 2ir
(2)
718 BELL SYSTEM TECHNICAL JOURNAL
In the transmission of signals by frequency modulation, the instantaneous
radian frequency deviation, 9'{t), is made to vary in proportion to the
signal amplitude, so that 6{i) then varies in proportion to the time integral
of the signal amplitude.
As a preliminary step to the discussion of the frequency detector itself,
we require a formula for the voltage drop across an impedance Z{f) when the
frequency-modulated current (1) flows through it. The point of view
usually adopted is to regard the impedance as a composite function of time,
viz., Z[f{t)], and to say that the voltage across it is
F(/) = imum = i(i)z [/o + ^ 0'(o] . (3)
This quasi-stationary viewpoint gives results that are nearly correct if the
rate of change, 9"(l)/2ir, of the variable frequency is not too large. The
magnitude of the error has been determined in a paper by Carson and Fry.^
In the present paper, impedance is a function of frequency that is inde-
pendent of time, as in the classic theory of linear systems. The frequent
use of the term "instantaneous frequency," as defined by (2), does not
imply a departure from this point of view.
In the following, H{l) is, in general, the voltage response as a function
of time, of a network to a unit impulse of current applied at time / = 0.
In the case of a two-terminal impedance element, II(i) is the voltage drop
across the element when a unit impulse of current is sent through it. Then,
if the frequency modulated current (1) flow through the impedance, the
voltage drop is
1/(0 = h [ cos [mil - r) + e{t- - T)]H(r) dr (4)
''0
where wo = 27r/o . 6(1) can have any form as a function of time. V(i)
can be written,
2 Jo
2 Jo
In the frequency detector problem, the result finally desired is the envelope
of the voltage wave. It will clarify the discussion to explain first what is
meant by an envelope. If the voltage is of the form
V{t) = c{t) cos [oiai + 0(/)]
= a{C) cos oifl/ — &(/) sin coo/
= \[a{l) + ib{l)] r"' + h{a{i) - ib(t)] £-'■"«' (6)
' Item 1 in the bibliography. See formula 21 in that paper.
TRA NSIENT RESPONSE OF A N FM RECEI VER 7 1 9
the complex function,
[Vm = c{t) .'*'" = a{l) + ib{f) (7)
is here called the "envelope function" of the voltage with respect to the
radian frequency wo , c{l) being a real amplitude modulation factor, which is
the envelope'' itself, as usually conceived, and exp [i<t>{t)\ a complex fre-
quency modulation factor, in which <^'(/) is the instantaneous deviation of
the radian frequency from the reference value, wo . If such a modulated
voltage wave is applied to an ideal linear detector, the output voltage across
the load circuit of the latter is the real envelope, c{t) = [a-(l) + 6-(0l' '•
This concept of an envelope function provides a convenient generalization
of modulation ideas. Both amplitude modulation and frequency modula-
tion vary the envelope function, but in different ways. In amplitude mod-
ulation, the real ma;:nitude, c(/), is varied while the angle is constant,
whereas, in frequency modulation, c is constant and it is the angle, <^(/), that
is varied.
It will be seen that (5) is in precisely the same form as (6), so that we can
write the envelope function of V(i) immediately, as follows:
[!'(/)] - a(l) + ib(0 = h [ e-'""'+''"-^' ff (r) rfr. (8)
The conjugate envelope function then is
[V(T)] = a(l) - ib{l) = h I e"'-"'-'''^'-^'//(r)rfT. (9)
The spectrum of the envelope function is also of interest. To obtain the
spectrum, which we shall call, Fo(/), we find the Fourier transform (hereafter
abbreviated, F.T.) of both sides of (8), viz,;
F,{J) = r [V{t)\e-'''' dl = h r e''^' [ e-""^'^''"-''-H{T)(lrdl. (10)
J— CO J—x Jo
It is permissible to reverse the order of integration of t and /, obtaining
Foif) = hj e~"">'H(r) j £-■-'+'■«<'-> rf; dr. (11)
The F.T. of h exp [i9(t)] will be designated, ^'(/), i.e.
^(f) = h r e''''^-'"' dl. (12)
J— w
* The "envelope", so defined, is an engineering concept and is not quite the same thing
as the envelope of mathematics, which is ahvays tangent to a curve or set of curves.
720 BELL SYSTEM TECHNICAL JOURNAL
Putting / — T in place of / in place of i as the variable of integration in (12)
we have
^(/) = he'"' r e-'"'+"t'-'' dL (13)
J — cc
Thus it is seen that the inner integral of (11) is equal to e^'"'^(f) and the
equation becomes
Foif) = nn f B{r)e-''"'^'">'' dr. (14)
Now the F.T. of E{l) is Z(/), i.e.
z(/) = r n{i)e~'''' dl. (15)
J— 00
Therefore
^(/ + /o)= r HiOe-'"''^'''" di. (16)
J— 00
This differs from the integral in (14) only in the lower limit of integration.
But since H{/) is the response to an impulse applied at time / = 0, H(t) =
for / < and the two integrals are therefore equal. Putting (16) in (14)
we have, finally
HI) - r HO + ibiOK'"" dt = m)zu + /o). (17)
J— M
The F.T. of the conjugate envelope function, a — ib, is
^o(-/) - r [a{l) - iime-'"' dl = ^(-f)Z{-f -\- /,) (18)
where symbols with the superbar denote the complex conjugates of unbarred
symbols. Since Z(f) is the F.T. of a real variable, B{l), it must assume con-
jugate values for positive and negative values of /, i.e., Z(f) = Z(—f) and
therefore Z{f + fn) = Z(— f + fo). Consequently, (IS) could be written
A(-/) = ^(-/)Z(/+/o) (19)
(17) and (18) are the final solutions in frequency functions corresponding
to the solutions (8) and (9) in tune functions. The formulas in frequency
functions have the advantage of compactness, which makes them easy to
remember.
We require also the F.T. of the voltage itself, 1 (t), which we shall call
F(f). From (6)
F(f) = r l/(Oe""' dt = \\ (a + j6)e-**^""""'V/
+ i f (a - i6)e~""^""* dH (20)
TRANSIENT RESPONSE OF AN FM RECEIVER 721
and from (17) and (18) this evidently is
PU) = h^(f - h)zU) + hH-f - foM-f) (21)
or, since Z(f) = Z{—f)
F(f) - mi) [*(/ - /o) + H-f - /o)]. (22)
Anyone familiar with the rules of Fourier transforms could write down this
frequency function in the first place and then proceed to find the time func-
tions by the reverse of the process just carried out. But the time functions
are more closely related to the physics of the problem and therefore provide
a more fundamental starting point for its solution.
It wih be appreciated that, although the above discussion has been phrased
to apply to the problem of finding the voltage drop across an impedance
when a frequency modulated current flows through it, the formulas also give
the current through an admittance when a frequency-modulated voltage is
applied across it. They also give the output voltage or current of a four-
terminal network when a frequency-modulated current or voltage is apphed
at the input. These various applications of the formulas obviously can be
made by placing definitions on Z and // appropriate to the particular
problem.
The next step is to assume suitable values of the impedance Z{f) and vari-
ous forms of the frequency modulation function 6 and to employ these
particular values in the general formulas 8, 9, 17 and 18.
Balanced Frequency Detector
The hnpedance can be defined, in general, as a rational algebraic function,
VIZ.;
_ (/to — ai)(fto — lh) ■ ■ ■ ( io j — Om) ^ Pjiui) /23'j
'^^'"-' " {m - A)(/7r - ^) . . ■ (io) - p.) Qiioy) ■
Writing the polynominals P and Q in this way as the products of their factors
exhibits the d's as the zeros of Z and the ^'s as the poles. The latter deter-
mine the frequencies of free vibration of the network. For the network to
be stable, the ^'s must aU have negative real parts. The o's and ^'s are
either real or occur in conjugate complex pairs. By the partial fraction
rule, the expression can be broken up into a series of simple fractions; thus
Z{i.) - -•--'' ^ + , ■'—,+■■■■ (2*)
liO — p\ i^ ^ pi
If the poles are all simple, the /I's are given by
4, ^ ^^^ . (25)
' Q'iPi)
722 BELL SYSTEM TECHNICAL JOURNAL
For the present purpose, only a pair of terms of (24) need be considered.
This will provide a specific solution of the balanced frequency detector for the
case, previously used for illustration, where the impedances are two simple
resonant circuits of parallel R, L and C. At the same time, this solution
can be extended to more complicated circuits by superposing a number of
such elementary solutions, as is clearly possible with the type of impedance
development indicated by (24).
The impedance of R, I. and C in parallel, written in the form of (23) and
(24), is
.. . i(ji 1 k k
Z(iw) = - -^,-. -^ = . — - +
C (iW — p\){J(i} — pi) i<i3 — Pl i03 — p'l
k k
(26)
7w 4" 7 /o) + T
where
— p-i=y = a-\-i^
— Pi ^ y = a ~ i^
^ 2cV^/3
""'IRC ^~ y W \R'C' - Vc..-«, ^0 - ^j^.
The voltage response of the circuit to a unit impulse of current is then
H{t) = ^^^— + ^^_ e'-' df = ke-'' + ke-~'\ t > 0. (27)
J-« \IL^ + 7 'CO + 7/
To find the envelope function of the voltage drop when the frequency
modulated current
/(/) = ^ (,'■"■>'*, -sen ^^-.-..^-mn^ ^jj
is applied to the circuit, we make use of the general formula, (17), which
states that the spectrum, or Fourier transform, of this envelope function is
^o(/) = / " Ht) + ibilW'"' dl = ^{f)Z(f + /o) (17)
TRANSIENT RKSFONSE 01- AN FM RECEIVER 723
where *( /) is the F.T. of // cxp \iB{l)\ iind Z{f) is the impedance, (26), in this
case. The envelope function then is
J- 50
= hke-''"'^'" /' ,<'"''*-^'^+''"^'(/t (28)
This result is obtained by employing the convolution formu!a^ which states
that if Fi and /''2 are F.T.'s of d and d , respectively, then the F.T. of FiF. is
J- CO •'- «:
(The upper limit of the integrals in (28) is / instead of » for the reason that
H(l) is zero for / < 0.) The result could have been obtained equally well
without using the Fourier transforms by substituting (27) in (8). When
8(t) is specified mathematically in the infinite interval (-«>, «) these
formulas give the resultant of the steady state and transient oscillations.
\'arious problems can be solved by specifying particular forms of variation
for d{l). Two examples follow: (1) where the instantaneous frequency,
e'{l), is an impulse and (2), where d'O) is a sinusoidal wave, as for elemen-
tary signal transmission.
Example 1: Impulse Modulation
Ignition interference comprises a sequence of sharp impulses, each of du-
ration very brief compared to the interval between them, so that the tran-
sient in the receiver produced by one impulse dies away before the next one
arrives. It is therefore sutruieiit to consider the disturbance caused by a
single impulse.
If the receiver is jierfectly lulled, an impulse produces, in the tuned cir-
cuits, a transient of the same nominal frequency^ as the signal carrier.
When superposed on the carrier, the interfering transient alters, or mod-
ulates, both the amplitude and phase of the carrier. The amplitude mod-
ulation is wiped out by the limiter, but the phase modulation remains to
produce noise in the output. The phase shift caused by the transient is a
random variable, because it depends ujion the time of arrival of the impulse,
and this is entirely fortuitous.
^ See pair 202 of Itfiii 10 in tlu- l)il)lioRr;i]ihy.
« By "nominal frcqiiuLU-y" is mennl the fretiuency as dctermint-d by counting zeros of
the wave. 'I'hf transient actually compris'.'S a spectrum of freciuencies spread over the
hand of the Uined circuits, of course.
. 724 BELL SYSTEM TECHNICAL JOURNAL
It is of engineering interest to determine the noise produced by a very
large impulse, exceeding greatly the amplitude of the signal carrier. When
such a large impulse arrives, it causes a sudden jump, or discontinuity, in
the phase of the carrier. The excursion of the instantaneous frequency
corresponding to the phase jump is indefinitely large and the problem ac-
cordingly cannot be solved satisfactorily by means of the usual assumption
of quasi-stationary frequency. The problem of large impulsive interference
provides, therefore, the principal justification for the more exact method of
analysis here employed. In the paragraph following, the problem is re-
stated in terms providing a suitable basis for mathematical analysis.
We assume, as before, that the Umiter is dehvering to the frequency de-
tector a steady carrier current of constant amplitude h and frequency /o .
At time ^ = a brief disturbance occurs specified by the statement that
the instantaneous frequency, e'{l), of the current suddenly executes an
impulse of moment 6. That is: e'{i) is zero at all times except at f = 0,
when it goes to infinity and back to zero again in such a way that the area
of the impulse so formed is 0. The carrier current amphtude then remains
constant but the phase, d{t), of the carrier takes a sudden jump of 6 radians
at / = 0. What is the voltage output of the frequency detector?
The general formula (28) gives directly tlie envelope function of the volt-
age across the impedance (26) for a phase function e{t). In this formula we
have now to put e{i) = before time / = and 9, after / = 0. We do this by
dividing the interval of integration into two parts, (— oo, 0) and (0, /); thus
a{t) -i- ib{i) = A*e-'*""+^'' {( e''"'»+^=^ dj ^ ^^ f e''"»+^'^ dr)
iwo + 7 j'wo + 7
Let the radian frequency interval by which the apphed frequency wo
is set off from the resonant frequency toc be
A = Wo — Wc (31)
as indicated on the curves of Fig. 2. When a/uc is small compared to unity,
as it is in practical circuits, ^ is very nearly equal to w^ . (See the formulas
following equations (26).) Then
iojo + 7 = im -\- a -\- iiOc = a ~ iA -\- 2ia)o
and
im -\- y = i<t}o -\- ct — i(ac = a -\- iA. (32)
TRANSIENT RESPONSE OF AN FM RECEIVER 725
From this it is evident that the first term of (30) contains the demodulation
sum product of frequency on the order of 2wo . This frequency will be
suppressed by the diode load circuit and consequently the second term of
(30) is an adequate representation of the envelope function. Therefore we
write
„(l} + ,m = M ':' ^^^ ^ + "- (33)
and with the above approximations this is very nearly equal to
One deduction that can be made immediately from this formula is that the
frequency of the oscillation in the output of the rectifier caused by the phase
jump at the input is A, the radian frequency interval by which the applied
carrier frequency differs from the resonant frequency. The oscillation is
heavily damped, however, because a, while being very small compared to
0),. , is comparable in magnitude with A in circuits commonly used.
The angle of the complex envelope function (34) represents merely
a phase shift of the carrier frequency wo . \\'e are interested only in the
magnitude of the function, viz.:
After some algebraic work, the desired formula comes out of (34) in the
following form:
1 - 2.:(/) sin (a/ + ^) + »r(/)]"" ^ ^ ^ ,
where
n:{l) ^ le-"' sin ~ (36)
The discussion so far has dealt with a single impedance (or network) and has
been concerned with obtaining formulas for the voltage across the imped-
ance, and the envelope thereof, when a frequency-modulated current is sent
through it. It is necessary now to refer to the construction of the balanced
frequency detector, which is the particular object of our study. Figure 1
shows two impedances having the variation with frequency sketched in
Fig. 2. The carrier current is driven through the two impedances in series
and linear rectifiers are connected across each in such polarity that their
low-frequency output voltages are opposed. We assume that the output
726 BELL SYSTEM TECHNICAL JOURNAL
voltage of each rectifier is the envelope of the voltage existing across its
associated impedance. Therefore, to find the total output of the balanced
frequency detector, we have to find the difference between the envelopes of
these voltages.
It is necessary to specify the two impedances more precisely. It appears
that the best operation is obtained if the frequency of the carrier is midway
between the resonant frequencies of the two impedances. That is
A -
Wo — lji\ = OJ-j — Uo
where wi , 0)2 are the resonant frequencies of Zi , Z5 , (previously written as
«,, for any impedance). Furthermore it appears that the two impedances
should have identical values of C and very nearly the same damping con-
stants. The design of the detector circuit is accordingly specified by
and then
Rx
= R^_= R
U
- i/co!c
u
= \lJ,c
al
^ Q!2
= a = 1/2RC
k^
- ^2
^ k ^ 1/2C
0)0
0,'!
Wo + A
ct'o — A
(fl/ojn « 1) ,^^.
All the quantities are assumed to be substantially constant over the signifi-
cant frequency range.
With the circuit constants so proportioned, it can be seen from (36) that
the envelope of the voltage across Z\ differs from that across Z3 only in the
sign of A. Therefore, the output voltage of the balanced frequency detector,
when the instantaneous frequency variation is an impulse of moment G at
/ = 0, is
\\{l) = c,{l) - r,(/)
2C
(a + A-
1 -\- 2m sin ( A/ - ® ) +
-J"
— I 1 — 2w sin [ A/ + :r 1 + w
(38)
/ >
= 0, / < 0, where w = le""' sin—.
TRANSIENT RESPONSE OF AN FM RKCEIVEH
727
On Fig. 5 is given a plot of this function for a value a/A = 1 of the rela-
tive damping. Calculations for other values of a; A show that the output os-
cillates only weakly for a/A =- -i anrl is nearly dead-heat for a/A - 2
iii
0.9
o.a
r0,7
0.5
o
>0.5
1-
0:0.4
D
°0.3
ID
>
i=0.2
<
^0.1
r^
\
V„CO--pi^ fCO IN VOLTS
WHERE h =CARRIER-CURRENT AMPLITUDE AT
FREQUENCY- DETECTOR INPUT
cr. A, R = DAMPING CONSTANT, HALF-BAND WIDTH. AND
RESISTANCE OF FREQUENCY- DETECTOR
CIRCUIT
' O 30 40 60 80 100 120 140 l60 t80 200 220 240 260 2B0 300 320 340 360
At IN DEGREES
Fig. 5— Transient response of a balanced I'rcriuency detccUir for a step O in the carrier
phase.
Example 2: Signal Reception
If the carrier is frequency-modulated by a signal
sit) = S cos ql (39)
and the frequency modulation factor of the transmitter is m, the phase of the
carrier wave is made to vary in accordance with the relation
e{i) = M / s{l) ill =- -- sin ql - .V sin ql
•'- X q
(40)
nS is then the radian frequency deviation of the transmitter and fiS/q,
which is the ratio of this frequency deviation to the frequency of the signal,
is commonly referred to as the "frequency deviation ratio." This factor,
which is denoted by .v in the al)()\-e equation, enters as a fundamental
parameter in all KM theory.
To find the envelope function uf the voltage wave produced across the
impedance (26), when the frequency-modulated carrier is received at the
frequency detector, we put (40) in (28). To effect the integration, the
expansion
„,.si.,/^ £ JUx)e'""
(41)
728 BELL SYSTEM TECHNICAL JOURNAL
is used, /n(.T) being the Bessel coefficient of the first kind, of the nth order and
with the argument x. For brevity, /„(x) will be written /„ . Also, the
first term of (28) is to be omitted, because, as was shown in the preceding
example, it represents frequency sum terms which are filtered out by the
diode output circuit. Then we have
a{t) + ib{t) = h'ke^""'^''' ( e''"'-'-"' E J J""' dr. (42)
J— CO »=— 00
When the integration is carried out, the result is
a{l) + ib{l) = hTc Z . f^- . = hk T. •'':' , . -■ (43)
To obtain the magnitude of o + ih, which is the envelope required, we multi-
ply the above Fourier series for a-\- ibhy that for a — ib, obtaining a double
summation, which can be written as follows:
c\l) = a\l) + b'(!) = <-o + E (cJ"" + c.e-'"^')
= flo + 2 2J fin cos nql
where a„ is the real part of the complex coefficient c„ and qq = co .
The coefficients are given by
(44)
Cn =
40 m=-=o (a — (A — hnq)[a + 7-A + i{m + )i)q\
(45)
40 m=-» (a + iA + imq)[a — JA — i(fn + n)q]'
Obtaining this result involves use of the relation
J^.(x) = (-rJnix). (46)
From (45)
^ J[_ "y Jm Jm+,\a + (a + mq){L + mq + nq)] ,
°" 4C^«f:^« [a2 + (A + mqYla'' + (A + »/7+"».^]' ^ ^
Finally, we have to obtain, as before, the difference between the en-
velopes of the voltages across the two impedances of the balanced frequency
detector; that is, we have to determine
V,{t) = c,{l) - ct{t) (48)
where ci(/) is given by (44) as it stands and C2{t) is obtained from the same
expression merely by reversing the sign of A. The complete solution for
TKANi;iE.\T RE^FOy.SE Of AX FM RECEIVER
729
the output voltage of the frequency detector, when a phase variation,
e'(l) = xq cos qt, is impressed on the carrier at the input, then is
Ko(/) = \ aa -\- 2 J^ an2 cos nqt
1/3
do
+ 2 ^ a„i cos nqt
11/2
(49)
where a,a is given by (47) and a„2 is given by the same formula with the
sign of A reversed.
An approximation that is permissible when the frequency swing xq does
not approach the available frequency range A is
VqU) = J] /I „ cos nql
(50)
where
A. -
fln2 ~ ^nl
V
Ofl
n odd; = 0, n even.
(51)
The table following gives the results of a computation of the first four
coefficients from formulas (45) and (51) for the case of a frequency deviation
ratio, x= 5, for q/2ir = 3000 cycles per second, for A/2-n- = 30,000 cycles
per second and for a/A = 7, 1 and 2.
Coefficient
«/A = .5
1.0
2.0
4o
Ai
A,
.848
.0312
.478
-.00438
.191
-.00281
Note: To obtain volts, multiply all values by
Rha
(a= + A"-y'"-
The coefi&cients for even values of n vanish, which confirms what can be
inferred from physical considerations, namely, that the balanced construc-
tion of the frequency detector eliminates the d.c. component and all even
harmonics. From the ratio of Aa to /h we obtain the following ratios, ex-
pressed in db's, of the third harmonic distortion to the fundamental signal
for the three circuit designs:
a/A 20 log 10 U 3/^1 1 1
.5
-28.7 db
1.0
-40.8
2.0
-36.6
The results for the sinusoidal signal, when considered in conjunction with
those for the impulse modulation, also permit certain conclusions regarding
signal-to-noise ratios for impulsive mterference in FM reception.
730 BELL SYSTEM TECHNICAL JOURNAL
Ratio of Noise to Signal
It may be helpful, in conclusion, to attempt a theoretical estimate of the
ratio of noise to signal in the audio output under the condition of severe
impulsive interference.
The ratio of the peak, value of the pulse to the signal amplitude at the
frequency detector output is given by the ratio of the peak values of /(/),
as plotted in figure 5, to the values of Ai in the table above. To obtain
a result of practical significance, however, the effect of the audio circuit
should be taken into account. In the absence of specific information on
the structure of this circuit, we assume that the peak value of a pulse at
its output is equal to the area, or moment, of the pulse at its input times
twice the audio cutoff frequency. This is true for an ideal "square cutofif"
filter and not seriously in error for actual circuits. The area of the largest
pulse at the frequency detector output is approximately 2A/{a -\- A^) and
the value of A i, the signal fundamental amplitude, can be approximated by
A, = 4^_^' (52)
a^ -\- A^
(For the example above, this approximation gives Ai = .8, .5 and .2 as
compared to the exact values, .848, .478 and .191.) In this way we arrive
at the following estimate of the peak ratio of noise to signal in the audio
output :
Max. value of largest pulse _ Wa , ,
Signal amplitude ir x q
where ojq is the cutoff- frequency of the audio circuit, q the signal frequency
and X the frequency deviation ratio. Then .r^ is the "frequency swing" of
the transmitter, i. e., the maximum departure of the instantaneous frequen-
cy from its mean value. It is to be noted thai this formula is free from the
detector circuit parameters, a, A, R, and indicates that, to a first approxi-
mation, at least, the maximum ratio of noise to signal depends only upon
the audio circuit cutoff frequency and the FM swing. Furthermore, this
establishes a ceiling for the interference that will not be exceeded no mat-
ter how large the impulses may be.
Bibliography
1. Variable Frequency Electric Circuit Theory with Application to the Theory of Fre-
quency-Modulation, John R. Carson and Thornton C, Frv in Bell System Technical
Journal, Vol. XVI, No. 4, October 1937.
2. The Detection of Frequency Modulated Waves, J, G. Chaffee in Froc. LR.E.. Vol. 23,
May 1935. (Bell Tel. Sys. Monograph B-863)
3. Effects of Tuned Circuits upon a Frequency Modulated Signal, Hans Roder in Proc
LR.E., Vol. 25, December 1937.
4. The Reception of Frequencv-Modulated Radio Signals, Victor J. Andrew in Proc
LR.E., Vol. 20, May 1932.'
TRANSIENT RESPONSE OF AN FM RECEIVER 731
.S. The Phase Discriminator, K. R. Sturley in Wireless Engineer, February 1944,
6. Radio Engineers Handbook, F. E. Terman, First Ed., pp. 585-588.
7. Motor Car Ignition Interference, C. C. Eaglesfield in Wireless Engineer, 23, 1946.
8. Interference Problems in Frequency Modulation, F. L. H. M. Stumpers in Philips
Research Reports, 2, 1947.
9. On the Calculation of Impulse Noise Transients in Frequenc\- Modulation Receivers,
F. L. H. M. Stumpers in Philips Research Reports, 2, 1947.'
10. Fourier Integrals for Practical Applications, George A. Campbell and Ronald M. Foster
in Bell System Technical Journal. October 1928 — Monograph B-584.