The relationship between transmission rate and source and channel signal-to-noise ratios (SNR's) is discussed for the transmission of a Gaussian source over a binary input, additive Gaussian channel, with a mean-squared distortion criterion. We point out that for any finite rate, and sufficiently high channel SNR, the fidelity criterion (reproduction SNR) is upper bounded by a function of the transmission rate. Thus, the performance becomes rate limited rather than power limited. This effect is not observed with the binary symmetric source, the binary-input Gaussian channel combination, or the Gaussian source, unconstrained-input Gaussian channel combination

Belongie, M.

Costa, M.

Pollara, F.

English

NASA Technical Reports Server

N95- 32223
TDA Progress Report42-121 May15,1995
Rate Considerations in Deep Space Telemetry
M. Costa, M. Belongie, and F. Pollara
Communications Systems Research Section
The relationship between transmission rate and source and channel signal-to-
noise ratios (SNRs) is discussed for the transmission of a Gaussian source over a
binary input, additive Gaussian channel, with a mean-squared distortion criterion.
We point out that for any finite rate, and sumciently high channel SNR, the fidelity
criterion (reproduction SNR) is upper bounded by a function of the transmission
rate. Thus, the performance becomes rate limited rather than power limited. This
effect is not observed with the binary symmetric source, the binary-input Gaussian
channel combination, or the Gauss/an source, unconstrained-input Gaussian channel
combination.
I. Introduction
The deep space communication channel uses binary phase shift keying (BPSK) modulation and is well
modeled as a binary input, additive white Gaussian noise (AWGN) channel model. It is usually accepted
that there is no bandwidth constraint in deep space communication application and that, for sufficiently
wide bandwidth usage, the full benefit of unconstrained bandwidth is essentially realized. While these
notions are correct, they must be viewed with caution. It does not necessarily follow that, for sufficiently
low overall transmission rate, there is little to be gained by further decreasing the rate. The interplay
between source and channel coding and the issue of coding complexity need to be considered. Depending
on the telemetry source and the available channel signal-to-noise ratio (SNR), there may be a significant
advantage in further decreasing the rate.
In this article, we review these notions in the context of a deep space communication system with
an independent identically distributed (i.i.d.) Gaussian source and a conventional BPSK, power-limited
channel, using mean-squared error (MSE) as a distortion criterion. While not an accurate model for
most deep space telemetry sources, the white Gaussian source is a useful reference model. Typical
telemetry data can be transformed by an (approximately) decorrelating orthogonal transformation, such
as the discrete cosine transform, producing data that can be approximated by parallel sources with white
(generalized) Gaussian distributions of different variances, one for each transform coefficient. Thus, the
combined source and channel coding of a white Gaussian source for transmission over the deep space
channel is a relevant exercise.
II. Preliminaries
The well-known equations governing transmission rate and source and channel SNRs were established
by Shannon in his seminal 1948 articles [1]. We refer to [2] as a source of notation. Figure 1 shows the
system under consideration.
https://ntrs.nasa.gov/search.jsp?R=19950025802 2020-06-16T06:11:51+00:00Z
r= rc, SOURCE SAMPLES/CHANNEL SYMBOL
rs
GAUSSIAN
SOURCE
A
SOURCE
SAMPLE:I
DISTORTION 5
T DECODEDSOURCE
I SAMPLES
USER
SOURCEENCODER_
(RsBITS/SOURCE
SAMPLE)
CHANNEL ENCODER
(Rc BITS/CHANNEL
SYMBOL)
DECODED
BITS
DECODER DECODER
CHANNELSYMBOLS
I AWGN CHANNEL INOISE -N(0,N 0/2)
I
py = Ey I
I
Fig. 1. Communication system model.
The capacity of a binary-input AWGN channel is given by
C(pzl ) = I - E_ [Iog2(I + e-2_)] (1)
where Pv = 2£u/No, £u is the available energy per channel symbol, No 2 is the two-sided noise spectral
density, and Eu denotes expectation over u, a random variable with distribution N(pu, pv).
The rate distortion function for an i.i.d. Gaussian source is given by
lln(6) = _ og 2 (2)
where 5 is the normalized MSE distortion. The reproduction SNR (RSNR) is given by 1/5.
III. Discussion
There are three variables of interest in this communication problem. They are
(1) 6, the normalized MSE distortion of reproduction at the receiver
(2) p_, the available channel SNR, given by p= = 2,f.x/No
(3) r, the overall transmission rate, measured in source samples per channel use
These quantities must satisfy the inequality
c(rp=)_> (3)
If the coding procedure is divided into a cascade of source and channel encoders, where the source is
first converted into a string of binary symbols, the rate r satisfies
lO
rc Rx
r -- -- (4)
r_ P_
where rs is the source code rate measured in bits per source sample, rc is the channel code rate in
information bits per channel use, R_ is the source rate in samples per second, and R_ is the channel rate
in channel uses per second. Considering that each bandwidth unit (Hertz) corresponds, by the Nyquist
sampling theorem, to two dimensions (channel uses) per second, we relate the bandwidth B to R_ by
B = RJ2.
Other channel SNRs of interest are Pb and p_, the signal-to-noise ratios available per information bit
and per channel use, respectively. We have selected p= for our considerations because it is desirable to
compare transmission schemes that use the same power and time to transmit each source sample. These
three SNRs are related by rpx = rcpb = py.
Substituting Eqs. (1) and (2) in Eq. (3), we can obtain the fundamental bound on RSNR given r and
Px:
where the distribution of u is now expressed as N(rp=,rp=). This bound is depicted in Fig. 2, where we
present plots of RSNR versus E=/No for different values of overall rate r. (We use E=/No instead of p= in
all the figures for consistency with [2] and other articles.)
In the limit as r --* 0, Eq. (5) becomes
1
- < e_ (6)
5-
"0
n:-
z
o9
rr
4O
35
30
25
20
15
I0
0
0
r= 0 r= 1/16
r= 1/4
r=l
4 6 8 10 12
E#No,d_
14
Fig. 2. Bounds on performance for • binary Input channel with fixed r.
11
Thus, as p= increases without bound, RSNR also may increase without bound. To increase p=, one
needs to alter the source transmission rate or the available power P. We have Px = P/Rx. Thus, p= can be
increased by reducing the source rate Rz. This in turn affects the overall rate, since r = Rx/Ry = R=/2B.
Alternatively, Px can be increased with an increase in P.
The noted unbounded growth in RSNR only occurs in the limit as r --* 0. For any positive value of r,
the upper bound on RSNR approaches a finite limit as p= increases. This occurs when p_ is large enough
to make the channel essentially noiseless. Since the channel is restricted to binary input, its capacity is
upper bounded by 1-bit-per-channel use. Thus, the RSNR is upper bounded by a function of the overall
rate: 1/6 < 2 (2/_). Since this bound can be arbitrarily smaller than the bound that prevails in the limit
as r ---, 0, Eq. (6), it is clear that the performance can greatly benefit from a decrease in overall rate (or
an increase in bandwidth when Rx is held constant).
As shown in [3], the binary input AWGN channel has essentially the same performance as the un-
constrained power-limited AWGN channel for low enough overall rates (e.g., less than 0.3 bit/channel
use) when used to communicate a binary symmetric source. Interestingly, the same observation cannot
be made for the case of communicating a Gaussian random variable, except in the limit as r _ 0. For
any positive value of r, which suggests a finite level of complexity, and sufficiently high Px, the binary
input channel will have its performance (RSNR) limited by rate rather than by power. This effect is not
observed in the unconstrained input AWGN case, where, for a fixed arbitrary rate, the upper bound on
RSNR grows to co as p= --* c_. Figure 3 compares, for various values of r, the unconstrained input and
binary input cases. (The dotted lines are asymptotes.)
m
-o
o:"
Z
if)
o"
4O
35
3O
25
2O
15
10
UNCONSTRAINED
r=0
I
r= 1/16 _ , • ,r
r= 1/4 , " /
r=l • ,
• I
t
• #
• B
0 2 4 6 8
ex/No,aB
BINARYINPUT
10 12 14
Fig. 3. Comparison of binary input channel and unconstrained channel for fixed r.
IV. Applicability
Under what circumstances might there be a lower bound on the overall rate r? This is a complicated
issue, but we can make a few observations. First, any real system must have some nonzero value of r.
Second, r clearly has some relationship to complexity, because r = rc/r,, and both lower-rate channel
12
codes and higher-rate source codes generally imply higher complexity. Thus, a constraint on r can be
seen as a constraint on overall complexity. However, we can also consider the two components, rs and
re, separately. Fixing rs explicitly puts an upper bound on RSNR, resulting in the bounds shown in
Fig. 4. For this case, there is no difference between the unconstrained and binary input channels. Fixing
rc results in curves as shown in Fig. 5. Although a difference is seen between the unconstrained and
binary input channels, the curves all have the same exponential shape. So, the interesting phenomenon
described for fixed values of r (i.e., the different limiting behavior for binary input and unconstrained
channels) depends on a simultaneous bound on r_ and rc by fixing their ratio.
To see what implications this phenomenon might have, we must consider for which combinations of r,
RSNR, and p= it occurs. For a fixed value of r, the intercept of the asymptotes, as illustrated in Fig. 3,
is approximately where the effect becomes significant. This intercept occurs at 6 = 2-2/r and Px -- 4/r.
So, for instance, if r = 1/4, the effect becomes significant for RSNR > 24 dB and p= > 9 dB. While these
SNRs are certainly within the range of interest, it is hard to imagine reasonable circumstances requiring
r _> 1/4. For r = 1/16, which is known to be quite feasible for deep space communication, the effect
becomes significant for RSNR > 96 dB and p_ > 15 dB. These SNRs are probably outside the range of
interest of most missions.
m
z
(/3
rr
4O
35
3O
25
20
15
lO
5
0
0
J ' ' ' I ' ' ' _ I ' ' J ' I ' i , , I ' l , , I_
rs<_6
rs<4
rs<2
' ' ' , I I I I I I I I = I I I = , , I J , , , I
2 4 6 8 10
_x/N0,dB
Fig. 4. Bounds on performance for binary input channel or unconstrained channel with
rs limtted.
V, Performance Bounds With Fixed Channel SNR
Complexity is not the only reason that r = 0 is impossible. For a fixed Pz, r ---* 0 implies py ---, 0. Thus,
even if the computational complexity of a very low-rate channel code or very high-rate source code is not
a concern, the low SNR of the channel symbols might be. Although in theory py can be arbitrarily small
as long as C(py) > rR(_), in practice there is a lower bound on pu below which any given receiver cannot
perform symbol synchronization. Performance curves at constant py are shown in Fig. 6 for both the
unconstrained and binary input channels. Since the curves are all exponential, we see that the differing
13
behavior between the unconstrained and binary input channels for fixed values of r is not due to a bound
on p_. It can also be seen from Fig. 6 that the performance difference between the unconstrained and
binary input channels is negligible for p_ < 0 dB.
40
35
3o
25
m
.o
z 20
IT,
15
10
UNCONSTRAINED
rc= 1/4
rc=1/2 \
rc=3/4 \
_=r=O
BINARY INPUT
0 2 4 6 8 10
Ex/N O,dB
Fig. 5. Bounds on performance for binary Input channel and unconstrained channel with
fixed re.
40
35
30
25
_ 2o
15
0
0 2 4 6 8 10
ExJNO, dB
Fig. 6. Bounds on performance for binary input channel end unconstrained
channel with fixed£y/N0.
14
References
[1] C. E. Shannon, "A Mathematical Theory of Communication," Bell System Tech-
nical Journal, vol. 27, pp. 379-423 and 623-656, 1948.
[2] S. J. Dolinar and F. Pollara, "The Theoretical Limits of Source and Channel
Coding," The Telecommunications and Data Acquisition Progress Report 42-102,
April-June 1990, Jet Propulsion Laboratory, Pasadena, California, pp. 62-72,
August 15, 1990.
[3] S. A. Butman and R. J. McEliece, "The Ultimate Limits of Binary Coding
for a Wideband Gaussian Channel," The Deep Space Network Progress Report
42-22, May-June 1974, Jet Propulsion Laboratory, Pasadena, California, pp. 78-
80, August 15, 1974.
15
N95- 32224
TDA ProgressReport42-121 May15,1995
An Efficient Implementation of Forward-Backward
Least-Mean-Square Adaptive Line Enhancers
H.-G. Yeh
SpacecraftTelecommunicationsEquipmentSection
T. M. Nguyen
CommunicationsSystemsResearchSection
An efficient implementation of the forward-backward least-mean-square
(FBLMS) adaptive line enhancer is presented in this article. Without changing the
characteristics of the FBLMS adaptive line enhancer, the proposed implementation
technique reduces multiplications by 25 percent and additions by 12.5 percent in two
successive time samples in comparison with those operations of direct implemen-
tation in both prediction and weight control. The proposed FBLMS architecture
and algorithm can be applied to digital receivers for enhancing signal-to-noise ratio
to allow fast carrier acquisition and tracking in both stationary and nonstationary
environments.
I. Introduction
Adaptive line enhancers (ALEs) are useful in many areas, including time-domain spectral estimation
for fast carrier acquisition [2-4]. For example, a fast carrier acquisition technique [2], 1 as shown in Fig. 1,
will be very useful for a deep-space mission, especially in a nonstationary environment or emergencies.
Figure 1 is the block diagram of an ALE in a digital receiver used for both acquisition and tracking. First,
the receiver is in the acquisition mode. Second, when the uplink carrier is acquired as indicated by the lock
detector, the switch is shifted to the tracking position and the tracking process takes over immediately.
With this acquisition scheme, the uplink carrier can be acquired by a transponder in seconds (as opposed
to minutes for the Cassini transponder). Although devised to support a space mission, the architecture of
the forward-backward least-mean-square (FBLMS) ALE and the associated algorithm proposed in this
article are also applicable to other systems, including fixed-ground and mobile communication systems.
Note that this proposed ALE scheme in the receiver needs a residual carrier, and does not work directly
in suppressed-carrier cases.
A conventional ALE system using a least-mean-square (LMS) algorithm is depicted in Fig. 2, where
z -1 represents a delay. The analysis of the ALE for enhancing the signal-to-noise ratio (SNR) to allow
fast acquisition is given in [2]. The block diagram of a FBLMS adaptive line enhancer is shown in Fig. 3.
The performance analysis of the FBLMS adaptive line enhancer is provided in [1]. The FBLMS adaptive
line enhancer algorithm enjoys approximately half the misadjustment of that of the LMS algorithm [1].
l T. M. Nguyen, H. G. Yeh, and L. V. Lam, "A New Carrier Frequency Acquisition Technique for Future Digital Transpon-
ders," to be published in a future issue of The Telecommunications and Data Acquisition Progress Report.
16
L_
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INPUT xk
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ERROR ak
RESPONSE y _
k
Fig. 2. The architecture of the conventional ALE.
x(n)
.d
ef (n)
7
lid eb (n)
Fig. 3. The structure of the FBLMS adaptive line enhancer.
However, it requires about twice the number of multiplications and additions of the LMS algorithm. In
this article, an efficient implementation of the fast FBLMS algorithm is presented. This fast algorithm
provides the same speed of convergence as that of the LMS algorithm and provides the same misadjustment
as that of the FBLMS adaptive line enhancer, but requires fewer multiplications and additions. The
computational reduction is achieved by grouping two successive predictor computations together and
computing weight adaption at every other sampling time [5]. By using a radix-2 structure to manipulate
time samples, redundant computations embedded in two successive time samples can be removed via a
new structure of the fast FBLMS algorithm.
This article is organized as follows. The FBLMS algorithm is reviewed in Section II. The fast FBLMS
algorithm is derived and proposed in Section III. The fast FBLMS algorithm implementation is given in
Section IV and simulation results are presented in Section V. Finally, the conclusion is given in Section VI.
II. Forward-Backward LMS Adaptive Line Enhancer Algorithm
The structure of the forward-backward LMS adaptive line enhancer [1] is shown in Fig. 3. The forward
and backward prediction errors are then defined, respectively, as follows:
18


Rate considerations in deep space telemetry

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