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

2025-04-27 2 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 10−4.

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

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 Q−1Rayleigh 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

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 <

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

w≥dfree.

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

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

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