# Pre-scaling and Codebook Design for Joint Radar and Communication Based on
Index Modulation

Shengyang Chen  Aryan Kaushik  and Christos Masouros  Shengyang Chen and
Christos Masouros are associated with the Department of Electronic and
Electrical Engineering, University College London, London U.K. (e-mail:
{shengyang.chen.20, c.masouros@ucl.ac.uk}).Aryan Kaushik is associated with
the School of Engineering and Informatics, University of Sussex, Brighton,
U.K. (e-mail: a.kaushik@ucl.ac.uk).This work was supported by the UK
Engineering and Physical Sciences Research Council (EPSRC) Grant number
EP/S026657/1, and the UK MOD University Defence Research Collaboration (UDRC)
in Signal Processing.This paper has been developed on master’s thesis of
Shengyang Chen.

###### Abstract

This paper develops an efficient index modulation (IM) approach for the joint
radar-communication (JRC) system based on a multi-carrier multiple-input
multiple-output (MIMO) radar. The communication information is embedded into
the transmitted radar pulses by selecting the corresponding indices of the
carrier frequencies and antenna allocations, providing two degrees of freedom.
Our contribution involves the development of a novel codebook based minimum
Euclidean distance (MED) maximization and a constellation randomization pre-
scaling (CRPS) scheme for efficient IM-JRC transmission. It can be inferred
that the IM approach integrating the CRPS scheme followed by the codebook
design maximizes the signal-to-noise ratio gain. The numerical results support
the effectiveness of the proposed approach and show enhanced bit error rate
performance when compared to the existing baseline.

###### Index Terms:

Joint radar and communication, index modulation, constellation randomization,
frequency/spatial agility.

##  I Introduction

The frequency competition between the communication and radar systems will
inevitably cause severe congestion within the finite spectrum of sub-6 GHz. To
save RF resources, the sub-6 GHz spectrum allocated to radar systems can be
made available for sharing between radar and wireless systems [1]. Going
beyond spectral co-existence, the joint radar-communication (JRC) systems
integrate the complete hardware platform and the transmitted signal waveform
of the two systems [2]. Recently the JRC systems have been advanced for
hardware efficient architectures and employing low resolution converters [3,
4] that are sought for in emerging fifth generation (5G) wireless
communications [5].

The JRC systems are classified into three categories [6]. One is
_communication-centric_ JRC which implements radar sensing as the secondary
function of the existing communication system, e.g., [7]. Second is _radar-
centric_ JRC which integrates wireless communication as the secondary function
of the existing radar system, e.g., [8]. Third is _joint JRC_ that offers
tunable trade-off between both the operations, e.g., [3]. The design concept
of the radar-centric JRC systems is to embed the communication information
into the transmitted radar pulses and meanwhile maintain the sensing
performance [9]. Furthermore, [10] classifies the information embedding
approaches for the radar-centric JRC systems into beam pattern modulation,
index modulation (IM), and fast-time modulation.

The IM approach which utilizes the randomness and uniqueness of the radar
system to convey the information outperforms the conventional methods with
enhanced data throughput and robust error performance [8]. It aims to generate
a comprehensive coding mechanism so that the blocks of information bits can be
mapped accordingly to the indices of function blocks (communication codewords)
within the JRC system [11]. Example function blocks can cover the spatial
position of the antennas in the multiple-input multiple-output (MIMO) system,
the carrier frequency in the multiple-carriers system, and the type of
waveform in the multi-waveform system. The work in [12] extends frequency
agile radar [13] into the multi-carrier MIMO transmission. [8] further extends
this model for the JRC systems by exploiting the frequency and antenna
agility. However, the exploitation of information embedding with more
effective signal processing approaches and enhanced error performance has not
been widely studied.

_Contributions_ : This paper implements an efficient IM based information
embedding approach in JRC systems that utilizes the spectral and spatial
randomness of the radar pulse signals. For that, the communication codewords
are embedded in the transmitted radar pulses by selecting the corresponding
indices of the carrier frequencies and antenna allocations. Firstly, we
propose a novel codebook based minimum Euclidean distance (MED) maximization
approach to select the best combinations of antennas and frequencies to form
our signal constellation. We then further augment performance through a
constellation randomization pre-scaling (CRPS) scheme for dual functionality
transmission. We show the benefits of integrating the CRPS scheme followed by
the codebook design that maximizes the signal-to-noise ratio (SNR) gain. Our
numerical evaluation reveals an enhanced bit error rate (BER) for the proposed
approaches when compared to the existing baseline.

##  II System Model

We consider IM-JRC, which is implemented based on a multi-carrier MIMO radar
(MCMR) [12] through IM. In MCMR, the collection of available carrier
frequencies is as $F=\\{f_{\textrm{c}}+m\Delta f|m\in(0,M-1)\\}$, where $M$ is
the number of carrier frequencies available at the transmitter (TX),
$f_{\textrm{c}}$ is the initial carrier frequency, and $\Delta f$ is the
frequency step. Let $L_{\textrm{R}}$ be the number of the TX antennas, with
$d=10\frac{c}{f_{\textrm{c}}}$ being the spacing between the neighboring
antenna elements and $c$ is the speed of light. Suppose that there are $N$
radar pulses repeatedly transmitted during each coherent processing interval
of MCMR. Thus, the radar pulses will be transmitted within the time instance
from $nT_{\textrm{r}}$ to $nT_{\textrm{r}}+T_{\textrm{p}}$, where
$n\in(0,N-1)$.

During operation, MCMR randomly chooses a subset of $K$ carrier frequencies
from $F$ for the $n$-th radar pulse as
$F_{\textrm{n}}=\\{\Omega_{\textrm{n,k}}|k\in(0,K-1)\\}$, where
$F_{\textrm{n}}\subset F$, and $\Omega_{\textrm{n,k}}$ is an individual
selected carrier frequency. Subsequently, these frequencies are randomly
allocated among the phased array antennas, so that each antenna utilizes a
single carrier frequency from $F_{\textrm{n}}$. Let $f_{\textrm{n,l}}$ be the
carrier frequency transmitted by the $l$-th antenna. If the same antenna
element also happens to transmit at $\Omega_{\textrm{n,k}}$, then
$\Omega_{\textrm{n,k}}=f_{\textrm{n,l}}$. In MCMR, the TX radar waveform is

$\phi(f,t)=\text{rect}(t/T_{\textrm{p}})\text{exp}(j2\pi ft).$ (1)

In addition, the radar signal emitted from each antenna element should be
multiplied with a weighting function to direct the antenna beam to the optimum
angle $\theta$ [8], given by

$w_{\textrm{l}}(\theta)=\text{exp}(j2\pi f_{\textrm{c}}ld\text{sin}\theta/c).$
(2)

To formulate the transmitted signal vector
$\mathbf{x}(n,t)\in\mathbb{C}^{L_{\textrm{R}}}$ of the entire antenna array
for the $n$-th radar pulse, define $\mathbf{x}_{\textrm{k}}(n,t)$ as the
subset of $\mathbf{x}(n,t)$ that transmits at frequency
$\Omega_{\textrm{n,k}}$. Then the finalized transmission model of MCMR is

$\mathbf{x}(n,t)\\!\\!=\\!\\!\sum_{k=0}^{K-1}\mathbf{x}_{\textrm{k}}(n,t)\\!\\!=\\!\\!\sum_{k=0}^{K-1}\mathbf{P}(n,k)\mathbf{w}(\theta)\phi(\Omega_{\textrm{n,k}},t\\!-\\!nT_{\textrm{r}}),$
(3)

where $\mathbf{P}(n,k)\in\\{0,1\\}^{L_{\textrm{R}}\times L_{\textrm{R}}}$ is
the diagonal matrix containing the spatial agility information. It depends on
the diagonal vector $\mathbf{p}(n,k)\in\\{0,1\\}^{L_{\textrm{R}}}$, whose
$l$-th element is $1$ if $f_{\textrm{n,l}}=\Omega_{\textrm{n,k}}$, and is $0$
otherwise. The parameter $\phi(\Omega_{\textrm{n,k}},t-nT_{\textrm{r}})$
denotes the transmitted waveform at time instance $t$, which carries the
frequency agility information.

For IM-JRC, considering a single pulse transmission, i.e., $N=1$, we simplify
notations as
$\mathbf{x}(t)=\mathbf{x}(n,t),\mathbf{x}_{\textrm{k}}(t)=\mathbf{x}_{\textrm{k}}(n,t),\mathbf{P}_{\textrm{k}}=\mathbf{P}(n,k)$
and $\mathbf{w}_{\textrm{k}}=\mathbf{w}(\theta)$. At the initial stage of
transmission, IM-JRC chooses a subset of $K$ carrier frequencies out of $F$.
The set of all possible frequency selections $\zeta$ is
$\zeta=\\{F^{(i)}|F^{(i)}\subset F,|F^{(i)}|=K,\,\,i=0,1,2,...\\}$, where
$F^{(i)}$ is an individual selection of frequencies. The cardinality of
$\zeta$ is $|\zeta|=\frac{M!}{K!(M-K)!}$. We define
$L_{\textrm{K}}=L_{\textrm{R}}/K$, where $L_{\textrm{K}}$ is assumed to be an
integer greater than $1$, i.e., each carrier frequency is transmitted by
precisely $L_{\textrm{K}}$ antenna elements. $P$ is the collection of all
potential antenna allocations, given by
$P\\!\\!=\\!\\!\\{\mathbf{P}_{\textrm{k}}^{(i)},k\in(0,K\\!-\\!1)|\text{trace}(\mathbf{P}_{\textrm{k}}^{(i)})\\!\\!=\\!L_{\textrm{K}},\,\,i\\!=\\!0,1,2,..\\}$,
where $\\{\mathbf{P}_{0}^{(i)},...,\mathbf{P}_{\textrm{{K-1}}}^{(i)}\\}$
represents one possible allocation pattern. The cardinality of $P$ is
$|P|=\frac{L_{\textrm{R}}!}{{(L_{\textrm{K}}!)}^{K}}$.

IM-JRC integrating the frequency and spatial index modulation and following
[14], the maximum number of bits that can be embedded in a single radar pulse
transmission is

$\text{log}_{2}|\zeta||P|=\text{log}_{2}\frac{M!}{K!(M-K)!}+\text{log}_{2}\frac{L_{\textrm{R}}!}{{(L_{\textrm{K}}!)}^{K}}.$
(4)

Assuming $L_{\textrm{C}}$ receiver (RX) antenna elements, we have

$\mathbf{y}_{\textrm{C}}(t)=\mathbf{H}\mathbf{x}(t)+\mathbf{n}_{\textrm{C}}(t)=\sum_{k=0}^{K-1}\mathbf{H}\mathbf{x}_{\textrm{k}}(t)+\mathbf{n}_{\textrm{C}}(t),$
(5)

where $\mathbf{y}_{\textrm{C}}(t)\in\mathbb{C}^{L_{\textrm{C}}}$ represents
the received JRC signal. $\mathbf{H}\in\mathbb{C}^{L_{\textrm{C}}\times
L_{\textrm{R}}}$ is the Rayleigh flat fading channel.
$\mathbf{n}_{\textrm{C}}(t)\in\mathbb{C}^{L_{\textrm{C}}}$ is the additive
white Gaussian noise (AWGN).

Before the sampling process, the received JRC signal
$\mathbf{y}_{\textrm{C}}(t)$ is firstly down-converted by a factor of
$e^{-j2\pi f_{\textrm{c}}t}$ to remove the high frequency component
$f_{\textrm{c}}$. After that, IM-JRC employs the Nyquist sampling theorem and
samples $\mathbf{y}_{\textrm{C}}(t)$ at the time instances of
$iT_{\textrm{s}}$, where $T_{\textrm{s}}$ is defined as the sampling period
and $i\in(0,\lfloor T_{\textrm{p}}/T_{\textrm{s}}\rfloor$). Assuming the input
bitstream is randomly and equally distributed, the optimum Nyquist sampling
period is given by $T_{\textrm{s}}=1/M\Delta f$. Considering
$L_{\textrm{T}}=\lfloor T_{\textrm{p}}/T_{\textrm{s}}\rfloor+1$ as the number
of sampled outputs per pulse and $c_{\textrm{k}}\in(0,M-1)$ being the
frequency index corresponding to the selected carrier frequencies
$\Omega_{\textrm{k}}$ which is
$c_{\textrm{k}}=(\Omega_{\textrm{k}}-f_{\textrm{c}})/\Delta f$. Recall (3),
and after sampling the finalized reception model of IM-JRC is

$\mathbf{Y}_{\textrm{C}}=\sum_{k=0}^{K-1}\mathbf{H}\mathbf{P}_{\textrm{k}}\mathbf{w}_{\textrm{k}}\boldsymbol{\Phi}_{c_{\textrm{k}}}^{T}+\mathbf{N}_{\textrm{C}},$
(6)

where $\mathbf{Y}_{\textrm{C}}\in\mathbb{C}^{L_{\textrm{C}}\times
L_{\textrm{T}}}$ and
$\mathbf{N}_{\textrm{C}}\in\mathbb{C}^{L_{\textrm{C}}\times L_{\textrm{T}}}$
are the sampled RX signal, and sampled AWGN signal, respectively.
$\boldsymbol{\Phi}_{c_{\textrm{k}}}\in\mathbb{C}^{L_{\textrm{T}}}$ is the
down-converted and sampled transmission waveform, computed as
$\boldsymbol{\Phi}_{c_{\textrm{k}}}={[1,e^{j2\pi c_{\textrm{k}}\Delta
fT_{\textrm{s}}\times 1},...,e^{j2\pi c_{\textrm{k}}\Delta
fT_{\textrm{s}}\times(L_{\textrm{T}}-1)}]}^{T}$. It is also regarded as the
baseband signal carrying the information of the frequency index
$c_{\textrm{k}}$ of a single radar pulse.

##  III Proposed Codebook Design and CRPS Schemes

In this section, we introduce the novel codebook design and CRPS schemes for
IM-JRC. Following [8], for IM-JRC the maximum-likelihood (ML) decoder shows
best BER performance in terms of estimating the frequency indices
$\\{c_{\textrm{k}}\\}$ and antenna allocation patterns
$\\{\mathbf{P}_{\textrm{k}}\\}$, which are previously embedded in the radar
pulses through IM at the TX. The ML detection for IM-JRC is expressed as

${\\{{c}_{\textrm{k}},{\mathbf{P}}_{\textrm{k}}\\}}_{k=0}^{K-1}={{}_{\\{{c}_{\textrm{k}},{\mathbf{P}}_{\textrm{k}}\\}}^{\text{arg
min}}\left\lVert\mathbf{Y}_{\textrm{C}}-\sum_{k=0}^{K-1}\mathbf{H}\mathbf{P}_{\textrm{k}}\mathbf{w}_{k}\boldsymbol{\Phi}_{c_{\textrm{k}}}^{T}\right\rVert_{F}^{2}},$
(7)

where $\left\lVert.\right\rVert_{F}$ denotes the Frobenius norm. Additionally,
we assume that the RX has _a priori_ knowledge of $K$, the desired pointing
angle $\theta$, and perfect channel state information (CSI).

###  III-A Codebook Design

When the available codewords, i.e., combinations of antenna and frequency
indices, are not equal to a power of $2$ [14], IM-JRC only utilizes a subset
of the available codewords for transmission. Defining the subset of codewords
as valid codewords $V$, then $|V|=2^{B}\leq|\zeta||P|$, where $B$ is the
maximum number of bits that can be conveyed in an individual radar pulse. In
general, the codebook design approach aims to locate the optimum $|V|$ valid
codewords. To define “optimum”, [15] pointed out that the BER performance of
the communication system is highly dependent on the MED between the
neighbouring constellation symbols, i.e., valid codewords, where the MED is
formulated as

$\textrm{MED}=d_{\textrm{min}}=_{i\neq
j}^{\text{min}}\left\lVert\mathbf{Y}_{\textrm{i}}-\mathbf{Y}_{\textrm{j}}\right\rVert_{F}^{2}.$
(8)

Consequently, the designed codebook for IM-JRC should comprise the subset of
valid codewords with maximum MED.

Figure 1: Illustration of Codebook Design Approach: (a) Original Codewords;
(b) Designed Codebook.

Fig. 1 demonstrates the proposed codebook design approach. In Fig. 1(a), there
are $C=|\zeta||P|=6$ codewords available for transmission
$\\{\textrm{Y}_{\textrm{0}},...,\textrm{Y}_{\textrm{5}}\\}$. Hence, IM-JRC can
convey a maximum of $\lfloor\text{log}_{2}(C)\rfloor=2$ bits information with
four valid codewords. The design philosophy is to locate the “worst” two
codewords and make them idle. In Step 1, IM-JRC computes for the current MED
and the corresponding neighbouring codewords, which are
$\textrm{Y}_{\textrm{0}}$ and $\textrm{Y}_{\textrm{1}}$. After that, the
system compares their second-minimum Euclidean distances with the remaining
codewords. In this example,
$d_{\textrm{Y}_{0}-\textrm{Y}_{3}}>d_{\textrm{Y}_{1}-\textrm{Y}_{3}}$; thus,
the system keeps $\textrm{Y}_{\textrm{0}}$ and removes
$\textrm{Y}_{\textrm{1}}$. In Step 2, IM-JRC repeats the procedure in Step 1,
and it decides to keep $\textrm{Y}_{\textrm{5}}$ and make
$\textrm{Y}_{\textrm{3}}$ idle. The remaining four valid codewords
$\\{\textrm{Y}_{\textrm{0}},\textrm{Y}_{\textrm{2}},\textrm{Y}_{\textrm{4}},\textrm{Y}_{\textrm{5}}\\}$
comprise the finalized codebook design. The designed codebook maximizes the
MED between the codewords, thereby reducing the probability of detection
errors and enhancing the BER performance.

###  III-B Constellation Randomization Pre-Scaling (CRPS)

A constellation randomization approach is presented in [16, 17] to improve the
performance of spatial modulation in the MIMO communication systems by
maximizing the system MED. The operating principle is to randomize the
original constellation diagram by multiplying codewords with a range of
randomly generated transmit pre-scaling (TPS) factors. We extend that
technique for IM-JRC systems, termed as constellation randomization pre-
scaling (CRPS).

Let $\mathbf{X}_{\textrm{C}}$ represent the original constellation diagram. At
the beginning of CRPS, a number of normalized TPS factors are generated, given
by $\boldsymbol{\alpha}=\\{\boldsymbol{\alpha}_{\textrm{d}}|d\in(1,D)\\}$,
where $\boldsymbol{\alpha}$ is the set containing all the TPS factors with the
cardinality of $D$. The parameter
$\boldsymbol{\alpha}_{\textrm{d}}\in\mathbb{C}^{L_{\textrm{R}}}$ is one
particular TPS factor, where each entry is randomized from the standard normal
distribution, and $\mathbb{E}[\mathbf{\alpha}_{\textrm{d}}]=1$ where
$\mathbb{E}[.]$ is the expectation operator. Defining
$\mathbf{A}_{\textrm{d}}\in\mathbb{C}^{L_{\textrm{R}}\times L_{\textrm{R}}}$
as the diagonal matrix of $\boldsymbol{\alpha}_{\textrm{d}}$. Then the
original constellation diagram is randomized by multiplying it with each of
the TPS matrices, such as
$\mathbf{X}_{\textrm{C-CRPS}}\\!\\!=\\!\\!\mathbf{A}_{\textrm{d}}\mathbf{X}_{\textrm{C}}\\!\\!=\\!\\!\sum_{k=0}^{K-1}\mathbf{A}_{\textrm{d}}\mathbf{P}_{\textrm{k}}\mathbf{w}_{\textrm{k}}\boldsymbol{\Phi}_{c_{\textrm{k}}}^{T}$,
which is the collection of the randomized constellation diagrams. As shown in
Fig. 2, the system MED varies accordingly with the randomization of the
constellation diagram. Consequently, there are possibilities for the new MEDs
to be greater than the original MED, where the possibilities increase
proportionally with the number of TPS factors. Later, IM-JRC determines the
optimal TPS factor which results in the maximum MED among the “pre-scaled”
constellation symbols, hence the optimum BER performance, for which

$\displaystyle\\{\boldsymbol{\alpha}_{\textrm{o}},\mathbf{A}_{\textrm{o}}\\}$
$\displaystyle\\!\\!=\\!\\!\text{arg}\max_{d}{{}_{\\{c_{k_{1}},\mathbf{P}_{k_{1}}\\}\neq\\{c_{k_{2}},\mathbf{P}_{k_{2}}\\}}^{\text{min}}}||\sum_{k=0}^{K-1}\mathbf{A}_{\textrm{d}}\mathbf{P}_{k_{1}}\mathbf{w}_{\textrm{k}}\boldsymbol{\Phi}_{c_{k_{1}}}^{T}$
$\displaystyle-\sum_{k=0}^{K-1}\mathbf{A}_{\textrm{d}}\mathbf{P}_{k_{2}}\mathbf{w}_{\textrm{k}}\boldsymbol{\Phi}_{c_{k_{2}}}^{T}||_{F}^{2},$
(9)

where $\\{\boldsymbol{\alpha}_{\textrm{o}},\mathbf{A}_{\textrm{o}}\\}$ are the
optimal TPS factor and matrix, respectively. As shown in [16], information of
$\\{\boldsymbol{\alpha}_{\textrm{o}},\mathbf{A}_{\textrm{o}}\\}$ can be made
available to the RX, or the RX can independently identify the TPS given an
agreed codebook and CSI. Accordingly, the ML decoder in (7) is modified as

${\\{{c}_{\textrm{k}},{\mathbf{P}}_{\textrm{k}}\\}}_{k=0}^{K-1}={{}_{\\{{c}_{\textrm{k}},{\mathbf{P}}_{\textrm{k}}\\}}^{\text{arg
min}}\left\lVert\mathbf{Y}_{\textrm{C}}-\sum_{k=0}^{K-1}\mathbf{H}\mathbf{A}_{o}\mathbf{P}_{\textrm{k}}\mathbf{w}_{k}\boldsymbol{\Phi}_{c_{\textrm{k}}}^{T}\right\rVert_{F}^{2}}.$
(10)

Figure 2: Illustration of CRPS: (a) Original Constellation Symbols; (b) Randomized Constellation Symbols. Method | Complexity order
---|---
IM-Codebook design | $[KL_{\textrm{R}}+(2K-1)L_{\textrm{T}}]L_{\textrm{R}}|\zeta||P|$
|
$+\frac{3L_{\textrm{R}}L_{\textrm{T}}+1}{2}\sum_{i=1}^{Q}(|\zeta||P|-i+1)(|\zeta||P|-i)$
| $+[(L_{\textrm{R}}+3)L_{\textrm{C}}L_{\textrm{T}}+1](|\zeta||P|-Q)N$
IM-CRPS approach | $[KL_{\textrm{R}}+(2K-1)L_{\textrm{T}}]L_{\textrm{R}}|\zeta||P|+[[L_{\textrm{R}}^{2}L_{\textrm{T}}$
| $+((3L_{\textrm{R}}L_{\textrm{T}}+1)(|\zeta||P|-1))/2]|\zeta||P|+1]D$
| $+[(L_{\textrm{R}}+3)L_{\textrm{C}}L_{\textrm{T}}+1]|\zeta||P|N$
TABLE I: Computational complexity of the proposed schemes. Figure 3: BER
performance of the proposed codebook design and CRPS schemes for $M=7$ and
$L_{\textrm{R}}=6$.

The overall computational complexity of IM with codebook design and CRPS
scheme is provided in Table 1, where $Q$ is the number of codewords to be
eliminated, expressed as $Q=|\zeta||P|-|V|=|\zeta||P|-2^{B}$ and
$B=\lfloor\text{log}_{2}|\zeta||P|\rfloor$. The complexity of the CRPS scheme
also depends on $D$, which is the cardinality of the randomly generated TPS
factors.

##  IV Simulation Results

We set the system parameters as, unless stated otherwise, $M$ = 7,
$f_{\textrm{c}}$ = 1.9 GHz, $\Delta f$ = 10 MHz, $L_{\textrm{R}}$ = 6,
$\theta$ = $\frac{\pi}{4}$, $c=3\times 10^{8}$ m/s, $T_{\textrm{r}}$ = 2 $\mu
s$, $T_{\textrm{p}}$ = 1 $\mu s$, $K$ = 2, $L_{\textrm{C}}$ = 4, and $D$ =
100. Under this condition, the JRC model can convey a maximum of $8$-bits
information during each radar pulse transmission. Additionally, the transmit
signal power is normalized to unity. The BER results are averaged over
$N=1\times 10^{5}$ radar pulses.

Fig. 3 shows the BER performance of the proposed schemes and compare it with
ML decoder baseline [8]. The codebook design scheme slightly outperforms the
ML decoder as it eliminates the idle communication codewords in the
constellation diagram. The CRPS scheme outperforms the codebook design
approach as we can observe that this scheme meets the BER levels of $1\times
10^{-3}$ and $1\times 10^{-4}$ at the SNRs of -15.4 and -12.1 dB, giving the
SNR gains of 2.5 and 2.1 dB, respectively.

Furthermore, in the case of codebook design followed by CRPS, the codebook
design firstly eliminates 164 idle codewords and enlarges the temporal MED;
after that, the CRPS scheme randomizes the remaining 256 valid codewords and
maximizes the average symbol distance. However, this randomization inevitably
decreases the system MED. For the case of CRPS followed by codebook design,
the CRPS scheme randomizes the overall 420 communication codewords and
maximizes the average symbol distance while leaving a set of lower symbol
distances. In the following, these undesirable distances are handled by the
codebook design approach during the elimination of the idle codewords. As a
result, the finalized constellation diagram is more suitable for the
communication transmission, which also accounts for the optimal BER
performance, e.g., the corresponding SNR gains at the BER levels of $1\times
10^{-3}$ and $1\times 10^{-4}$ are both 4.4 dB.

The data rate being the maximum number of bits embedded per radar pulse,
firstly we investigate in Fig. 4 the impact of varying $M$ on the BER
performance of the optimal integrated CRPS and codebook design scheme.
Following the Nyquist sampling period $T_{\textrm{s}}=\frac{1}{M\Delta f}$, it
can be observed that when $M$ increases, the sampling period narrows, and the
reliability of the decoding process is enhanced accordingly, which accounts
for the BER performance enhancement of the IM-JRC model. This result also
supports the trade-off between data rate and BER performance. We can also
observe that the optimal proposed scheme outperforms ML decoder baseline for
all the cases. Similarly Fig. 4 observes the BER performance with varying
number of TX antennas $L_{\textrm{R}}$. The optimal integrated CRPS and
codebook design approach again outperforms the ML decoder baseline for
different $L_{\textrm{R}}$ values.

Figure 4: BER performance of the integrated CRPS and codebook design for
varying $M$ ($L_{\textrm{R}}=6$) and varying $L_{\textrm{R}}$ ($M=7$).

##  V Conclusion

This paper proposes the novel and efficient codebook based MED maximization
and CRPS schemes to employ IM in JRC systems with spectral and spatial
agility. The proposed approaches outperform the state-of-the-art ML decoder
baseline, and the integration of the CRPS scheme followed by the codebook
design leads to the best optimal solution in terms of BER performance. In the
specific case, employing CRPS followed by codebook design outputs the maximum
SNR gain of around 4.4 dB when compared to the baseline approach.

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# Quantum Autoencoders for Learning Quantum Channel Codes

Lakshika Rathi1, Stephen DiAdamo1∗, and Alireza Shabani2 ∗Corresponding
author<EMAIL_ADDRESS>1Cisco Quantum Lab, Garching bei München, Germany
2Cisco Quantum Lab, Los Angeles, USA

###### Abstract

This work investigates the application of quantum machine learning techniques
for classical and quantum communication across different qubit channel models.
By employing parameterized quantum circuits and a flexible channel noise
model, we develop a machine learning framework to generate quantum channel
codes and evaluate their effectiveness. We explore classical, entanglement-
assisted, and quantum communication scenarios within our framework. Applying
it to various quantum channel models as proof of concept, we demonstrate
strong performance in each case. Our results highlight the potential of
quantum machine learning in advancing research on quantum communication
systems, enabling a better understanding of capacity bounds under modulation
constraints, various communication settings, and diverse channel models.

###### Index Terms:

Channel coding, quantum communication, quantum machine learning, quantum
Shannon theory, quantum channel capacity, classical-quantum communication.

##  I Introduction

In classical coding theory, machine learning (ML) has emerged as a powerful
tool for generating communication codes that nearly achieve channel capacity
for various channel models. Promising results using ML have been demonstrated
for learning channel codes [1, 2, 3, 4]. Specifically, in these works,
autoencoders have been proposed as an alternative model for communication
systems. An autoencoder is a type of neural network used for unsupervised
learning. It consists of an encoder neural network that can compress the input
data into a lower-dimensional representation, and a decoder neural network
that reconstructs the original input from the compressed representation. The
network learns to minimize the reconstruction error.

In these past works, the authors map the original models introduced by Claude
Shannon [5] to layers of a neural network representing the encoder, the noisy
channel, and the decoder. By framing the training problem as a classification
problem, the neural network can learn how to overcome the channel noise while
simultaneously compressing the data that is transmitted. Moreover,
particularly in [2], the cost function is set such that the autoencoder is
encouraged to increase the mutual information of the code to better train for
a capacity achieving code.

In this work, our objective is to apply the insights gained from previous
studies to the realm of quantum communication. Quantum autoencoder models have
been swiftly adopted across various research domains [6, 7, 8, 9, 10, 11]. In
these contexts, quantum autoencoders draw inspiration from variational quantum
algorithms [12] which iteratively update parameters to learn how to compress
and decompress the Hilbert spaces encompassing quantum systems. By operating
within a compressed, latent space, these models achieve more efficient
computations within smaller Hilbert spaces. This becomes particularly valuable
during the NISQ era of quantum computing, where resource conservation is of
high importance

Figure 1: The circuit model for classical communication. A message
$s\in\mathcal{M}$ is encoded into a series of qubits $\ket{s_{i}}$ and input
to a parameterized encoder. Next, the channel effects $\mathcal{N}$ are
applied. Once through the channel, a parameterized decoder is applied, and a
series of outputs $\hat{s}_{i},i\in[n]$ are collected.  Figure 2: An example
of a parameterized information-pooling circuit, where $U_{\pi_{i}}$ are
arbitrary rotation gates parameterized by $\pi_{i}\in\mathbb{R}^{3}$.  Figure
3: The circuit model for EA-classical communication. A parameterized circuit
generates entanglement using learned entanglement resources. A message
$s\in\mathcal{M}$ is fed into the encoder and half of the qubits are encoded
by a parameterized encoder. Next, the channel effects are applied to half of
the qubits. Once through the channel, a parameterized decoder is applied to
the whole system, and a series of outputs $\hat{s}_{i},i\in[n]$ are collected.
The red dashed line represents the system separation. Figure 4: The circuit
model for quantum communication. A maximally entangled quantum state is taken
as input where part of the system is fed through a parameterized encoder, the
noise model, and a parameterized decoder. The density matrix representing the
final state is taken as output. The red dashed line represents the system
separation where some of the state remains with the sender.

In information theory, channel capacity theorems are often proven non-
constructively, leaving the challenging task of determining a capacity-
achieving code open-ended. This holds true for quantum communication as well,
where, for example, the Holevo capacity outperforms classical capacity [13]
for certain channels but joint-detection receivers (JDRs) that are easy to
produce is still a challenging problem [14, 15]. By using quantum autoencoders
for training communication systems, we believe it can lead to easier-to-build
JDRs that can allow for higher communication rates approaching the theoretical
limits. This is highly aligned with the objective of future communication
systems, especially in 6G networks. 6G networks aim to incorporate quantum
features into the communication networks to enhance performance. Using our
framework, we can, for example, learn the optimal channel codes under the
constraints of what 6G networks can do.

In previous literature, autoencoders have been extensively used in a classical
setting to generate channel codes for noisy classical channels [1, 2, 4].
While these concepts have been explored in the classical domain, their
application to quantum communication remains unexplored. Hence, in this work,
we investigate this unexplored direction. Our approach involves developing a
framework to analyze quantum channel codes and using it to evaluate the
performance of learned codes across various channels. We focus on three
communication settings: 1) Classical-quantum communication; 2) Incorporating
shared entanglement resources into the model for entanglement-assisted (EA)
communication; and 3) Quantum communication. Our investigations yield
promising results, with the models efficiently learning encoding and decoding
parameters and approaching theoretical capacities in all cases.

##  II Channel Coding with Quantum Autoencoders

We explore the performance of parameterized circuits training for learning
classical, and EA-classical, and quantum capacities for various channels. Each
communication setting has a slightly different configuration but there is a
general structure that allows us to easily modify the framework to
accommodate.

In the classical communication setting, the communication process is the
following: a sender chooses a classical message from a codebook
$s\in\mathcal{M}$, which is then encoded into a quantum state $\rho_{s}$ using
a unitary operation $E_{\theta}$. The parameterized operation $E_{\theta}$ has
a particular circuit structure and a $k$-length vector
$\theta\in\mathbb{R}^{k}$ defines the operations. The encoded state $\rho_{s}$
is transmitted through a channel $\mathcal{N}$, a completely positive, trace
preserving map. The resulting state $\mathcal{N}(E_{\theta}(\rho_{s}))$ is
processed by a decoder $D_{\phi}$, which is parameterized by
$\phi\in\mathbb{R}^{k^{\prime}}$, where $k^{\prime}$ can be different than
$k$. The measurement of the state yields a classical message $\hat{s}$
approximating $s$ and defines a conditional distribution $p(\hat{s}|s)$. This
setting is illustrated in Fig. 1. In some cases, we introduce redundant qubits
to test repetition codes, for example. In these cases, a parameterized
“pooling” circuit (Fig. 2) is incorporated in the decoder, placed between
$D_{\phi}$ and the measurements. To analyze the quality of the learned channel
code, we calculate the respective mutual information using input-output values
over the channel and compare it to the known channel capacities.

For EA communication, additional qubits are introduced to the system and are
entangled with the communication system before communication occurs. In this
case, we assume the entanglement is distributed in a noiseless way, but we do
not presume which state the entanglement was in to start with. Pairs of qubits
are therefore firstly entangled using parameterized entangling operations with
a gate set forming $K_{\lambda}$, where $\lambda$ is the parameter vector. For
the encoding step, the classical message is fed into the encoder such that
controlled gates can manipulate the sender’s half of the entangled state. The
framework applies channel noise to the sender’s half and the other half of the
system does not experience the channel noise. The receiver then uses the total
system for message decoding. This setting is illustrated in Fig. 3. Again to
analyze the quality of the learned EA channel code, we calculate the
respective mutual information.

The quantum capacity of a quantum channel is defined in a different way than
classical capacities are defined. In this case, it is not a direct measure of
which rate quantum states are being transmitted, but rather the direct value
can be thought of as a measure of how well a quantum channel preserves
entanglement. To therefore compute the quantum capacity of a quantum channel,
the communication task is to firstly generate a maximally entangled bi-partite
system and then apply the encoding, channel model, and decoding to one half of
the system. The value in which to maximize is no longer the mutual
information, but rather the quantum coherent information defined as
$I(A\rangle B)_{\rho}\coloneqq S(B)_{\rho}-S(AB)_{\rho}$, where $S(X)_{\rho}$
is the von Neumann entropy of state $\rho$ restricted to subsystem $X$. To
simulate this in the framework, we apply parameterized encoding and decoding
gates to the subsystem that undergoes channel noise as depicted in Fig. 4.

In this initial work, we have set up the framework for learning quantum
channel codes and analyzed learned codes for different channels. In
particular, we study the quantum bit-flip, phase-flip, depolarizing, and
amplitude damping channel for $p=1/2$ and $p=1$. The Kraus operations modeling
the behavior for these channels are the following:

  1. 1.

Bit-flip:

$\displaystyle K_{0}=\sqrt{1-p}\begin{bmatrix}1&0\\\ 0&1\\\
\end{bmatrix},K_{1}=\sqrt{p}\begin{bmatrix}0&1\\\ 1&0\\\ \end{bmatrix}$ (1)

  2. 2.

Phase-flip:

$\displaystyle K_{0}=\sqrt{1-p}\begin{bmatrix}1&0\\\ 0&1\\\
\end{bmatrix},K_{1}=\sqrt{p}\begin{bmatrix}1&0\\\ 0&-1\\\ \end{bmatrix}$ (2)

  3. 3.

Depolarizing:

$\displaystyle K_{0}=\sqrt{1-p}\begin{bmatrix}1&0\\\ 0&1\\\
\end{bmatrix},K_{1}$ $\displaystyle=\sqrt{p/3}\begin{bmatrix}1&0\\\ 0&-1\\\
\end{bmatrix}$ (3) $\displaystyle K_{2}=\sqrt{p/3}\begin{bmatrix}0&1\\\ 1&0\\\
\end{bmatrix},K_{3}$ $\displaystyle=\sqrt{p/3}\begin{bmatrix}0&-i\\\ i&0\\\
\end{bmatrix}$

  4. 4.

Amplitude Damping:

$\displaystyle K_{0}=\sqrt{1-p}\begin{bmatrix}1&0\\\ 0&\sqrt{1-\gamma}\\\
\end{bmatrix},K_{1}$ $\displaystyle=\begin{bmatrix}0&\sqrt{\gamma(1-p)}\\\
0&0\\\ \end{bmatrix}$ $\displaystyle
K_{2}=\sqrt{p}\begin{bmatrix}\sqrt{1-\gamma}&0\\\ 0&1\\\ \end{bmatrix},K_{3}$
$\displaystyle=\begin{bmatrix}0&0\\\ \sqrt{\gamma p}&0\\\ \end{bmatrix}$ (4)

We configure the model by combining a parameterized encoder, channel, and
parameterized decoder, along with optional components such as the pooling
layer. This configuration can be trained using the following software tools.
Our framework leverages the JAX interface [16] of the Pennylane software
library [17] for quantum simulation. For parameter updating, we employ a
variant of the Adam optimizer [18]. In the classical communication case, we
employ the average cross-entropy loss [19] as the cost function. In the
quantum communication case, we use the trace distance between the output state
of the reference system. The training flow is depicted in Fig. 5. Once the
model is trained for a specific channel, a set of test messages is used to
evaluate the code. In the classical case, we compute the mutual information,
while in the quantum case, we calculate the quantum coherent information.

Figure 5: A flow diagram of how the framework executes. For classical message
transmission a training collection of messages $\\{m_{i}\\}_{i=1}^{k}$ are
selected and the initial parameters
$(\theta_{0},\phi_{0},\lambda_{0},\pi_{0})$ are set. The data is fed into the
parameterized circuit which outputs either classical measurement results or a
density matrix. The cost function is computed and the parameters are updated.
This repeats $n$ times. After $n$ iterations, the final parameters are used to
evaluate the code against a test set of messages, when applicable for
classical communication.

##  III Training Results

In this section, we present the results of training the circuit models and the
evaluation of the final trained codes in their ability to achieve capacity. We
explore the classical and quantum communication scenarios.

###  III-A Classical Communication

Classical communication over quantum channels can be done in a variety of ways
and therefore there are four main capacities to consider when no additional
entanglement resources are used: 1) Separable-state encoding and decoding
$C_{ss}$; 2) Separable-state encoding and joint measurement decoding $C_{sj}$,
also known as the Holveo capacity [20]; 3) Entangled-state encoding and
separable measurement decoding $C_{es}$; and 4) Entangled-state encoding and
joint decoding $C_{ej}$ [21]. Depending on the desired model, our framework
allows to test any of the cases.

For the demonstration of our framework in the classical communication setting
without entanglement, we have tested various code models for the (a) Bit-flip
channel, (b) Depolarizing channel, and (c) $p=1$ amplitude damping channel.
The results of training the code are shown in Fig. 6(a)-(c). For the bit-flip
and depolarizing channel, the framework was able to train the circuit
parameters to produce a code that meets the channel capacity. We can see that
by adding an encoding operation to the bit-flip channel, the Holveo capacity
is exceeded since it becomes possible to encode in the $X$ basis. For the
depolarizing channel, encoding at the sender’s side does not enhance the
capacity of the channel in this case and the Holveo capacity is equal to
$C_{ej}$.

The $p=1$ amplitude damping channel is a non-unital channel, which means that
the accessible information between channel uses is non-additive, making the
determination of channel capacity more challenging. For the $p=1$ amplitude
damping channel, the Holveo quantity serves as a lower bound for the capacity.
Initially, without encoding on the sender side and using a standard basis
embedding, the output of our framework produced a code with a sub-optimal
capacity, lower than the lower bound in the single qubit case. To further
explore this, we introduced repetition and pooling techniques (see Fig. 2) in
the encoding and decoding scheme. Although the repetition code uses the
channel more times, the circuit with pooling yields a single output bit. We
compute the mutual information (without regularization) using this single bit
and observe that for $\gamma$ values greater than 0.5, repetition and pooling
improve the results, where the solid lines in the plot represent the no-
encoding scheme. When we introduce a parameterized encoding circuit
(represented by the dashed lines) with the same pooling scheme, we observe a
significant improvement in the mutual information of the code. We present
known upper and lower bounds for the total capacity of the channel using the
dash-dotted lines.

###  III-B Entanglement-Assisted Classical Communication

When considering entanglement-assisted (EA) communication, important
considerations include determining the type of entanglement to be used (e.g.,
Bell states, GHZ states, etc.) and how these states should be encoded with
classical information. In our framework, we delegate these complex decisions
to the training process. The model in this case, as depicted in Fig. 3,
parameterizes the initial entanglement generation and then the classical
encoding step. Then, part of the system gets the noise effects from the
channel and some remains noiseless. Upon arrival, the receiver decodes the
total state also using a parameterized circuit. We analyzed the EA capacity
for three channels: (a) Phase-flip, (b) Depolarizing, and (c) $p=1/2$
amplitude damping. The results in Fig. 6(d)-(f) show that the framework is
able to train the parameterized circuit model to find the optimal parameters
that nearly meet or do meet the channel capacity in each case. In these cases,
the channel models are unital channels and so their capacities are known. They
are found by using standard super-dense coding which our trained model
successfully reproduced.

In these results, we assumed that the stored entanglement remains noiseless
while the sender makes transmissions, but we also test the case where a small
amount of depolarizing noise affects the stored qubit. In Fig. 6(d)-(f),
represented by the orange and green crosses, we also show the result of
depolarizing noise applied to the idle qubit to simulate the effects of
memory. We assume the sender’s qubit undergoes a depolarizing channel with
$p_{i}=0.05$ and $p_{i}=0.10$ during transmission. We see that the noisy
memory diminishes the rate and that the framework could still train the model
to achieve the maximum rate. We show the results in Fig. 6(d)-(f) with red
crosses.

###  III-C Quantum Communication

The final communication scenario we considered for our framework is the
quantum communication setting. In this scenario, we focused on transmitting
parts of maximally entangled states and evaluating the preservation of
entanglement. Based on the configuration in Fig. 4, we set the model for
training. In quantum communication, channels that are “degradable” have a
single-letter quantum capacity formula [22], making them easier to analyze.
The phase-flip and $p=1/2$ amplitude damping channels are such channels, where
the depolarizing channel is not. We show the training results for determining
the quantum capacity for those channels in 6(g)-(i). Notably, for degradable
channels, a single maximally entangled state meets the capacity, and our
framework successfully reproduces this result. However, the capacity for the
depolarizing channel is generally unknown. We explore various methods of
transmitting entanglement over this channel. The blue crosses in Fig. 6(h)
show the single unit of entanglement. The red and black crosses show the
regularized performance of transmitting $n-1$ parts of an $n$-qubit GHZ state.
As observed, the performance is worse, but intriguingly, for larger values of
$p$, this form of entanglement outperforms the single-qubit case, which is
already a known fact. To highlight this advantage, we provide a zoomed-in plot
within (b) on a logarithmic scale, revealing that the 4- and 5-qubit cases
remain positive for larger values of $p$. As in the entanglement-assisted
case, we show the effects of memory noise for at the sender in (g) and (i) on
the capacity.

In summary, there is much more to learn about quantum capacities, and our
framework serves as a valuable tool for deeper investigations. The
distribution of entanglement is a fundamental aspect of future quantum
networks, and understanding the ultimate rates of entanglement distribution
holds significant importance.

##  IV Conclusion and Outlook

In conclusion, we have developed a quantum machine learning framework for
training quantum channel codes over different channel models and constraints.
Our framework has demonstrated its effectiveness in learning capacity-
achieving codes for qubit channels. We have modeled classical, entanglement-
assisted classical, and quantum capacities, and in each case, our framework
has shown strong performance. The trained models have achieved capacities that
are close to the known limits for unital channels, enabling straightforward
analysis of various error-correcting codes. Furthermore, our framework has
allowed us to observe quantum effects, such as super-additivity, with just a
few lines of software code.

Although we have covered a vast subset of theory in this work, we strongly
believe that quantum machine learning holds tremendous potential for future
quantum communication systems, particularly in the era of 6G networks where
strong constraints will have to be imposed to include quantum features. Our
framework can be extended in numerous ways to explore various communication
scenarios and channel capacities, including private and zero-error capacities,
as well as adaptability to multiple input-output channels. Additionally, we
foresee promising outcomes by exploring different pooling strategies [23] to
facilitate efficient joint-detection receiver designs. Furthermore, while this
work focuses on qubits, conducting an in-depth study of continuous variable
communication can yield novel insights for optical communication, and is a
project we are currently undertaking. In conclusion, our research introduces a
new domain of investigation, employing ML techniques for quantum
communication, and establishes the foundation for future explorations.

## Acknowledgements

The authors thank Bing Qi, Hassan Shapourian, and Ionel Miu for helpful
discussions.

Figure 6: The training results for various channel codes for classical, EA-
classical, and quantum communication produced by the framework for various
channel models. In plots (a) and (b), we show with green circles how adding an
encoding layer affects the code capacity. In plot (c), we show the results of
using a repetition code with a pooling layer. In the plots with orange and
green curves from (d)-(i), we show how memory decoherence on the sender’s side
affects the code capacity using depolarizing noise with idler probability
$p_{i}$. In (h), we show how GHZ states of varying sizes displays super-
additivity in the zoomed-in plot.

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