Spectral analysis applied to digitally coded phase modulated radio frequency carrie

Woodbury, R. C.

English

NASA Technical Reports Server

' , 
A 
N A T l O N A L  A E R O N A U T I C S  A N D  S P A C E  A D M I N I S T R A T I O N  
Technicaf Report No. 32-893 
Spectral Analysis Applied to a Digitally 
Coded PM RF Carrier 
R. C. Woodbury 
Microfiche (MF) L 
ff 653 July 65 
N66 34406  - .  
UCCESSION NUMBER1 
I 7  
{PA'GESI w- 773// 
(NASA CR O R  TUX O R  AD NUMBER1 
677 
(CATEGORY1 
J E T  P R O P U L S I O N  L A B O R A T O R Y  
C A L I F O R N I A  INSTITUTE OF TECHNOLOGY 
PAS AD EN A. CALI FOR N I A 
August 15, 1966 
https://ntrs.nasa.gov/search.jsp?R=19660025116 2020-03-16T20:15:19+00:00Z
N A T I O N A L  A E R O N A U T I C S  A N D  S P A C E  A D M I N I S T R A T I O N  
Technical Report No. 32-893 
Spectral Analysis Applied to a Digitally 
Coded PM RF Carrier 
R. C. Woodbury 
E. L. Sheldon, Manager 
Environmental and Dynamic Testing Section 
JET P R O P U L S I O N  L A B O R A T O R Y  
CALIFORNIA INSTITUTE OF TECHNOLOGY 
P A S A D E N A .  C A L I F O R N I A  
August 15, 1966 
Copyright  0 1966 
Jet Propulsion Laboratory 
California Institute of Technology 
Prepared Under Contract N o .  NAS 7-100 
National Aeronautics & Space Administration 
JPL TECHNICAL REPORT NO . 32-893 
CONTENTS 
1 . Introduction . . . . . . . . . . . . . . . . . . . . .  1 
II . Square-Wave PM Carrier . . . . . . . . . . . . . . . .  2 
5 
7 
V . Conclusion . . . . . . . . . . . . . . . . . . . . . .  7 
Nomenclature . . . . . . . . . . . . . . . . . . . . . .  8 
References . . . . . . . . . . . . . . . . . . . . . . .  8 
111 . Even-Harmonic Energy vs Square-Wave Symmetry . . . . . .  
IV . Even-Harmonic Energy vs Trapezoidal-Wave Symmetry . . . .  
Appendix . . . . . . . . . . . . . . . . . . . . . . . .  9 
FIGURES 
1 . Typical PM calibration curve. sinusoidal modulation . . . . . . .  4 
2 . Square-wave PM line spectra (parameter: modulation index Dl . . .  5 
3 . Voltage ratio of fundamental to second harmonic in db as a 
function of percentage deviation from semiperiod . . . . . . . .  6 
6 4 . Voltage ratio of fundamental to second harmonic in db . . . . . .  
B-1 . Waveform geometry vs second-harmonic distortion . . . . . . .  11 
J P L  TECHNICAL REPORT NO. 32-893 
* 
ABSTRACT 
To ensure a high degree of confidence in calibrating a digitally coded 
phase-modulated ( PM ) system, it is desirable to calibrate directly in 
terms of the digital modulation. The complexity of the spectra resulting 
from the single-tone case suggests that the spectra from a digitally 
modulated PM system must be extremely complex and are, therefore, 
not usable for direct calibration. 
This Report demonstrates that the contrary is true, by investigating 
the spectrum of a square-wave PM carrier. It is shown that the spectra 
obey simple relationships which permit accurate calibration of a PM 
sys tem. 
The geometric tolerances of the square- and trapezoidal-wave modu- 
lation functions are investigated in terms of harmonic content. It is 
shown that the symmetry and the rise-time of these functions provide 
a sensitive indication of the performance of a PM system. 
1. INTRODUCTION 
Calibrating and establishing the performance criteria of a digitally coded PM 
communication system is intimately related to a knowledge of the spectral char- 
acteristics defining this class of modulation. 
This Report derives the spectral characteristics of a square-wave phase- 
modulated (PM) RF camer and also demonstrates the geometric tolerances for 
the modulation function in terms of harmonic content. 
It is of particular interest to note the general simplicity of the square-wave 
PM spectra in contrast to the relative complexity of the single-tone PM case. The 
unique properties of the square-wave PM spectra provide a simple means of 
accurately calibrating a digitally mded PM system in terms of the clocking 
frequency of the digital system. 
1 
JPL TECHNICAL REPORT NO. 32-893 
II. FOURIER ANALYSIS OF SQUARE-WAVE PHASE MODULATION 
A PM function can be expanded into a Fourier series 
such that (Ref. 1) 
By Euler's formula 
Therefore 
where 
The function 9 ( t )  is a square wave and may be expressed 
analytically as 
To avoid integrating through a finite discontinuity, C,, 
must be evaluated as the sum of two integrals: 
Evaluating Part A by substituting for + ( t )  in Eq. (2) 
its value of +p, we have 
By Euler's formula 
e-inT = cos np - j sin nr = ( -  1)" for all n 
Therefore 
A =-[1- (-1)"I i2nn 
Part B may be evaluated by substituting for ( t )  in 
Eq. (2) the value - P :  
2 
e-iS 
B = -  -[1 - (-l)"] j 2 n ~  
Since C, is the algebraic sum of the 
values for A and B ,  
for all n 
corresponding 
PI 
The spectral distribution of the square-wave PM wave 
can now be obtained by substituting in Eq. (1) the derived 
value of C,,.  Thus 
Equation (3) can be evaluated for k n ,  ultimately leading 
to 
When n is even, 
When n is odd, 
1 - ( - 1 ) n  2 
n#O - - n n 
and 
( - 1 ) n  = -1 
J P L  TECHNICAL REPORT NO. 32-893 
When n is zero, 
1 -  ( - 1 ) n  0 - - 
n 0 
Equation (4) cannot be evaluated for C,, = 0, the RF 
carrier, because the indeterminate 0/0 occurs. Also, 
L'Hospital's rule fails because of the appearance of the 
In (- 1) when 1 - ( - 1)" is differentiated. Therefore, it be- 
comes necessary to revert to the exponential form of C,; 
applying L'Hospital's rule we find 
The complete square-wave PM wave can be precisely 
stated as follows: 
If Eq. (5) is viewed on a spectrum analyzer, the absolute 
magnitude of the nth term will appear as: 
21 sin p 
linl = zcosp + -7n (n is odd) 
Equations (4) and (5) establish several facts, eat-. of 
1. When n is an even integer, all even sidebands dis- 
appear because of the term 1 - (- 1)" 
2. When p is an odd multiple of 7/2, the carrier ampli- 
tude is zero. The spectrum envelope of the sidebands 
describes the hyperbola K/'n 
3. When j3 is an even multiple of ~ / 2 ,  all sidebands 
are zero and the carrier is a maximum 
which has been substantiated experimentally: 
4. Energy in the carrier is transferred to the sidebands 
as a cosinusoidal function of p and the total energy 
remains constant 
It is interesting to note that there is a unique cor- 
respondence between the spectrum of a square-wave 
balanced modulated carrier and a square-wave PM camer 
for values of the modulation index p equal to odd multi- 
ples of 7/2. 
The surprising simplicity of the square-wave PM camer 
can be appreciated by comparing Eq. (5) to the expression 
for single-tone PM: 
where In@) is an nth-order Bessel function. It is im- 
mediately evident from Eq. (6) that to calibrate a PM 
system using sinusoids involves the uncertainty inherent 
in equating sidebands and determining zeros in accord- 
ance with ],, (8). Also, the bandwidth required for a con- 
stant depends upon the modulation frequency and, if  
inadequate, may introduce subtle calibration errors. 
Figure 1 is a typical sinusoidal PM system calibration 
curve which illustrates the technique. 
In calibrating a digitally coded PM system, it is 
desirable to calibrate in terms of a square-wave. Only in 
this manner can the system performance be established 
with a high degree of certainty. 
Equation (5) indicates that the modulus of the unit 
carrier vector, ejOot, varies cosinusoidally with the modu- 
lation index. Thus, p can be accurately calculated from 
trigonometric tables by simply noting the ratio of the 
unmodulated carrier to its modulated value. A crosscheck 
can be obtained by multiplying the value of the un- 
modulated carrier by 2/7 and the sine of the modulation 
index, and comparing this number with the amplitude 
of the first sideband. 
Figure 2 is a plot of Eq. (5) for modulation indices 
7/2, 7/6, and 7/18. Although the sideband spectra are 
discrete, the envelope contour has been shown for 
clarity. It should be noted that for symmetrical square- 
wave modulation, the sidebands are composed of odd 
harmonics. 
The two essential conditions that a communication 
system must satisfy in order that the modulation shall be 
geometrically similar to the demodulated product are 
that the system must be linear and nonselective with 
regard to frequency (Ref. 2). Equation (5) uniquely 
describes a square-wave PM waveform. This uniqueness 
is predicated on the preservation of amplitude and phase- 
angle components. Square-wave modulation is a partic- 
uiarIy good method of investigatiug the chaiacteristics of 
3 
J P L  TECHNICAL REPORT NO. 32-893 
31.9 Mc I 14 
12 
Y 
0 
0, 
Q 
I 
0 
Y 
0 
0) a 
u) 
0 > 
- IO 
= 
' 8  z 
l- 
-J 
3 
0 
= 6  
W z 
u) 
0 Y
0 
a 
n 
0 
0 
0 4  
2 
0 
I I 
CRYSTAL 
OSCILLATOR 
a PM channel, For example, any asymmetry in the sys- 
tem will cause the appearance of even harmonics. If 
this asymmetry is caused by nonuniform phase-shift, 
then the total R F  power will not change because the 
effect of phase distortion is to remove part of the energy 
from the beginning of the signal and replace it later in 
the signal. Thus phase distortion cannot modify the en- 
ergy in the signal. If the RF power changes with an in- 
crease of the modulation index without introducing 
appreciable even-harmonic distortion, the bandwidth of 
x 5  X 6 0  7 IF POWER CHANGES) - MULTIPLIER MU LT I P L I ER 
the system is symmetrical yet inadequate. In this case the 
demodulated waveform will be trapezoidal. 
In summary, the use of a square-wave to calibrate a 
PM system provides an accurate method of calibration 
through the linear measurement of line spectra which 
behave according to simple relationships. A high degree 
of confidence can be achieved by observing the constancy 
of the RF power level and the absence of even-harmonic 
or sideband components. 
4 
J P L  TECHNICAL REPORT NO. 32-893 
1.0- 
-0.8 
'*I -7w1 -%I - 
LOWER SIDEBANDS 
I 
YI -'I 00 *I *I 501 7 9  
CARRIER I UPPER SIDEBANDS 
Fig. 2. Square-wave PM line spectra (parameter: modulation index p) 
111. EVEN-HARMONIC ENERGY VERSUS SYMMETRY OF A SQUARE-WAVE 
In order to calibrate a PM system using a square-wave, 
it is necessary to estimate the effect of waveform s p -  
metry in terms of even-harmonic components. The follow- 
ing derivation provides this information. 
The voltage ratio of the nth to the (n + 1) th harmonic, 
expressed in db, is 
I n  I
A rectangular wave of unit amplitude can be defined 
I as (see Fig.-3): 
where 
where the absolute magnitude of the nth harmonic as 
displayed by a spectrum analyzer is and 
5 
I 
JPL TECHNICAL REPORT NO. 32-893 
4 1  2 sin - T xt, ,  - 
- secT - xt, ,  at0 2 sin - cos - T T 
PERCENT DEVIATION FROM SEMIPERIOD, ( Z A t / T )  1100) 
Fig. 3. Voltage ratio of fundamental to second harmonic 
in db as a function of percentage deviation 
from semiperiod 
1 I I 
Of particular interest is the voltage ratio of the funda- 
mental to the second harmonic (n = l): 
Formula (7) was evaluated for a unit period T within 
the interval 0 4 t,, TIT Tabulated values may be 
found in Appendix A, 
Figure 3 is a graph of the voltage ratio, in db, of the 
fundamental to the second harmonic as a function of 
the percentage deviation from the semiperiod T/2. It 
should be noted that the fundamental frequency in this 
case will appear as the first sideband of the PM carrier. 
Figure 3 indicates that 2% asymmetry will result in a 
second-harmonic component which is 30 db down from 
the fundamental. 
IV. EVEN HARMONIC ENERGY VERSUS SYMMETRY AND 
RISE TIME OF A TRAPEZOIDAL WAVE 
Although the rectangular wave represents the design 
goal of a digital system, it very often happens that thtx 
best approximation that can be achicwd is in the form 
of a trapezoid. Tlic following analysis investigates thc 
effect of symmetry and rise time of a trapezoidal wavc 
in terms of harmonic contmt. 
The nth harmonic of a trapezoidal w a v ~  is given by 
(see Fig. 4) 
xn (t ' l  + t l )  1 2T c,, = [-I(Tn)'tl sin?][ sin 
The ratio of the ntli to the (ti + 1) t h  harnionic is 
6 
J P L  TECHNICAL REPORT NO. 32-893 
0 6  
0 
X 
s s  
9 
f 4  
I 
1 3  
8 
f 
z *  g z r  
s 
8 0  
6 
0 
n VOLTAGE RATIO W FUFIMMENTAL TO SECOND HARMONIC, db 
Fig. 4. Voltage ratio of fundamental to second harmonic 
in db 
As with the rectangular wave, the ratio of the funda- 
mental to the second harmonic (n = l) is of primary 
value. Thus 
2 C ,  = [,e,+][ sec @" T + tl) 1 C 
where 
0 I (to + t l )  L T / 2  
The voltage ratio of the fundamental to the second 
harmonic, expressed in db, is 
db = 2ologlo [ set%][ sec T 
A plot of 100 [ ( T j 2 )  - (to + t , ) ] / ( T / 2 )  versus Eq. (S), 
with T taken as unity, is given by Fig. 4. The parameter 
t l / T  is a measure of the rise-and-fall time of the trapezoid. 
Figure 4 indicates that less than 2% asymmetry is per- 
mitted before the ratio of the fundamental to the second 
harmonic exceeds 30 db when the rise time is zero (the 
rectangular case). When the rise-time is 20% of the period 
( t , /T  = 0.2), asymmetry of 2.5% is permitted before the 
above ratio is exceeded. 
Appendix B contains tabulated values giving the ratio 
of the fundamental to the second harmonic in db as a 
function of the percentage deviation from the semiperiod 
100 [ (T /2 )  - (to + t , ) ]  /( T / 2 )  in terms of the parameters 
t , ,  and t,). It should be noted that ( t ,  + to) are restricted 
to the range 0 4 (tl + to) A T/2.  Figure B-1 illustrates 
the table; t ,  was not extended beyond 0.2 because, in 
general, digital systems will seldom exhibit values of 
t , / T  exceeding 0.06. 
V. CONCLUSION 
The Fourier analysis of a square-wave PM RF carrier leads to a 
remarkably simple expression in which the carrier and sideband 
energy vary mutually as the cosine and sine of the modulation index. 
The resultant line spectra are indistinguishable from that of a square- 
wave function. This is in contrast to the multiplicity of line spectra 
resulting from single-tone modulation. 
The fact that the spectra of the square-wave PM carrier are the 
same as that of the square-wave modulation modified by the sine of 
the modulation index provides an easy method of calibrating and 
confirming the operation of the PM system. 
7 
JPL TECHNICAL REPORT NO. 32-893 
The existence of even-harmonic components indicates that the 
modulation is asymmetrical. A tolerance for symmetry can be estab- 
lished in terms of the ratio of the fundamental to the second harmonic. 
If 30 db is arbitrarily established, 2% asymmetry can be tolerated for 
the square-wave. The tolerance for the trapezoidal case appears to 
be less critical because 2.5% asymmetry is possible for a trapezoid 
whose rise-time is 20% of its period. 
N O M E N C L A T U R E  
C. Amplitude of nth Fourier term (harmonic) t ,  Rise-and-fall time of a trapezoidal wave 
db Decibel 
e Base to napierian logarithms, 2.71828 
T Repetition period 
/3 Maximum phase excursion expressed in radians 
i Instantaneous value of RF spectra 
1 Maximum amplitude of RF carrier 
j Complex operator: j' = - 1  
(commonly referred to as modulation index: 
p = A 0  = A*,/*,, 
;T 3.14 
+ A phase constant In ( p )  nth-order Bessel function 
I 2 ~ f :  angular frequency of fundamental com- 
ponent of a square-wave modulating function, n An integer 
t Time, sec rad/sec 
t , ,  Time duration of flat-top region of a waveform ( - I ( ,  Angular frequency of RF carrier, rad/sec 
REFERENCES 
1 . Cuccia, G. L., Harmonics, Sidebands and Transients in Communication Engineering, 
New York: McGraw-Hill Book Company, Inc., 1952, p. 308. 
2. Cherry, C., Pulses and Transients in Communication Circuits, New York: Dover 
Publications, Inc., 1950, p. 147. 
8 
JPL TECHNICAL REPORT NO. 32-893 
APPENDIX A 
Square-Wave 
0.500 
0.499 
0.490 
0.480 
0.460 
0.440 
0.420 
0.400 
0.300 
0.200 
0.100 
0.000 
0.0 
0.2 
2.0 
4.0 
8.0 
12.0 
16.0 
20.0 
40.0 
60.0 
80.0 
100.0 
infinity 
49.897 
30.056 
24.038 
18.037 
14.544 
12.086 
10.2M) 
04.614 
01 440 
00.432 
00.Ooo 
At = 1/2 - tn 
T = 1.00 
~ 
APPENDIX B 
1. Trapezoidal-Wave 
to t r  
0.00 0.04 0.06 0.10 0.12 0.20 0.08 - 
x db db db 7. db db db db 
03.68 
04.10 
04.58 
05.13 
05.75 
04.45 
07.26 
08.18 
09.25 
10.52 
12.04 
13.92 
16.38 
19.88 
25.88 
infin 
0.00 
0.02 
0.04 
0.06 
0.08 
0.10 
0.12 
0.14 
0.16 
0.18 
0.20 
0.22 
0.24 
0.26 
0.28 
0.30 
00.00 
00.01 
00.06 
00.15 
00.27 
00.43 
00.63 
00.86 
01.14 
01.46 
01.84 
02.26 
02.74 
03.29 
03.91 
04.61 
100 
96 
92 
88 
84 
80 
76 
72 
68 
64 
60 
56 
52 
48 
44 
40 
00.13 
00.22 
00.34 
00.50 
00.70 
00.93 
01.21 
01.53 
01.91 
02.33 
02.81 
03.36 
03.98 
04.68 
05.48 
06.4 1 
92 
88 
84 
80 
76 
72 
68 
64 
60 
56 
52 
48 
44 
40 
36 
32 
88 
84 
80 
76 
72 
68 
64 
40 
56 
52 
48 
44 
40 
36 
32 
28 
- 
84 
80 
76 
72 
68 
64 
60 
56 
52 
48 
44 
40 
36 
32 
28 
24 - 
80 
76 
72 
68 
64 
60 
56 
52 
48 
44 
40 
36 
32 
28 
24 
20 - 
76 
72 
68 
64 
60 
56 
52 
48 
44 
40 
36 
32 
28 
24 
20 
16 - 
60 
56 
52 
48 
44 
40 
36 
32 
28 
24 
20 
16 
12 
08 
04 
00 - 
00.87 
01 .w 
01.30 
01.58 
01.90 
02.27 
02.70 
03.18 
03.72 
04.34 
05.05 
05.85 
06.77 
07.85 
09.1 1 
10.63 
01.26 
01.50 
01.78 
02.10 
02.47 
02.89 
03.37 
03.92 
04.54 
05.24 
06.05 
06.97 
08.04 
09.31 
10.83 
12.72 
00.31 
00.43 
00.59 
00.78 
01.02 
01.30 
01.62 
01.99 
02.42 
02.90 
03.44 
04.06 
04.77 
05.57 
06.49 
07.57 
00.55 
00.71 
00.91 
01.14 
01.42 
01.74 
02.1 1 
02.54 
03.02 
03.57 
04.18 
04.89 
05.69 
06.62 
07.69 
08.95 
Parameter: t1 
Independent vorioble: to 
'h = 100 [[T/2) - 
T = 1.00 
9 
J P L  TECHNICAL REPORT NO. 32-893 
28 
24 
20 
16 
12 
08 
04 
02 
00 
10.27 
12.15 
14.61 
18.10 
24.1 1 
infin 
APPENDIX B (Cont’d) 
I. Trapezoidal-Wave (Cont’d) 
db 010 
25.20 
26.53 
26.83 
28.12 
30.05 
32.55 
36.07 
42.09 
46.54 
56.10 
3.5 
3.0 
2.9 
2.5 
2.0 
1.5 
1 .o 
0.5 
0.3 
0.1 
1.8 
0.4025 
0.4050 
0.4055 
0.4075 
0.4100 
0.4125 
0.4 150 
0.4 175 
d b  % 
25.48 3.5 
26.81 3.0 
27.1 1 2.9 
28.39 2.5 
30.33 2.0 
32.83 1.5 
36.35 1 .o 
42.37 0.5 
-~ 
tu db 
0.00 I 0.04 0.06 0.10 0.12 0.20 0.08 - 
010 
24 
20 
16 
12 
08 
04 
00 
__ 
__ 
- 
% db d b  d b  d b  db % 
0.32 
0.34 
0.36 
0.38 
0.40 
0.42 
0.44 
0.46 
0.48 
0.49 
0.50 
05.41 
06.34 
07.41 
08.68 
10.20 
12.08 
14.54 
18.03 
24.04 
30.05 
infin 
28 
24 
20 
16 
12 
08 
04 
00 
10.47 
12.36 
14.82 
18.31 
24.32 
infin 
20 
16 
1 2  
08 
04 
00 
12.52 
14.98 
18.47 
24.47 
infin 
16 
12 
08 
04 
00 
~ 
1 2  
08 
04 
00 
__ 
15.17 
18.67 
24.67 
infin 
08.83 
10.35 
12.24 
14.70 
18.19 
24.19 
infin 
Parameter: t ,  
Independent variable: t8) 
= 100 [ ( T / ~ I  - (tal + t , ) i  /(TIZ) 
T = 1.00 
db = 20 log,<\ [sec wt,/Tl [sec (tal + tl) /T] 
I I. Trapezoidal- Wave Intermediate Values 
t, = 0.06 f, = 0.04 
db d b  t,, 
0.4825 
0.4850 
0.4855 
0.4875 
0.4900 
0.4925 
0.4950 
0.4975 
0.4985 
0.4995 
0.4425 
0.4450 
0.4455 
0.4475 
0.4500 
0.4525 
0.4550 
0.4575 
25.27 
26.60 
26.90 
28.19 
30.13 
32.62 
36.14 
42.16 
3.5 
3 .O 
2.9 
2.5 
2.0 
1.5 
1 .o 
0.5 
0.4225 
0.4250 
0.4255 
0.4275 
0.4300 
0.4325 
0.4350 
0.4375 
25.35 
26.69 
26.99 
28.27 
30.21 
32.71 
36.23 
52.25 
2.9 
2.5 
1.5 
2’o I 
1 .o 
0.5 1 
I, 0.08 71 2 .o 
1 .o 
0.5 
f ,  = 0.12 
db 
25.83 
27.17 
27.46 
28.75 
30.69 
33.19 
36.71 
42.73 
010 
0.3825 
0.3850 
0.3855 
0.3875 
0.3900 
0.3925 
0.3950 
0.3975 
25.63 
26.97 
27.27 
28.55 
30.49 
32.99 
36.51 
42.53 
3.5 
3.0 
2.9 
2.5 
2.0 
1.5 
1 .o 
0.5 
0.3625 
0.3650 
0.3655 
0.3675 
0.3700 
0.3725 
0.3750 
0.3775 
10 
JPL TECHNICAL REPORT NO. 32-893 
to’Cl20 1.84 db 
A& 
I 
t,’ 0.00 00% 
to=0.50 infinity 
APPENDIX B [Cont’dl 
+ 
II. Trapezoidal-Wave Intermediate Values (Cont’dl 
t,, 
0.2825 
0.2850 
0.2855 
0.2875 
0.2900 
0.2925 
0.2950 
0.2975 
0.2985 
0.2995 
0.2999 
t, = 0.20 
db 
27.04 
28.38 
28.67 
29.94 
31.89 
34.39 
37.92 
43.93 
48.38 
57.92 
71.87 
x 
3.5 
3.0 
2.9 
2.5 
2.0 
1.5 
1 .o 
0.5 
0.3 
0.1 
0.02 
1 1 
to= 0.20 12.04 db 
t ,  = 0.20 0.02% 
to=O.299 71.8 db 
11 


Spectral analysis applied to a digitally coded PM RF carrier

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