Document text
Spectra of Digital FM
By R. R. ANDERSON and J. SALZ
(Manuscript received March 29, 19(55)
Formulas are derived for the spectral density function of an ensemble of
continuous-phase, constant-envelope FM waves. The modulation signals are
random time series of the form J^„a„g(t — nT), where g{t) is an arbitrary
pulse of finite duration rT, r ^ 1. The a n 's are independent random vari-
ables possessing identical but otherwise arbitrary probability distribution.
The derived results are general and are presented in terms of averages of
elementary functions. When the a n 's are discrete random variables, both con-
tinuous and discrete spectra are treated, and conditions in terms of the
modidation parameters are given under which discrete spectral lines are pres-
ent. Several of our specialized formulas are applicable in the study of mul-
tilevel FM data transmission systems as well as in pulse frequency
modulation.
I. INTRODUCTION
Progress in analysis of multilevel frequency shift keying (FSK) has
lagged behind that of binary. Inherent difficulties associated with an
increase in the number of levels are partly responsible, but activity also
has been inhibited by the general impression that multilevel FSK is in-
ferior to differential phase modulation with the same number of levels
operating in the same bandwidths.
Recent work, Ref. 1, has evolved design principles showing a possibil-
ity of substantial improvements in multilevel FM performance over
that formerly thought to be typical. Also there are many existing analog
channels, e.g., in microwave radio relay, which operate by FM. Attempts
to send digital data efficiently over such channels force consideration of
the multilevel FSK problem.
An important item in the statistical description of an information-
bearing signal is the spectral density, which defines the average power
density of the signal as a function of frequency. In addition to furnish-
ing an estimate of bandwidth requirements, the spectral density is
1165
1166 THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUG. 1965
critically important in optimization procedures for minimizing the effect
of channel noise subject to a constraint on mean total transmitting signal
power. Evaluation of mutual interference between channels also requires
knowledge of spectral distribution.
From the practical point of view the most interesting case of digital
FM is that in which the phase is continuous at the transitions, as may
be obtained at the output of a keyed oscillator. The memoiy thus intro-
duced makes the analysis far from trivial. So far as is known, the binary
case is the only continuous-phase FSK problem hitherto covered in the
literature. The present paper gives a complete analytic solution for a
general set of parameters.
An interesting feature is the extent to which sharp spectral peaks occur
near the discrete signaling frequencies. These peaks can, under certain
conditions, become delta functions indicating steady sine-wave com-
ponents. Such components make the design of optimum niters difficult
because the best results usually demand sharply tuned suppression of
the corresponding regions at the transmitter and complementary high
gain peaks at the receiver. Furthermore, the interference produced in
other channels by untreated peaked spectra can be inordinately severe.
One important result of the analysis is an establishment of conditions
on the frequency spacing relative to signaling rate such that spectral
peaking does not occur.
In this paper we derive compact formulas for the spectral density
function of an ensemble of continuous-phase, constant-envelope FM
waves. The frequency of the wave is switched every T seconds by a
known signal. The phase of the wave is so adjusted as to maintain con-
tinuity at the transitions. For example, when the baseband signal is a
rectangular pulse of T seconds in duration, the frequency of the wave
during each interval T may be one out of many different frequencies
picked at random. In general, the baseband signal is not time limited to
T seconds. This case will arise when the original time limited signal is
passed through a filter.
The random signal whose spectral density we wish to study has the
following standard representation:
S(t) = A cos Let + cod Z an I g(t' - nT)dt' + *>>
I «=° '• i (1)
where co c and oid are arbitrary angular frequencies. The angle <p is a uni-
formly distributed random variable (r.v.) on [0,27r] and the a„'s are in-
DIGITAL FM SPECTRA 1167
dependent r.v.'s with arbitrary but identical probability distribution.
The symbol A is an arbitrary real amplitude.
The only restriction on the baseband signal g(t) is that
9(0 - /*(<). O^tZrT (2)
\0, everywhere else
where r is an arbitrary positive integer. (We naturally require that g(t)
be integrable over this interval.) When the a n 's are binary r.v.'s and
g(t) is a rectangular pulse of T seconds duration, spectra and correlation
functions have recently been derived by Bennett and Rice, Ref. 2. Salz,
Ref. 3, extended Bennett's and Rice's results to include arbitrary a„'s
possessing arbitrary probability distributions.
In our treatment the distribution of the a„'s as well as g(t) is entirely
arbitrary. For instance when g(t) is a rectangular pulse and the a„'s are
discrete r.v.'s, the wave (1) represents the ensemble of multilevel FM
waves. If the a„'s correspond to the samples of speech taken every T
seconds, we have pulse amplitude modulation via frequency shift key-
ing. Many other applications depending on the choice of the a„'s and
g(t) may be cited, and are covered in our results.
Our method of attack on the problem is direct. We calculate the seg-
mented Fourier transform of (1), obtain its magnitude squared, average
over all random variables, divide by the length of the segment, and then
evaluate the limit as the length increases without bound. After consider-
able amount of bookkeeping, we obtain general formulas. We then
specialize the formulas, and investigate some interesting representative
cases. The general formula for the continuous spectrum is given in (31).
Equation (40) is the general formula for the discrete as well as the con-
tinuous spectrum.
II. GENERAL DEVELOPMENT
We found it easier to work with the complex representation of (1).
Therefore let
S(t) = j[z(t) +z*(t)] 0£*g * (3)
where
(n=« »t—nT \
wj + ^XiJ g(t')dt').
n=0 J-nT )
(4)
The symbol * denotes the complex conjugation.
1168 THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUG. 1965
Choose a finite interval [0,NT]. Over this interval the Fourier trans-
form of z(t) is
h~N-l r(k+l)T
Z(u,NT) = £ / z(t)e~ ut dt
k=0 J kT
= J* £ Q*
fr=0
where
r(k+\)T n = oo I .t-nT \
Qk = dlexpi(u) c — u)t II exp ilu^a,, I g(t')dt'\ . (6)
J kT »=0 { J-nT )
Set t — kT = y above to obtain
Qk = e ikT ^~ u) f T dy exp i (co c - u>)y ff P n , k (y) (7)
''O n-0
where
Pn,k(y) = exp i<o} d a n j_ g(t')dt'\. (8)
Since g (t) is time limited to rT, where r is a positive integer, it follows
that we can write P„,k(y) for ^ y ^ T 7 , as;
exp zco d a„ / <7r (0 dM , ^ w ^ & — r
ta*a» J o g(t')dt'\ , k - r + 1 S n :g k w
1 w > k.
With this representation, we can take the product in (7) running
from n = to n=k — r outside the integral sign since P n ,k(y) in this
range of n does not depend on y. P n ,k{y) depends on y only in the range
k — r + 1 ^w^/c. Making use of these facts, we write for Qk
Q k - e ikTv exp ilctr £" a„|F(v,a fc _ r+ i • • • a,) (10)
where
F (v,(lk-r+l • ■ • OLk)
rT n=fc f p U -(n-k)T \
-J f II expila) d a n ] g r {t')dt'\dy (u)
= /^ e** exp »/]£ a*_ r+m 7[2/ - (m - r) T]\ dy,
DIGITAL FM SPECTRA 1169
and
r*
a T = u d g T {t)dt
v = w c — to (12)
ri
V(t) = co, [ g(t)dt.
The segmented Fourier transform of the original signal (3 ) is
S( U ,NT) = \ [Z{u,NT) + Z c (co,iVT)]
(131
[A=AT-1 k=N-\ "I
e* E Qk + e'* E Qck\
k=Q k=0 J
= A
2
where Z e (u>,NT) is the Fourier Transform of Z*(co,N) given as
k=N—l r(k+l)T
k=N—l />(*+!) r
Z c (co,iVr) = E / z*U)e- iul dt
(14)
= ^<r v E Q*,
fc=AT-l
L
and
, (k+l)T
Qck = I dtexp —i(u c + u)t
J/cT
(15)
n=w ( <• t— n z |
•II exp — i{o) d a n \ g{t')dt'\.
n=0 { J-nT )
The magnitude squared of S(w,NT) averaged with respect to the
r.v. <p is
2 rk,s=N-\ k,a=N-l
. 2 rk,s=N-l k,8=N—l
(\S(co,NT)f), =^-\ E Q*Q.*+ E QcsQck* .
4 |_ fc,g=0 fc,«=0
(16)
The symbol ( • ) denotes the averaging operator.
From the definition of Q k in (10) we obtain
QkQs* = e iTHk - 8) F( v ,a k - r+1 • • • a k )F*(p,a,- r+1 ■ ■ • a.)
•exp*<a r E a « - E a » [ • ( 17 )
We observe that Q c ,Qck* equals Q*Q«* with co c — co replaced by u> c + «.
1170 THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUG. 1965
It is thus sufficient to continue our calculation using only the first sum
in (16).
If we let the first term in (16) be W + (u) and the second term W-{w),
the desired power spectrum G(o>) is by definition
G(«) = lim {2/TN){(W+ («)). + <TT_(co)> a } (18)
where the ensemble average is taken over the collection of r.v.'s a =
(ao , ai • • • a N ).
We now proceed to calculate the respective averages. From (16)
<IM«)>. =x*E * <&&*>.
4 k, s =o
= ~2
[ E <Q*Q.*>.] (19)
a 2 k=N-l
+ 4- £ <l Q*l 2 >«-
4 t=0
The symbol "Re[-]" denotes the real part.
To facilitate the evaluation of the averages, we rearrange the double
sums above in the following manner:
k,t=N-l s=N—2 k=N—l
E <Q*Q.*>. = E E <&&*>.
fc,8 = 8 = fe— 8 + 1
- E «M>.*>.+ <&+#.*>.
a=N—2
8=0
+ • • • + (Qs + rQ.*) a + iT <Q*Q.*>,
it=S + f+l
*=AT— 2 s=AT-3
(20)
= E <Q.+iQ.*>.+ E <Q 8+2 Q8*) a
8=0 8=0
+ ■■•+' E «M?.*>.
s=0
8=JV-r— 2 i=JV— 1
+ E E <Q*Q.*>..
8=0 A=8+r+l
Using the explicit representation of F and F* in 11, we obtain from 17
DIGITAL FM SPECTRA
1171
(21
(Qs + iQ**)a=fM, 1 ^j^r
= e Tv '{F (v,a a+j - r+1 ■■■ a a+j )
■F*{v,a,- r+l ••■ a,) exp i \a r J^ a„\)
[ Tl =j_r4-l J a
Jo •'o
• (expi<£ a s+j -r+mV[y - (m — r)T]
- (Z as-r+mVhj' - (m - r) n + a r f) a._ r+n j>
Using elementry manipulations we obtain
r T rT m=T—j
SM = e 4Tfj / dydy'e*^ U C a {V[y - (m - r)T]
J J m=l
- [y f - (m- r+j)T\) (22)
• "ff C{V[y - (m - r)T}\ U C*{V[y' - (m - r)T] - a r ).
The function
C a (s) = (e ia8 ) a = f e ial dF(a)
(23)
is the characteristic function of the r.v. a, and F(a) its probability
distribution.
We next calculate in the same manner as above
(QkQ.*)a = e iTHk -' ) (F(p i a^r + i ■■■ a k )F*{v,a,- r+l •■• a.)
n=Ar— r
•exp i {a T T] a„,
„=g_ r +l J /
\
(24)
= e ' r ^-"<F(,,a t _ r+1 ••■ a k )\(F >,a._ r+1 ••• a.)
\\
n=«-r+l )/a
■expilcxr 2 a »r/ C«* ' '(«»■)
1172 THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUG. 1965
when
k > s + r.
When k = s we have
(|0*| 2 >.= (\F(v,a k . r+1 ---a k )\\. (25)
From the definition of F or F* in (11) we obtain explicitly
(| F(.,a*_ r+1 • • • a t )| 2 ) a = f fdydy'e***-*"
Jo J o , v
26
m—r
•II C a {V[y - (m - r)T] - V[y' - (m - r)T]},
m-l
rT m=r
(F(v,a k - r+1 ••• a*)>. = / e**U C a {V[y - (m - r)T])dy (27)
•"0 m=l
and
(F*(v,a a - r+ i ■ ■ ■ a*) expilotr £ a n f) a
a-r+l
= / e-^II Ca*{7[7/ - (m - r)T) - a r \dy.
JO m=l
(28)
We now observe that the various averages in (22)-(25) are independent
of the indices k and s and therefore when we divide (19) or (20) by N
and take the limit as N approaches infinity we obtain
lim i (W + («) > a = ^ (I ^(a) I 2 >a + ir ^ ITifM
N :„N" " x //a 4 M v ' ' /a 2 l£S
+ <^(a)) a (^(a) expzL £ a n l) a (29)
I »-«-r+l J
t=N-r— 2 k=N-l
jV-»oo iV »=0 fc=s+r+l
where we set F*(a) = /^(^at-r+i • • ■ a k );f Jr (v) is defined in (22).
The limit in (26) can readily be evaluated provided | C a {a r )\ < 1.
This is the case when we have only continuous spectra. When | C a (a r )\ =
1, the evaluation of the limit is more involved. But this latter is precisely
the case when discrete spectra appear, which we shall study in a forth-
coming section. For the moment we proceed to evaluate the limit when
the modulus of the characteristic function evaluated at a r is less than
unity.
DIGITAL FM SPECTRA 1173
In this case, from (29) we obtain
, a=N— r-2 t=.V-l
lim Tr L L « Ca («r)
iV-ec JM s=0 k=s+r+l
= e iT " r \hn E [C a («r) exp (iTv)] 1 (30)
AT-»oo i=l
e L a (a r )
1 - e iT *C„(a r )
Using the definition of the spectrum in (18) and applying the explicit
representation of the averages computed in (22)- (30), we obtain finally
the positive image spectrum
m= r
U C a {V[y - (m - r)T] - V[y - (m - r)T]
I 2
+ yRe|g (Equation 22)
+ i , 7V< a ; ; / dye
1 — 6" 7 "C a (a r ) JO
■UC a {V[y- (m - r)fl| f d</<f
TO=1 "O
i»=r ]
•II <V{F[y- (m - r)T] - a r ] .
(31)
Although the final formula may appear rather complicated at first,
under close scrutiny it will be observed that the formula is in a con-
venient form for numerical calculation by a digital computer. At most a
double integral on a finite dimensional plane needs to be evaluated. We
will later demonstrate, by using a few interesting examples, how the
numerical work can be carried out, and the results will be exhibited
graphically.
III. SINGULAR CASES
So far we have considered only continuous spectra. In order to arrive
at the result of (31) we had to sum the series in (30), and that series
converges only when the magnitude of the characteristic function evalu-
1174 THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUG. 1965
ated at a r is less than unity. This turns out to be the requirement for
the spectrum to contain no lines.
Whenever the magnitude of the characteristic function, evaluated at
a r 7* 0, is unity we are no longer justified in using the results in (30)
since the series diverges. This behavior suggests the presence of discrete
spectral lines associated with periodicities in the original random process.
Mathematically, this result can only occur if the r.v.'s a are discrete
and have a definite relationship. The characteristic function of a con-
tinuous r.v. must satisfy | C„(s)| < | C„(0)| when s ^ 0. We proceed
to identify the conditions on the a n which give rise to a characteristic
function with unit modulus and therefore spectral lines.
Loeve, Ref. 4, shows that if | C a (s)\ = 1 for s ^ 0, the form of C a (s)
must be
fc=oo
C a (s) = T,P ak exp (isa k ) (32)
A-=0
where P„ k ^ 0, E*=" P ak = 1 and the random variables a* must satisfy
a k = (2w/s)k + b (33)
where b is an arbitrary constant.
Thus if | C a (a r )\ = 1 the r.v.'s must be integral multiples of one another
plus an arbitrary constant common to each of them.
Using (33) in (32) we see that the characteristic function becomes
exp (ibs). We set C a = exp (ibs) and evaluate the following limit:
A = e iTvr lim i- E E e^"-*
*" N a=0 * =8+r+1 (34)
(l=N , l=N ) W
JV-»oo [1=1 1\ J=l )
where
Let
7 = Tv + ba r .
A exp iy
i=i I — A
exp iy
(35)
= i -r- In (1 — A exp iy) , A < 1.
dy
The limit of A x as A — > 1 is the first sum in (34). Obviously this limit
DIGITAL FM SPECTRA
1175
does not exist in the ordinary sense. However the "distribution" limit,
denoted by lim (D) , Ref. 5, does exist. Barnard, Ref. 6, has shown that
lim (Z)) In (1 - A exp iy)
- 4 - 1 " (36)
= In | sin ( 7 /2 ) | + In 2 + i[(y/2) - irR^iy) - 2tM]
where M is an arbitrary integer, and
RiAy) = Zm(7 - 2irn) - IX -7 - 2im)
n=0 n=l
R r (y) - ftr(7 - ■■)
1, 7^0
tt ^>-\0 f 7<0.
He also proved that the right side of (36) constitutes a properly denned
generalized function or a distribution.
When (36) is differentiated with respect to 7 we obtain
i — Jlim In (1 — A exp iy)\
*w- / (37)
= ^COt| -i+7rE5(7 " 2tt?0
-v — — n
where 5 ( • ) is the well known dime delta-function.
The limit of the right hand sum in (34) approaches zero since this
sum is proportional to the derivative of the first sum divided by N.
Since the first sum is a generalized function or a distribution so is its
derivative. Consequently in the distribution sense the limit is zero.
When the characteristic function is of the form (32), which implies
that the r.v.'s satisfy (33), we observe that
C a \V[ij + pT] - a r \ = e- iba 'C a {V[y + pT]\ (38)
f p = r — m.
From this we obtain
UC a *\V[y + pT] - ar \ = e bari l[ C*\y + pT] (39)
m = l ">=1
Applying (38), (39) in (31), and replacing
"'CaM
1 - e iT 'CaM
1176 THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUG. 1965
by (37), we obtain for the continuous as well as the discrete spectral
density
<?+(*) - £ f f dydy'e^-^
•II C a [V[y + pT\ - V[y + pT]|
I 2 ( | , T p^r-1
/ dye* II
'0 p =0
+ % Re ( \\ dye* U C.{ V[y + P T] } (40)
•[l + J yT (* Z 5(7 - 27m) - ^COt |)J
y=2-i •)
+ g /*«J.
By recalling the definition of 7 in (34 ) we see that spectral lines occur
when
Tv + ba r = 27m, (41)
or
v = (2tji/T) - biptr/T)
and the minimum spacing Ap between the lines are given by
2t(u + 1) 2irn 1
Ap = 2ttA/ =
y T r
IV. SPECIAL CASES
In this section we select several special cases, believed to be of general
interest, and exhibit them graphically.
The first case we want to explore is that in which g(t) is a rectangular
pulse of unit height and the a„'s are discrete, that is, a frequency shift
keying system. For the binary case, the two frequencies are referred to
as mark and space frequency, and each is located co d from the carrier,
co c . For the multilevel case the frequency spacing is uniform. In reference
to our general formula (31), the following parameters apply:
r = 1
(42)
V(£) = co,,£, ai = Ud T.
When (42) is applied in (31) and after considerable manipulation we
obtain the specialized formula for the one sided continuous density
DIGITAL PM SPECTRA
1177
G(u>) = — \ {\ F(co — w c — aw d f) c
+ 2 Re
[ exp (iT(u
L 1 - C a l
i>) ) (F (« — C0 C — Cdd) )|
- C a (a> d T) exp (*T(» e - co))
(43)
where
and
Let
•{F*(ic — oj c — aw d ) exp (io) d aT)) a >,
w -^(-«»^
| Ca(a*T)| < 1.
= (to - a e )T/2ir
7 = (to — u c — aa),i)T/2,
(44)
(45)
and write (43) as
2A 2
(His) =
T
T - iy sin 7
2 e —
+ 2 Re
" ->'f> i T e -* sin A /£ -*> !E_? -**-
6 \2 e 7 /a\2 e 7
1 _ e-wC.iaaT)
w = 2 + Re lr
/,
- C a faT)e-*«
]
(40)
where
/i =
/sin" 7
/2 = / s iEJV-\ .
(47)
For binary frequency shift keying, the frequency deviation parameter,
fc, may be defined as the ratio of frequency shift (the difference between
mark and space frequency) to the signaling frequency (the sum of the
number of marks and the number of spaces in one second). That is,
1178 THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUG. 1965
The same definition holds for multilevel with frequency assignments
which make the frequency spacing uniform and equal to that for the
binary case. The frequencies nearest to the carrier are located at o» c ±
ud , the intermediate frequencies are at co c ± (2n — 1 )aid , and the ones
furthest from the carrier are at u c ± (N — 1 )o>d , where N is the number
of levels. Thus, the frequency band of the power spectrum will increase
approximately with N for constant k. The frequency band occupied
can be kept approximately the same by letting k decrease with N.
In this example, the random variables a„ are discrete and may be
represented as
a n = 2n-(N + l), n = 1,2, ••• ,2V. (49)
The argument of F in (43) is
y n = (co - co c - a n o> d )T/2 = (/3 - a n k/2)r, (50)
and the equations in (47 ) become
r _ i v f sin T- T
r r -h.+r,) -i
|_i - CaT*"']
cos (y n + 7m) — C a cos (y n + 7m — 2tt/3)
1 + CJ - 2C a cos 2tt/3
(51)
r 2 1 vr • / i n sin 7„ sin 7 m
h = W2 g g exp -t( 7 „ + 7m) — —
iv n=l m=l 7n 7m
Since we can alternatively write
On = ±(2n - 1), n = 1,2, ••• , N/2
we have that
CafaT) = Caikir) = J^P r (a n ) exp (iaddnT)
= (1/iV) £ [e** (2 »- u + e -*'«»-»] (52)
n=l
Nil
= (2/N) j>osM2n - 1).
n=l
The complex terms from (46) are
r -i(y n +y m )
B = Re
(53)
DIGITAL FM SPECTRA
1179
We can now write the JV-lcvel normalized spectral density as
0(0) J f [1 sin' 7,. , I v » sin y n sin y m ~\ (r ,,
A^T'N^i |_2 y,r + N £* 7« 7»< J ' W
where 7,, and # are given by (50) and (53).
Using several values of the two parameters — N, the number of levels
and k, the deviation — we have calculated numerically the spectral
densities from the relations given above, and plotted them against the
normalized frequency /3 = (co — oi c )T/2ir. On this scale, oyj occurs at
fc/2 for all values of N.
A large number of spectra are presented to indicate the way the
shape changes as the frequency deviation varies. For binary FM, these
are given in Fig. 1. We point out that the spectra for the binary cases
0.80
0.64
0.48
0.32
0.16
uj
\k=0.5
(a)
0.6 \
0.6§\/.
0.7 \y
0.80
0.64
0.48
0.32
0.16
0.5
1.0
(c)
k=i.o
_^
0.64
0.48
0.32
0.16
2.0
0.80
0.64
0.48
0.32
0.16
1
(b)
k 7
9 J J
0.95
V£k
0.5
1.0
0.5 10 1.5 2.0
NORMALIZED FREQUENCY
0.5
(g)-(i> c )T
2ir
1.0
1.5
2.0
(d)
1 1.5
A'A
k=i.i J
?>fl
I.P^
1.5
2.0
Fig. 1 — Spectral density for 2-level FM.
1180 THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUG. 1965
are the same as those given by Bennett and Rice, op cit, except that the
origin of the frequency scale has been shifted from the lower (space)
frequency to midband.
The spectra for 4- and 8-level FM are given in Figs. 2 and 3, respec-
tively. The multilevel cases show considerable similarity to the binary
ones. For small values of k, the spectra are narrow and decrease smoothly
towards zero. In particular, Fig. 4 shows the spectra for k = 1/N, and
these three are nearly identical. As k increases towards unity the spec-
trum widens, and as predicted, there tends to be concentration of power
about the a priori chosen frequencies. This concentration is especially
marked in the range 1 — 1/2N < k < 1 + 1/2N. Atk= 1, there is a
spectral line at the frequency f , and its odd multiples. As k increases
from unity the concentration at a>a is again broadened, and reduced
in intensity. We attempt to show these features in several plots as a
function of k.
Fig. 5 shows the decrease in spectral density at zero (mid-band) fre-
quency with increasing k. For higher level systems the zero-frequency
level is less for any value of k, but the decrease with k is slower.
The position of the spectral peaks, as a function of k for the 8-level
system is shown in Fig. 6. Other level systems show similar behavior.
For k = 1, the a priori chosen frequencies are (measured from the car-
rier) at ± (2n — l)/2, and the delta functions in the spectral density
occur at these same frequencies. For k < 1, the peaks of the spectral
density are no longer delta functions, and they occur nearer the carrier
than the chosen frequencies. They are further from the carrier for
k > 1.
An interesting phenomenon is observed for the cases where k is the
reciprocal of the number of levels. For these relations the principal por-
tion of the spectrum is confined to a relatively narrow band. These
curves have approximately the same shape as seen from the curves in
Fig. 4 and the following table:
No. of Spectral Density at Freq =
Levels k .25 .5 .75
2 0.500 0.810 0.500 0.090 0.00
4 0.250 0.750 0.470 0.117 0.011
8 0.125 0.735 0.430 0.124 0.013
The program was extended to calculate the power in the continuous
portion of the spectrum. For all values of N, and for k away from unity
this power is J, and for k = 1, this power is \ — 1/2N. Thus the total
power in the spectral lines is 1/2N. Clearly the power in each line is l/2iV 2
since they are assumed to have uniform likelihood of occurrence. It is
DIGITAL FM SPECTRA
1181
(a)
\k=o
.25
0.35\
0.40JLX
0.50 \
v^
080
0.5
0.75
0.32
(b)
0.8 0.7 .
k=0.6
U
~T~~—i_L
T" v 0.85
0.9
\^C\}
(c)
k = 0.95
u
_^
2.0
0.80
0.64
0.5
0.32
0.16
0.75 1.50 2.25 3.00
0.80
0.64
0.32
0.16
(e)
k=1.05
-J
.50 2.25 3.00 1.0
(io-u) c )T
NORMALIZED FREQUENCY —
2*7T
1.5
2.0
(d)
k = i.o
0.75 1.50 2.25
3.00
(f)
k=i.i
vk 2
1.5
2.0
3.0
4.0
Fig. 2 — Spectral density for 4-level FM.
1182 THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUG. 1965
0.80
0.64
0.48
0.32
0.16
0.125
0.100
u3 0.075
0.050
(a)
\k = 0.12
5
-"k 0,2
. \\0-3
0.4
X. n :
^
^S
2.0
0.025
0.125
0.100
0.075
0.050
0.025
\
0.125
0.100
0.075
0.050
0.025
3.0 4.0
0.125
k=0.6
07 \
(b)
a 0.8 jr^^
i\ o,9 n n
1 0.95 *— ""
W
2.0 3.0
k = 0.995
(c)
0.100
0.075
0.050
0.025
(e)
1
J
I
"\
I
*~\
\
\
k=
1.05
* \
2.0 3.0 4.0
NORMALIZED FREQUENCY
1.0
((U-(Oc)"T
21T
2.0
4.0
(d)
-• — ■
k = i
.0
\
1.0 2.0 3.0 4.0 1.0 2.0 3.0 4.0
O.I^S
0.100
k=i.i
(0
1.2
1.5 A /
0.050
0.025
3.0 4.0
Fig. 3 — Spectral density for 8-level FM.
DIGITAL FM SPECTRA
1183
o
uj 0.4
0.2
\*- — 2- LEVEL
^ 4-LEVEL
^Q---8-LEVEL
k = l/2
k = l/4
,k = l/fl
,-2 LEVEL
•£ „4 LEVEL
Sfsjfi LEVEL
NORMALIZED FREQUENCY
1.5
(W-tt> c )T
20
Fig. 4 — Spectral density for k = 1/No. of levels.
also very easy to show from (40) that this is the expected division of
power between the continuous spectrum and the discrete spectral lines.
We also thought it interesting to exhibit spectral shapes when the
a„'s have a gaussian probability distribution. This situation may arise
in pulse frequency modulation with baseband amplitude samples possess-
ing gaussian probability densities. In this case, the probability density
of the a„'s is
pr ^ 1 ( a " 2 \ I T li - fell ft*
p(an) = v^ * exp v 2?J = v^~a exp L~ — %r~ J ' (55)
and the characteristic function is
CaUT) = exp [- ^i^J = exp (-2 M 2 ),
H = oiaTcr/2,
y = fiir — an/a.
where
and
(56)
1184 THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUG. 1965
25
;.\o
« i.o
0.5
\ 2 -LEVEL
\ \ 4_L
8-LEVEL\.
evel\
0.5 1.0
NORMALIZED DEVIATION, k
1.5
Fig. 5 — Spectral density at midband — Discrete multilevel case.
Equation (51) can be written as
(7 - fa)*
1 /""sin 2 ? I
7l = V5^L~ exp L
2 M < ] d7 '
1 r sin 7 _, 7 r (7 - frp' i ,
(57)
Using elementary reductions I\ is written as
-=4@ fl jC(- 1 {-H-S(--<S)> "
DIGITAL FM SPECTRA
1185
Let
where
Then
where
t = (xz/2) — iy
k = iy
k = x + iy
X = \/2 M, V = irP/V2 M-
T 2{Ax - By) . e z cos 2tt/3 - 1
li z ~r s
(59)
(60)
(61)
5
4
/
&
Z
w
d
a
UJ
tr
U-
Q
S3
_i
<
5
cr
o
z
*:
2
*2
o
z
o
/ j /
1-
1 / r\ —
m
o
Q.
t
5 1.0
NORMALIZED DEVIATION, k
Fig. 6 — Position of spectral peaks — Discrete multilevel case.
1186 THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUG. 1965
A = e'" 2 Re
B = e - " Im
In this same manner,
h =
e dt
dtj
- I dz exp ( — i/Sir2 -^r) rf? -
(62)
2tt J_
Sin 7 -iy -iy.
and
7 2 2 =
A 2 - £ 2 - 2iAfl
Substituting (61) and (64) into (46) we obtain
A 2 T
--1
2x a L
2 (Ax - Py) + e * 2 cos 2r/3 - 1
+
(A 2 - B 2 ) (e~ x - cos 2t/3) + 2AB cos 2tt^
]•
(63)
(64)
(65)
cosh x 2 — cos 2tt/3
In this case the deviation is controlled by the parameter n = udTc/2.
In Fig. 7 we display the spectra for several values of this parameter.
When y. = 0, the spectrum is a delta function at /3 = (midband ) . As
ix increases the spectrum widens, approximately as /x and the midband
value decreases approximately as n~ , for small n, and as yT for larger
values. We show these two trends in Figs. 8 and 9.
The values of spectral density at /3 = 0, together with the asymptotes,
are shown in Fig. 8. Two estimates of the width are shown in Fig. 9.
From the definition of the Gabor bandwidth, Ref. 7,
"li
G(/3)/3 2 d/3
(To =
I
Jo
[ G(p)dp
-Jo
(66)
We note that o Q is very nearly equal to n/ir. That is, the standard devia-
tion of the power spectrum is the same as the standard deviation of the
input times the normalized deviation frequency :
og = o- duiT/2r). (67)
Another estimate of the width of the power spectrum comes from the
value of /3 at which the density has fallen by e, namely
DIGITAL FM SPECTRA
1187
2.5
20
I
5
Z
uj
0.2
0.4
1.0
0.8
0.6
0.4
0.2
0.5
1
(a)
L.
3.1
k
25
V^
s^O-5
06
0.8
(b)
v=l
-1.5
^
v^O
25
0.20
0.15
0.10
0.05
(c)
\/A=7T
21T
15 2.0 2.5
NORMALIZED FREQUENCY
2.0
((Q-U) C )T
2-7T
6.0
Fig. 7 — Spectral density for gaussian distribution.
ok = /- at
G(0)
(08)
At high values of a-, where the spectral density curves appear more
nearly gaussian, a E approaches a i; .
The gaussian case spectral density curves were also integrated to ob-
tain the power. We obtain \ in all cases, thus providing a check on our
work. It is interesting to note that even for n as low as 0.5, 98 per cent
of the power lies within 3cr of midband.
V. SUMMARY OF CURVES
Spectra are presented for 2, 4, and 8 equally-spaced uniformly dis-
tributed frequencies and for normally distributed frequencies. The gen-
t 2
0.2
\
\
\
v
\
\
^
\
N
VS.
w
0.01 0.02 0.05 0.1 0.2 0.5 1.0 2 5
DEVIATION OF FM SIGNAL, /I
Fig. 8 — Spectral density at midband — Gaussian case.
o
5 i.o
0.02
/ /
/
7 J
/
/ J
/ /
/ /
/ G
A
/
/
7
/
0.01 0.02
0.05 0.1 0.2 0.5 1.0 2
DEVIATION OF FM SIGNAL,/*
Fig. 9 — Standard deviation of spectral density function — Gaussian case.
1188
DIGITAL FM SPECTRA 1189
eral trend of the curves, as a function of the frequency deviation, is
shown in Figs. 5, G, 8, and 9. As expected, the band occupied by a sig-
nificant portion of the spectrum increases with the deviation.
For the discrete multilevel case, the frequency deviation parameter
is k = u,iT/t. For k = \/N the spectral density functions for different
N arc nearly identical. They are relatively narrow and decrease smoothly
to zero. Line spectra occur at the a prion chosen frequencies when k
is an integer.
For the gaussian case the deviation parameter is n = o-o> rf T/2. For large
n, the shape of the spectral density approaches a gaussian curve with a
standard deviation of <ru,,T '/2ir. For lower ju, the curves are slightly
narrower with correspondingly longer tails. The maximum value for
each m, which occurs at = 0, approaches Y^' for small y. and 2/ttm for
large values.
REFERENCES
1. Salz, J., Performance of Multilevel Narrow-Band FM Digital Communication
Systems, to be published.
2. Bennett, W. R., and Rice, S. 0., Spectral Density and Autocorrelation Func-
tions Associated with Binary Frequency Shift Keying, B.S.T.J., 4«, Sept.
3. Salz, j., Spectral Density Function of Multilevel Continuous Phase FM,
IEEE Trans. Information Theory, July, 1965.
4. Lodve, M., Probability Theory, D. Van Nostrand Company, pp. 201-202, 1960.
5. Lighthill, M. J., An Introduction to Fourier Analysis and Generalized Functions,
Cambridge University Press, 1959. _
0. Barnard, R. D., A Note on a Special Class of One-Sided Distribution bums,
B.S.T.J., U, Nov., 1965.
7. Gabor, D., Theory of Communication, J. Inst. Elect. Engrs., 98 III, p. 429.