CRC-Aided Short Convolutional Codes and RCU Bounds for Orthogonal Signaling Jacob Kingy William Ryany and Richard D. Wesel

2025-04-27 0 0 362.75KB 6 页 10玖币
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CRC-Aided Short Convolutional Codes and RCU
Bounds for Orthogonal Signaling
Jacob King, William Ryan, and Richard D. Wesel
Department of Electrical and Computer Engineering, University of California, Los Angeles, Los Angeles, CA 90095, USA
Zeta Associates, Aurora, Denver, CO 80011, USA
Email: jacob.king@ucla.edu, ryan-william@zai.com, wesel@ucla.edu,
Abstract—We extend earlier work on the design of convo-
lutional code-specific CRC codes to Q-ary alphabets, with an
eye toward Q-ary orthogonal signaling. Starting with distance-
spectrum optimal, zero-terminated, Q-ary convolutional codes,
we design Q-ary CRC codes so that the CRC/convolutional
concatenation is distance-spectrum optimal. The Q-ary code
symbols are mapped to a Q-ary orthogonal signal set and
sent over an AWGN channel with noncoherent reception. We
focus on Q= 4, rate-1/2 convolutional codes in our designs.
The random coding union bound and normal approximation
are used in earlier works as benchmarks for performance
for distance-spectrum-optimal convolutional codes. We derive
a saddlepoint approximation of the random coding union bound
for the coded noncoherent signaling channel, as well as a normal
approximation for this channel, and compare the performance
of our codes to these limits. Our best design is within 0.6dB of
the RCU bound at a frame error rate of 104.
I. INTRODUCTION
A. Background
Phase coherency between transmitter and receiver is nec-
essary for optimal reception. However, phase coherency can
be difficult to achieve in practice, so orthogonal signaling
with noncoherent reception is often used. The most common
examples of orthogonal signal sets are Q-ary Hadamard
sequences and Q-ary frequency shift keying (QFSK) [1]. We
will assume the latter throughout this paper. Non-coherent
FSK signaling is of practical importance. It is currently used
in Bluetooth [2]. More recently the LoRa standard has adopted
noncoherent QFSK signaling [3] [4].
For values of Qgreater than 8, noncoherent QFSK loss
is small compared to coherent QFSK. In addition, for large
values of Q, noncoherent QFSK performs nearly as well as
BPSK signaling, at the expense of bandwidth. With these facts
in mind, developing good codes for noncoherent QFSK is very
important for contexts in which phase coherency is difficult or
impossible. This occurs when there is a high relative velocity
between the transmitter and the receiver or when the receiver
must be very simple or inexpensive. A natural code choice
for QFSK is a code based on a Q-ary alphabet so that code
symbols are directly mapped to modulation symbols.
This research is supported by Zeta Associates Inc. and National Science
Foundation (NSF) grant CCF-2008918. Any opinions, findings, and conclu-
sions or recommendations expressed in this material are those of the author(s)
and do not necessarily reflect views of Zeta Associates Inc. or NSF.
Binary convolutional codes concatenated with binary CRC
codes have been shown to perform very well on BPSK/QPSK
channels [5] [6] [7]. Following [5], we design Q-ary cyclic
redundancy check (CRC) codes to be concatenated with
optimal, Q-ary, zero-state-terminated convolutional codes
(ZTCC), where zeros are appended to the end of the CRC
word to force the convolutional encoder to terminate in the
zero state. We denote this concatenated code by CRC-ZTCC.
The Q-ary CRC code design criterion is optimization of
the distance spectrum of the concatenation of the CRC code
represented by g(x)and the convolutional code represented
by [g1(x)g2(x)], where each polynomial has Q-ary coeffi-
cients. With all operations over GF(Q), this concatenation is
equivalent to a Q-ary convolutional code with polynomials
[g(x)g1(x)g(x)g2(x)], which is ostensibly a catastrophic
convolutional code. However, rather than applying a Viterbi
decoder to this resultant code, we employ the list Viterbi
algorithm (LVA) [8]. The LVA produces a list of candidate
trellis paths in the original convolutional code trellis, ordered
by their likelihoods, and then chooses as its decision the most
likely path to pass the CRC check.
Design of codes for noncoherent orthogonal signaling has
been done for long messages in [9]–[12]. Here, we analyze
Q-ary CRC-ZTCC codes for short messages. Optimal Q-ary
convolutional codes for orthogonal signaling were described
by Ryan and Wilson [13]. We design distance-spectrum
optimal (DSO) CRCs for two of the codes in [13].
Since the pioneering work of Polyanskiy et al. [14], the
random coding union (RCU) bound has been used as a
measure of the performance quality of short-message binary
codes. The RCU bound is very difficult to calculate, but Font-
Segura et al. [15] derived a saddlepoint approximation for the
RCU bound that is more practical to calculate. In this paper
we extend their work to the noncoherent QFSK channel. We
also include here the normal approximation to the RCU bound
for its simplicity. A converse sphere packing bound was also
presented by Shannon [16] as a lower bound on error rate for
finite blocklength codes, and revisited by [17].
B. Contributions
This paper designs DSO Q-ary CRC codes for two 4-ary
ZTCCs selected from [13] and we apply their concatenation
to the noncoherent 4-FSK channel with list Viterbi decoding.
We also derive a saddlepoint approximation of the RCU
arXiv:2210.00026v1 [cs.IT] 30 Sep 2022
bound for the special case of the noncoherent QFSK channel.
The performances of the codes designed are compared to
their respective RCU bounds and the normal approximation.
Applying these techniques to larger values of Qis an area for
future work.
C. Organization
Section II details the channel model and properties of the
noncoherent QFSK channel. Section III then describes the
design criteria for optimal CRC-ZTCC concatenated codes, as
well as the algorithm for finding optimal CRCs. Section IV
then shows the equations for the saddlepoint approximation
for the RCU bound and derives the relevant equations for the
noncoherent QFSK channel. Finally, Section V presents the
performance of optimal CRC-ZTCC codes compared to the
RCU bound.
II. CHANNEL MODEL
Our discussion here of the noncoherent QFSK channel fol-
lows [18]. For a message symbol x∈ {1,2, ..., Q}, the trans-
mitter takes x=iand transmits the corresponding duration-
Tsignal si(t, φ) = Acos(ωit+φ), where A=p2Es/T so
that the energy of the signal is Es,φis uniform over [0,2π),
and the frequencies ωi/2πare separated by a multiple of the
symbol rate to ensure mutual orthogonality among the signals
si(t).
The detector receives the signal r(t) = si(t, φ) + n(t),
where n(t)is zero mean AWGN with power spectral density
N0/2. The detector consists of Qpairs of correlators, with the
jth pair correlating r(t)against 2
N0sj(t, 0) and 2
N0sj(t, π
2).
The two correlator outputs are then squared and summed, and
a square root is taken of the result. We denote this root-sum-
square of the two values by yj. The vector y= [y1, ..., yQ]T
is the soft decision output of the detector.
If i6=j, the correlation of si(t, φ)and sj(t, 0) is 0 due
to orthogonality, and the same is true for the correlation
of si(t, φ)and sj(t, π
2). As such, the value yjwill be the
root-sum-square of two zero-mean Gaussian random variables
with variance σ2= 2Es/N0. Thus, yjwill have a Rayleigh
distribution with parameter σ2= 2Es/N0. If i=j, however,
the Gaussian random variables that are root-sum-squared will
not be zero mean. As a result, yjwill instead have a Rice
distribution with parameters µ= 2Es/N0and σ2= 2Es/N0.
The Rayleigh and Rice distributions are as follows:
fRayleigh(yj|x=i) = yj
σ2exp "y2
j
2σ2#(1)
fRice(yi|x=i) = yi
σ2I0(yi) exp y2
i+µ2
2σ2(2)
where I0(.)is the zeroth-order modified Bessel function of
the first kind.
The optimal decoding metrics, i.e., log likelihoods, for the
AWGN channel involves the logarithm of a Bessel function
[18] which is clearly impractical. In practice, the "square-law
metric" y2
iis generally used instead of the optimal metrics [9],
[13], [19]. We found that yiperforms better than the square-
law metric y2
isuggested in these papers, and yiis a better
approximation for the optimal metric. Optimal decoding for
noncoherent QFSK is an area for future attention.
Given the message symbol x=i, the received vector y
has a density function that is the product of one Rice density
function, corresponding to yi, and Q1Rayleigh density
functions, corresponding to all yjfor j6=i. This yields the
following transition probabilities for the noncoherent QFSK
channel:
W(y|x=i) = QQ
k=1 yk
σ2QI0(yi) exp "µ2+PQ
k=1 y2
k
2σ2#(3)
III. CRC/CONVOLUTIONAL CODE DESIGN FOR QFSK
The asymptotic (in signal-to-noise ratio, SNR) codeword-
error rate (also, frame-error rate) for a length-nblock code
with minimum distance dmin on the QFSK/AWGN channel
is union upper bounded [13] as
Pcw <
n
X
d=dmin
N(d)P2(d)(4)
where N(d)is the number of weight-dcodewords in the code
and P2(d)is the pairwise error probability for two codewords
at distance d. Asymptotically in SNR, P2(d)decreases with
increasing d[13] so that, from the bound, codes should be
designed with dmin as large as possible. Also from the bound,
for each d, the multiplicities N(d)should be as small as
possible. Codes satisfying these criteria are called distance-
spectrum optimal (DSO).
Ryan and Wilson [13] have found optimal non-binary
convolutional codes for small memory. These codes are op-
timal in the sense of maximizing the free distance dfree and
minimizing the information symbol weight at each weight
wdfree.
In 2015, Lou et. al. [5] showed the importance of designing
CRC codes for specific convolutional codes. An optimal CRC
should minimize the frame error rate (FER) of the CRC-ZTCC
concatenated code based on the union bound above on FER.
These CRCs are called distance-spectrum-optimal CRCs. It
can also be shown that, at high signal-to-noise ratios, this
is equivalent to maximizing the minimum Hamming distance
dmin of the concatenated code, and minimizing the number
of codewords N(dmin)at dmin and weights near dmin.
This criterion is very similar to the criterion for the optimal
convolutional codes in [13].
In this paper, we adapt the methods in [5] to find DSO
CRCs for 4-ary convolutional codes. We consider a memory-
2 (ν= 2) and a memory-4 (ν= 4) code presented in
[13]. The convolutional code generator polynomials g1and
g2can be found in Table I and the optimal CRC polynomials
are in Table II, with the x0coefficient appearing on the
left. These polynomials are elements of GF (4)[x], with
摘要:

CRC-AidedShortConvolutionalCodesandRCUBoundsforOrthogonalSignalingJacobKingy,WilliamRyany,andRichardD.WeselDepartmentofElectricalandComputerEngineering,UniversityofCalifornia,LosAngeles,LosAngeles,CA90095,USAyZetaAssociates,Aurora,Denver,CO80011,USAEmail:jacob.king@ucla.edu,ryan-william@zai.com,w...

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