This paper presents a computational technique to determine the occupied bandwidth for a class of PCM/PSK/PM signals. The signals considered here employ ...
CLOSED FORM EXPRESSIONS FOR COMPUTING THE OCCUPIED BANDWIDTH OF PCM/PSK/PM SIGNALS
Tien Manh Nguyen Jet Propulsion Laboratory 4 8 0 0 Oak Grove Drive Pasadena, California 91109 ABSTRACT where Sd(f) is the well-known power spectral density for an equiprobable NRZ binary data. This paper presents a computational technique The first term shown in Eqn ( 3 ) is the residual to determine the occupied bandwidth for a class carrier spectral component, and the second term of PCM/PSK/PM signals. The signals considered is the data component. here employ either a squarewave or sinewave subcarrier. Closed form expressions for the The CCSDS also recommends that the subcarrier frequency-to-bit rate, n, to be an integer [2], occupied bandwidth as a function of the i.e. , subcarrier frequency-to-bit-rate ratio with modulation index as a parameter are derived n = f / R I where n is integer and R, using linear-fitting techniques. (4) ig tie data rate
1. INTRODUCTION
If we let p% be the percentage of power containment in the data component, and BW, be the occupied bandwidth with p% power contained in it, then, from [3], we can show that BW, related to p% through the following relationship BWl cos2(m) + sin2(m) S1,(f)df = p% (5) -BW,
The topic of occupied bandwidth for the space telemetry signals, employing residual carrier modulation technique, has been discussed in the past [l]. However, the results presented in [l] are only applicable for a specific case of interest, e.g, 99 % power containment with the subcarrier frequency-to-bit-rate ratio, n, equals to 3 and high modulation index. This report generalizes the results presented in [I] to include all practical cases. Closed form expressions are obtained for the occupied bandwidth as a function of n for a practical range of modulation index. For space applications, the typical signal recommended by the CCSDS is the residual carrier signal using the subcarrier for the data [2]. The subcarrier is used to separate the data from the residual carrier. Mathematically, the space telemetry signal can be expressed by S(t)
= ( 2
Asin[o,t + md(t)P(t)]
(1)
where A is the rms voltageroc is the angular carrier center frequency in rads/sec, m is the modulation index in radian, d(t) is the NRZ binary valued data sequence, and P(t) is the subcarrier waveform. Expanding Eqn (1) we obtain S(t) = J2 A[sin(o,t)cos(mP(t))
+ d(t)cos(o,t)sin(mP(t))1
where SIN(f) is the normalized power spectral density of S(f) (See Equation ( 3 ) ) which is given by SIN(f) = 4 A2sin2(m)C ( S f f 2k-1 f,J + 7r2 kt1
-'L(w
S
f-f + 2k-1 f ,U
)
*2+
It is noted that the spectral density SIN(f)has taken f = f, as its origin. If we denote M as the occupied bandwidth with p% power containment-to-bit rate ratio, i.e.,
M
=
BW,/R,
(7)
then we can show that, from Eqns (5) and (6), the p% power containment in the data channel is
(2)
The subcarrier waveforms recommended by the CCSDS are the squarewave and sinewave for deep space and near earth missions, respectively. In the followings we will derive the occupied bandwidth for these cases. Approximate closed form expressions for the occupied bandwidth are obtained using linear-fitting technique.
Eqn ( 8 ) can be used to compute the occupied bandwidth for Category B missions for any values of p%, n and m. From [3], the occupied bandwidth is defined for p% = 99%. 2.2 OCCUPIED BANDWIDTH FOR PCM/PSK/PM WITH
SINEWAVE
When P(t) is a sinewave subcarrier, the power spectral density is given by [l]
2. MATHEMATICAL MODELS AND ANALYSES 2.1 OCCUPIED BANDWIDTH FOR PCM/PSK/PM WITH
S2(f)
SQUAREWAVE
=
A2JO2(m)6(f-f,) +A2 C Jk2(m)(6(f-fC-kf,,)+ k even
When P(t) is a unit power squarewave subcarrier 6(f-fC+kf,,)) + A2 C J 2(m) (Sd(f-f,-hf,,)+ of frequency f,,. It has been shown in [l] that h odd the power spectral density for this case is (9) given by The first term shown in Eqn (9) is the residual S(f) = A2cos2(m)6(f-f,) + (3) spectral component, the second term is the intermodulation loss component, and the third 4 A2sin2(m)C S, ( f-fc-( 2k-1) f,,l+Sd ( f;fc+ (2k-1) term is the data component. Ir' ktl (2k-1)
fscu 414
CH3044-5/91/0000-0414$01 .OO 0 1991 IEEE -
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Let BW, be the occupied bandwidth with p% power BW, = (20.7n + 7.0)RS, 0.7 5 m < 0.9 rads containment, and L be the ratio of the occupied bandwidth-to-subcarrier frequency ratio, i.e., BW, = (41.7n + 9.5)Rs, 0.9 5 m < 1.4 rads For
PCM/PSK/PM
with
(15) (16)
sinewave Subcarrier and
2 < n < 40,
Similarly, Using (9) we can show that the occupied bandwidth with p% power containment in the signal is found to be L L C J,'(m)+ C J2(m)a,(L) = p% J;(m)+2 (11) k even h od$
BW,
=
BW,
=
+ 1.2)RS, 0.5 (n + 2.5)Rs, 0.8 (n
BW, = (n
+
< m < 0.8 rads
(17)
5 m < 1.1 rads
(18)
5.2)Rs, 1.1 5 m < 1.6 rads
where ah(L) is defined as
4.
(19)
CONCLUSION
Simple closed form expressions for computing the occupied bandwidth for a class of PCM/PSK/PM signals have been derived. It is found that the occupied bandwidth for squarewave subcarrier at low modulation index Note that, From Eqns (4) and (10), f and Lf is about 10 to 15 times larger than the are defined as nRs and the occupied %andwid& with p% power containment, respectively. Thus, sinewave. from Eqns (4) and (lo), the occupied bandwidth ACKNOWLEDGEMENT BW,-to-bit rate ratio is given by (13) The work described in this paper was carried out at the Jet Propulsion Laboratory, Eqn (11) can be used to calculate the occupied California Institute of Technology, under contract with the National Aeronautics and bandwidth for PCM/PSK/PM signal with sine- wave subcarrier as a function of n for 99% power Space Administration. containment and a practical range of modulation REFERENCES index.
M, = BWJR, = Ln
Tien M. Nguyen, "Occupied Bandwidth of Square waveform/Sine Waveform as Subcarrier in Residual Carrier Systems,'I JPL-IOM 3396-86-144, 10/11/1986.
3. NUMERICAL RESULTS
Using Eqns (8) and (11), the computation of the occupied bandwidth for space telemetry signals has been performed on a digital computer. The closed form expressions for the occupied bandwidth are then obtained using linear curve fitting technique. For PCM/PSK/PM with squarewave subcarrier and 2 < n < 20, the results for 99% power containment in the signal are : BW, = (12.6n + 9.3)Rs, 0.5 < m < 0.7 rads Ftgure
1.
800'00~
Occupled FuncLlon
CCIR, 1982, Recommendation 328-5, "Spectra and Bandwidth of Emissions,11 Volume I, Recommendations and Reports of the CCIR, Geneva, Switzerland.
(14)
B o n d u l d L h w i t h 99 16 P o u e r Containment o s o o f n f o r V a r l o u s Values o f Modulotlon Indox
A: 0.90 $ m < 1 . 4 r a d s B: 0.70 S m < 0 . 9 r a d s
c:
Consultative Committee for Space Data Systems (CCSDS), Recommendations for Space Data System Standards, Radio Frequency and Modulation Systems, 401.0-B-1, Blue Book, January 1987.
F i y u r n 2 . o o c u p Led B a n d u LdLh FuncLlon
0.55 S m < 0.7 r a d s
n o
on-
600,00
D DEEP SPACE MISSIONS 400.00
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for
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1.10 5 m < 1 . 6 0 : 0.00 I m < 1 . 1 C: 0.50 < m < 0.8 A:
5
of
rads rads rods
Values
lnman L os c o f Modulatlon Index
