The effect of finite interleaver size on bit error rate (BER) performance of coded 8-DPSK is determined by means of computer simulations. The losses evaluated in this way include the SNR degradation due to the timing and frequency errors of the symbol synchronizer and the automatic frequency control (AFC) of the receiver. BER measurements are presented using a conventional 2/3 rate convolutional 8-state trellis-code for typical Rayleigh and Rician fading channels. It is shown that for a Rician channel with a Rician parameter of 7 dB, a Doppler spread of 100 Hz and a data rate of 2400 bps, an interleaver with size 16 x 16 symbols performs nearly as well as a very large interleaver. It is also shown that for very fast Rayleigh channels, the BER-curves flatten out at large SNR

Edbauer, Franz

English

NASA Technical Reports Server

N88-25719
INTERLEAVER DESIGN FOR TRELLIS-CODED DIFFERENTIAL 8-PSK MODULATION WITH
NON-COHERENT DETECTION
DR. FRANZ EDBAUER, Institute of Communication Technology, German Aerospace Research
(DFVLR), Fed. Rep. of Germany.
GERMAN AEROSPACE RESEARCH (DFVLR)
Institute of Communication Technology
Oberpfaffenhofen
W.-Germany
ABSTRACT
Trellis-coded differential 8-DPSK with non-coherent detection is considered to be an
efficient modulation scheme for satellite mobile communication. Theoretical analysis of such
systems often assumes perfect, i.e., infinite interleaving. In this paper the effect of finite
interleaver size on bit error rate (BER) performance of coded 8-DPSK is determined by
means of computer simulations. The losses evaluated in this-way include the SNR-degra-
dation due to the timing- and frequency errors of the symbol synchronizer and the AFC
(automatic frequency control) of the receiver. BER-measurements are presented using a
conventional 2/3-rate convolutional 8-state trellis code for typical Rayleigh- and Rician fad-
ing channels. It is shown that for a Rician channel with Rician parameter of 7 dB, a Doppler
spread of 100 Hz and a data rate of 2400 bps, an interleaver with size 16x16 symbols per-
forms nearly as well as a very large interleaver. It is also shown that for moderately fast
Rayleigh channels the BER-curves flatten out at large SNR. This is due to the fact, that the
random phases of the fading channel in rare cases coincide with valid phase sequences of
coded symbols, thus producing error events, even without any additive noise disturbance.
For such channels an additional code design criterion would be to minimize the error floor.
INTRODUCTION
Trellis-coded modulation provides coding gain without expanding bandwidth
[UngerbSck, 1982-1. Most applications in the past have been concerned with coherent sys-
tems. For fading channels non-coherent modulation schemes like trellis-coded differential
octal phase shift keying (coded 8-DPSK) with differentially coherent detection are probably
more suitable since they do not require the derivation of a coherent reference carrier. Coded
8-DPSK has been investigated by [Simon and Divsalar 1987, McLane et al. 1987, Edbauer
1987, Makrakis et al. 1987]. Theoretical analysis of interleaved systems often assumes ideal,
i.e., infinite interleaving, to make the channel to appear memoryless. In this paper the effect
of finite interleaver size on BER-performance is determined by means of computer simu-
lations for a Rayleigh and a Rician channel, which are considered to be typical of mobile
satellite communications.
MODULATOR FOR CODED 8-DPSK SIGNALS
The modulator is shown in Fig. 1. The input bits are passed through a rate 2/3 trellis
encoder which generates at its output coded 3-bit tuples. These are written into a block
interleaver row by row. The interleaver consists of I rows and r columns and its size is
denoted by I x r. The 3-bit tuples are read out column by column and mapped into coded
8-PSK symbols a',, (la'. I = 1) in the complex signal plane according to the set partitioning
method [UngerbSck, 1982-1. The symbols a'n are then differentially phase encoded producing
coded 8-DPSK symbols c.
271
https://ntrs.nasa.gov/search.jsp?R=19880016335 2020-03-20T06:53:17+00:00Z
Fig. 1. Generation of Interleaved Trellis-Coded 8-DPSK Signals
cn = Cn_la" n. (1)
These are filtered by a pulse shape filter with impulse response h(t), supplying
= _cih(t --iT), (2)x(t)
i
where T is the symbol period. ×(t) is converted to the radio frequency and transmitted.
DIGITAL RECEIVER WITH SOFT DECISION 8-PSK VITERBI DECODER FOR CODED 8-DPSK
SIGNALS
A simplified block diagram of the receiver in the complex baseband is drawn in Fig. 2.
Coarse AFC, AGC, filtering and interleaver synchronisation are not shown. We assume
Nyquist signaling with zeros of the filter impulse response at multiples of the symbol period,
satisfying Nyquist's first criterion. For the computer simulation the full Nyquist filter is placed
on the transmitter side.
The receiver input signal is written in the complex baseband in the form
y(t) = A(t)x(t) + w(t). (3)
A(t) is a complex-valued stochastic process which describes multiplicative frequency non-
selective fading. For Rayleigh- and Rician channels A(t) is a complex Gaussian process. The
envelope IA(t)l has a Rayleigh- or Rician distribution, w(t) is additive complex Gaussian
noise. The input signal is sampled once per modulation interval by an A/D-converter pro-
viding the complex sample
Yn = AnXn + Wn. (4)
x, is the sample of x(t) at time tn =nT-_ , where n =0,1,2 ..... and z , with
-T/2 < _ < T/2, determines the sampling point within a symbol period. The noise samples
w, can be assumed to be complex Gaussian and uncorrelated. The average bit energy to
spectral noise power density ratio is
Eb E{IAI 2}
- (5)
No 4# 2 '
where E{ } denotes expectation and (_ is the variance of the noise w, in each dimension. The
average signal power E{IAI 2} can be determined by the AGC of the receiver whose time
constant is large with respect to the fading period. In the following the normalization
E{IAI 2} = 1 is assumed.
The receiver performs the following signal processing functions: differentially coherent
8-DPSK symbol detection, hard 8-PSK detection, automatic frequency control (AFC), symbol
synchronisation (timing) and soft decision Viterbi decoding.
Differential symbol detection is performed by a one-delay symbol detector whose output
is
Zn = Y'nY'n-1 = h('c)a'n + nn + Vn, (6)
272
r___ SYMBOL-
[SYNCHR.
GAUSSIAN • •
NOISE ISAMPLE
FADING fl w(t) I_TTIME
liA,,lil £'n,1
CODED v v r'_(t; L__._.J r_
8-DPSK
s°FToEcl B,TSCON,G,, z --, , OUTPO,
_1 LEAVER _I_VITERBI-
Zn I I I DECODER /
[._ _- -- REAL SIGNAL
"_" n an ( ..... I
_ _. o-for*, j -- COMPLEX SIGNAL
_ DETECTOR I
Fig. 2. Simplified Block Diagram of the Digital Receiver with Soft Decision Viterbi
Decoder for Interleaved Trellis-Coded 8-DPSK Modulation
where y'. is the back-rotated version of the received 8-DPSK symbol y. after frequency offset
compensation (see Eq. (10)), and ()* denotes the conjugate complex value, n. is additive
complex noise and v. is the intersymbol interference term which vanishes for exact timing
= 0).
A t
The hard 8-PSK decision a .'of z. is given by that symbol an which is closest to z.:
A I
a n*- minlz n - a'nl.
a n
(7)
A o
a. is needed for decision-directed AFC and decision-directed symbol synchronisation.
An error signal for the radian frequency offset Z%{o.of the receiver input signal y. is given
by
A *
eafc,n = Im{z n • a'n}. (8)
A
/_(o., being the estimate of A_o., is updated by a first order loop
A A eafc,n
AO}n = /_O_n--1Jr
Talc (9)
where Tarc is the AFC time constant for high SNR and small frequency errors, The frequency
A
Offset is compensated by back-rotating Yn with angle _n, yielding Y'n • The rotated signal then
reads
with
A
Yn" = Yn exp( -- J(/)n) (10)
A
_)n = _n-1 + A(°n" T. (11)
Symbol synchronisation for coded 8-DPSK [Edbauer, 1987] is achieved by using an
approach ._imilar to J_Mueller, 1976]. It can be shown, that an error signal for the timing error
= nT - t., where t. is the sampling time of the A/D-converter for symbol n, is given by
p t* A *
et,n = T.Re{y nY n-1 -- a'nlY'n-112} . (12)
Eq. (12) has the property that it is invariant against a constant phase rotation, thus enabling
non-coherent opreration. For a first order loop one obtains
273
^ A 4o_et,n
tn+ 1 = tn + T + _ (13)
where ¢ is the ratio of loop noise bandwidth to symbol rate, and _ is the slope of the average
loop S-curve at the origin • = O.
Decoding of the trellis code is performed~by a Viterbi decoder which uses the unquan-
tized deinterleaved symbol detector outputs z. of Eq. (6) for metric computation. The Viterbi
algorithm is used to determine that coded 8-PSK symbol sequence {an}whose quadratic
Euclidean distance is minimum with respect to the differentially detected noisy 8-PSK symbol
sequence {zn}- The metric increment over one trellis branch therefore is
qn= - - 2 (14)
where a, is the coded 8-PSK symbol attributed to that branch. We note that Eq. (14) is not the
=optimum metric in the maximum likelihood sense, since the noise nn of the 8-DPSK detector
output (Eq. (6)) is non-Gaussian and correlated.
The digital signal processing and decoding methods, as applied here, are explained in more
detail in FEdbauer, 1987-1.
COMPUTER SIMULATION RESULTS
Computer simulations of the proposed coded 8-DPSK system have been performed to
determine the effect of finite interleaver size on BER-performance. Symbol synchronisation
and AFC have been included in the simulation programs as described above. Two typical
mobil satellite fading channels, a Rician and a Rayleigh channel with a Doppler spread of
100 Hz, have been chosen for the simulation. The data rate, having been used, was 2400 bps,
which may allow sufficient digital speech quality for aeronautical applications. The Rician
parameter K, i.e., the power ratio of the direct line of sight signal to the scattered non-co-
herent component, was 7 dB. The choice of channel parameters and of the data rate was
mainly motivated by aeronautical channel measurements rNeul et al., 1987]. In the simu-
lation, samples An (Eq. (4)) of the complex multiplicative fading were computed by digitally
low-pass filtering independent Gaussian random numbers. The Doppler spread of channels,
generated in this way, is twice the bandbwidth of these filters. Pulse shape filtering was
performed by a raised cosine Nyquist filter with roll-off of 40 % and impulse response length
of 9 symbol periods. A conventional 8-state UngerbOck type convolutional code
[Ungerb6ck 1982, Fig. 9] without feed-back was used as trellis code. Since the interleaver
size should be the only varying parameter the decision delay of the Viterbi decoder was kept
constant and equal to M =16 symbols.
Usually the block interleaver size is chosen in that way that the time for transmission of
a column - i.e. the number of rows - is of the order of expected fade durations, and that the
number of columns is equal to the decoding decision delay. In the simulation even smaller
interleaver sizes were selected since short delays are desirable especially for speech tran-
smission.
The BER results for the Rician channel with the interleaver size as parameter are shown
in Fig. 3. It can be seen that a small interleaver with size 16x16 is only 0.5 dB worse than the
large 100x30 interleaver at all EJNo values of the figure. The smaller 8x8 interleaver would
also be acceptable giving a total delay of 120 ms between transmitter and receiver for digital
2400 bps speech transmission. No further improvement was possible by using an interleaver
larger than 100x30. If no interleaver were used the losses would be 12 dB and 6 dB at BER
of Pb -- 10-4 and Pb = 10-3 with respect to the largest interleaver. For comparison reasons
the theoretical and the computer simulation BER-curves of uncoded 4-DPSK with differen-
tially coherent detection are also drawn in Fig. 3.
BER-measurements for a Rayleigh channel with the same Doppler spread of 100 Hz and
interleaver sizes 8x16 and 8x8 are shown to the right of Fig. 3. It is seen that the BER-curves
flaten out at large EJNo. This error floor is caused by the phase variations of the fading
channel which in rare cases coincide with the phases of coded symbols of an incorrect path,
even without any additive noise disturbance, The error floor depends on the trellis code, the
274
IEe
5
3
2
IE-I
5
3
2
_-_ 1E-2
L,J
e:5
_3
_2
_ 1E-3
5
3
2
IE-4
5
3
2
IE-5
8
CHANNEL: .... THEORY
RICE K=7dB (SLOW FADING,
PERFECT SYNCH.)
%"'_" _'_0 0 o DATA RATE: 2400bps
DOPPLER SPREAD: 10OHz
a=O .003
Tofc :50ms
TRELLIS CODE: 8 STATES (CONVENT.)
+_ DECODING DELAY: M_16
_/\ _ ",_. DN_ODED
&_+ _ ,,, 4-OPSK
8x16"_ _ "'.\
4x8_ "" ,X_'CODED 8-DPSK'NTERLEAVER
8x4 i/ ",
coD_o6-D.S_ ",
lb 1_ 2b 2_ 3e 3_ 4e
MEAN BITENERGY/NOISEDENSITY Eb/No, dE]
IEe
5
3
2
IE-I
5
3
2
IE-2
_s
_3
i,r
_ IE-3
5
3
2
1E-4
5
3
2
1E-5
CHANNEL: .... THEORY
RAYLEIGH (SLOW FADING.
PERFECT SYNCH.)
_UNCODEO 4-OPSK
"-tl ,
CO;EO\_, -,%
e-o_J< ".S ",,,x,,x
"'. o
_ "'" _o-- NO INTERLEAViE.R
,.'ERL R
_A_ 8x16 ,
DATA RATE: 2400bps
DOPPLER SPREAD: IOOHz
=:e.ee3 ",
Tofc:5Oms
TRELLIS CODE: 8 STATES (CONVENT.)
DECODING DELAY: M= 16
• , • , . , . , , , . .
s _ _s 2_ 2s 3e 3b 4b 4b se
MEAN BITENERGY/NOISEDENSITY Eb/No, dB
Fig. 3. Computer Simulation Results of BER-Performance of Coded 8-DPSK Modulation for
Various Interleaver Sizes over a Rician and a Rayleigh Channel
interleaver size and, most significantly, on the ratio of the Doppler spread to the symbol rate.
For the slower Rayleigh channel with Doppler spread of 20 Hz and similar signal processing '
no error floor was observed above P_ = 10 -s [Edbauer, 1987,1. It was shown by computer
simulation that the effect of the AFC on the error floor is negligible. This was done by setting
the AFC output to a constant which corresponded to the Doppler'frequency of a vehicle
moving with constant speed.
An important aspect in system design for a fading environment is the proper choice of
the trellis code. [Simon and Divsalar, 1987,1 have analytically derived the following code
design criteria for the slow fading fully, interleaved Rayleigh channel with differential
detection. The code should have:
a.) large length of the shortest error event path, where equal symbols of the correct and
the incorrect path do not contribute to the length,
b.) large product of the branch distances of this path,
c.) large minimum free Euclidean distance.
A criterion concerning the error floor does not yet exist, since this would require consider-
ation not only of amplitude- but also of phase variations of the channel.
In view of these criteria the convolutional 8-state trellis code, used in this work, is not
optimum. It was designed for the Gaussian channel [UngerbOck, 1982-1, mainly by consider-
ing criterion c.). The code has two minimum free distance paths of lengths 3 and 4 with dis-
tance 2.14 for the unit radius constellation. But the code also has one path with length 2 and
distance 2.45, which does not meet criterion a.) very well. If an 8-state code with shortest
path length of 3 would exist, some performance improvement would be possible. We
therefore intend to extend this work to other codes, e.g. those proposed by rSimon and
Divsalar, Globecom'87,1.
275
REFERENCES
Ungerb6ck, G. 1982. "Channel Coding with Multilevel/Phase Signals". IEEE Trans. on Inf.
Theory, Vol. IT-28, No. i, Jan. 1982, pp. 55-76.
Simon M.K., Divsalar D. 1987. The Performance of Trellis Coded Multilevel DPSK on a Fad-
ing Mobile Satellite Channel. Proc. of Internat. Con[ on Communications ICC'87, pp.
21.2.1.-21.2.7.
McLane, P.J., Wittke, P.H., Ho, P.K., Loo, C. i987. PSK and DPSK Trellis Codes for Fast Fad-
ing, Shadowed Mobile Satellite Communication Channels. Proc. of Internat. Confer-
ence on Communications ICC'87, pp. 21.1.1.-21.1.4.
Edbauer, F. 1987. Coded 8-DPSK Modulation with Differentially Coherent Detection - an Effi-
cient Modulation Scheme for Fading Channels. Conf. Record of GLOBECOM'87, Vol.3,
pp. 42.2.1-42.2.4.
Makrakis, D., Mathiopoulos, P., Feher, K. 1987. Trellis-Coded Modulation for Noncoherent
Detection Schemes, Part I1: Differential Detection. Proc. of The 1Oth Symposium on
Information Theory and Its Applications SITA'87, pp. EEl-2-1 - EEl-2-5.
Mueller K.H., MiJller M 1976. Timing Recovery in Digital Synchronous Data Receivers. IEEE
Trans. on Communications, Vol. COM-24, No. 5, pp. 516-531.
NeuI, A.,Hagenauer, J., Papke, W., Dolainsky, F.,Edbauer, F. 1987. Aeronautical Channel
Characterization Based on Measurement Flights. Conf. Record of GLOBECOM'87,
Vol.3, pp. 42.3.1-42.3.6.
Simon, M.K., Divalar, D. 1987. Multiple Trellis Coded Modulation (MTCM) Performance on a
Fading Mobile Satellite Channel. Conf. Record of GLOBECOM'87, Vol.3, pp.
43.8.1-43.8.6.
276


Interleaver design for trellis-coded differential 8-PSK modulation with non-coherent detection

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server