# Compressed Shaping:
Concept and FPGA Demonstration

Tsuyoshi Yoshida, Koji Igarashi, Magnus Karlsson, and Erik Agrell This work
was presented in part at OFC 2019[32] and ECOC 2019[33].T. Yoshida is with
Information Technology R&D Center, Mitsubishi Electric Corporation, Kamakura,
247-8501, Japan. He also belongs to Graduate School of Engineering, Osaka
University, Suita, 565-0871, Japan (e-mail:
Yoshida.Tsuyoshi@ah.MitsubishiElectric.co.jp).K. Igarashi is with Graduate
School of Engineering, Osaka University, Suita, 565-0871, Japan.M. Karlsson is
with the Dept. of Microtechnology and Nanoscience and E. Agrell is with the
Dept. of Electrical Engineering, both at Chalmers University of Technology,
SE-41296 Gothenburg, Sweden.This work was partly supported by “Massively
Parallel and Sliced Optical Network (MAPLE),” the Commissioned Research of
National Institute of Information and Communications Technology (NICT), Japan
(project no. 20401).Copyright (c) 2021 IEEE. Personal use of this material is
permitted. However, permission to use this material for any other purposes
must be obtained from the IEEE by sending a request to pubs-
<EMAIL_ADDRESS>
###### Abstract

Probabilistic shaping (PS) has been widely studied and applied to optical
fiber communications. The encoder of PS expends the number of bit slots and
controls the probability distribution of channel input symbols. Not only
studies focused on PS but also most works on optical fiber communications have
assumed source uniformity (i.e. equal probability of marks and spaces) so far.
On the other hand, the source information is in general nonuniform, unless
bit-scrambling or other source coding techniques to balance the bit
probability is performed. Interestingly, one can exploit the source
nonuniformity to reduce the entropy of the channel input symbols with the PS
encoder, which leads to smaller required signal-to-noise ratio at a given
input logic rate. This benefit is equivalent to a combination of data
compression and PS, and thus we call this technique _compressed shaping_. In
this work, we explain its theoretical background in detail, and verify the
concept by both numerical simulation and a field programmable gate array
(FPGA) implementation of such a system. In particular, we find that compressed
shaping can reduce power consumption in forward error correction decoding by
up to 90% in nonuniform source cases. The additional hardware resources
required for compressed shaping are not significant compared with forward
error correction coding, and an error insertion test is successfully
demonstrated with the FPGA.

###### Index Terms:

Coding, data compression, distribution matching, entropy, implementation,
modulation, optical fiber communication, probabilistic shaping, source coding.

##  I Introduction

Traffic demands are growing with deployments of mobile communication systems
for the 5th generation and beyond, which is further accelerated by the
pandemic [1]. Optical fiber communications take a key role in the
communication infrastructure because of its high capacity. In the past, the
modulation formats used in optical fiber communications were binary, e.g.,
on–off keying, binary, or quaternary phase-shift keying without forward error
correction (FEC) or with hard-decision FEC [2]. However, the latest 400 Gb/s
standards [3, 4] utilize 16-ary quadrature amplitude modulation (QAM) with
soft-decision (SD) FEC under bit-interleaved coded modulation (BICM)[5, 6, 7,
8]. Furthermore, constellation shaping [9], or more specifically,
probabilistic shaping (PS) [10, 11], has attracted wide research interest due
to its capacity-approaching performance [12, 13, 14, 15]. Especially reverse
concatenation, where the shaping encoding (also known as _distribution
matching_ , DM)[16, 17, 18, 19, 20, 21], is done outside the FEC encoding [22,
23, 24], made PS deployable in terms of implementation capability.

An optimal encoder, which minimizes the rate loss in the conversion process,
can be theoretically achieved in two steps if the block length is large
enough; by applying source coding (often called “data compression”) first and
channel coding (i.e., constellation shaping and FEC coding) next. Information-
theoretic coding and modulation techniques have realized significant
performance improvements in recent years, thus almost closing the gap to the
Shannon channel capacity [25]. In contrast, coding for dynamically variable
source information[1] has rarely been investigated for fiber-optic
communication systems, which aggregates massive user traffic in frames. In the
standard [26], simple bit scrambling (flipping bits by the exclusive OR
operation with a pseudorandom bit sequence (PRBS)) has been implemented to
balance the mark (logic ‘$1$’) and space (logic ‘$0$’) counts instead of
applying any source coding. Often we tend to assume the source bits as just
uniformly distributed and independent, although the true source entropy before
bit scrambling is variable and dependent on the user traffic, e.g., due to the
existence of idle frames in the media access control (MAC) protocol [26].

Data compression and shaping are almost inverse operations, i.e., the former
converts a nonuniform information sequence into a shorter uniform one, while
the latter does the opposite. Simultaneous realization of data compression and
shaping is not only an interesting research topic but also a key technique for
more efficient communications in practice. Thus in this paper, we propose and
investigate _compressed shaping_ , which combines the benefits from data
compression and shaping. Similar ideas have been studied in the context of
joint source–channel coding in communication theory [27, 28], but an
application to fiber-optic communication is presented here, for the first time
to the best of our knowledge.

Compressed shaping is enabled by a shaping encoding that is sensitive to the
source entropy. As data compression allocates short bit patterns to frequently
occurring source words, compressed shaping allocates amplitudes with small
energy to such frequent source words. This compression feature is similar to
burst signalling in time-domain multiple access [29], i.e., optical power
variation depending on the traffic. Compressed shaping is a fixed-length to
fixed-length conversion, and the average energy of channel input symbols is
reduced for source information sequences having a small entropy. We do not
need an operational mode change such as updating the source statistics based
on prior knowledge, although this would be a kind of data compression. Our
previously proposed look-up table (LUT)-based hierarchical DM [19] and
following works [30, 31] can be applied for this purpose by reordering the LUT
entries without significantly increased complexity. The proposed technique can
reduce the rate losses associated with source and channel coding compared with
state-of-the-art DM schemes such as constant-composition DM (CCDM) [16], or
reduce the power consumption in the FEC at a given information rate and
signal-to-noise ratio (SNR) by relaxing the FEC performance. We also report a
field programmable gate array (FPGA) implementation, and the hardware
demonstration results for compressed shaping $16$\- and $64$-QAM at system
throughputs of $153$ and $113\,\mathrm{Gb/s}$, respectively.

This is an evolutional work of [32, 33, 34], which are here extended by
providing a more detailed theoretical background of compressed shaping. The
system throughput is increased by separating the clock domain into data and
controlling because the control circuitry was the bottleneck in the logical
circuitry when making the clock frequency faster. The accuracy of the power
consumption estimation is improved by introducing a dynamic simulation. Even
if we consider shaping encoding only, there have been very few other reports
on FPGA implementations [35, 36]. Furthermore, there are neither any reports
on static/dynamic power consumption estimates nor any demonstrations by FPGA
implementations including both shaping encoding and decoding, except [33].

The rest of the paper is organized as follows. The principle of compressed
shaping is explained in Sec. II, and its numerical simulations are shown in
Sec. III. An FPGA implementation example of compressed shaping is presented in
Sec. IV, and the hardware-based demonstration results are summarized in Sec.
V. Finally, Sec. VI concludes the paper.

##  II Compressed shaping—basic principles

While conventional PS systems employ full bit-scrambling and assume source
uniformity, the proposed compressed shaping scrambles the sign bit only and
applies source-sensitive amplitude shaping to realize better performance in
the case of small source entropy. In this section, we firstly review the
historical source uniformity assumption in conventional systems and discuss
rate loss under either uniform or nonuniform source conditions. Then we
compare compressed shaping with bit-sequence data compression and PS. Finally,
we show the system model and characterize entropy bounds in compressed shaping
systems.

###  II-A Rate loss in conventional systems

Optical fiber communication systems gather many MAC frames from client
traffic. When investigating coding and modulation techniques, source
uniformity is usually assumed. It is because a $50\%$ mark ratio is produced
by line coding or bit scrambling, even if the true source information is
nonuniform. This may be a reasonable model under the idealized assumption of a
stable, uniform traffic flow. However, due to dynamically variable traffic
volumes [1], there will be many idle frames. These are encoded by 64B/66B line
coding into the zero codeword except for the sync header and block type field
[26]. Due to the existence of such idle frames, the source bits
$S_{i}\in\\{0,\,1\\}$ can have more than $50\%$ ‘$0$’s, which makes the binary
source entropy $\mathbb{H}(S_{i})<1$, where $\mathbb{H}(\cdot)$ denotes
entropy. For example, when an 8-byte idle signal is encoded by 64B/66B, a
66-bit block consists of the 2-bit sync header $01$, the 8-bit block type
field $00011110$, and eight 7-bit control bits $0000000$ for the 8-byte idle
signal, which results in a source mark ratio of $5/66=7.6\%$.

In conventional systems, the (serial) source bits $S_{i}$ are parallelized to
form a source bit sequence $[S_{1}\ldots S_{k_{\mathrm{bs}}}]$, after which
the bit scrambling converts $[S_{1}\ldots S_{k_{\mathrm{bs}}}]$ into a
scrambled bit sequence $[U_{1}\ldots U_{k_{\mathrm{bs}}}]$. The sequence
length $k_{\mathrm{bs}}$ is chosen to match the applied bit-scrambling
protocol. An arbitrary (randomly selected) bit in this scrambled sequence has
the distribution
$P_{U}=(1/k_{\mathrm{bs}})\sum_{i=1}^{k_{\mathrm{bs}}}P_{U_{i}}$, which is
assumed to be uniform and $\mathbb{H}(U)=1$, even when the corresponding
(serial) unscrambled bit distribution
$P_{S}=(1/k_{\mathrm{bs}})\sum_{i=1}^{k_{\mathrm{bs}}}P_{S_{i}}$ yields an
entropy $\mathbb{H}(S)<1$. The bit scrambling is essential in binary
modulation for maintaining direct current levels in electronic devices and for
recovering the clock signal at the receiver. Non-PS QAM systems also utilize
bit scrambling to maintain the $50\%$ mark ratio. In general, neither $S_{i}$
nor $U_{i}$ ($i=1,2,\ldots,k_{\mathrm{bs}}$) is ensured to be identical and
independent distributed (i.i.d.), so $\mathbb{H}([S_{1}\ldots
S_{k_{\mathrm{bs}}}])\leq k_{\mathrm{bs}}\mathbb{H}(S)$ and
$\mathbb{H}([U_{1}\ldots U_{k_{\mathrm{bs}}}])\leq
k_{\mathrm{bs}}\mathbb{H}(U)$ by the concavity of entropy. However,
$[U_{1}\ldots U_{k_{\mathrm{bs}}}]$ is assumed to be i.i.d. and uniform in
most research works on channel coding.

In PS, the goal is to reduce symbol entropy at a given information rate in
order to reduce the required SNR for quasi-error-free operation over channels
approximated by the Gaussian channel. The scrambled bit sequences
$[U_{1}\ldots U_{k_{\mathrm{bs}}}]$ are rearranged into sequences of length
$k$, and each such length-$k$ bit sequence is mapped into a sequence of
amplitudes $[A_{1}\ldots A_{n}]$. The sequence lengths $k$ and $n$ are
selected to match the PS scheme, regardless of $k_{\mathrm{bs}}$. Since
$U_{i}$ is in general not i.i.d., neither is $A_{i}$, and thus
$\mathbb{H}([A_{1}\ldots A_{n}])\leq n\mathbb{H}(A)$, where
$P_{A}=(1/n)\sum_{i=1}^{n}P_{A_{i}}$. On the other hand, $A_{i}$ is assumed to
be i.i.d. in a mismatched (memoryless) receiver.

The PS performance is typically quantified with a rate loss

$\displaystyle R_{\mathrm{loss}}(U,A)$ $\displaystyle=$
$\displaystyle\mathbb{H}(A)-\frac{k}{n},$ (1)

where $R_{\mathrm{loss}}(\cdot,\cdot)$ denotes a rate loss in a sequence
conversion, and $R_{\mathrm{loss}}(U,A)\geq 0$ for i.i.d. $U_{i}$. The CCDM
[16], which has been the state-of-the-art PS coding, has an almost negligible
rate loss when $n$ is sufficiently large, e.g., $1000$–$10000$. The CCDM
generates a fixed probability mass function (PMF) $P_{A}$ of output amplitudes
$A$ regardless of the DM encoder input bit sequence and its statistics. For a
fixed PMF $P_{A}$ with a nonconstant-composition DM, $U_{i}$ is required to be
uniformly distributed in general, which can be achieved by bit scrambling.

###  II-B Rate loss in source to amplitude conversion

The CCDM shows negligible rate loss $R_{\mathrm{loss}}(U,A)$ in (1) with a
sufficiently large block length in PS for a uniform source, but not for
nonuniform sources, in which case the channel input symbol entropy or average
symbol energy can be further reduced by exploiting nonconstant-composition DM.

During a conversion process from source bit sequence to amplitude symbol
sequence, bit scrambling acts against the symbol entropy reduction because it
maximizes binary entropy ($\mathbb{H}(U)=1$) even if $\mathbb{H}(S)<1$.
Instead, we propose to exploit this source nonuniformity. In the case of
$\mathbb{H}(S)<1$, the rate loss in the conversion from a nonuniform source
bit sequence $[S_{1}\ldots S_{k}]$ into an amplitude sequence $[A_{1}\ldots
A_{n}]$ can in general be expressed as

$\displaystyle R_{\mathrm{loss}}([S_{1}\ldots S_{k}],A)$ $\displaystyle=$
$\displaystyle\mathbb{H}(A)-\frac{\mathbb{H}([S_{1}\ldots S_{k}])}{n},$ (2)

which is bounded as

$\displaystyle R_{\mathrm{loss}}([S_{1}\ldots S_{k}],A)$ $\displaystyle\geq$
$\displaystyle R_{\mathrm{loss}}(S,A)$ (3) $\displaystyle
R_{\mathrm{loss}}(S,A)$ $\displaystyle=$
$\displaystyle\mathbb{H}(A)-\mathbb{H}(S)\frac{k}{n}.$ (4)

Obviously, CCDM is not optimum (with or without bit scrambling) because of the
constant $P_{A}$ and $\mathbb{H}(A)$ regardless of the source distribution.
The general rate loss bound $R_{\mathrm{loss}}(S,A)$ in (4) becomes
significantly larger than $R_{\mathrm{loss}}(U,A)$ in (1) under a small
$\mathbb{H}(S)$. A nonconstant-composition DM could be source sensitive, i.e.,
realizing a smaller output entropy $\mathbb{H}(A)$ for a smaller input entropy
$\mathbb{H}(S)$, resulting in a smaller $R_{\mathrm{loss}}(S,A)$ compared with
CCDM. Even in such cases, at least the sign bits should be bit-scrambled for
the direct current level management and clock recovery. To enhance the
performance under nonuniform source information, data compression is another
option, at the expense of, possibly large, digital signal processing circuit
resources, since it has to adapt to the time variation of $\mathbb{H}(S)$.

###  II-C Proposed bit sequence conversion

This section explains the principle of the proposed compressed shaping
compared to well-known data compression and PS. Tab. I shows small examples of
bit sequence conversions using (a) data compression, (b) PS, and (c)
compressed shaping.

Tab. I(a) assumes a nonuniform source input bit sequence
$\boldsymbol{S}=[S_{1}S_{2}]$ with a fixed length $k=2$. The bit sequence
conversion is given by Huffman coding [37], which allocates a short output
string to a high probability input word. The output bit sequence
$\boldsymbol{C}$ have variable lengths $n_{\boldsymbol{C}}$ from $1$ to $3$ in
this example. Then the fixed input length $k=2$ is shortened to
$n_{\mathrm{avg}}=\mathbb{E}[n_{\boldsymbol{C}}]=1.7$ on average, where
$\mathbb{E}$ denotes expectation. Such conversion is useful to reduce the
required storage size after conversion. Huffman coding is an invertible data
compression; however, it is usually not suitable for high throughput data
communications due to issues of latency and required storage size in the
variable-length conversion process.

Tab. I(b) assumes a uniform source input bit sequence
$\boldsymbol{U}=[U_{1}U_{2}]$ with a fixed length $k=2$, which is converted
into an amplitude sequence $\boldsymbol{A}=[A_{1}A_{2}A_{3}]$ with a fixed
length $n=3$ and an amplitude element $A\in\\{1,\,3\\}$. Candidates of output
amplitudes are sorted in ascending order of average energy per codeword
$E_{\boldsymbol{A}}=||\boldsymbol{A}||^{2}_{2}/n$. The output amplitude
sequence $\boldsymbol{A}=111$ has the smallest $E_{\boldsymbol{A}}$ of $1$,
and $\boldsymbol{A}=113$, $131$, and $311$ have the second smallest
$E_{\boldsymbol{A}}$ of $11/3$. Amplitude sequences $\boldsymbol{A}=133$,
$313$, $331$, and $333$ are not chosen as output strings in this codebook
because of their large $E_{\boldsymbol{A}}$. Then the average output energy
$E_{\mathrm{avg}}=\mathbb{E}[E_{\boldsymbol{A}}]=3$. When we use a codebook
with uniform output amplitudes of $1$ and $3$ with a length of $2$ (i.e.,
$11$, $13$, $31$, and $33$), $E_{11}=1$, $E_{13}=E_{31}=5$, $E_{33}=9$, and
$E_{\mathrm{avg}}=5$. Compared with such a uniform output amplitude case,
$E_{\mathrm{avg}}$ is reduced by $2.2\,\mathrm{dB}$ with this exemplified PS,
leading to a small required SNR at a given information rate over the Gaussian
channel. To shape the output amplitude probabilities (i.e., to reduce the
output amplitude entropies), we thus need more output bit slots than input bit
slots.

Tab. I(c) exemplifies the proposed compressed shaping, which assumes a
nonuniform source input bit sequence $\boldsymbol{S}=[S_{1}S_{2}]$ as in the
case of Tab. I(a). The source bit sequence $\boldsymbol{S}$ with a fixed
length $k=2$ is converted into an amplitude sequence
$\boldsymbol{A}=[A_{1}A_{2}A_{3}]$ with a fixed length $n=3$. When generating
the codebook, we need two sortings: (i) the input bit sequences
$\boldsymbol{S}$ are sorted in descending order of probability
$P_{\boldsymbol{S}}$, and (ii) the output amplitude sequence $\boldsymbol{A}$
is sorted in ascending order of average energy $E_{\boldsymbol{A}}$. By
allocating output amplitude sequences with a small energy to input bit
sequences with a high probability, the average output energy
$E_{\mathrm{avg}}$ becomes $7/3$, which is $1.1\,\mathrm{dB}$ further less
than the one in Tab. I(b). Both conventional PS and compressed shaping are
fixed-length to fixed-length conversions in these examples, and are therefore
suitable for high-throughput data communication which arranges data into
fixed-length frames. We do not need any adaptation of the codebook to
different source statistics.

TABLE I: Bit sequence conversions.

(a) Data compression

Input | Output
---|---
Bits | Probability | Bits | Probability | Length
$\boldsymbol{S}$ | $P_{\boldsymbol{S}}$ | $\boldsymbol{C}$ | $P_{\boldsymbol{C}}$ | $n_{\boldsymbol{C}}$
00 | 0.50 | 1 | 0.50 | 1
01 | 0.30 | 01 | 0.30 | 2
10 | 0.15 | 001 | 0.15 | 3
11 | 0.05 | 000 | 0.05 | 3
Average | 1.7

(b) PS

Input | Output
---|---
Bits | Probability | Amplitudes | Probability | Avg. energy
$\boldsymbol{U}$ | $P_{\boldsymbol{U}}$ | $\boldsymbol{A}$ | $P_{\boldsymbol{A}}$ | $E_{\boldsymbol{A}}$
00 | 0.25 | 111 | 0.25 | 1
01 | 0.25 | 113 | 0.25 | 3.67
10 | 0.25 | 131 | 0.25 | 3.67
11 | 0.25 | 311 | 0.25 | 3.67
Average | 3

(c) Compressed shaping

Input | Output
---|---
Bits | Probability | Amplitudes | Probability | Avg. energy
$\boldsymbol{S}$ | $P_{\boldsymbol{S}}$ | $\boldsymbol{A}$ | $P_{\boldsymbol{A}}$ | $E_{\boldsymbol{A}}$
00 | 0.50 | 111 | 0.50 | 1
01 | 0.30 | 113 | 0.30 | 3.67
10 | 0.15 | 131 | 0.15 | 3.67
11 | 0.05 | 311 | 0.05 | 3.67
Average | 2.33

###  II-D Proposed system model

Fig. 1 shows the system model for the proposed compressed shaping. Here we
exemplify using hierarchical DM [19] for the shaping encoder/decoder, but
generally any nonconstant-composition DM can be used. The source information
bits are parallelized into a source bit sequence $\boldsymbol{S}=[S_{1}\ldots
S_{k_{\mathrm{bs}}+k}]$, which is separated into a source sign bit sequence
$\boldsymbol{S}_{\mathrm{s}}=[S_{\mathrm{s},1}\ldots
S_{\mathrm{s},k_{\mathrm{bs}}}]$ and a source amplitude bit sequence
$\boldsymbol{S}_{\mathrm{a}}=[S_{\mathrm{a},1}\ldots S_{\mathrm{a},k}]$. The
source sign bit sequence $\boldsymbol{S}_{\mathrm{s}}$ is bit-scrambled into
$\boldsymbol{U}_{\mathrm{s}}=[U_{\mathrm{s},1}\ldots
U_{\mathrm{s},k_{\mathrm{bs}}}]$ by taking the exclusive OR with a PRBS, to
balance the numbers of ‘$0$’s and ‘$1$’s. The source amplitude bit sequence
$\boldsymbol{S}_{\mathrm{a}}$ is processed by a bit-flipping function, which
flips all input bits if there are more ‘$1$’s than ‘$0$’s and adds a parity
bit ‘$1$’, otherwise just adds a parity bit ‘$0$’, because a large mark ratio
is not desirable in the compressed shaping scheme.111There may be situations
with a predominance of ‘$0$’s (due to many idle frames) or ‘$1$’s (due to an
alarm indication signal) [26]. In both cases, the source entropy is small.
Then the bit-flipping encoding output bit sequence
$\boldsymbol{F}=[F_{1}\ldots F_{k+1}]$ contains at least as many ‘$0$’s as
‘$1$’s. The sequence $\boldsymbol{F}$ is then processed in a hierarchical DM,
which consists of hierarchically connected small look-up tables (LUTs) as
shown in Fig. 2 [19, 39]. There are $L$ layers and $T_{\ell}$ LUTs in each
layer $\ell$. Each LUT in a layer $\ell$ receives $s_{\ell}$ bits from the
input interface of the DM and $r_{\ell}$ bits from layer $\ell+1$, and it
transmits $r_{\ell-1}$ bits to each of $t_{\ell-1}$ LUTs in layer $\ell-1$, in
total $u_{\ell}=t_{\ell-1}r_{\ell-1}$ transmitted bits. To determine the one-
to-one correspondence of input and output words in each small LUT, we sort the
input words in descending order of the number of ‘$0$’s and the output
amplitudes in ascending order of average energy, as in the small example shown
in Tab. I(c). The output amplitude sequence from the hierarchical DM is
$\boldsymbol{A}=[A_{\mathrm{c},1}\ldots A_{\mathrm{c},n}]$, where each element
is a two-dimensional vector
$A_{\mathrm{c},i}\in\\{1,\,3,\,\ldots,\,2^{m_{\mathrm{a}}/2-1}\\}^{2}$,
$m_{\mathrm{a}}$ is the number of bit tributaries for a two-dimensional
amplitude, and $P_{A_{\mathrm{c}}}=(1/n)\sum_{i=1}^{n}P_{A_{\mathrm{c},i}}$.
The amplitude sequence $\boldsymbol{A}$ is represented by a bit sequence
$\boldsymbol{B}_{\mathrm{a}}=[B_{\mathrm{a},1}\ldots
B_{\mathrm{a},nm_{\mathrm{a}}}]$. From $\boldsymbol{B}_{\mathrm{a}}$ and
$\boldsymbol{U}_{\mathrm{s}}$, an FEC parity bit sequence
$\boldsymbol{B}_{\mathrm{fp}}$ is generated by a systematic FEC encoder with
an FEC code rate $R_{\mathrm{c}}$. Then $\boldsymbol{U}_{\mathrm{s}}$ and
$\boldsymbol{B}_{\mathrm{fp}}$ are concatenated into a sign bit sequence
$\boldsymbol{B}_{\mathrm{s}}=[B_{\mathrm{s},1}\ldots
B_{\mathrm{s},nm_{\mathrm{s}}}]$, where $m_{\mathrm{s}}$ denotes the number of
sign-bit tributaries. The number of elements in $\boldsymbol{S}_{\mathrm{s}}$
or $\boldsymbol{U}_{\mathrm{s}}$ is
$k_{\mathrm{bs}}=n(m_{\mathrm{s}}R_{\mathrm{c}}-m_{\mathrm{a}}(1-R_{\mathrm{c}}))$
and that in $\boldsymbol{B}_{\mathrm{fp}}$ is
$(1-R_{\mathrm{c}})n(m_{\mathrm{s}}+m_{\mathrm{a}})$. Finally channel input
QAM symbols $\boldsymbol{X}=[X_{\mathrm{c},1}\ldots X_{\mathrm{c},n}]$ are
generated from $\boldsymbol{B}_{\mathrm{s}}$ and
$\boldsymbol{B}_{\mathrm{a}}$, where
$P_{X_{\mathrm{c}}}=(1/n)\sum_{i=1}^{n}P_{X_{\mathrm{c},i}}$.

The channel output QAM symbols $\boldsymbol{Y}=[Y_{\mathrm{c},1}\ldots
Y_{\mathrm{c},n}]$ are demapped into an L-value sequence
$\boldsymbol{L}=[L_{1}\ldots L_{(m_{\mathrm{s}}+m_{\mathrm{a}})n}]$ by
memoryless bit-metric decoding [12], where $Y_{\mathrm{c},i}\in\mathbb{R}^{2}$
and $\mathbb{R}$ denotes the real number set. The receiver-side processing
reverses the one on the transmitter side. QAM symbol demapping and FEC
decoding are performed to recover the scrambled source sign bit sequence
$\hat{\boldsymbol{U}}_{\mathrm{s}}$ and the shaped amplitude bit sequence
$\hat{\boldsymbol{B}}_{\mathrm{a}}$. The recovered scrambled source sign bit
sequence $\hat{\boldsymbol{U}}_{\mathrm{s}}$ is bit-descrambled into the
source sign bit sequence $\hat{\boldsymbol{S}}_{\mathrm{s}}$. The shaping
decoder, which is a hierarchical DM decoder here, converts
$\hat{\boldsymbol{B}}_{\mathrm{a}}$ into the flipped bit sequence
$\hat{\boldsymbol{F}}$. The bit-flipping is terminated based on the parity
bit, i.e., if the parity bit is ‘$1$’, all input bits are flipped and the
parity bit is removed, otherwise the parity bit is just removed, to obtain the
source amplitude bit sequence $\hat{\boldsymbol{S}}_{\mathrm{a}}$. Finally,
the source sign bit sequence $\hat{\boldsymbol{S}}_{\mathrm{s}}$ and the
source amplitude bit sequence $\hat{\boldsymbol{S}}_{\mathrm{a}}$ are
concatenated into the source information bit sequence $\hat{\boldsymbol{S}}$.

We here summarize the entropy and rate loss in this compressed shaping system.
The entropy of the channel input symbol is given by

$\displaystyle\mathbb{H}(X_{\mathrm{c}})$ $\displaystyle=$ $\displaystyle
m_{\mathrm{s}}+\mathbb{H}(A_{\mathrm{c}}).$ (5)

If $S_{1},\ldots,S_{k_{\mathrm{bs}}+k}$ are i.i.d. with a distribution
$P_{S}=P_{S_{\mathrm{a}}}=\frac{1}{k_{\mathrm{bs}}+k}\sum_{i=1}^{k_{\mathrm{bs}}+k}P_{S_{i}}$,
then the rate loss in the sequence conversion from $S_{\mathrm{a}}$ to
$A_{\mathrm{c}}$ is, similarly to (4),

$\displaystyle R_{\mathrm{loss}}(S_{\mathrm{a}},A_{\mathrm{c}})$
$\displaystyle=$
$\displaystyle\mathbb{H}(A_{\mathrm{c}})-\mathbb{H}(S)\frac{k}{n}.$ (6)

In order to characterize the performance in Sec. III, we here define the
minimum entropy of a channel input symbol as

$\displaystyle\mathbb{H}_{\mathrm{LB}}$ $\displaystyle=$ $\displaystyle
m_{\mathrm{s}}+\mathbb{H}(S)\frac{k}{n}.$ (7)

Note that the essential function of compressed shaping is the shaping
encoder/decoder based on a nonconstant-composition DM. A small example of this
is shown in Tab. I(c). Once the shaping encoder/decoder functions are
determined, they will not be adapted even if the source statistics would vary
dynamically. This feature helps to ensure a low-complexity compressed shaping.
The other additional functions in Fig. 1 are optional. For an implementation
with the standardized framing using 64B/66B encoding and the alarm indication
signal, the bit-flip encoding/decoding is required. To support bipolar
signaling such as QAM for coherent systems, the bit scrambling/descrambling of
the sign bit is required to maintain the DC level at zero. With some
adjustments, we can apply compressed shaping to 400ZR[3] and openROADM[4].
These employ 256B/257B transcoding after the 64B/66B encoding. In the
transcoding, most bits are transparently output, while several bits are
shortened or changed. Thus the source nonuniformity from the 66-bit blocks are
retained. Before the symbol mapping, the sign and amplitude bits must be
properly distributed, and the bit scrambling should be modified to avoid
breaking the source nonuniformity. These requirements are reasonably minor in
a practical implementation.

Figure 1: System model for the proposed compressed shaping. Figure 2:
Schematic of hierarchical DM encoding [39, Fig. 2].

##  III Simulations

To verify the concept of compressed shaping, we performed numerical
simulations based on the system model in Sec. II-D. In this section,
nonuniform, independent source information bits $S$ were generated for
simplicity, by independent, uniformly distributed pseudorandom numbers from
the Mersenne twister. For a given target source mark ratio $P_{S}(1)$, the
uniformly distributed pseudorandom numbers ranging from $0$ to $1$ were
binarized with a threshold level $1-P_{S}(1)$, i.e., generating a logic ‘$0$’
for a random number $0$ to $1-P_{S}(1)$ and a logic ‘$1$’ otherwise. In this
simulation, we set a static source mark ratio $P_{S}(1)$ in a simulation batch
for a short period.

Fig. 3 shows the average two-dimensional symbol energy $E_{\mathrm{2d}}$ as a
function of the lower bound entropy of channel input symbols
$\mathbb{H}_{\mathrm{LB}}$ in (7) for various PS-QAM formats and source mark
ratios $P_{S}(1)=0.5$, $0.4$, $0.3$, $0.2$, $0.1$, and $0.05$, where the
minimum Euclidean distance $d_{\mathrm{LB}}=2$. For compressed shaping with
hierarchical DM, we employed $8$-, $16$-, $32$-, $64$-, and $128$-QAM as base
constellations. The PS overhead was around $7\%$ in each case when assuming
the use of a rate-$5/6$ FEC, and the PS codeword length (number of QAM
symbols) was $256$, $256$, $192$, $128$, and $128$ for $8$-, $16$-, $32$-,
$64$-, and $128$-QAM, respectively. The PS codeword length equals the block
length of compressed shaping. The PS encoder input block consists of several
$100$ information bits, and the source mark ratio is averaged over each block.
This number of several $100$ bits is small enough to avoid over-averaging of
dynamic variations of the source mark ratio due to idle MAC frames. Such
granular base constellation and shallow shaping help to avoid excessive
increases of peak-to-average power ratio and power consumption [38, 39] and
penalties from nonideal FEC performance [40]. As the $8$-QAM constellation,
$\mathcal{C}_{3}$ in [41] was used to make the constellation symmetric around
the imaginary axis,222The signal points are $(\pm 1,\,0)$, $(\pm 1,\,\pm 2)$,
and $(\pm 3,\,0)$. so that the uniformly distributed FEC parity bits could be
placed on the sign bits without changing $P_{|X_{\mathrm{c}}|}$. The bit
labelling for $128$-QAM was based on [42, Fig. 3]. For comparison, CCDM-based
PS-$16$-QAM and PS-$64$-QAM were also evaluated, using the same PS overhead
and codeword length as with compressed shaping. We also evaluated the
performance of PS-$4096$-QAM with an ideal Maxwell–Boltzmann input
distribution and perfect data compression.

With CCDM, the energy $E_{\mathrm{2d}}$ in Fig. 3 is constant for various
$P_{S}(1)$ cases, because its output PMF $P_{A_{\mathrm{c}}}$ does not depend
on incoming bits. With compressed shaping, on the other hand,
$E_{\mathrm{2d}}$ decreases with decreasing $P_{S}(1)$ (and $\mathbb{H}(S)$)
for each base constellation, although there are significant performance gaps
to the ideal case (black solid line in Fig. 3), especially for high-order QAM.
This gap comes from the fixed compressed shaping regardless of the source
statistics. While an adaptive compressed shaping would bring the performance
closer to the ideal case, it would lead to a high complexity and is out of
scope of this work. Fig. 4 exemplifies the PMF $P_{A}$ for compressed shaping
$64$-QAM with source mark ratios $P_{S}(1)=0.5$–$0.05$. Higher source
nonuniformity (i.e., smaller $P_{S}(1)$) makes deeper PS. Under such source
nonuniformity, we observed reduced $P_{A}(3)$, $P_{A}(5)$, and $P_{A}(7)$ and
increased $P_{A}(1)$ compared with the uniform source case. The transmitted
PMFs of one-dimensional amplitudes $A$ and two-dimensional channel input
symbols $X_{\mathrm{c}}$ at a given source mark ratio $P_{S}(1)$ are denoted
as $P_{A(P_{S}(1))}$ and $P_{X_{\mathrm{c}}(P_{S}(1))}$, respectively.

Figure 3: Average two-dimensional symbol energy $E_{\mathrm{2d}}$ for various
PS-QAM formats and source mark ratios $P_{S}(1)$ in the range $0.5$–$0.05$ as
a function of the lower bound entropy of channel input symbols
$\mathbb{H}_{\mathrm{LB}}=m_{\mathrm{s}}+\mathbb{H}(S)k/n$. Figure 4: An
example of a one-dimensional amplitude PMF $P_{A}$ by compressed shaping
$64$-QAM, for various source mark ratios $P_{S}(1)$. In situations with
dynamically variable $P_{S}(1)$, the demapper assumes a constant
$P_{S}(1)=0.5$, as explained in the text.

We then simulated the required SNR over the Gaussian channel with the DVB-S2
low-density parity check code [43] having a code rate of $5/6$, with a maximum
number of decoding iterations of $50$. Fig. 5 shows the simulated required SNR
with the maximum number of FEC decoding iterations. The required SNR is
defined at an asymmetric information [44, 45]333The asymmetric information is
equivalent to normalized generalized mutual information [46, 15] and
achievable FEC rate [14] under matched decoding. of $0.87$, which corresponds
to a pre-FEC BER of about $3.4\cdot 10^{-2}$. We observed no residual bit
errors in more than $2.5\cdot 10^{6}$ information bits after FEC decoding in
the SNR regime larger than the required SNR for each case. We simulated both
_mismatched_ and _matched_ decoding. The true transmitted two-dimensional
symbol PMF is denoted by $P_{X_{\mathrm{c}}(P_{S}(1))}$ and the transmitted
symbol PMF assumed in the soft demapping by $Q_{X_{\mathrm{c}}(P_{S}(1))}$. In
the matched case, $Q_{X_{\mathrm{c}}(P_{S}(1))}$ was set to
$P_{X_{\mathrm{c}}(P_{S}(1))}$ for all $P_{S}(1)$, while in the mismatched
case, $Q_{X_{\mathrm{c}}(P_{S}(1))}$ was set to a fixed PMF of
$P_{X_{\mathrm{c}}(0.5)}$ regardless of the true $P_{S}(1)$. This is because
under dynamically variable source situations, the true transmitted symbol PMF
$P_{X_{\mathrm{c}}(P_{S}(1))}$ is hard to track in deployable systems. As
shown in Fig. 5, the required SNR can be reduced by compressed shaping, in
contrast to the fixed required SNR by CCDM and uniform QAM with bit scrambling
due to the fixed $P_{X_{\mathrm{c}}}$ ($=P_{X_{\mathrm{c}}(0.5)}$). The SNR
penalty by the mismatch between $P_{X_{\mathrm{c}}(P_{S}(1))}$ and
$Q_{X_{\mathrm{c}}(P_{S}(1))}$ is not significant except at very small
$P_{S}(1)$.

The better performance of compressed shaping compared with conventional PS can
be converted into lower power consumption. We quantified the power consumption
in FEC decoding, because it dominates the power consumption among all coding
functions. Fig. 6 shows the relative power consumption of the FEC decoding for
PS-QAM with $P_{S}(1)=0.5$ (circle), $0.4$ (square), $0.3$ (diamond), $0.2$
(triangle), $0.1$ (cross), or $0.05$ (plus), so there are six curves for each
QAM order with compressed shaping. The power consumption is assumed to be
proportional to the average number of decoding iterations, which is almost
proportional to the toggle rate in logical circuitry. The vertical axis in
Fig. 6 is normalized by the maximum number of decoding iterations, i.e., $50$.
The soft demapping was assumed to be mismatched as in Fig. 5, i.e.,
$Q_{X_{\mathrm{c}}(P_{S}(1))}=P_{X_{\mathrm{c}}(0.5)}$. While CCDM consumes a
fixed power even if $P_{S}(1)$ is reduced (this is the same for non-PS
signaling with bit scrambling, but not shown in Fig. 6), compressed shaping
significantly reduces the power to about $10\%$ for highly nonuniform source
probabilities, because of the smaller required SNR, which leads to a smaller
number of decoding iterations than with conventional bit-scrambled PS.

While in this work, the source mark ratio is assumed to be constant, dynamic
variations thereof were simulated in [34, Sec. 5.4]. Source mark ratios were
linearly varied, e.g., from $20$% to $40$% or from $10$% to $50$%, and
compared with a constant $30$% as a benchmark. For a larger range of
variation, the performance improved slightly, even if the average source mark
ratio over many blocks was the same. Note that additional complexity in
compressed shaping is not significant when hierarchical DM is employed for
shaping encoding/decoding, which will be shown in the next section.

Figure 5: Simulated required SNR for compressed shaping QAM under matched
(dotted lines) and mismatched decoding (solid lines). In the mismatched case,
we set the transmitted PMF assumed in the demapper to the one for a uniform
source ($P_{S}(1)=0.5$), for every source mark ratio $P_{S}(1)$. Dashed and
long-dashed lines correspond to CCDM and uniform QAM, respectively. Figure 6:
Simulated relative power consumption of the FEC decoding, which is assumed to
be proportional to average number of decoding iterations, as a function of SNR
for PS-QAM. Solid and black dashed lines correspond to compressed shaping and
CCDM, respectively.

##  IV FPGA implementation

We implemented compressed shaping in a single FPGA chip on an evaluation board
Xilinx® Virtex® Ultrascale+™ VCU118 XCVU9P. Fig. 7 shows the functional block
diagram of the implemented circuitry. The source generator outputs source
information bits based on a given target mark ratio $P_{S}(1)$. The schematic
of the source generator is illustrated in Fig. 9. Each source information bit
is selected from one of three possible candidates; 0) the logic bit ‘$0$’, 1)
a PRBS of length $2^{31}-1$ bits, and 2) the logic bit ‘$1$’. The mask signal,
used for the selection, is generated from the target source mark ratio
$P_{S}(1)$, which is given by the user. If $P_{S}(1)\leq 0.5$, the mask signal
takes on values $0$ or $1$ such that the average fraction of logic ‘$1$’s in
$S$ is $P_{S}(1)$, and if $P_{S}(1)>0.5$, the mask similarly is $1$ or $2$.
Fig. 9 shows an exemplified mask signal when the target $P_{S}(1)$ is $0.4$.
For simplicity, we classified source bits into 20 groups (32 bits per group),
provided a ‘$0$’ mask window for four groups, and slided the window in every
clock cycle.

The bit-flipping encoder counts the numbers of ‘$0$’s and ‘$1$’s. When the
number of ‘$1$’s is larger than that of ‘$0$’s, it flips all bits at the clock
cycle and adds a parity bit ‘$1$’. Otherwise it just adds a parity bit ‘$0$’.
The shaping encoding/decoding is realized by hierarchical DM having a total
codeword length of $404$ bits for the shaped two-dimensional amplitudes. The
number of shaped information bits per two-dimensional amplitude $k/n$ for
compressed shaping employing hierarchical DM is generally flexible, and is in
this implementation fixed to $372/202$ and $372/101$ for $16$-QAM and
$64$-QAM, respectively, with compressed shaping.444A flexible choice of
$(n,\,k)$ in an FPGA implementation of hierarchical DM is left as potential
future work. The LUT contents can be reconfigured in software or firmware
without increasing the RAM size. Before the receiver-side processing, we have
an error insertion function, which inserts bit errors based on a given bit
error rate (BER) before the shaping decoder. The shaping decoder is also
implemented with a hierarchical DM decoder, and the bit-flipping decoder
recovers the source bits. When the parity bit is ‘$1$’, it flips all bits at
the clock cycle and removes the parity bit. Otherwise, it just removes the
parity bit.

There are several monitoring functions, i.e., the PMF $P_{A_{\mathrm{c}}}$ at
the shaping encoder output, the assumed post-FEC BER, and the system output
BER. Note that hierarchical DM mainly consists of LUTs, which are implemented
with random access memory (RAM). Because it is sensitive to unwanted bit-
flipping due to radiation-induced soft errors, we implemented soft error
protection circuitry.

Figure 7: Block diagram of FPGA implementation of compressed shaping.

Figure 8: Schematic of source generation function. The mask signal selects the
output source bit from logic ‘$0$’, ‘$1$’, or a bit given by the PRBS based on
the target source mark ratio $P_{S}(1)$.

Figure 9: Example of mask signals for given clock cycle indices and group
indices for a target source mark ratio $P_{S}(1)=0.4$. In each clock cycle
index or each group index, $20\%$ of the generated bits are set directly to
‘$0$’ and $80\%$ are taken from a uniform PRBS, which implies $20+80/2=60\%$
‘$0$’s on average.

The clock domain was initially single and the clock frequency
$f_{\mathrm{clk}}$ for the fitting (FPGA synthesis) was $90\,\mathrm{MHz}$
[33]. Later we found a bottleneck in making the clock frequency higher inside
the soft error protection circuitry (consisting of flip-flops and a selector
tree for refreshing RAM contents), so we separated the clock domain into one
for data processing and one for control. We then achieved
$f_{\mathrm{clk}}=240\,\mathrm{MHz}$ for the data processing for higher
throughput. Assuming a suitable FEC concatenation (not implemented here), the
system throughputs for compressed shaping 16- and $64$-QAM would be $57$ and
$42\,\mathrm{Gb/s}$ at $f_{\mathrm{clk}}=90\,\mathrm{MHz}$, and $153$ and
$113\,\mathrm{Gb/s}$ at $f_{\mathrm{clk}}=240\,\mathrm{MHz}$, respectively.
The number of bits per PS codeword were the same in both cases, i.e., there
were half as many PS-$64$-QAM symbols as PS-$16$-QAM symbols.

Tab. II shows the utilized hardware resources at a clock frequency of the data
signals $f_{\mathrm{clk}}=90$ or $240\,\mathrm{MHz}$. Fig. 10 depicts the
utilized area of the FPGA chip having three dies. The used resource elements
were mainly located in the left and right dies, and the center die was mainly
used for connection between the two dies. The _register_ elements were mainly
used by the soft error protection circuitry for storing the RAM contents. Out
of the about 290,000 utilized _LUT as logic_ elements, $70\%$ were provided to
external functions such as source generator and BER/PMF monitors, $10\%$ were
for bit-flipping encoding, and the rest was for other combinational logics.
The data processing of the hierarchical DM used totally $3.5\,\mathrm{Mb}$
_block RAM_ elements. No _ultra RAM_ or _DSP slice_ elements were used.

As a benchmark, 400 ZR FEC was implemented in $50$ Xilinx® Virtex® Ultrascale™
FPGAs [47]. The clock frequency was $125\,\mathrm{MHz}$ and the system
throughput was $200\,\mathrm{Gb/s}$ with external functions, i.e., source
generation, mapping, demapping, interleaver, de-interleaver, and noise
loading. Even considering that this reference includes many external
functions, compressed shaping utilizes a very small amount of hardware
resources.

Tab. III shows the estimated dynamic power consumption for compressed shaping
$64$-QAM at $f_{\mathrm{clk}}=240\,\mathrm{MHz}$. While we estimated a static
power consumption based on a default toggle rate of $12.5\%$ in our previous
report [33], we now improved the estimation accuracy by taking realistic node
switching activities into account based on register transfer level simulation
waveform (a so-called switching activity interchange format) over $4000$ clock
cycles, where the source mark ratio $P_{S}(1)$ was set to $0.5$.

The blocks of Tx data and Rx data in Tab. III are essential for data
communications. Among them, the preprocessing (including bit-flipping
encoding) and DM encoder core in the Tx data block, and the DM decoder core in
the Rx data block, consumed most of the power. The DM encoder and decoder
cores mainly consisted of block RAMs for LUTs. The bit-flipping encoding
counted the logic ‘$1$’s using adders, leading to a relatively large power
consumption. Other pre- and postprocessing functions including the lane
reorder consumed little power. Note that the power consumption of either the
Tx or the Rx data block was smaller than that of external functions, e.g.,
source generation and BER monitoring. We had nonnegligible power consumption
in the control blocks. They had soft error protection functions for the DM
encoder and decoder cores by holding copies of the entire RAM contents in
registers and refreshing the RAM intermittently. These powers can be expected
to be less in an ASIC implementation by making the activation ratio small.

TABLE II: Utilization of key resources in FPGA for compressed shaping at $f_{\mathrm{clk}}=90$ or $240\,\mathrm{MHz}$. Category | Element | Available | Utilization
---|---|---|---
|  |  | $90\,\mathrm{MHz}$ | $240\,\mathrm{MHz}$
System logic | LUT as logic | $1182\mathrm{k}$ | $24.17\%$ | $24.64\%$
cell | Register | $2364\mathrm{k}$ | $22.49\%$ | $21.38\%$
Memory | Block RAM | $75.9\,\mathrm{Mb}$ | $4.65\%$ | $4.65\%$
| Ultra RAM | $270.0\,\mathrm{Mb}$ | Not used | Not used
DSP slice |  | $6840\,\mathrm{slices}$ | Not used | Not used
Figure 10: Utilized area (green) of a single FPGA chip for compressed shaping. TABLE III: Simulated dynamic power consumption of compressed shaping $64$-QAM at $f_{\mathrm{clk}}=240\,\mathrm{MHz}$. Block | Function | Power (mW)
---|---|---
External | Source generation | $589$
functions | Error insertion | $11$
| BER monitor | $638$
| Clock generation | $99$
| Sub-total | $1\,337$
Tx data | Pre-processing | $232$
| DM encoder core | $456$
| Post-processing | $78$
| Delay adjustment | $46$
| Sub-total | $812$
Rx data | Pre-processing | $44$
| DM decoder core | $562$
| Post-processing | $34$
| Delay adjustment | $35$
| Sub-total | $675$
Tx control |  | $320$
Rx control |  | $295$
Tx/Rx control |  | $167$
Total |  | $3\,613$

##  V Hardware demonstration

We demonstrated compressed shaping by employing the FPGA evaluation board at
$f_{\mathrm{clk}}=90\,\mathrm{MHz}$. First, the transmitter-side functions
were verified. Histograms of the two-dimensional amplitude $A_{\mathrm{c}}$
for compressed shaping $16$\- and $64$-QAM were measured over $10^{10}$ two-
dimensional amplitude samples for various source mark ratios $P_{S}(1)$. The
obtained histograms were interpreted as PMFs $P_{A_{\mathrm{c}}}$ and their
entropies were computed. Fig. 11 shows $\mathbb{H}(A_{\mathrm{c}})$ as a
function of $P_{S}(1)$. The entropy $\mathbb{H}(A_{\mathrm{c}})$ is maximum
for $P_{S}(1)=0.5$. The two-dimensional PS rate losses
$R_{\mathrm{loss}}(U,A_{\mathrm{c}})$ are $0.034$ and $0.064\,\mathrm{bpcu}$
for the exemplified PS-$16$-QAM and PS-$64$-QAM schemes, respectively. When
$P_{S}(1)$ deviates from $0.5$, i.e., $\mathbb{H}(S)$ decreases,
$\mathbb{H}(A_{\mathrm{c}})$ also decreases monotonically. Because of the bit-
flipping encoding, $\mathbb{H}(A_{\mathrm{c}})$ is symmetric around
$P_{S}(1)=0.5$. In contrast, the CCDM-based schemes show a constant
$\mathbb{H}(A_{\mathrm{c}})$, which is independent of the source distribution.
Scaled binary entropy functions
$\mathbb{H}(S)\max_{P_{S}(1)}\mathbb{H}(A_{\mathrm{c}})$ are also depicted in
Fig. 11 with black dotted lines for the two cases. The gap between
$\mathbb{H}(A_{\mathrm{c}})$ and
$\mathbb{H}(S)\max_{P_{S}(1)}\mathbb{H}(A_{\mathrm{c}})$ is a data compression
rate loss. The data compression rate loss is ideally zero but is larger here
for smaller $\mathbb{H}(S)$, due to the nonideal simple processing of
compressed shaping. Here the bit sequence conversion loss in compressed
shaping $R_{\mathrm{loss}}(S_{\mathrm{a}},A_{\mathrm{c}})$ in (6) is given by
the sum of rate losses in PS and data compression. Regardless of the rate loss
increase in data compression, $\mathbb{H}(A_{\mathrm{c}})$ itself is reduced
to help reduce the required SNR substantially.

Figure 11: Measured two-dimensional amplitude entropy
$\mathbb{H}(A_{\mathrm{c}})$ for compressed shaping as a function of the
target source mark ratio $P_{S}(1)$ based on $>10^{10}$ two-dimensional
amplitude samples. The dashed lines show the binary entropy function scaled to
the maximum entropies of the two-dimensional amplitudes for comparison.

Next, the receiver-side functions were verified. We turned the error insertion
function on to provide sparse bit errors between the transmitter and the
receiver. As reported in [19], sparse errors creates the worst system output
BER at a given post-FEC BER. We swept the assumed post-FEC BER from $10^{-5}$
to $10^{-15}$ and examined $P_{S}(1)$ of $0.3$, $0.5$, and $0.7$ for
compressed shaping $16$\- and $64$-QAM. After the receiver-side processing,
the system output BER was measured. Fig. 12 shows the system output BER as a
function of assumed post-FEC BER in the back-to-back error insertion test. The
BER increase due to compressed shaping decoding was only around $10$ times
because hierarchical DM can partially decode correctly even if there are
incoming bit errors. As predicted in [19], the BER increase factor is
significantly smaller than for other DM techniques. For example, CCDM decoding
having a PS codeword length of around $400$ bits results in more than $100$
times higher BER, though we could not implement CCDM in the FPGA due to its
high complexity. We performed $70$ hours of long-term measurement only in the
case of $P_{S}(1)=0.5$, due to time constraints. The number of observed bit
errors after compressed shaping decoding were more than $250$. The system
output BER was $1.5\cdot 10^{-14}$ at a post-FEC BER of $1.6\cdot 10^{-15}$.
This proves that the proposed compressed shaping does not cause an error floor
or excessive error increase, so the required post-FEC BER remains around
$10^{-16}$ to satisfy a required system output BER of $10^{-15}$.

Figure 12: Measured system output BER for compressed shaping as a function of
assumed post-FEC BER in back-to-back error insertion tests.

##  VI Summary

We proposed and demonstrated compressed shaping, i.e., the application of
hierarchical DM to simultaneous source data compression and probabilistic
shaping, which is an example of joint source–channel coding. Under a reduced
source entropy, compressed shaping reduces channel input symbol entropy,
symbol energy, and required SNR. Simulation results showed its smaller
required SNR, as well as reduced power consumption compared with CCDM. We
implemented compressed shaping, which are mainly hierarchical DM
encoding/decoding, into a single FPGA and estimated the dynamic power
consumption based on simulated waveforms. The system throughput reached $153$
and $113\,\mathrm{Gb/s}$ for compressed shaping $16$\- and $64$-QAM,
respectively. Real-time evaluation results showed expected performance in both
encoding and decoding. Compressed shaping works at a very small BER of around
$10^{-15}$ without any error floor, and its decoding increases the BER only
around $10$ times, which is small compared to other DM schemes.

The time-varying symbol energy in compressed shaping causes practical issues
with respect to electrical amplitude, optical power, and SNR control in fiber-
optic communication systems, although such a variation per channel can be
statistically relaxed by multiplexing many channels. One example can be found
in a wavelength-division-multiplexed system operated at a given total launch
power. Without any control of the average symbol energy per channel, the
average optical power in some channels can temporarily be reduced when the
source entropy becomes small. This leads to a larger average optical power and
thus higher nonlinearity impairments in other copropagating channels. Analyses
of such issues and development of appropriate control methods are deferred to
a future work. We can also potentially explore the adaption of compressed
shaping to mass trends of source and channel variations.

## Acknowledgment

We thank Kyo Inoue of Osaka University for assistance in the research.

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