Telemetry signals from several channels are aligned in time and combined by the Real-Time Combiner (RTC) in order to increase the strength of the total signal. In this article, the impact of the timing jitter in the RTC on the bit/symbol error rate is investigated. Equations are derived for the timing jitter loss associated with the coded and uncoded channels. Included are curves that depict the bit-symbol error rate vs. E sub b/N sub 0 and E sub s/N sub 0 for some typical telemetry conditions. The losses are typically below 0.1 dB

Mileant, A.

Simon, M. K.

English

NASA Technical Reports Server

TDA Progress Report 42-87 July-September 1986 
Real-Time Combiner Loss 
M. K. Simon and A. Mileant 
Telecommunications Systems Section 
Telemetry signals from several channels are aligned in time and combined by the Real- 
Time Combiner (RTC) in order to increase the strength of the total signal. In this article, 
the impact of the timing jitter in the RTC on the bitlsymbol error rate is investigated. 
Equations are derived for the timing jitter loss associated with the coded and uncoded 
channels. Included are curves that depict the bitlsymbol error rate us. E, IN, and E, /N, 
for some typical telemetry conditions. The losses are typically below 0.1 dB. 
1. Introduction 
The Real-Time Combiner (RTC) is part of the Baseband 
Assembly (BBA). Its purpose is to time align baseband signals 
from different antennas in order to increase the strength of the 
total combined signal. Figure 1 represents a block diagram of 
the overall downlink telemetry process of signal combining, 
subcarrier demodulation, symbol synchronization and data 
recovery. 
If there were no timing jitter in the RTC, the maximum 
signal-to-noise ratio at the output of the RTC would be the 
sum of the SNRs at its input. However, timing jitter will 
decrease the output SNR from its maximum value. This 
decrease in combined SNR will in turn increase the bit/ 
symbol error rate of the recovered data. In Ref. 1 the degrada- 
tion in the RTC's output SNR was estimated. In this analysis, 
the degradation in bit/symbol error rate performance will be 
estimated for the rate 1/2, constraint length 7 convolutional 
code and the uncoded data cases. Expressions are derived for 
the timing jitter loss. 
II. Impact of the Timing Jitter on the 
RTC's SNR 
As shown in Fig. 2, the operation of the RTC can be visu- 
alized as a weighted sum of L streams of binary data. At the 
discrete time instant i ,  the RTC produces at its output 
L 
x i  = C ai(sji + nj i )  
j = 1  
where ai is the j t h  weighting factor, sii is the signal amplitude 
of the i th  sample of the j th  signal withE{s..} = 4 a n d  nji is 
the amplitude of the additive thermal noise in the j t h  channel. 
This noise is modeled as a zero-mean Gaussian process with 
variance of =NoiBj ,  where Bi is the noise bandwidth at the 
j t h  input to the RTC. The noise processes in each of the chan- 
nels are statistically independent of each other. 
J !  
The transmitted data symbols are recovered by adding up 
all the samples belonging to  each individual symbol. Let N, be 
179 
https://ntrs.nasa.gov/search.jsp?R=19870005916 2020-03-20T12:23:45+00:00Z
?~ 
1 
the number of Nyquist samples per symbol, then. as shown in 
Fig. 2, at the end of each symbol the RTC produces 
Assuming perfect time alignment of the combined signals, 
inserting Eq. (1) in Eq. (2) and using the above definitions, the 
first two moments of the k t h  symbol, yk , will be 
L L 
With imperfect timing in the RTC loops, the first two 
moments of the k t h  symbol conditioned on E = (e1, e 2 , .  . . 
eL ) will be 
E{Ojc)2}  = N,’ (aj 4)’ + N, a; U; (4 )  
j =  1 j =  1 
The first term in Eq. (4)  represents the expected value of the 
signal energy in the kth symbol, and the second term, the 
corresponding noise variance. 
Imperfect timing in the RTC loops decreases the effective 
energy of the detected symbol. Since the RTC’s loop update 
time is much larger than the symbol time, the alignment error 
is constant during many symbols. Let ei be the normalized 
alignment error defined as follows 
E .  I G 4 I/N,, ( 5 )  
where Si is the alignment error of the j t h  channel relative to 
the master channel (j = 1) and N,, is the period of the subcar- 
rier waveform (both 6 j  and N,, are in units of Nyquist samples 
and Si is a random variable with the probability density 
function (pdf) given by Eq. (75) of Ref. 1). Using that equa- 
tion, the pdf of ej  will be 
where pi = E{4Sj/N, ,} ,  0% = (4/N,,)2 u2 and U(.) is the 
unit step function. The closed-loop variance of the timing 
jitter, uhj,  is given by Eq. (53) of Ref. 1 .  It can be shown that 
for data modulating a squarewave subcarrier, the variance of 
ei will be 
Ni 
u2. €1 = BL { [erf(vl)erf(vj)] -’ - 1}/(2B1) (7) 
where BL is the bandwidth of the RTC loop and vi G (Si/ 
2uj)1/2. 
180 
j =  1 j= 1 
(9) 
Note that e l  = 0 by definition. 
It can be shown (see Ref. 1 ,  Eq. [66]) that optimum signal 
combining is obtained when the weighting factors ai are 
related as follows. 
Since scaling all a’s by a constant factor does not change the 
SNR at the output of the RTC, we set a1 = 1.0 and readily 
obtain the optimum values for all the remaining weighting 
coefficients. Namely, we make 
where m ‘ S  IS  and y. 
i - 1  i I oi/u;. 
With these preliminary steps completed, we are now ready 
to evaluate the conditional SNR at the RTC’s output. Let 
R,  A Ek/Nk be the SNR at the end of the k t h  symbol (see 
Eq. (1 0)). Then, using Eq. (9),  R ,  conditioned on E is 
N, (Ea J”I.,)’ 
t 
Ea; NoBi 
where all summations are from j = 1 to j = L and, again, e l  = 0 
by definition. 
Using the definitions of mi and yi, the optimum values of 
ai (Eq. ( 1 2 ) )  and simplifying, the conditional symbol SNR can 
be expressed in terms of S ,  and B, , namely 
N,S, [ ( C : Y ~ / ~ ~ ) ~  - ( Z : Y ~ / ~ ~ ) ( C Y ~ E ~ / ~ ~ ) ]  
Rk(E) = NoB1 ( w j I m i )  
where 
R, 2 2Esl /No = 2Rs 
and 
L 
B :Yi/mj 
j =  1 
Note that > 1 when L > 1. Equation (1 5) was obtained using 
the following relations: N, = 2B, /r  and S , / r  = ES1 where E,, 
is the symbol energy in the first channel and r is the symbol 
rate. Note that with no timing jitter in the RTC, the output 
SNR is 
Rk = RbP for all k ( 1  7) 
So, 
is larger than the SNR in channel 1. 
represents the factor by which the ideal combined SNR 
111. Evaluation of the Loss Due to Timing 
Jitter in the RTC 
Timing jitter in the RTC loops has the effect of decreasing 
the effective SNR of the combined signal. This decrease of 
SNR in turn increases the bitlsymbol error rate of the recov- 
ered data. In order to avoid confusion of terms, we define 
degradation factor as the factor by which the combined SNR 
is decreased due to  timing jitter in the RTC, whereas we define 
jitter loss (like “radio loss”) as the factor by which the SNR 
has to  be increased in order to  achieve a specified bit/symbol 
error rate. The evaluation of the degradation factor (decrease 
in combined SNR) requires averaging over E of the conditional 
SNR in Eq. (15). This evaluation was done in Ref. 1. In this 
analysis, the jitter loss, namely, the degradation in bit error 
rate performance, is determined. 
The evaluation of the jitter loss requires two steps. First, 
the particular coding scheme has to be specified. Then, averag- 
ing on the bit/symbol error rate is performed over the timing 
jitter pdf, Eq. (6 ) .  In our analysis two telemetry cases will be 
considered: (a) the rate 1 / 2 ,  constraint length 7 convolu- 
tionally coded and (b) uncoded data. 
A. Convolutional Code 
For the rate 112, constraint length 7 convolutional code, 
an expression for bit error rate has been empirically found 
and is given as follows: 
Cl exp (C, E, / N o ) ,  E, /No > 0.8989 
1 / 2 ,  Eb/No < 0.8989 
(18)  
‘b = fb(Eb/NO) = 
where C, = 85.7469 and C, = -5.7230. The term Eb/No is the 
bit energy to  noise spectral density ratio at the input to  the 
Maximum Likelihood Decoder (MCD). Note that E,/No = 
2E,/No = R, of Eq. (15). 
6. Uncoded Data 
For uncoded data the symbol error rate is given by 
P, = f,(Es/No) = (1/2)erfc[(Es/N0)1’2 1 (19) 
where erfc(x) is the complementary error function. Note that 
now E,/No = Rb / 2  of Eq. ( 1 5 ) .  
Since the data rates considered in our analysis are higher 
than the RTC’s bandwidth, or, in other words, the loop timing 
error process tii is slow and essentially constant over a bit/ 
symbol time interval, the average bit/symbol error rate is 
estimated using the so called “high-rate model,” namely 
b ,  for coded channel 
s , for uncoded channel 
where Rk(e) is given by Eq. (15) and p ( e j )  by Eq. (6 ) .  
For the coded channel, the average bit error rate as given 
above can be obtained in a closed form as follows. Assuming 
that the RTC has doppler compensation, pi in the pdf of the 
normalized timing jitter (Eq. [6 ]  ) will be zero. This is our first 
simplification in the evaluation of Eq. (20).  A second simpli- 
fication results in assuming that the contribution of the right- 
most term in the numerator of Eq. (15) is much smaller than 
181 
that of the middle term (i.e., E { e j )  > E  { ( E ~ ) ~ ) ) .  By discard- 
ing the right-most term in the numerator of Eq. (15) we will 
obtain the upper bound of Pb. Inserting now these simplified 
Eqs. (6) and (15) with (18) in Eq. (20), integrating and sim- 
plifying, we finally obtain the average bit error rate at the 
output of the MCD, namely 
where 
I .  I = exp (vT)erfc(v.) J I (22) 
and R ,  is given by Eq. (16a). 
For the uncoded case, the average degraded symbol error 
rate can be obtained by first approximating the complemen- 
tary error function of Eq. (18) by erfc(x) = exp(-x2)/.\/;x. 
Then we expand this function in a Taylor series about ei = 0 
0 = 2,  . . . , L ) .  Finally, we take the expected value of the series 
expansion and keep the first few terms. Performing the above 
steps we find after some algebra the desired E', which for 
L = 3 becomes 
where again R, G E,,/N, (Eq. [ I6a] )  and hi 
L = 2, we simply set A, = 0 in the above equation. 
yiui/mi.  For 
When, in the region of specified (desired) bit/symbol error 
rate, both curves of P;: vs. SNR (the ideal and that degraded by 
the RTC's timing jitter) are parallel, the timing jirter loss can 
be obtained from the following expressions. 
For the coded channel 
L, = 10 log,, [Rb P/(Eb /No  )ref1 , dB (25) 
where (Eb/No)ref = In (Pb/Cl)/C2, (In(*) denotes natural 
logarithm) and Pb is the degraded bit error rate obtained from 
Eq. (2 1). 
For the uncoded channel 
where (Es/No)ref = [erfc-' (2P,)] ', erfc-' is the inverse com- 
plementary error function and P, is the degraded symbol error 
rate obtained from Eq. (24). 
IV. Conclusion 
In the above analysis the impact of the timing jitter in the 
RTC on the bit/symbol error rate was evaluated. Equations 
were obtained for the degraded average bit/symbol error rate 
vs. SNR in the reference channel for the rate 1/2, constraint 
length 7 convolutionally coded and uncoded data. The result 
of the above analysis are illustrated in Figs. 3 through 6 for 
several typical telemetry conditions. The losses are typically 
below 0.1 dB. 
Reference 
1. Simon, M. K.,  and Mileant, A., Performance Analysis of the DSN Baseband Assembly 
(BBA) Real-Time Combiner (RTC), JPL Publication 84-94, Rev. 1, Jet Propulsion 
Laboratory, Pasadena, Calif., May 1, 1985. 
182 
BASEBAND ASSEMBLY IBBA) 
 
I .  4 RECEIVER# 1 l+d I ,  SIGNAL FROM ANTENNA 1 
I 
I SYNCHRONIZER 
I 
,-+=Id{ SIGNAL FROM ANTENNA 2 ~ ~ ~ I N E R  REAL-TIME H ~"0",4P,"I',", DEMODULATOR 1 
I 
(DSA) SIGNALS w- 
I 
SIGNAL FROM ANTENNA L 
RECEIVER # L 
I L:yF 4"""". 1 
CONVOLUTIONAL 
DECODER 
(MCDI 
DETECTED 
DECISION 
UNCODED LOGIC 
Fig. 1. Downlink process of signal combining, subcanierdemodulation, symbol synchronization and decoding of telemetry data 
REAL-TIME COMBINER 
I NTEG RATE-AND- 
DUMP PROCESS 
S2i + "2i Iy, 
Ti + " Li 
5 I= 1 
CODED 
DATA C 'b 
MCD - 
vk 
UNCODED 
I= 1 
U L d  
Fig. 2. Simplified representation of the signal combining and decoding process 
183 
0.3 I I I I I I 1 
MISSION: VOYAGER CHANNEL: CODED 
DATA RATE: 40000 s/s B1 = 3.75 X lo6 Hz 
mi = 4.0 Yi = 1 .o I r 4 . 1 2  x 10-2 g 0.2 2 s 
-1 0 1 2 3 4 5 
Eb/N@ dB 
Fig. 3. The RTC loss vs master antenna €,,/No, L = 2 
0.3 I I I I I I 
MISSION: VOYAGER CHANNEL: CODED 
DATA RATE: 40000 s/s 81 = 3.75 X lo6 Hz I <L = 0.1058 Hz 
0 I 1 d 
-1 0 1 2 3 4 5 
Eb/No, dB 
Fig. 4. The RTC loss vs master antenna Eb/No, L = 4 
0.3 1 1 I I I I 
CHANNEL: UNCODED 
81 = 135 X 1s 
MISSION: PIONEER 
DATA RATE: 1024 s/s 
9 = 4.0 yi = 1.0 g 0.2 c 
EL = 4.12 X lo-’ Hz 
4.0 x 10-4 
3.90 x 10-3 
0 
2 4 6 8 10 12 
€,/NO, dB 
Fig. 5. The RTC loss vs master antenna €*/No, L = 2 
0.3 I I I I I I 
MISSION: PIONEER CHANNEL: UNCODED 
DATA RATE: 1024sis B1 = 135X lo3  
r q  = 4.0 yi = 1.0 g 0.2 
2 4 6 8 10 12 
ES/No, dB 
Fig. 6. The RTC loss vs master antenna €*/No, L = 3 
184 


Real-time combiner loss

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server