Phase-Equivariant Polar Coded Modulation

Marvin Geiselhart, Marc Gauger, Felix Krieg, Jannis Clausius and Stephan ten Brink

Institute of Telecommunications, Pfaffenwaldring 47, University of  Stuttgart, 70569 Stuttgart, Germany

This work is supported by the German Federal Ministry of Education and Research (BMBF) within the project Open6GHub (grant no. 16KISK019).

MLmaximum likelihood
BPbelief propagation
BPLbelief propagation list
LDPClow-density parity-check
BERbit error rate
BPSKbinary phase shift keying
AWGNadditive white Gaussian noise
LLRlog-likelihood ratio
MAPmaximum a posteriori
FERframe error rate
BLERblock error rate
SCLsuccessive cancellation list
SCsuccessive cancellation
BI-DMCBinary Input Discrete Memoryless Channel
CRCcyclic redundancy check
CA-SCLCRC-aided successive cancellation list
BECBinary Erasure Channel
BSCBinary Symmetric Channel
3GPP3rd Generation Partnership Project
eMBBenhanced Mobile Broadband
CNcheck node
VNvariable node
GenAlgGenetic Algorithm
CSIChannel State Information
OSDordered statistic decoding
MWPC-BPminimum-weight parity-check BP
FFGForney-style factor graph
MBBPmultiple-bases belief propagation
URLLCultra-reliable low-latency communications
mMTCmassive machine-type communications
DMCdiscrete memoryless channel
SGDstochastic gradient descent
NNneural network
5Gfifth generation mobile telecommunication
SCANsoft cancellation
AEDautomorphism ensemble decoding
CCDFcomplementary cumulative distribution function
VVPEViterbi–Viterbi Phase Estimation
QAMquadrature amplitude modulation
QPSKquaternary phase shift keying
PSKphase shift keying
DPSKdifferential phase shift keying
PATpilot assisted transmission
PMpath metric
XORexclusive or

For short-packet, low-latency communications over random access channels, piloting overhead significantly reduces spectral efficiency. Therefore, pilotless systems recently gained attraction.
While blind phase estimation algorithms such as VVPE can correct a phase offset using only payload symbols, a phase ambiguity remains.
We first show that the remaining phase rotations in a polar coded QAM transmission with gray labeling are combinations of bit-flips and automorphisms. Therefore, the decoder is equivariant to such phase rotations and, by smartly selecting the frozen bits, one can jointly decode and resolve the phase ambiguity, without the need for pilot symbols or an outer code. Our proposed system outperforms pilot-assisted transmissions by up to 0.8 dB and 2 dB for QPSK and 16-QAM, respectively.

§ INTRODUCTION

Recent trends in wireless networks, such as URLLC and mMTC, create an urgent need for the robust and efficient transmission of short packets. These scenarios typically involve a random access channel and due to the sporadic transmissions the receiver must re-estimate the channel for every transmission. Conventional systems rely on dedicated pilot symbols in the packet for this purpose, which is called PAT. However, pilot symbols attribute a non-negligible overhead in the short-package systems, reducing the spectral efficiency.

Consequently, systems with smaller or no piloting overhead (also called “non-coherent communication”) received increased attention.
A common approach is a two-stage estimation process.
The first stage is a blind channel estimator that operates on all received symbols and is not aware which data symbols have been transmitted.
The second stage is the resolution of  the remaining phase ambiguity using a few pilot symbols or the error-correcting code.
For instance, in [1], an LDPC code is used to resolve the phase ambiguities. In [2], the authors propose to use polar codes with an outer CRC code for this purpose. Similarly, in [3], a polar codes were proposed for joint decoding and channel estimation of a fading channel.

In this work, we consider joint decoding and phase estimation by exploiting the symmetries of polar codes. In contrast to [2], our approach poses little constraints on the polar code and does not require an outer code. Moreover, any polar decoder can be used, e.g.,

a single low-complexity SC decoder. However, a list decoder together with a CRC can still be used to enhance the performance.

Polar code symmetries, namely their affine automorphism group  [4, 5, 6] have created interest due to applications in cryptoanalysis [7] and low-latency permutation-aided decoder architectures [5, 8]. Moreover, in [9], an automorphism corresponding to a quasi-cyclic shift by $N/4$ has been proposed to simplify the detection in a broadcast scenario.

§ PRELIMINARIES

§.§ System Model

We consider a transmitter consisting of a linear channel encoder described by the generator matrix $\Gm$, encoding $K$ information bits $\uv$ into a codeword $\cv = \uv \Gm$ of length $N$. Assuming QAM, information is encoded in both amplitude and phase of the transmit signal. The codeword is split into groups of $M_\mathrm{b}=\log_2(M)=\log_2(|\mathcal{K}|)$ bit, each mapped onto a complex symbol from the constellation $\mathcal{K}$. Fig. <ref> shows examples of constellations for QPSK and 16-QAM as considered in this work. The mapping between bits $\mathbf{c}$ and symbols $\mathbf{x} = \operatorname{map}(\mathbf{c})$ corresponds to gray labeling.[For the proposed application in Section <ref>, the gray labeling as in Fig. <ref> must be used, i.e., without an interleaver.]
The signal $\xv$ is modulated to passband, transmitted over the channel, and at the receiver demodulated to the complex baseband again. An incorrectly assumed initial carrier phase at the receiver leads to a rotation of the constellation symbols in the complex plane. The equivalent baseband channel is modelled as

\begin{equation}
y_i = x_i \cdot \mathrm{e}^{j\phi} + n_i
\end{equation}

where $\phi \sim \mathcal{U}(0,2\pi)$ is a random uniform phase offset and $n_i \sim \mathcal{CN}(0,2\sigma^2)$ is complex AWGN with noise power $N_0 = 2\sigma^2$.

The estimation and correction of the phase offset $\phi$ is the task of the receiver and is covered in Section <ref> and <ref>. The demapper obtains LLR for each coded bit from the phase corrected received vector $\yv'$.
Under the assumption of equally likely transmission symbols, the MAP rule for the  $k$-th bit of a symbol is

\begin{equation}
\ell\left(y_i \right) \triangleq \ell\left( c_{i,k} \vert y_i \right) = \ln{\frac{\sum_{\hat{x}\in\mathcal{K}_{k,1}}p(y_i\vert x=\hat{x})}{\sum_{\hat{x}\in\mathcal{K}_{k,0}}p(y_i\vert x=\hat{x})}},
\end{equation}

where $\mathcal{K}_{k,0}$ and $\mathcal{K}_{k,1}$ denote the subsets of constellation points with the $k$-th bit equal to $0$ and $1$, respectively.

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Constellation diagrams with gray labeling.

§.§ Polar Codes

Polar codes are based on the principle of channel polarization [10]. For a polar code of length $N=2^n$, the Plotkin transformation is applied $n$ times to obtain $N$ polarized synthetic channels. $K$ polarized “reliable” channels are designated as the information set $\mathcal{I}$ carrying the information, while the remaining $N-K$ “unreliable” channels are frozen bits $\mathcal{F}$ and fixed to 0. Hence, the generator matrix is $\Gm$ are those rows of $\Gm_N = \left[\begin{smallmatrix} 1 & 0 \\ 1 & 1 \end{smallmatrix}\right]^{\otimes n}$ corresponding to $\mathcal{I}$, where $(\cdot)^{\otimes n}$ denotes the $n$-th Kronecker power.
A partial order on the reliability of the synthetic channels induces the affine automorphism group of the code [4, 11], i.e., the set of codeword bit permutations that map the code onto itself.
A common decoder for polar codes is the SC decoder [10], where the information bits are decoded sequentially in natural order. An extension is SCL decoding that keeps a list of the $L$ most promising decoding paths [12].

§.§ Blind Phase Estimation

§.§.§ Maximum Likelihood

Assuming $N_\mathrm{c}$ independent, uniformly distributed transmitted symbols, the log-likelihood function [13] of the phase angle $\phi$ is

\begin{equation}
\ell(\phi) = \sum_{i=0}^{N_\mathrm{c}-1} \ln{ \sum_{x\in\mathcal{K}} \exp{\left(-\frac{1}{2\sigma^2}{\vert y_i - x \mathrm{e}^{j\phi} \vert}^2 \right)} }.
\end{equation}

Consequently, the ML estimate of $\phi$ is

\begin{equation}
\hat{\phi} = \arg\max_{\phi \in [0,2\pi)} \ell(\phi).
\end{equation}

If the constellation exhibits a rotational symmetry, the estimation can only be unique up to this symmetry.
In case of the fourfold-symmetric QPSK and 16-QAM, the phase decomposes into $\phi = \phi_\mathrm{i} + \phi_\mathrm{f}$ with the integer part $\phi_\mathrm{i}=m\frac{\pi}{2},\,m \in \{0,1,2,3\}$ and the fractional (fine) phase shift $\phi_\mathrm{f} \in [0,\nicefrac{\pi}{2})$. Only $\phi_\mathrm{f}$ can be estimated blindly, while $\phi_\mathrm{i}$ has to be recovered by different methods.

§.§.§ Viterbi–Viterbi Algorithm

In [14], the calculation of a phase offset of an $M$-PSK was presented.
The VVPE takes advantage of the inherent symmetry of the PSK constellation.
Independent of the transmitted symbols, multiplying the phase angle by $M$ rotates all symbols onto the same point, directly yielding an estimate of $M\phi$.
The blind fine estimation can be calculated to be

\begin{equation}
\hat{\phi}_\mathrm{f} = \frac{1}{M}\operatorname{arctan}\left(
\frac{
\sum_{i=0}^{N_\mathrm{c}-1}\operatorname{Im}\left(\frac{y_i^{M}}{\lvert y_i \rvert^{M-1}}\right)
% \frac{1}{N_\mathrm{c}}\cdot
\sum_{i=0}^{N_\mathrm{c}-1}\operatorname{Re}\left(\frac{y_i^{M}}{\lvert y_i \rvert^{M-1}}\right)
\right).
\end{equation}

However, multiplying the phase by $M$, along with the final operation of dividing the $\arctan$ function by $M$, gives rise to an $M$-fold ambiguity in the phase estimate [14].

§.§.§ Rice–Rice–Cowley Algorithm

To estimate the phase offset of higher QAM modulations, the blind estimation scheme needs to be adapted.
In [15], the authors treat an 16-QAM as a combination of three individual QPSK in order to perform the phase estimation.
We denote this estimation scheme as RRC estimation. In a first step, the received symbols are classified into the three rings by their amplitude.
The phases of the eight symbols on the innermost and outermost ring are jointly estimated, as they stem from the same (scaled) QPSK constellation with the same phase offset.
Using this first estimate, the remaining eight symbols on the middle circle are partitioned into two sub-constellations whose offsets are again estimated by VVPE. With this additional information, the initial phase estimation can be improved.

§ PHASE-EQUIVARIANT POLAR CODED MODULATION

§.§ Symmetries of Polar-Coded Modulation

Lemma 1 (2-Cyclic Polar Codes):
Let $\mathcal{C}$ be a polar code with information set $\mathcal{I}$ such that $i | 1 \in \mathcal{I} \; \forall i \in \mathcal{I}$, where “$|$” denotes the bitwise or operation. Then for all $\cv \in \mathcal{C}$, also $\cv' = [c_{(i+1)\operatorname{mod} 2}]_i \in \mathcal{C}$. In other words, if for each even information bit index $i$ also $i+1$ corresponds to an information bit, then the code is quasi-cyclic with block-size 2, i.e., the permutation

\begin{equation}
\tau = (1,0,3,2,5,4,\dots,N-1,N-2)
\end{equation}

is an automorphism of $\mathcal{C}$. We call such a polar code 2-cyclic. Note that all polar codes constructed for DMC exhibit this property, as the synthetic channel $i+1$ is always more reliable than $i$, for even $i$ [10].

Proof. The statement is a relaxed version of <cit.> for weakly decreasing codes. Intuitively, $\tau$ affects solely the very first stage of the polar factor graph, so only the $N=2$ component codes have to be regarded. The swap of the two coded bits is an automorphism, if each component code is either rate-0, rate-1 or a repetition code, i.e., $\mathcal{I}_{N-2} \in \{ \varnothing, \{1\}, \{0,1\}\}$.$\qed$

Furthermore, we know that an automorphism on the codeword bits (represented by the $N\times N$ permutation matrix $\Tm$) corresponds to a linear transformation $\Sm$ of the message bits <cit.>

\begin{equation}
\uv \cdot \Gm = \cv \quad \Longleftrightarrow \quad  \uv \cdot \Sm \cdot \Gm = \cv \cdot \Tm. \label{eq:scramble}
\end{equation}

For polar codes and the 2-cyclic shift $\tau$, we find this transformation matrix as [17]

\begin{equation}
\tilde{\Sm} = \Gm_N \Tm \Gm_N = \left[\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}\right] \otimes \Id_{N/2} . \label{eq:polarscramble}
\end{equation}

Note that in this case $\tilde{\Sm}$ is also of dimension $N \times N$ and includes also the frozen bits. $\Sm$ is obtained by selecting only the rows and columns corresponding to $\mathcal{I}$.

Theorem 1 (Rotational Symmetry): Let $\mathcal{C}$ be a 2-cyclic polar code with $N\ge4$ and information set $\mathcal{I}$. Furthermore, let $\mathcal{X} = \{\xv=\operatorname{map}(\cv),  \cv \in \mathcal{C}\}$ be the set of all complex transmit signals under one of the mappings in Fig. <ref> and let $\mathcal{R}=\{N-2,N-1\}$ for QPSK and $\mathcal{R}=\{N-4,N-3\}$ for 16-QAM. Then

\begin{equation}
\xv \mathrm{e}^{-j\frac{\pi}{2}} \in \mathcal{X} \quad \forall \xv \in \mathcal{X},
\end{equation}

In other words, a mapped codeword is transformed to another mapped codeword under the rotation $\mathrm{e}^{-j\frac{\pi}{2}}$.

For QPSK, the rotation of $\pi/2$ corresponds to swapping adjacent bits and then flipping every other bit, i.e.,

\begin{equation}
(c_0,c_1,c_2,c_3,\dots) \mapsto (\bar{c}_1,c_0,\bar{c}_3,c_2,\dots).
\end{equation}

This is a combination of the addition of row $N-2$ of $\Gm_N$ and the 2-cyclic automorphism $\tau$ from Lemma 1. From the requirements of Lemma 1, also $N-1$ must be in $\mathcal{I}$.
For 16-QAM with the labeling from Fig. <ref>, the $\pi/2$-rotation implies

\begin{equation}
(c_0,c_1,c_2,c_3,\dots) \mapsto (\bar{c}_1,c_0,c_3,c_2,\dots),
\end{equation}

corresponding to $\tau$ from Lemma 1 followed by the addition of row $N-4$ of $\Gm_N$. Again, then also $N-3$ must be in $\mathcal{I}$. $\qed$

Remark: A similar statement holds for the rotation by $\pi$ in case of the BPSK. Here, observe that this corresponds to flipping all bits, i.e., the addition of the all-one codeword. This codeword is the last row of $\Gm_N$ and thus part of the code if $N-1 \in \mathcal{I}$.

§.§ Joint Decoding and Phase Ambiguity Resolution

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Block Diagram of joint decoding and phase estimation at the receiver.

Based on Theorem 1, we propose a method for pilotless transmission that resolves the phase ambiguities of the blind phase estimator after polar decoding. We again let $\mathcal{R}=\{N-2,N-1\}$ for QPSK and $\mathcal{R}=\{N-4,N-3\}$ for 16-QAM and define a vector of rotation discriminating bit channels $\uv_\mathrm{R}$ that allows a recovery of the phase. In particular, $\uv_\mathrm{R,QPSK}=(u_{N-2},u_{N-1})$ and $\uv_\mathrm{R,16-QAM}=(u_{N-4},u_{N-3})$ for QPSK and 16-QAM, respectively.
Notice that if at the transmitter $\uv_\mathrm{R}=\zerov$, a phase rotation by multiples of $\pi/2$ results in a nonzero estimate $\hat{\uv}_\mathrm{R}$ at the receiver, namely $\hat{\uv}_\mathrm{R}$ indicates the number of $\pi/2$-rotations in binary. This property directly follows from Theorem 1, as a single rotation flips bit $\uv_{\mathrm{R},0}$ and a double rotation flips bit $\uv_{\mathrm{R},1}$. However, due to the 2-cyclic automorphism $\tau$, also other bits of the estimate $\hat{\uv}$ are affected, which must be undone by applying the matrix $\Sm$ from Eq. <ref> on $\hat{\uv}$ (or, $\tilde{\Sm}$, respectively on the full $\hat{\uv}$). Our proposed joint estimation and decoding scheme can thus be summarized as follows:

* Design a 2-cyclic polar code $\mathcal{C}$, where $\mathcal{R} \subseteq \mathcal{F}$. A practical approach is to use a code design with $K+2$ information bits and freeze the bits $\mathcal{R}$ afterwards.

* At the transmitter, encode using $\mathcal{C}$, followed by either QPSK or 16-QAM mapping according to Fig. <ref>, i.e., $\xv = \operatorname{map}(\uv \Gm)$.

* At the receiver, perform blind fine phase estimation on the received symbols $\yv$, e.g., using ML or VVPE and correct the fine phase offset $\hat{\phi}_\mathrm{f}$ to obtain $\yv'$ which is aligned to the constellation up to rotations of $\frac{m\pi}{2}$.

* Demap and decode $\yv'$ using any polar decoder using information set $\mathcal{I}'=\mathcal{I} \cup \mathcal{R}$. The information bit estimates corresponding to $\mathcal{I}$ are denoted by $\hat{\uv}'$, and the phase recovery bits corresponding to $\mathcal{R}$ by $\hat{\uv}'_\mathrm{R}$.

* Recover the remaining phase rotation as $\hat{\phi}_\mathrm{i} = \frac{m\pi}{2}$ with $m=2\hat{u}'_{\mathrm{R},1}+\hat{u}'_{\mathrm{R},0}$. The total phase rotation is $\hat{\phi} = \hat{\phi}_\mathrm{i}+\hat{\phi}_\mathrm{f}$.

* Resolve the phase ambiguity in the message: $\hat{\uv} = \hat{\uv}' \cdot \Sm^{m}$. Note that $\Sm^2 = \Id$ and, thus, $\Sm^{m}=\Sm^{\hat{u}'_{\mathrm{R},0}}$, which can be implemented using at most $K/2$ binary additions.

Fig. <ref> depicts a block diagram of the proposed receiver.

Remark: Again, the same idea can be applied to BPSK modulation. In this case, $\mathcal{R}=\{N-1\}$ and only phase ambiguities of $\pi$ have to be considered. As no permutation of the coded bits happens, no correction matrix $\Sm$ has to be applied, futher simplifying the system.

§.§ List and Ensemble Decoding

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[singlearrow] (flip2) –node[above, xshift=.1cm]  $\hat{\uv}_{L-1}$ (check2);
[singlearrow] (check2) –node[above]  (ml.west |- check2);
[singlearrow] (check2) –node[right]  (fail2);

[doublearrow] (s1) |- (m0);
[doublearrow] (s1) |- (m1);
[doublearrow] (s1) |- (m2);

[singlearrow] ([yshift=.8cm] decoder.east |- flip0) –node[above]  $\mathrm{PM}_0$ ([yshift=0.8cm] ml.west |- flip0);
[singlearrow] ([yshift=.8cm] decoder.east |- flip1) –node[above]  $\mathrm{PM}_1$ ([yshift=0.8cm] ml.west |- flip1);
[singlearrow] ([yshift=.8cm] decoder.east |- flip2) –node[above]  $\mathrm{PM}_{L-1}$ ([yshift=0.8cm] ml.west |- flip2);

() at (5.3,-3.5) $\vdots$;
() at (9.7,-3.5) $\vdots$;
() at (12,-3.5) $\vdots$;

(uhat) at (15,-2.5) $\hat{\uv}$;
[singlearrow] (ml.east |- uhat) – (uhat);

Modified receiver structure for list or ensemble decoding without blind phase estimation.

The symmetry from Theorem 1 is a polar code property and thus, the proposed scheme works with any polar decoder. However, when an outer code is required in the polar decoding, such as CRC-aided SCL, the system has to be modified. In general, the permutation involved in the $\pi/2$ rotation is not an automorphism of the concatenated code and, hence,  $\hat{\uv}'$ is not a codeword of the outer code. For this reason, the outer code membership must be checked after resolving the phase ambiguity for each candidate in the list.

With this addition, the error-rate performance of the system (up to the 2 bit rate loss for $\uv_\mathrm{R}$) is only limited by the accuracy of the blind fine phase estimator. For short blocklengths, this performance limitation may not be desired. Therefore, we propose a second system compatible with SCL decoding without blind phase estimation.

Fig. <ref> shows the block diagram of the modified receiver using list decoding. While the standard SCL decoder uses only a single input, variants have been proposed that initialize all $L$ paths with different $\yv$, generated from codeword permutations [18, 19]. The decoding paths from the different inputs compete with each other whenever the decoding paths branch and only the $L$ most promising paths based on a PM are kept. In this work, we initialize the $L$ paths with the different hypotheses for $\phi$, ranging from $0$ to $\pi/2$. At the decoder output, the message candidate $\hat{\uv}_l$ with the smallest final PM is declared as the final decoding result. For this message, the remaining phase ambiguity is resolved as presented in Section <ref>. When an outer CRC code is used, the ambiguity is resolved for each candidate before removing the candidates failing the CRC check and PM-based selection.

Another decoding option is an ensemble decoder, i.e., a collection of low-complexity independent (e.g., SC) decoders [20]. Compared to SCL decoding, all decoding paths are separate and no branching is performed, reducing the complexity and decoding latency at the cost of a slight error-rate performance degradation at the same list size [19].

§ RESULTS

§.§ Baseline System

The error-rate performance of the proposed system is evaluated using polar codes according to the 5G design [21].
As reference, we compare to multiple possible baseline systems:

* A scenario where $\phi$ is perfectly known at the receiver, corresponding to an AWGN channel only.

* A PAT system without any blind estimation, i.e., $\phi$ is only estimated from $p$ pilot systems. To make the comparison fair, the polar code has been shortened by $p\cdot M_\mathrm{b}$ bit such that this system has the same overall rate. The number of pilots $p$ is optimized for best BLER performance in each case. This case is denoted by “$p$-PAT”.

* A hybrid PAT system with blind estimation, i.e., $\phi_\mathrm{f}$ is estimated using a blind estimator and the remaining phase ambiguity is resolved using $p$ pilots. Like in the previous system, rate-matching has been optimized. This case is denoted by “$p$-PAT+ML/VVPE/RRC”, depending on the blind estimation algorithm used.

§.§ Error Rate Performance

grid style=dotted,gray,
legend columns=1,
legend pos=south west,
legend cell align=left,
xlabel=$E_\mathrm{s}/N_0$ in dB,
legend image post style=mark indices=,
legend style=fill opacity=0.7, text opacity=1,
mark size=2pt,
mark options=solid,

[color=red,line width = 1pt,dashed,mark=triangle]
table[col sep=comma]
0.00, 9.751e-01
0.50, 9.516e-01
1.00, 8.750e-01
1.50, 7.802e-01
2.00, 6.133e-01
2.50, 4.238e-01
3.00, 2.478e-01
3.50, 1.182e-01
4.00, 5.253e-02
4.50, 1.709e-02
5.00, 5.062e-03
5.50, 1.176e-03
6.00, 2.526e-04
6.50, 4.613e-05

[color=apfelgruen,line width = 1pt,solid,mark=square]
table[col sep=comma]
0.00, 9.039e-01
0.50, 8.589e-01
1.00, 6.887e-01
1.50, 5.254e-01
2.00, 3.458e-01
2.50, 2.360e-01
3.00, 1.141e-01
3.50, 5.203e-02
4.00, 1.938e-02
4.50, 5.786e-03
5.00, 1.285e-03
5.50, 2.811e-04
6.00, 5.207e-05

[color=mittelblau,line width = 1pt,solid,mark=o]
table[col sep=comma]
0.00, 8.688e-01
0.50, 7.692e-01
1.00, 6.286e-01
1.50, 4.438e-01
2.00, 3.011e-01
2.50, 1.575e-01
3.00, 6.711e-02
3.50, 2.690e-02
4.00, 7.898e-03
4.50, 2.099e-03
5.00, 4.231e-04
5.50, 7.085e-05

[color=black,line width = 1pt,solid]
table[col sep=comma]
0.00, 7.365e-01
0.50, 6.026e-01
1.00, 4.099e-01
1.50, 2.686e-01
2.00, 1.370e-01
2.50, 6.110e-02
3.00, 2.424e-02
3.50, 7.490e-03
4.00, 1.921e-03
4.50, 4.835e-04
5.00, 9.569e-05

AWGN only

[-latex, line width=1pt, color=black] (axis cs: 5.38, 1.7e-3) –node[right,yshift=0cm, xshift=0.6cm, color=black]  $0.8$ dB (axis cs: 4.58, 1.7e-3);

[-latex, line width=1pt, color=black] (axis cs: 5.2, 7e-4) –node[right,yshift=0cm, xshift=0.2cm, color=black]  $0.3$ dB (axis cs: 4.85, 7e-4);

BLER performance comparison for QPSK based transmission of $K=64$ information bits over $N_\mathrm{c}=64$ complex channel uses (code rate $R=\nicefrac{1}{2}$, SC decoding).

grid style=dotted,gray,
legend columns=1,
legend pos=south west,
legend cell align=left,
legend style=fill opacity=0.7, text opacity=1,
xlabel=$E_\mathrm{s}/N_0$ in dB,
legend image post style=mark indices=,
mark size=2pt,
mark options=solid,

[color=red,line width = 1pt,dashed,mark=triangle]
table[col sep=comma]
7.00, 1.000e+00
7.50, 1.000e+00
8.00, 1.000e+00
8.50, 9.980e-01
9.00, 9.961e-01
9.50, 9.808e-01
10.00, 9.324e-01
10.50, 8.361e-01
11.00, 6.662e-01
11.50, 4.715e-01
12.00, 2.839e-01
12.50, 1.539e-01
13.00, 7.474e-02
13.50, 3.205e-02
14.00, 1.323e-02
14.50, 4.592e-03
15.00, 1.465e-03
15.50, 4.699e-04
16.00, 1.271e-04
16.50, 3.069e-05

[color=apfelgruen,line width = 1pt,dashed,mark=square]
table[col sep=comma]
7.00, 1.000e+00
7.50, 1.000e+00
8.00, 1.000e+00
8.50, 9.980e-01
9.00, 9.658e-01
9.50, 9.107e-01
10.00, 7.779e-01
10.50, 5.657e-01
11.00, 3.459e-01
11.50, 1.710e-01
12.00, 7.516e-02
12.50, 2.466e-02
13.00, 5.680e-03
13.50, 1.167e-03
14.00, 1.908e-04
14.50, 2.393e-05

[color=apfelgruen,line width = 1pt,solid,mark=square]
table[col sep=comma]
7.00, 1.000e+00
7.50, 1.000e+00
8.00, 9.980e-01
8.50, 9.942e-01
9.00, 9.678e-01
9.50, 8.839e-01
10.00, 7.282e-01
10.50, 5.288e-01
11.00, 3.117e-01
11.50, 1.463e-01
12.00, 6.062e-02
12.50, 1.690e-02
13.00, 3.672e-03
13.50, 7.087e-04
14.00, 1.042e-04
14.50, 1.146e-05

[color=mittelblau,line width = 1pt,dashed,mark=o]
table[col sep=comma]
7.00, 1.000e+00
7.50, 1.000e+00
8.00, 9.980e-01
8.50, 9.846e-01
9.00, 9.678e-01
9.50, 8.791e-01
10.00, 7.175e-01
10.50, 5.282e-01
11.00, 2.989e-01
11.50, 1.497e-01
12.00, 6.231e-02
12.50, 1.812e-02
13.00, 4.526e-03
13.50, 9.203e-04
14.00, 1.366e-04
14.50, 1.625e-05

[color=mittelblau,line width = 1pt,solid,mark=o]
table[col sep=comma]
7.00, 1.000e+00
7.50, 9.980e-01
8.00, 9.980e-01
8.50, 9.903e-01
9.00, 9.587e-01
9.50, 8.729e-01
10.00, 7.011e-01
10.50, 4.946e-01
11.00, 2.725e-01
11.50, 1.218e-01
12.00, 4.746e-02
12.50, 1.261e-02
13.00, 2.784e-03
13.50, 4.538e-04
14.00, 6.764e-05

[color=black,line width = 1pt,solid]
table[col sep=comma]
7.00, 1.000e+00
7.50, 9.980e-01
8.00, 9.931e-01
8.50, 9.702e-01
9.00, 9.215e-01
9.50, 7.843e-01
10.00, 5.877e-01
10.50, 3.681e-01
11.00, 2.001e-01
11.50, 8.889e-02
12.00, 2.880e-02
12.50, 7.614e-03
13.00, 1.609e-03
13.50, 2.877e-04
14.00, 3.894e-05

AWGN only

BLER performance comparison for 16-QAM based transmission of $K=192$ information bits over $N_\mathrm{c}=64$ complex channel uses (code rate $R=\nicefrac{3}{4}$, SC decoding).

Fig. <ref> shows the BLER performance of PAT and the proposed joint estimation and decoding for QPSK transmission of $K=64$ bits over $N_\mathrm{c}=64$ channel uses. All systems use SC decoding. The baseline PAT system requires $p=10$ pilot symbols for best performance, while blind VVPE gains $0.5$ dB by reducing the required pilots to 5. Our proposed pilotless equivariant system outperforms this hybrid PAT by another $0.3$ dB, with a remaining gap to the case without phase offset of another $0.5$ dB at BLER of $10^{-3}$.
In Fig. <ref> the same setup is used with 16-QAM to achieve a target spectral efficiency of $3$ bit per channel use (bpcu) ($K=192$, $N_\mathrm{c}=64$). Both RRC and ML blind phase estimation are considered. The proposed joint system with ML blind phase estimation performs within $0.15$ dB of the AWGN case. Moreover, the joint method outperforms the hybrid PAT by $0.1$ dB; in both cases, RRC estimation loses roughly $0.15$ dB versus the ML estimator. Here, the optimal PAT system would require $p=5$ pilots and $2$ dB higher SNR for a  BLER of $10^{-3}$.

§.§ List Decoding

grid style=dotted,gray,
legend columns=1,
legend pos=south west,
legend cell align=left,
xlabel=$E_\mathrm{s}/N_0$ in dB,
legend image post style=mark indices=,
mark size=2pt,
mark options=solid,

[color=red,line width = 1pt,dashed,mark=triangle]
table[col sep=comma]
5.00, 9.789e-01
5.50, 9.204e-01
6.00, 8.481e-01
6.50, 7.533e-01
7.00, 6.259e-01
7.50, 4.638e-01
8.00, 3.329e-01
8.50, 2.523e-01
9.00, 1.548e-01
9.50, 8.729e-02
10.00, 4.844e-02
10.50, 2.364e-02
11.00, 9.946e-03
11.50, 4.475e-03
12.00, 1.735e-03
12.50, 5.970e-04
13.00, 1.844e-04
13.50, 6.609e-05

[color=red,line width = 1pt,solid,mark=triangle]
table[col sep=comma]
5.00, 9.788e-01
5.50, 9.375e-01
6.00, 8.671e-01
6.50, 7.991e-01
7.00, 7.131e-01
7.50, 5.751e-01
8.00, 4.251e-01
8.50, 2.767e-01
9.00, 1.672e-01
9.50, 8.271e-02
10.00, 4.194e-02
10.50, 1.852e-02
11.00, 7.028e-03
11.50, 2.781e-03
12.00, 9.518e-04
12.50, 3.408e-04
13.00, 9.278e-05

[color=apfelgruen,line width=1pt,dashed,mark=+,mark phase=1,mark repeat=2]
table[col sep=comma]
5.00, 8.613e-01
5.50, 7.790e-01
6.00, 6.964e-01
6.50, 5.895e-01
7.00, 4.777e-01
7.50, 3.851e-01
8.00, 2.754e-01
8.50, 1.992e-01
9.00, 1.398e-01
9.50, 9.109e-02
10.00, 6.050e-02
10.50, 3.751e-02
11.00, 2.165e-02
11.50, 1.296e-02
12.00, 7.016e-03
12.50, 3.730e-03
13.00, 1.904e-03
13.50, 1.009e-03
14.00, 5.010e-04
14.50, 2.249e-04
15.00, 9.728e-05

[color=apfelgruen,line width=1pt,solid,mark=x,mark phase=2,mark repeat=2]
table[col sep=comma]
5.00, 8.613e-01
5.50, 7.790e-01
6.00, 6.964e-01
6.50, 5.895e-01
7.00, 4.777e-01
7.50, 3.851e-01
8.00, 2.754e-01
8.50, 1.992e-01
9.00, 1.398e-01
9.50, 9.109e-02
10.00, 6.050e-02
10.50, 3.751e-02
11.00, 2.165e-02
11.50, 1.296e-02
12.00, 7.016e-03
12.50, 3.730e-03
13.00, 1.904e-03
13.50, 1.009e-03
14.00, 5.010e-04
14.50, 2.249e-04
15.00, 9.728e-05

[color=apfelgruen,line width=1pt,dashed,mark=square,mark phase=1,mark repeat=2]
table[col sep=comma]
5.00, 8.528e-01
5.50, 7.582e-01
6.00, 6.810e-01
6.50, 5.730e-01
7.00, 4.507e-01
7.50, 3.445e-01
8.00, 2.391e-01
8.50, 1.714e-01
9.00, 1.156e-01
9.50, 7.074e-02
10.00, 4.634e-02
10.50, 2.712e-02
11.00, 1.573e-02
11.50, 8.311e-03
12.00, 4.320e-03
12.50, 1.854e-03
13.00, 6.688e-04
13.50, 1.911e-04
14.00, 4.821e-05

[color=apfelgruen,line width=1pt,solid,mark=diamond,mark phase=1,mark repeat=2]
table[col sep=comma]
5.00, 9.073e-01
5.50, 8.322e-01
6.00, 7.434e-01
6.50, 6.201e-01
7.00, 4.800e-01
7.50, 3.394e-01
8.00, 2.257e-01
8.50, 1.384e-01
9.00, 9.060e-02
9.50, 6.016e-02
10.00, 3.928e-02
10.50, 2.483e-02
11.00, 1.562e-02
11.50, 8.284e-03
12.00, 3.942e-03
12.50, 1.719e-03
13.00, 6.458e-04
13.50, 1.887e-04
14.00, 5.051e-05

[color=magenta,line width=1pt,dashed,mark=*]
table[col sep=comma]
5.00, 8.372e-01
5.50, 7.631e-01
6.00, 6.680e-01
6.50, 5.504e-01
7.00, 3.918e-01
7.50, 2.724e-01
8.00, 1.609e-01
8.50, 9.089e-02
9.00, 4.791e-02
9.50, 2.190e-02
10.00, 9.609e-03
10.50, 3.628e-03
11.00, 1.212e-03
11.50, 3.820e-04
12.00, 1.221e-04
12.50, 3.493e-05

[color=magenta,line width=1pt,solid,mark=*]
table[col sep=comma]
5.00, 9.427e-01
5.50, 8.912e-01
6.00, 8.128e-01
6.50, 7.145e-01
7.00, 5.372e-01
7.50, 3.903e-01
8.00, 2.509e-01
8.50, 1.469e-01
9.00, 7.693e-02
9.50, 3.473e-02
10.00, 1.355e-02
10.50, 4.425e-03
11.00, 1.332e-03
11.50, 3.131e-04
12.00, 8.216e-05

[color=mittelblau,line width=1pt,dashed,mark=o]
table[col sep=comma]
5.00, 8.446e-01
5.50, 7.427e-01
6.00, 6.151e-01
6.50, 4.898e-01
7.00, 3.376e-01
7.50, 2.061e-01
8.00, 1.187e-01
8.50, 7.121e-02
9.00, 3.617e-02
9.50, 1.689e-02
10.00, 7.416e-03
10.50, 2.743e-03
11.00, 1.073e-03
11.50, 3.281e-04
12.00, 1.132e-04
12.50, 3.295e-05

[color=mittelblau,line width=1pt,solid,mark=o]
table[col sep=comma]
5.00, 8.934e-01
5.50, 8.073e-01
6.00, 6.942e-01
6.50, 5.277e-01
7.00, 3.682e-01
7.50, 2.281e-01
8.00, 1.275e-01
8.50, 5.827e-02
9.00, 2.182e-02
9.50, 6.631e-03
10.00, 1.731e-03
10.50, 3.919e-04
11.00, 7.774e-05

[color=black,line width=1pt,dashed]
table[col sep=comma]
5.00, 7.800e-01
5.50, 6.454e-01
6.00, 5.045e-01
6.50, 3.476e-01
7.00, 2.481e-01
7.50, 1.589e-01
8.00, 9.089e-02
8.50, 5.098e-02
9.00, 2.467e-02
9.50, 1.086e-02
10.00, 4.416e-03
10.50, 1.461e-03
11.00, 4.951e-04
11.50, 1.404e-04
12.00, 3.594e-05

[color=black,line width=1pt,solid]
table[col sep=comma]
5.00, 7.740e-01
5.50, 6.392e-01
6.00, 4.751e-01
6.50, 3.140e-01
7.00, 1.887e-01
7.50, 9.675e-02
8.00, 4.003e-02
8.50, 1.427e-02
9.00, 3.932e-03
9.50, 9.021e-04
10.00, 1.886e-04
10.50, 2.676e-05

(legend) at (axis description cs:0.98,0.98);

[-latex, line width=1pt, color=black] (axis cs: 12.55, 3e-4) –node[above,yshift=0cm, xshift=.4cm, color=black]  $2$ dB (axis cs: 10.6, 3e-4);

[-latex, line width=1pt, color=black] (axis cs: 13, 2e-4) –node[below,yshift=0cm, xshift=.1cm, color=black]  $1.3$ dB (axis cs: 11.7, 2e-4);

fill opacity=0.7,
text opacity=1,
matrix of nodes,
align =left,
column sep = -5,
inner sep= 2,
anchor=north east,
column 1/.style=anchor=base west,
mark options=solid,
] at (legend)
no CRC     CRC-7

5-PAT     <ref>     <ref>

1-PAT+RRC     <ref>     <ref>

1-PAT+ML     <ref>     <ref>

Joint, ens.     <ref>     <ref>

Joint, list     <ref>     <ref>

AWGN only     <ref>     <ref>

BLER performance comparison for 16-QAM based transmission of $K=64$ information bits over $N_\mathrm{c}=32$ complex channel uses (code rate $R=\nicefrac{1}{2}$, SC-based list and ensemble decoding with $L=8$).

Fig. <ref> compares the performance of PAT and joint estimation and decoding for the case of list and ensemble decoding with $L=8$. Systems with and without a 7-bit CRC are examined; the CRC generator polynomial is $g(z) = z^7 + z^3 +1$ [22].
The considered scenario is the transmission of $K=64$ information bits over $N_\mathrm{c}=128$ complex channel uses, i.e., a spectral efficiency of $2$ bpcu.  In this scenario, the main source of error is in the phase estimation. Hence, the CRC does not improve the performance of the hybrid PAT systems and shows a small gain in case of standard PAT, which outperforms the hybrid approach. The proposed joint estimation and decoding without any blind estimation however, outperforms PAT by $1.3$ dB and $2$ dB, with and without CRC, respectively. As expected, the ensemble decoder cannot profit from the outer CRC code, in contrast to the list decoder. However, if no CRC is used, the ensemble decoder performs very similar to the list decoder and should hence be preferred in this case due to the lower implementation complexity and latency.

§ CONCLUSION

In this paper, we presented a joint estimation and decoding framework for channels with phase offset. The system works by exploiting phase-equivariant symmetries of polar-coded transmission with QPSK and 16-QAM. An extension for list and ensemble decoding has been proposed, significantly outperforming PAT in BLER performance.

While this work considered only single-shot processing, extensions to iterative estimation and decoding (e.g., using soft-in/soft-out decoders) may also profit from the phase-equivariant properties. Additionally, future work may focus on other channels, such as block fading channels or multipath channels. Lastly, different outer codes can be considered that respect the 2-cyclic polar code automorphism (similar to [23]) and thus, may be checked before resolving the phase ambiguities to reduce implementation complexity.

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# ConQX: Semantic Expansion of Spoken Queries for Intent Detection based on
Conditioned Text Generation

Eyup Halit Yilmaz
Aselsan Research Center
Ankara, Turkey
<EMAIL_ADDRESS>
Cagri Toraman
Aselsan Research Center
Ankara, Turkey
<EMAIL_ADDRESS>

###### Abstract

Intent detection of spoken queries is a challenging task due to their noisy
structure and short length. To provide additional information regarding the
query and enhance the performance of intent detection, we propose a method for
semantic expansion of spoken queries, called ConQX, which utilizes the text
generation ability of an auto-regressive language model, GPT-2. To avoid off-
topic text generation, we condition the input query to a structured context
with prompt mining. We then apply zero-shot, one-shot, and few-shot learning.
We lastly use the expanded queries to fine-tune BERT and RoBERTa for intent
detection. The experimental results show that the performance of intent
detection can be improved by our semantic expansion method.

##  1 Introduction

In human-to-machine conversational agents, such as Amazon Alexa and Google
Home, _intent detection_ aims to identify user intents that determine the
command to be executed. _Spoken query_ , also called as _utterance_ , can be
classified into a set of pre-defined user intents Tur et al. (2010).

Intent detection is a challenging task due to the noisy, informal, and limited
structure of spoken queries. Detection models may suffer from the problems of
sparsity, ambiguity, and limited vocabulary. Recent state-of-the-art language
models, such as GPT-2 Radford et al. (2019) and GPT-3 Brown et al. (2020)
based on the Transformer architecture Vaswani et al. (2017), incorporate
domain-independent large corpora in training. They have the capability of
coherent text generation when the task is prompted in natural language. The
clarification of short spoken queries can be done by generating coherent and
semantically related text. For instance, the given query “what is amzn worth"
is expanded to “what is amzn worth _what is Amazon’s stock worth_ " that
clarifies _stock worth_ , as well as solving the ambiguity in _amzn_.

Transformer-based text generation does not always produce meaningful text
segments Shao et al. (2017a). For instance, the given query without
conditioning “has my card application processed yet?" is expanded to “has my
card application processed yet? _If you are not yet with us_ " that gets a
trivial text segment given in italic. The reason would be that the model does
not know the context of card application, such as banking or membership card.
However, the input can be conditioned with a better prompt “[I am a bank
customer], has my card application processed yet?" that gets additional
context as bank customer. The input would then be expanded to a non-trivial
text segment in the context of bank cards.

In order to solve the problems regarding the noisy and limited structure of
spoken queries, we propose a novel method, called ConQX, for semantic
expansion of spoken queries with conditioned text generation. The method name
refers to Conditioned spoken Query eXpansion. Specifically, we employ a
Transformer-based language model, namely GPT-2, to generate semantically
related text segments. We condition the input query to set up a structured
context for generating text segments. For conditioning, we mine useful prompts
that provides structured context, as we call _prompt mining_. We then append
the generated text segments to existing spoken queries, and fine-tune state-
of-the-art language models, namely BERT Devlin et al. (2019) and RoBERTa Liu
et al. (2019), for the downstream task of intent detection.

Conditioned expansion aims to describe the task to the model in natural
language and provide a number of ground truth demonstrations of the task at
inference time. To exploit conditioned expansion, we examine zero-shot, one-
shot, and few-shot learning Brown et al. (2020).

Traditional semantic expansion methods rely on keyword-based expansion, which
utilizes proximity in a semantic space regardless of contextual coherence Roy
et al. (2016). However, the models using contextual word embeddings, such as
BERT, are shown to benefit from natural language queries that keep the grammar
structure and word relations Padaki et al. (2020); Dai and Callan (2019).
Transformer-based text generation can output more coherent natural language
queries, compared to keyword-based expansion Radford et al. (2019) that mostly
adapts to improve the performance of retrieval algorithms Claveau (2020).

The language model can be adapted to the downstream task with natural language
prompts to achieve competitive performance. The design of an input prompt is
important, since different writings of the task can affect the performance
significantly Jiang et al. (2020). Although there are some efforts for the
automation and standardization of prompt generation Jiang et al. (2020); Gao
et al. (2020), they do not consider long text generation tasks, as in the case
of semantic expansion. We employ prompt mining on a set of manually generated
conditioning prompts and experiment with zero-shot, one-shot, and few-shot
learning to further adapt the language model to the task of semantic
expansion.

##  2 Conditioned Query Expansion

Given a set of spoken queries and input prompts, we use a pre-trained
Transformer-based language model, GPT-2 Radford et al. (2019), to generate
coherent text. A spoken query is placed in manually generated natural language
prompts that are determined with prompt mining, and given as input to GPT-2
with zero/one/few-shot learning. The generated text segment is appended to the
end of the original query to obtain the expanded query. BERT Devlin et al.
(2019) and RoBERTa Liu et al. (2019) are then fine-tuned for the task of
intent detection.

###  2.1 Text Generation

Our method for semantic expansion is based on language modeling Bengio et al.
(2003), formulated in Equation 1, where $q$ is a spoken query and $p(q)$ is
the maximum likelihood probability of document estimation based on a sequence
of tokens.

$p(q)=\prod_{i=1}^{n}p(s_{i}|s_{1},...,s_{i-1})$ (1)

ConQX employs the conditional probability of estimating semantic expansion,
$q^{\prime}$, given the original query within the context of the input prompt,
$p(q^{\prime}|q)=p(s_{n+1},...,s_{n+k}|s_{1},...,s_{n})$, where $q$ has the
length of $n$ tokens, and $q^{\prime}$ has $k$ tokens. The likelihood is
estimated by an auto-regressive language model that considers the
distributions of previously generated tokens for next token prediction.

Figure 1: An example expansion process is illustrated for zero/one/few-shot
learning. The input spoken query (labeled in yellow and italic) is inserted
into quotations in a set of prompts. The generated text is then extracted to
obtain the expanded query (labeled in red and bold). True demonstration(s) are
also given in the prompts for one/few-shot learning.

There are several methods to utilize auto-regressive text generation. Greedy
search predicts the next token that has the highest probability of occurrence.
However, greedy search does not generate coherent text due to repetitive
results Shao et al. (2017a). We apply top-_k_ sampling that Fan et al. (2018)
predicts the next token from the most likely _k_ tokens to provide coherent
and diverse text.

###  2.2 Zero/One/Few-shot Learning

Pre-trained language models are trained over large and domain-independent
corpora. When used for a downstream task, such as semantic expansion in our
case, they need conditioning to deduce the task and generate contextually
related text. Zero-shot expansion aims to achieve this conditioning by
inserting spoken queries into input prompts that contain natural language
descriptions of the task without any demonstrations of the desired output.

In one-shot expansion, the input prompt contains a ground-truth demonstration
of the semantic expansion task. The language model is expected to infer the
semantic expansion task more easily, compared to zero-shot expansion. Lastly,
few-shot learning provides a number of true demonstrations to increase the
performance of task inference.

Figure 1 shows an example spoken query for semantic expansion with
zero/one/few-shot learning. The ability of task inference is known to be
available in the large models in terms of the number of parameters, such as
GPT-3 Brown et al. (2020); but also observed in smaller models, such as GPT-2
Schick and Schütze (2020).

###  2.3 Prompt Mining

Determination of the proper input prompt for conditioning the language model
can be achieved through prompt mining. The prompts provide additional context
for the task inference of the language model. We manually generate a set of
prompts that differ in text length, formality of the language, syntactic
structure, and context. We apply empirical evaluation on the prompts, such
that the classification performance is used to select a prompt after 10-fold
leave-one-out cross-validation. We provide a subset of these prompts in Table
1.

Table 1: Prompt mining examples. [INP] is an input prompt, [EXP] is an expanded text segment. |  Input Prompt
---|---
1 |  [INP]. I would like to [EXP]
2 |  Spoken queries are generally short and need to be expanded. For example, [INP] is hard to process and can lead to poor quality results. The query may be rewritten as “[EXP]
3 |  Voice Assistant: “How can I help you?" User: [INP] Voice Assistant: “Sorry, I didn’t understand." User: “[EXP]
4 |  In conversational search, spoken queries are short and need to be expanded. For example, [INP] is hard to process. The query may be rewritten as “[EXP]
5 |  I am a bank customer and I need support, [INP]. My intention is [EXP]

We provide a short prompt (the first), as well as a longer one (the second)
that aims to exploit the ability of Transformer to model long-term
dependencies with the Attention mechanism. The third prompt introduces a
syntactic structure to condition the model, imitating a dialog. The fourth
prompt is written in a more formal language, while the others in a daily
language. The last one has additional context information as banking. Note
that some of the prompts end with a quotation mark that enforces the language
model to generate an example language; while the others generate expansions in
the form of sentence completions.

Table 2: The details of the datasets used in this study. Details | Banking | CLINC | SNIPS
---|---|---|---
Train samples | 10,003 | 18,000 | 13,084
Test samples | 3,080 | 4,500 | 700
Number of intents | 77 | 150 | 7
Avg. length (tokens) | 12.27 | 9.38 | 11.24

##  3 Experiments

###  3.1 Datasets

We use three publicly available datasets for intent detection; namely Banking,
CLINC, and SNIPS. Banking Casanueva et al. (2020) has 77 intents about
banking, which is challenging due to subtle differences among classes. CLINC
Larson et al. (2019) is a balanced dataset with 150 intents. SNIPS Coucke et
al. (2018) is another balanced dataset covering seven intents. The details of
the datasets are given in Table 2. We apply no preprocessing.

Table 3: Comparison of ConQX with the baselines for intent detection in terms of the weighted F1 score. The means of 10-fold cross-validation are reported. The bold score is the highest. $\bullet$ indicates statistically significant improvement at a 95% interval in pairwise comparisons between the highest method and baselines marked with $\circ$. Expansion Method | Banking | CLINC | SNIPS
---|---|---|---
BERT | RoBERTa | BERT | RoBERTa | BERT | RoBERTa
Without expansion | 0.908$\circ$ | 0.923 | 0.959 | 0.964 | 0.974 | 0.979
Bag-of-words (kNN) | 0.909$\circ$ | 0.922 | 0.953$\circ$ | 0.957$\circ$ | 0.969$\circ$ | 0.972$\circ$
Transformer (GPT-2) | 0.912 | 0.923 | 0.954 | 0.964 | 0.976 | 0.981
ConQX (zero-shot) | 0.920$\bullet$ | 0.928 | 0.960 | 0.965$\bullet$ | 0.978 | 0.983$\bullet$
ConQX (one-shot) | 0.920$\bullet$ | 0.928 | 0.959 | 0.961 | 0.983$\bullet$ | 0.981
ConQX (few-shot) | 0.916 | 0.925 | 0.962$\bullet$ | 0.962 | 0.981 | 0.976

###  3.2 Experimental Design

Query expansion is conducted on NVIDIA 2080Ti GPU with 12 GB memory; BERT
fine-tuning uses the same infrastructure as well. Query expansion takes
approximately an hour to complete in the 10-fold setting. The details of the
compared methods are given as follows.

  * •

Without expansion: As a baseline method, we fine-tune BERT-base Devlin et al.
(2019) and RoBERTa-base Liu et al. (2019) with default parameters by
Huggingface Wolf et al. (2019), but without expansion.

  * •

Bag-of-words (kNN): As a baseline expansion method, we consider that the
expanded words are independent (bag-of-words), and use GloVe Pennington et al.
(2014) word embeddings. We sample _k_ =1 nearest neighbor of each input token
in the embedding space, and append them to the input, using scikit-learn
Pedregosa et al. (2011).

  * •

Transformer (GPT-2): As a baseline expansion method, GPT2-large with 774M
parameters by Huggingface Wolf et al. (2019) is used for text generation,
given the original query with no input prompt Radford et al. (2019). The
number of generated tokens is approximated to the number of input tokens.

  * •

ConQX (zero-shot): Our semantic expansion method with zero-shot learning. Both
train and test instances are expanded, and fine-tuning is done using the
expanded queries. For top-k sampling in GPT-2, we experiment
$k\in(10,50,100)$, and select empirically by F1 score on the test set.

  * •

ConQX (one-shot): Our method with one-shot learning. A single true
demonstration of semantic expansion is provided.

  * •

ConQX (few-shot): Our method with few-shot learning. Multiple true
demonstrations of semantic expansion are provided (we use four demonstrations
for the sake of efficiency).

We report the weighted average F1 score for intent detection with leave-one-
out 10-fold cross validation. We use scikit-learn Pedregosa et al. (2011) for
evaluation metrics. Any improvements over the baselines are statistically
validated by the two-tailed paired t-test at a 95% interval.

###  3.3 Experimental Results

We compare the effectiveness of baselines and our method for intent detection
in Table 3. The results show that ConQX improves the effectiveness of intent
detection in all datasets, compared to all baselines. Although, the gap
between baselines and ConQX is not too wide, we show that the differences are
statistically significant in some cases. We show that conditioned text
generation is a promising approach for semantic expansion of spoken queries,
and its performance can be improved by additional _prompt mining_. ConQX with
zero/one-shot learning lead to better improvement in most cases, showing that
hand-crafted true demonstrations could cause noise in few-shot learning, i.e.
prompt mining can generate better demonstrations. GPT-2 without conditioned
text generation does not always improve effectiveness, showing the need of
conditioned text generation. kNN-based expansion also deteriorates
effectiveness in some cases, possibly due to the fact that the neighbor words
do not clarify the context of short queries.

We analyze prompts differing in length, formality of language, and syntactic
structure using punctuation. Longer prompts tend to result in more coherent
and informative expansions. However, depending on the dataset and classifier,
shorter prompts may be more suitable. The role of different prompts are
application-dependent, formally written prompts are favored by formal domains
such as banking. Syntactic structures, such as quotation marks, make
irrelevant text filtered out and result in less noisy expansions.

##  4 Conclusion and Future Work

We propose conditioned query expansion using Transformer-based language
models. We show that the performance is increased in all datasets from
different domains, with proper selection of parameters (zero/one/few-shot and
prompts). ConQX is thereby a promising method for similar tasks that can
benefit from semantic expansion.

Future work would have more observations on other models and sampling
strategies, such as beam search Shao et al. (2017b). We improve the
performance with hand-crafted prompts, but _prompt mining_ and _promp
engineering_ are novel research areas. We plan to focus on tuning parameters,
such as zero/one/few-shot learning and a systematic way to generate prompts
Jiang et al. (2020).

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# A Novel Method to Eliminate the Symmetry Dependence of Fiber Coils for Shupe
Mitigation

Tugba Andac Senol
Department of Physics
Nanotechnology Research Center (NANOTAM)
Bilkent University
Ankara
<EMAIL_ADDRESS>
Onder Akcaalan
Nanotechnology Research Center (NANOTAM)
Bilkent University
Ankara
<EMAIL_ADDRESS>
Aylin Yertutanol
Nanotechnology Research Center (NANOTAM)
Bilkent University
Ankara
<EMAIL_ADDRESS>
Ekmel Ozbay
Department of Physics
Nanotechnology Research Center (NANOTAM)
Department of Electrical and Electronics Engineering
Institute of Materials Science and Nanotechnology
Bilkent University
Ankara
<EMAIL_ADDRESS>

###### Abstract

It is a well-known fact that interferometric fiber optic gyroscopes (IFOGs)
are easily distorted by thermal effects and distortion results in the
degradation of the performance of these sensors. Changing the fiber coil
geometry, increasing the winding symmetry, adding fiber buffer layers around
the fiber coil, using different modulation methods for multifunctional
integrated optic chips (MIOCs), and using special types of fibers, such as
photonic crystal fibers (PCFs), are some alternative solutions for preventing
this degradation. This paper, theoretically and experimentally, investigates
not only how different types of fiber coil winding methods behave under
different rates of temperature change but also presents a novel method, to the
best of our knowledge, to eliminate the Shupe effect, without violating the
simplest IFOG scheme. This method rules out the importance of the winding
symmetry epochally and the need of any extra treatment for the fiber coil to
increase the thermal performance of the system. Regardless of the symmetry of
the fiber coil winding, the rate error due to the Shupe effect can be reduced
to about $\pm$$0.05^{\circ}/$h for any rate of temperature change with this
new method according to the experimental results.

_Keywords_ IFOG $\cdot$ coil $\cdot$ Shupe $\cdot$ symmetry $\cdot$ trimming

##  1 Introduction

Applications, such as guidance, navigation, and control systems in air and
land vehicles, both in industry and military fields, require compact, cost-
effective, and reliable inertial navigation systems equipped with a highly
sensitive gyroscope that uses the Sagnac Effect [1]. The effect that Sagnac
unearthed led to the advent of IFOG with improvements of low-loss optical
fibers, as well as solid-state semiconductor light sources and detectors [2].
Moreover, the fact that IFOGs are solid-state devices and have no moving
parts, the ease of achieving higher sensitivity by increasing the number of
wraps in the fiber coil used [3], their tendency for miniature manufacturing
[4], long lifetime [5], very quick turn-on time [5], high reliability [3, 6,
7] and high precision are just some of the advantages that have ensured their
outdistancing the other types of gyroscopes, e.g., ring laser gyroscopes
(RLGs) [5, 7, 8, 9].

Figure 1:  Rate vs. temperature graph of the IFOG comprising the fiber coil
with the hexadecapole pattern obtained at a rate of 0.2°C/min temperature
change horizontally placed in the (a) +x direction, (b) horizontally reversed
placed in the -x direction, (c) total phase shift of the same IFOG in both
directions, (d) compensated absolute rate errors in both directions compared
with the result obtained from the IFOG comprising a perfectly trimmed
hexadecapolar fiber coil at a rate of 0.2°C/min temperature change.

An ideal IFOG should be able to only measure the Sagnac phase shift. However,
the Sagnac effect is not the only source of measurable phase shift in
practice. Environmental effects such as temperature [10], magnetic field [11,
12], or vibration [13] create differences in the optical paths and can cause
some undesired nonreciprocal phases apart from the Sagnac phase shift. Because
of the difficulty of distinguishing the real Sagnac phase shift from the other
nonreciprocal phase shifts, reducing these nonreciprocal errors caused by the
environmental effects becomes a necessity and the success of this reduction
eventually determines the quality of these fiber optic rotation sensors.

As the sensing element, the fiber coil takes the lead in affecting its
performance. In particular, the thermal sensitivity of the fiber coil reveals
the sensitivity of the IFOG. Special winding methods have been proposed to
overcome this performance limitation [14, 15, 16, 17]. The quadrupole winding
pattern is one of the most widely used techniques with proven performance
[14]. It helps to reduce the non-reciprocity caused by the false rotation
signal when the counter-propagating light waves do not follow the same paths
through the fiber segments in the fiber coil exposed to a different rate of
temperature changes. However, since this winding method is no longer
sufficient to meet the performance demands of IFOGs, further work is required
to increase the winding symmetry of the fiber coil and eventually improve the
thermal performance of the IFOG [18]. Apart from the winding methods,
mechanical studies [19] and different modulation techniques [20, 21] have also
been carried out to reduce the temperature effects. Using special types of
fibers such as photonic crystal fibers (PCFs) in fiber coils to decrease the
temperature dependency of the refractive index is also offered as an
alternative to solve the problems caused by the environmental effects on IFOGs
[22].

In the present paper, we study the rate errors due to the thermal
characteristics of different types of fiber coils both theoretically and
experimentally. We bring in the trimming method in addition to the usage of
different winding methods for the very same purpose. We verify the model
experimentally as well with different rates of temperature change.
Furthermore, in noticing the rate error due to the Shupe effect behaving
independently in the direction of the real rotation during the temperature
analyses, we developed a novel method in which the Shupe effect can be easily
measured and integrated into the IFOG systems independent of temperature
variation and fiber coil winding symmetry.

##  2 Methods and Analysis

Beams traveling uneven paths create undesirable nonreciprocal phase shifts
along with the real phase shift coming from the rotation that is to be
measured and by considering this phase shift, the angular error of thermally
induced non-reciprocity, $\phi_{shupe}(t)$, for the IFOG can be calculated by
[10]

$\displaystyle\phi_{shupe}(t)=\frac{n_{c}}{4NA}(\frac{dn_{c}}{dT}+\alpha
n_{c})(\int_{0}^{L/2}dl(2l-L)\times[\Delta T(l,t)-\Delta T(l^{\prime},t)]),$
(1)

where $n_{c}$ = $1.46$ is the refractive index of the fiber core, $N=48$ is
the number of turns in the fiber coil, $A$ is the area of the fiber coil with
a diameter of $10$ cm, $dn_{c}/dT$ = $10^{-5}$$/^{\circ}C$ is the temperature
dependence of the refractive index of the fiber, $\alpha=5x10^{-7}/^{\circ}C$
is the coefficient of linear thermal expansion of fiberglass and $L$ = $1037$
m is the total length of the fiber coil. Here, $l$ and $l^{\prime}$ represent
the locations of the two rotating beams at the same time and $\Delta T(l,t)$
and $\Delta T(l^{\prime},t)$ represent the temperature differences between the
locations $l$ and $l^{\prime}$ simultaneously.

The simplified version of the equations, as the simulated and experimental
works of thermal effects for the winding methods of four different fiber coils
(dipole (AB), quadrupole (ABBA), octupole (ABBABAAB), and hexadecapole
(ABBABAABBAABABBA)), are presented in the supplementary material.

Figure 2: Photograph of the experimental setup

As seen in Supp. Eqs. (2) and (3), the rate error due to the Shupe effect is
an absolute error that is independent of the direction of the real rotation.
The phenomenon of the Shupe effect being independent of real rotation allows
us to easily calculate the rate error due to this effect by simply reversing
the fiber coil axis horizontally, such as changing its axis from the +x to the
-x. By considering a constant rotation such as the Earth’s rotation applied to
the system; the total phase shifts for the +x and -x axes can be shown as in
Eqs. (2) and (3) ,respectively. When these phase shifts are added up to each
other as shown in Eqn. (4), the twofold pure rate error due to the Shupe
effect can be easily found.

$\displaystyle\phi_{total}^{+x}=\phi_{rotation}+\phi_{shupe_{1}}$ (2)
$\displaystyle\phi_{total}^{-x}=-\phi_{rotation}+\phi_{shupe_{1}}$ (3)
$\displaystyle\phi_{total}^{+x}+\phi_{total}^{-x}=2\phi_{shupe_{1}}$ (4)

Figure 3:  (a) Absolute rate errors of the IFOG comprising the hexadecapolar
fiber coil obtained at a rate of 0.2°C/min., 1°C/min., and 4°C/min.
temperature change, (b) compensated absolute rate errors of the hexadecapolar
fiber coil, respectively, compared with the result obtained from the IFOG
comprising a perfectly trimmed hexadecapolar fiber coil at a rate of a
0.2°C/min. temperature change.

To prove this phenomenon experimentally, we used a hexadecapolar fiber coil
wound by using PM fiber with 80 $\mu$m cladding and 168 $\mu$m coating
diameter on a spool with an average diameter of 85 mm consisting of 48 layers
in total and each layer had 80 turns of fiber as was also used in the model
(see supplementary material). The total length of the fiber coil was
approximately 1037 m and was obtained by an optical time domain reflectometer
(OTDR) measurement (YOKOGAWA AQ7270). We constructed an IFOG by splicing a
three-in-one homemade multifunctional integrated optic chip (MIOC) fabricated
with annealed proton exchange (APE) method to the fiber coil that was also
spliced to a homemade broadband ASE light source with a central wavelength of
1537 nm coupled to a 3-dB optical coupler. APE-MIOC was acting as a $>$40-dB
polarizer, a 3-dB coupler and a phase modulator. Once the loop was completed,
we implemented a photodetector to the system to convert the interfered optical
signal. We packed and placed only the fiber coil with a temperature sensor in
a magnetic shield to prevent nonreciprocal errors coming from magnetic field
changes. First, we ran the test while the fiber coil was horizontally placed
in the +x axis over different temperatures in the climatic chamber. Then, we
horizontally reversed the fiber coil and repeated the same test. The rotation
rate of Earth was measured as +9.6°/h in Ankara, Turkey via the constructed
IFOG with horizontally placed fiber coil and -9.6°/h via the constructed IFOG
with horizontally reversed placed fiber coil. The rate data were read with the
aid of software by using a closed-loop modulation technique and presented as a
function of temperature assessed between -40°C and +60°C operating temperature
in the chamber with a 0.2°C/min. rate of temperature change.

Figure 1 shows the experimental rate vs. temperature data obtained from the
IFOG comprising the fiber coil placed horizontally in the +x direction (a),
horizontally reversed in the -x direction (b) and the total phase shift in
both directions (c) calculated by using Eqn. (4) from both tests respectively.
To compare the rate errors due to the Shupe effect, the Earth rotation is
subtracted from the real rates, and, therefore, the compensated absolute rate
errors that are reduced to $\pm$$0.05^{\circ}/$h are presented with the result
obtained from the IFOG comprising a perfectly trimmed hexadecapolar fiber coil
(d). The comparison between these three results shows good agreement with each
other.

In IFOG systems, it is not possible to simultaneously reverse the fiber coil
axis horizontally. Therefore, a separate setup has been prepared to see if the
presented phenomenon will still be valid when two different fiber coils, one
of which is horizontally reversed, are used in the setup, as shown in Figure 2
and similarly as explained herein above. Then, the equations turn to:

$\displaystyle\phi_{totalC1}^{+x}=\phi_{rotation}+\phi_{shupe_{C1}}$ (5)
$\displaystyle\phi_{totalC2}^{-x}=-\phi_{rotation}+\phi_{shupe_{C2}}$ (6)
$\displaystyle R_{C1,C2}=\phi_{shupe_{C1}}/\phi_{shupe_{C2}}$ (7)
$\displaystyle\phi_{totalC1}^{+x}+\phi_{totalC2}^{-x}=\phi_{shupe_{C1}}+\phi_{shupe_{C2}}$
(8)

Figure 4:  (a) Absolute rate errors of the IFOG comprising the quadrupolar
fiber coil obtained at a rate of 0.2°C/min., 1°C/min., and 4°C/min.
temperature change, (b) compensated absolute rate errors of the quadrupolar
fiber coil, respectively, compared with the result obtained from the IFOG
comprising a perfectly trimmed hexadecapolar fiber coil at a rate of a
0.2°C/min. temperature change.

If the real rotation $\phi_{rotation}$ is known, the ratio $R_{C1,C2}$ between
$\phi_{shupe_{1}}$ and $\phi_{shupe_{2}}$ can be easily found by using Eqs.
(5), (6), (7). Here, two different IFOGs comprising hexadecapolar fiber coils
were tested at 0.2°C/min., 1°C/min., and 4°C/min. rates of temperature change
and the ratio $R_{C1,C2}$ is found to be 3.015, 3.003, and 3.024, respectively
for each rate of temperature change, under the Earth rotation. This proves
that the ratio of $R_{C1,C2}$ is constant for any rate of temperature change,
even if two different fiber coils are used. The uncompensated absolute rate
errors are presented for one of the two different hexadecapolar fiber coils at
0.2°C/min., 1°C/min., and 4°C/min rates of temperature in Figure 3(a). The
error rate increases with respect to the rate of temperature accordingly.
However, the absolute rate error can be compensated down from $\pm
2^{\circ}/$h (as max. rate error) to $\pm$$0.05^{\circ}/$h as shown in Figure
3(b) by using Eqs. (7), and (8). The result obtained from the IFOG comprising
the perfectly trimmed hexadecapolar fiber coil at a rate of a temperature
change of 0.2°C/min. is also shown in Figure 3(b), and labeled as 0.2°C/min.*,
for comparing the results. The data obtained from the reverse axis are
provided in the supplementary material (Supp. Figure 3(b)).

To prove the validity of the claim about the elimination of the symmetry
dependence of the fiber coils, we used one fiber coil with a hexadecapole
winding pattern from the previous setup and one different fiber coil wound
with a quadrupole winding pattern that is known to be more sensitive to the
Shupe effect. The ratios $R_{C1,C2}$ have also been calculated for these fiber
coils and they were found to be 5.167, 5.203, and 5.189, for a 0.2°C/min.,
1°C/min., and 4°C/min. rate of temperature change, respectively. Figure 4(a)
shows the uncompensated absolute rate error for the quadrupolar fiber coil at
0.2°C/min., 1°C/min., and 4°C/min. rate of temperature change. The absolute
rate error can be decreased from $\pm 3^{\circ}/$h (as max. rate error) to
$\pm$$0.05^{\circ}/$h after the compensation as shown in Figure 4(b). The
result obtained from the IFOG comprising the perfectly trimmed hexadecapolar
fiber coil at a rate of a temperature change of 0.2°C/min. is also shown in
Figure 4(b), and labeled as 0.2°C/min.*, here for comparing the results. The
data obtained from the reverse axis are provided in the supplementary material
(Supp. Figure 4(b)).

Figure 5:  Schematics of the configuration of the symmetry-independent-winding
IFOG (SIW-IFOG).

Based on the agreement that we observed between the real and the compensated
results, a new configuration that consists of four-axes, namely -x (as Shupe),
+x, +y, and +z, is designed and a novel system called symmetry-independent-
winding IFOG (SIW-IFOG) is proposed to compensate for the errors caused by the
Shupe effect in Figure 5. The ratios for each different fiber coil pair
$R_{S,X}$, $R_{S,Y}$, and $R_{S,Z}$ can be calculated by using Eqs. (5), (6),
and (7). Then, we have a constant ratio between the real rate errors coming
from the Shupe effect for each different fiber coil pair, such as:

$\displaystyle\centering R_{S,i}=\phi_{shupeS}/\phi_{shupe(i)}\@add@centering$
(9)

where $i$ represents the axes as X, Y, and Z according to the position of the
gyroscope and the ratio $R_{S,i}$ is independent of the rate of temperature
change as shown in Figure 3 and 4.

In the system, the ratio $R_{S,X}$ between CoilS (-x axis) and CoilX (+x axis)
will be the checkpoint to calculate how much the real rate errors are coming
from the Shupe effect. By adding these two total rates, -x (Eqn. (11)) and +x
(Eqn. (12)), the real rotations will cancel each other out and the total real
rate errors will emerge, as in Eqn. (13). The known ratio $R_{S,X}$ reveals
the real rate errors for the +x and -x axes. In addition, when the real rate
error is known for CoilS, by using the ratio for remaining axes, $R_{S,Y}$,
and $R_{S,Z}$, the real rate errors can also be calculated for the y and z
axes easily.

$\displaystyle R_{S,X}=\phi_{shupeS}/\phi_{shupeX}\rightarrow known$ (10)
$\displaystyle\phi_{total}^{-x}=\phi_{rotation_{-x}}+\phi_{shupeS}$ (11)
$\displaystyle\phi_{total}^{+x}=\phi_{rotation_{+x}}+\phi_{shupeX}$ (12)
$\displaystyle\phi_{total}^{-x}+\phi_{total}^{+x}=\phi_{shupeS}+\phi_{shupeX}$
(13)

##  3 Conclusion

In conclusion, starting from the studies aiming to understand the thermal
effects that cause phase differences other than the Sagnac effect on different
fiber coils, we experimentally showed the independence of the Shupe effect on
the real rotation by simply repeating the test with a single hexadecapolar
fiber coil in reverse directions. We experimentally calculated a ratio between
the real rate errors of the two IFOGs comprising different hexadecapolar fiber
coils with the help of the tests conducted at a different rate of temperature
changes and showed that this ratio is constant for any rate of temperature
change. We also reinforced our argument by repeating the test with a
quadrupolar fiber coil that is known to be more sensitive to the Shupe effect
in order to prove the claim about the elimination of the symmetry dependence
of the fiber coil for the Shupe effect. Based on the outcomes of these
studies, we proposed a novel configured complete system called SIW-IFOG that
eliminates the winding symmetry dependence of the performance of the IFOGs and
compensates the errors 40 times reduced for the hexadecapolar fiber coil and
60 times reduced for the quadrupolar fiber coil down to $\pm$$0.05^{\circ}/$h
that is caused by the Shupe effect. In future studies, the effectiveness of
this new method will be examined for rate errors due to the Kerr effect as
well as the Faraday effect under temperature change and vibration, which are
independent of the direction of the real rotations.

## Acknowledgments

This work is partially produced from the Ph.D. studies of T.A.S. One of the
authors (E.O.) also acknowledges partial support from the Turkish Academy of
Sciences.

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