BER performance and spectral properties of interleaved convolutional time hopping for UWB impulse radio

Abstract

Interleaved convolutional time hopping (ICTH) is a recently proposed coding/modulation scheme for UWB impulse radio. ICTH is based upon a low-rate convolutional code, with optimized distance properties, combined with multilevel pulse-position modulation (PPM). In this context, bit-interleaved coded modulation (BICM) is used to decorrelate the bit errors and, more importantly, to obtain a flat spectrum. We present analytical bounds for the BER (bit-error-rate) performance of ICTH and an analysis of its spectral properties.

Full text

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BER Performance and Spectral Properties of Interleaved
Convolutional Time Hopping for UWB Impulse Radio
David Laney∗†, Gian Mario Maggio†‡, Frederic Lehmann†and Lawrence Larson†
†Center for Wireless Communications, UC San Diego
9500 Gilman Drive, La Jolla, CA 92093-0407
‡STMicroelectronics, Inc., Advanced System Technology, San Diego (CA)
Abstract—Interleaved convolutional time hopping (ICTH) is a
recently proposed coding/modulation scheme for UWB impulse
radio. ICTH is based upon a low-rate convolutional code, with
optimized distance properties, combined with multilevel pulse-
position modulation (PPM). In this context, bit-interleaved coded
modulation (BICM) is used to decorrelate the bit errors and,
more importantly, to obtain a flat spectrum. In this paper, we
present analytical bounds for the BER (bit-error-rate) perfor-
mance of ICTH and an analysis of its spectral properties.
I. INTRODUCTION
Over the last decade, there has been a great deal of inter-
est in communications based on impulse radio. These sys-
tems make use of ultra-short duration pulses which yield ultra-
wideband (UWB) signals characterized by low power spectral
densities [1], [2]. UWB systems are particularly promising for
short-range wireless communications as they potentially com-
bine reduced complexity with low power consumption, low
probability-of-intercept (LPI) and immunity to multipath fad-
ing. Namely, in [3], it was first demonstrated that UWB sig-
nalssufferlittlefromfadingand, therefore, onlyasmallfading
margin is required to guarantee reliable communications. Ex-
isting UWB communication systems employ pseudo-random
noise (PN) time hopping combined with modulation schemes
like PPM (pulse-position modulation) or PAM (pulse ampli-
tude modulation) for encoding the digital information.
Recently, it has been suggested to use aperiodic (chaotic)
codes in order to enhance the spread-spectrum characteris-
tics of UWB systems by removing the spectral features of
the transmitted signal, thus resulting in LPI [4]. In partic-
ular, the pseudo-chaotic time hopping (PCTH) scheme ex-
ploits concepts from symbolic dynamics [5] to generate aperi-
odic spreading sequences that, in contrast to fixed (periodic)
pseudo-noise sequences, depend on the input data. PCTH
combines pseudo-chaotic encoding with multilevel pulse-
position modulation. The pseudo-chaotic encoder operates on
the input data like a convolutional encoder [6]. Its output is
then used to generate the time-hopping sequence resulting in
a random distribution of the inter-pulse intervals, thus a noise-
like spectrum.
*Contact author: Tel.
laney@ece.ucsd.edu. Also with HRL Laboratories, Malibu (CA).
(858) 822-4017, Fax (858) 534-1483, E-mail
On the other hand, interleaved convolutional time hopping
(ICTH) has been first proposed in [7] as an alternative to the
pseudo-chaotic time hopping scheme. Basically, in ICTH the
shiftregisterpresentinPCTHisreplacedbyanequivalent-rate
convolutional code with optimized distance properties. The
addition of bit-interleaved coded modulation (BICM) has the
twofold effect of: i) decorrelating the bit errors for the opti-
mal operation of the Viterbi decoder, and ii) randomizing the
output of the convolutional code for spectral whitening pur-
poses. We recall that BICM separates encoding and modula-
tion by means of an interleaver. BICM was first introduced
by Zehavi [8] in order to increase the diversity of a coded
modulation on the Rayleigh fading channel to the minimum
Hamming distance of the code. In [9], Caire et al. showed
that bit interleaving induces a negligible capacity loss on the
AWGN (additive white Gaussian noise) channel. At the re-
ceiver, the detected bits are first deinterleaved and then used
by the decoder to recover the best estimate of the transmitted
information.
In this work we derive analytical bounds for the BER of
ICTH and compare its performance to PCTH as well as the
signal statistics (specifically the frequency of slot usage) and,
more importantly, the spectrum of the transmitted signal.
The paper is organized as follows. In Sec. II we recall the
basics of PCTH. Then, in Sec. III we introduce the ICTH
scheme. Analytical bounds for both schemes are derived in
Sec. IV. Then, in Sec. V we compare the BER bounds vs.
the simulation results. Finally, in Sec. VI we present some re-
sults illustrating the statistics of the transmitted signal and its
spectral properties.
II. PSEUDO-CHAOTIC TIME HOPPING
In this section, we recall the basics of the PCTH scheme [4]
andreinterprettheoperationoftheBernoullishiftmaputilized
in PCTH as a convolutional code.
A. Review of Basics
We start by recalling some useful concepts about the shift
map and its symbolic dynamics. Symbolic dynamics may be
defined as a “coarse-grained” description of the evolution of a
dynamical system [5]. The idea is to partition the state space

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and to associate a symbol to each partition. Consequently, a
trajectory of the dynamical system can be analyzed as a sym-
bolic sequence. A simple example of a chaotic map is the
Bernoulli shift [10], defined as:
xk+1= 2xk
mod 1
(1)
The state, x, can be expressed as a binary expansion:
x = 0.b1b2b3... =
∞
?
j=1
2−jbj
(2)
with bjequal to either “0” or “1”, and x ∈ I = [0,1). For
this map, a Markov partition [5] can be selected by splitting
the interval I = [0,1) into two subintervals: I0= [0,0.5) and
I1= [0.5,1). Then, in order to obtain a symbolic description
of the dynamics, the binary symbols “0” and “1” are associ-
ated with the subintervals I0and I1, respectively.
In PCTH, the Bernoulli shift (1) is approximated by means
of a finite-length (M-bit) shift register, R. Multiplication by 2
in Eq. (1) corresponds to a left shift (b2goes to b1, etc.), while
the modulo one operation is realized by discarding the most
significant bit (MSB). At each clock impulse the most recent
bit of information is assigned the least significant bit (LSB)
position in the shift register, while the old MSB is discarded.
In the PCTH scheme, the output of the pseudo-chaotic en-
coder is used to drive a pulse position modulator. Namely,
each pulse is allocated, according to the pseudo-chaotic mod-
ulation, within a periodic frame of period TF. In other words,
only one pulse is transmitted within each symbol period, TF.
If the pulse occurs in the first half of the frame a “0” is being
transmitted, otherwise a “1”. Each pulse can occur at any of
N = 2M+1discrete time instants, where M is the number of
bits in the shift register. The PCTH receiver comprises a pulse
correlator, matched to the pulse shape, followed by a pulse
position demodulator (PPD). In the simplest case, the binary
information may be retrieved by means of a threshold discrim-
inator at the output of the PPD. In general, though, maximum-
likelihood sequence estimationmaybeperformed byusingthe
Viterbi algorithm, as described in [4].
B. Bernoulli Shift Map as a Convolutional Code
From the viewpoint of information theory, the shift register
implementing the Bernoulli shift may be seen as a form of
convolutionalcoding[5]. Thememoryofthestructureisgiven
by the shift register which stores the last M input bits. Each
input bit causes an output of (M +1) bits; thus the overall rate
is 1/(M + 1). This is shown schematically (for M = 7) in
Fig. 1.
In this work, for simplicity, we consider an implementation
of the Bernoulli shift map witha 2-bit shiftregister resulting in
a rate 1/3 code (M = 2). Accordingly, the constraint length
of the equivalent convolutional code is k = (M + 1) = 3
corresponding to s = 2M= 4 states.
Input?
Output?
Fig. 1.
M = 7 bits.
Equivalent convolutional encoder to the Bernoulli shift map, for
By inspection, one can write down the generator matrix in
the standard octal form as:
GB3= [4,2,1]T
(3)
Using standard methods [6] the transfer function of GB3is
found to be,
TB3(W,X,Y ) =
W3XY3
1 − WXY3− W2XY3
(4)
where the exponent of W is the number of branches traversed,
the exponent of X is the Hamming weight of the input bits and
the exponent of Y is the Hamming weight of the output bits of
the code. This may be verified by analyzing the trellis or state
diagram of the code and calculating the ratio of polynomials
from the signal flow graph corresponding to path deviating
from the zero state and returning to it. Correspondingly, the
free distance of the code is dfree= 3 and there exists one path
with this distance.
III. INTERLEAVED CONVOLUTIONAL TIME HOPPING
Fig. 2 shows a simplified block diagram of the ICTH
scheme. Basically, ICTH may be thought as derived from the
PCTH scheme by replacing the shift register implementing the
Bernoulli shift map with a distance optimized convolutional
code of the same rate and the same number of states. In this
work, in accordance with the previous section, we consider a
rate 1/3 convolutional code with s = 4 states. One such code,
found by computer search, has the following generator matrix
in octal [6]:
GI3= [5,7,7]T
(5)
The transfer function corresponding to GI3is,
TI3(W,X,Y ) =
W3XY8− W4X2Y10 + W4X2Y8
1 − (WXY2+ W2XY2+ W3X2Y2− W3X2Y4).
The free distance of TI3is dfree= 8 and there are three paths
with this distance. An implementation of the encoder is easily
obtained from the generator matrix. In our ICTH scheme, the
convolutional encoder is followed by a bit interleaver whose

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Pulse?
Correlator?
N-PPD?
b?(j)?
k?
Symbol/Bit?
Converter?
^?
Block?
De-Interleaver?
Hard Viterbi?
Decoder?
Pulse?
Generator?
b?(j)?
k?
Convolutional?
Encoder?
Block?
Interleaver?
N-PPM?
Bit/Symbol?
Converter?
Fig. 2. Simplified block diagram of the ICTH scheme with hard Viterbi decoding. In this work, the bit interleaver is realized by a block interleaver.
function is to remove the correlation introduced by the mod-
ulation, according to BICM theory [8]. In particular, in [9] it
is shown that an interleaving depth, L, equal to 4-5 times the
constraint length of the code is sufficient. We emphasize that
in the context of ICTH the interleaver has a randomizing ef-
fect on the output of the convolutional encoder, thus resulting
in a transmitted signal with a flat spectrum (see Sec. VI), a
desirable property for UWB impulse radio.
IV. BER ANALYTICAL BOUNDS
In this section we derive analytical bounds, in the hard de-
coding case,both for PCTH and ICTH. Again, for simplicity,
we consider a rate 1/n convolutional codes, with n = 3. The
modulation used in each case is 8-PPM, where each symbol
transmitted on the channel is mapped to a triplet of encoded
bits. The bit interleaving is realized by a block interleaver with
interleaving depth equal to L = 6K symbols (that is 6K × n
bits), where K is the constraint length of the code. Referring
to Fig. 2, note that in the case of hard decoding, the Viterbi
decoder is fed directly from the de-interleaver after the most
likely transmit symbol has been selected by the pulse-position
demodulator (PPD) based on the observed channel energies in
each slot.
Without loss in generality, due to the linearity if convolu-
tional codes, we can assume the all-zero sequence was trans-
mitted. Say the decoder is comparing the distance between
the received path, lR, and the all-zero path, l0, with the dis-
tance between lRand some other path lD. Let the Hamming
distance between l0and lD, H(0,D) = d. If the Hamming
distance between the received sequence and the all-zero se-
quence H(0,R) <1
to the all-zero path than lD, resulting in the correct path be-
ing chosen. Since the all-zero path is being transmitted, the
Hamming weight of the received path is also the number of
errors experienced over the channel. If H(0,R) ≥1
then the wrong path will be selected. The probability that the
wrong path is selected for d odd in this pairwise comparison
is then,
?
2(d + 1), then the received path is closer
2(d + 1),
P2(d) =
d
?
k=d+1
2
d
k
?
pk(1 − p)d−k
where p is the probability of bit error over the channel. This
is simply the binomially distributed probability that between
1
2(d + 1) and d errors occur over the channel. If d is even,
the incorrect path is chosen when H(0,R) >
H(0,R) =
zero path and the competing path lD. So, an average of1
these cause an error. Then the probability of error for d even
is,
?
A long information sequence results in many paths that di-
verge from the all-zero path and remerge at any particular
node. The number of such paths depends on the length of
the information sequence causing the probability of error to
also be dependent on the length of the information sequence.
An upper bound can be expressed as the union bound of the
pairwise error probabilities,
1
2d. When
1
2d, there is a tie between the distance to the all-
2of
P2(d) =
d
?
k=d
2+1
d
k
?
pk(1 − p)d−k+1
2
?
d
d
2
?
pd/2(1 − p)d/2
Pe=
∞
?
d=dfree
adP2(d)
(6)
where adis the number of paths Hamming distance, d, from
the all-zero sequence and dfree is the free distance of the
code. The values of ad are just the coefficients in the ex-
pansion of T(Y ) = T(W,X,Y )|W=X=1. Note that some
values of ad = 0 where there is no term in the expansion
of T(Y ) ∝ Yd. The probability of information bit error
can be found by observing that the exponent of X in each
term of T(X,Y ) = T(W,X,Y )|W=1is equal to the num-
ber of ones in the information sequence (and the number of
errors from the all-zero information sequence). This is the
number of bit errors experienced along the path. By tak-
ing the derivative and expanding the result in a Taylor series
((1 − x)−1= 1 + x + x2+ x3+ ...)), we find
∂TB3(X,Y )
∂X
X=1
????
= βdfreeYdfree+ βdaYda+ βdbYdb+ ...
(7)
and the probability of bit error is
Pb=
∞
?
d=dfree
βdP2(d)
(8)
where again some values of βd are 0 due to the absence
of terms (exponents of X with value d in the expansion of
T(X,Y )).

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A. PCTH
For the 3-bit Bernoulli shift, evaluating Eq. (4) at W=1 re-
sults in,
TB3(1,X,Y ) = TB3(X,Y ) =
XY3
1 − 2XY3
(9)
and applying Eq. (7) we find the first three terms of the ex-
pansion to be characterized by the following exponents and
coefficients
[dfree,da,db] = [3,6,9]
[βdfree,βa,βb] = [1,4,12]
resulting in a probability of bit error upper-bounded by,
Pb≤ 1 · P2(3) + 4 · P2(6) + 12 · P2(9).
For N-ary orthogonal signals (N = 2n),
?∞
(10)
p =
2n−1
2n− 1·
1
√2π
−∞
?
1 − (1 − Q(y))N−1?
e−(y−√2S0)2
2
dy.
(11)
B. ICTH
For the rate 1/3 distance optimized code, the first three
terms of the expansion are
????
[dfree,da,db] = [8,10,12]
∂TI3(X,Y )
∂X
X=1
= 3Y8+ 15Y10+ 58Y12+ ...
(12)
resulting in
[βdfree,βa,βb] = [3,15,58].
The probability of bit error is upper-bounded by,
Pb≤ 3 · P2(8) + 15 · P2(10) + 58 · P2(12) + 33 · P2(14)
(13)
In both cases, a lower BER bound may be obtained by con-
sidering the first term (free distance) in the expansions.
V. COMPARISON WITH SIMULATIONS
Fig. 3(a) shows the BER bound and the Monte Carlo sim-
ulation results for the rate 1/3 PCTH scheme. Note that the
bounds and simulation results converge at high signal-to-noise
ratio.
Fig. 3(b) shows the BER bound and Monte Carlosimulation
results for the rate 1/3 ICTH scheme. Note that ICTH exhibits
a gain of about 1.5 dB over PCTH @ BER=10−3. This is a di-
rect consequence of the fact that, unlike PCTH, ICTH is based
upon convolutional codes with optimized distance properties.
012345678910
10
−5
10
−4
10
−3
10
−2
10
−1
10
0
10
1
Eb/N0
(a)
BER
Upper bound
Lower bound
Simulation
012345678910
10
−5
10
−4
10
−3
10
−2
10
−1
10
0
Eb/N0
(b)
BER
Upper bound
Lower bound
Simulation
Fig.3. Simulatedvs. analyticalBERperformanceofthea3-bitschemeusing
hardViterbidecodingwithaninterleavingdepthofL = 18, inthepresenceof
AWGN: (a) PCTH, and (b) ICTH. Note the ∼ 1.5 dB improvement exhibited
by ICTH over PCTH @ BER=10−3.
VI. STATISTICS OF THE TRANSMITTED SIGNAL
AND POWER SPECTRAL DENSITY
This section deals with the statistics of the transmitted sig-
nal, in terms of slot usage distribution, and the corresponding
spectral properties. The histogram in Fig. 4(a) shows the rela-
tive frequency of occupation of the time slots in the 256-PPM
case (n=8) for ICTH, without interleaving. The corresponding
PSD (power spectral density) is shown in Fig. 4(b). On the
other hand, Fig. 5 shows similar plots for ICTH with n = 8
with interleaving.
From Fig. 4, in particular, it is visible the effect of the struc-
ture imposed by the code on the slot usage distribution and the
corresponding PSD.The interleaver has a smoothing effect on
the spectrum (see Fig. 5(b)) and also the slot usage distribution
tends to become uniform, as it can be evinced from Fig. 5(a).

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1326496128160192224256
0
0.01
0.02
0.03
0.04
Slot Number
(a)
Relative Frequency
00.10.20.30.40.50.60.70.80.91
−35
−30
−25
−20
−15
−10
−5
Normalized Frequency
(b)
PSD
Fig. 4. (a) Histogram showing the slot usage distribution, and (b) PSD of the
transmitted signal for ICTH, without interleaving.
VII. CONCLUSIONS
Interleaved convolutional time hopping (ICTH) is a novel
coding/modulation scheme for UWB impulse radio, derived
from pseudo-chaotic time hopping (PCTH). ICTH relies upon
a low-rate convolutional code, with optimized distance prop-
erties, and uses BICM. In this paper we developed an expres-
sion for the error bound for PCTH and ICTH based on the
polynomials describing the codes. ICTH exhibits a significant
improvement of the BER performance over PCTH. Moreover,
the bit interleaver introduced because of BICM has a desirable
smoothing effect on the spectrum.
ACKNOWLEDGMENTS
D. Laney acknowledges financial support under a fellow-
ship from HRL Laboratories, Malibu (CA). Finally, the au-
thors would like to thank Prof. Milstein, Dr. Shamain and
Dr. Reggiani for stimulating discussions.
1 32 6496128160192224256
0
0.001
0.002
0.003
0.004
0.005
Slot Number
(a)
Relative Frequency
00.10.20.30.40.50.60.70.80.91
−35
−30
−25
−20
−15
−10
−5
Normalized Frequency
(b)
PSD
Fig. 5. (a) Histogram showing the slot usage distribution, and (b) PSD of the
transmitted signal for ICTH, with an interleaving depth L = 16 symbols.
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