# Why Neural Networks Work

Sayandev Mukherjee and Bernardo A. Huberman
CableLabs
{s.mukherjee<EMAIL_ADDRESS>
###### Abstract

We argue that many properties of fully-connected feedforward neural networks
(FCNNs), also called multi-layer perceptrons (MLPs), are explainable from the
analysis of a single pair of operations, namely a _random projection_ into a
higher-dimensional space than the input, followed by a _sparsification_
operation. For convenience, we call this pair of successive operations
_expand-and-sparsify_ following the terminology of Dasgupta. We show how
expand-and-sparsify can explain the observed phenomena that have been
discussed in the literature, such as the so-called Lottery Ticket Hypothesis,
the surprisingly good performance of randomly-initialized untrained neural
networks, the efficacy of Dropout in training and most importantly, the
mysterious generalization ability of overparameterized models, first
highlighted by Zhang et al. and subsequently identified even in non-neural
network models by Belkin et al.

##  1 Introduction

The recent successes of deep learning models in achieving state-of-the-art
performance in various tasks has increased the importance of understanding
many intriguing aspects of their observed behavior. Among them are:

  1. 1.

Why neural networks perform better than the classical kernel machine ML models
like support vector machines and random forests on the same problems;

  2. 2.

Why neural networks are curiously vulnerable to adversarially crafted inputs
that are imperceptibly different (to a human) from inputs which do not result
in erroneous output;

  3. 3.

Why randomly-initialized neural networks, even without training, can perform
fairly well;

  4. 4.

Why vanilla stochastic gradient descent without regularization is usually
sufficient to train a neural network;

  5. 5.

Why a trained neural network can be drastically pruned without loss of
performance;

  6. 6.

Why training a neural network with dropout leads to better performance of the
trained model;

  7. 7.

Why a neural network so overparameterized that it can memorize a training set
with arbitrary labels can nevertheless generalize well if trained on a
training set with correct labeling, even though it still learns the training
set with zero error. In other words, a neural network seems to _learn_ if
there is some underlying structure to the training set it is given, and if
there is no such structure, then it simply _memorizes_ the training set.

The final property in particular, has excited a lot of research ever since it
was highlighted by Zhang et al. in [5, 6].111Subsequently this property was
also reported in non-neural network ML models by Belkin et al. [7].
Explanations for this behavior have included attempts to define new metrics
for “effective complexity” of ML models (see, for example, [8]),
extrapolations to infinitely-wide neural networks [9], and the use of methods
from statistical mechanics to study the dynamics of the interacting neurons of
a large neural network [10].

In this paper, we will argue that the key to the explanation of this
phenomenon lies in something that while somewhat mysterious, it offers
excellent heuristic insights – the properties of a particular mathematical
operation that is called _expand-and-sparsify_ [1]: a _random projection_ to a
_higher-dimensional_ space than the input, followed by a _sparsification_
step.

We believe that the components of a successful neural network solution for a
given use case are:

  1. 1.

Architecture selection: how many layers, how many nodes in these layers, the
connections amongst these layers, and the activation functions of these nodes.
In the present work we restrict ourselves to MLPs where all nodes in a given
layer are connected to all the nodes in the next higher layer, and only the
next higher layer, and all nodes except possibly those in the output layer
have ReLU activations (1).

  2. 2.

Parameter initialization: The initial values assigned to the parameters of the
MLP.

  3. 3.

Model training, i.e., parameter optimization via iterative gradient descent on
a training dataset.

Nearly all the attention of researchers to date has been focused on model
training: its dynamics, both asymptotic (see the Neural Tangent Kernels of
Chizat and Bach [9]) and non-asymptotic (see the renormalization group
approach of Roberts and Yaida [10]), and numerous papers trying to explain why
vanilla stochastic gradient descent on mini batches even without
regularization yields models with excellent performance on both the training
data and new, unseen, test data.

We argue in the present work that this emphasis on training as opposed to
architecture and parameter initialization is misplaced. The reported strong
performance of uninitialized neural networks with random parameter selection
[3] and the fact that a good initialization allows for considerable pruning of
a network without noticeable degradation in performance (part of what is
called the Lottery Ticket Hypothesis [2]) suggest that the heavy lifting in
neural network modeling is done by the architecture and parameter
initialization rather than the numerical optimization step of model training.

In the present paper, we will attempt to explain all the above observed
phenomena as arising from the interesting and profound properties of the
expand-and-sparsify operation performed by the layers of the MLP architecture
with suitable parameter initialization. We argue that in a successful model
design, the choice of architecture (number of layers, and number of neurons
per layer) and parameter initialization has a zeroth-order impact on model
performance even before commencing any training, and that the model training
step has only a first-order impact on model performance because it is only a
perturbation of parameter values. As an aside, we interpret the Dropout trick
during training [4] as an attempt to compensate for a non-optimal choice of
architecture and/or parameter initialization, implying that a better choice of
architecture and initialization can allow us to dispense with the need for
Dropout during training.

We will begin with a summary of the principal known results about the expand-
and-sparsify operation, and their implications for neural network training,
almost all of which were recently published by Dasgupta and Tosh in [1]. But
first, we need to establish some background.

##  2 Literature Survey

The usefulness of random features (obtained from random linear projections of
input vectors) in machine learning (ML) models for classification and
regression has been known for a long time, going back to the design of the
Random Forest (RF) [11, 12]. It was extended to large-scale (at the time)
kernel machines in [13]. However, all these random projections were into
lower-dimensional spaces than that of the input, so the feature vectors
obtained from such random projections were dense vectors of smaller dimension
than the input vectors. Theoretical support for the usefulness of such
dimensionality reduction through random projection is provided by the Johnson-
Lindenstrauss lemma [14], which states that “large sets of vectors in a high-
dimensional space can be linearly mapped in a space of much lower (but still
high) dimension with approximate preservation of distances” [15]. In other
words, input vectors that are close in terms of Euclidean distance will remain
close in the lower-dimensional space after random projection. This fact has
been used to infer that random projections (into lower-dimensional, often just
two- or three-dimensional, space) implement a form of _locality sensitive
hashing_ (LSH).

Recently, however, a different form of LSH was discovered in a biological
neural network, viz., the olfactory system of the fruit fly [16]. Although it
also involves a random projection, the projection is not directly into a low-
dimensional space. Rather, the random projection is into a space of dimension
_higher_ than the input, after which the projected vector is then sparsified
by retaining only the highest few percentile of its values and setting all the
rest to zero. Proofs of the LSH property were given in [16, Lem. 1] and [17,
Thm. 3].222The latter result, however, is proved for a sparsification yielding
the same number of significant entries as the dimension of the input, and
relies on an approximation of the Binomial by the Normal distribution whose
accuracy was not validated in the paper.

The above is an example of a simple expand-and-sparsify operation where the
random feature vector is larger in dimension than the input vector, but is
also sparse with a constant number of nonzero entries. A more sophisticated
expand-and-sparsify operation would sparsify the entries in the dense higher-
dimensional (than the input) vector obtained by random projection by comparing
them against a threshold (that is different from one entry to another, and
could even be input-dependent) and retaining only those that exceed the
threshold. Note that this sparsification operation leaves us with a number of
nonzero entries in the projection that is a random variable and no longer
constant.

This new expand-and-sparsify operation was proposed and analyzed for
biological networks in [18], then abstracted and studied in more detail in the
context of ML by Dasgupta and Tosh in [1]. Since this more sophisticated
expand-and-sparsify operation corresponds to the action of a layer of
(artificial) neurons with ReLU activation, the analysis in [1] will be the
starting point for our discussion in the present work.

##  3 The Expand-and-Sparsify Operation

###  3.1 Almost-orthogonality in high dimensions

Before we review known results for the expand-and-sparsify operation, let us
begin by reviewing an even more basic result [19, Thm. 3.4] which provides the
heuristic justification for the expansion random projection in expand-and-
sparsify: there exist sets with an exponential (in $n$) number of almost-
orthogonal vectors (i.e., vectors whose inner products with one another are
all small) in $n$-dimensional Euclidean space. Moreover, it turns out that
such sets are typical in the following sense in high-dimensional Euclidean
space [20]: “With probability close to one an exponentially large number of
random vectors are pairwise almost orthogonal and do not span an arbitrarily
selected normalized vector if coefficients in linear combinations are not
allowed to be arbitrarily large.”

###  3.2 Random projection with almost-orthogonal vectors

It follows that one can simply select an exponentially large number $d$ of
vectors $\bm{w}_{1},\dots,\bm{w}_{d}$ at random (say, independent and
identically distributed, i.i.d., with directionality uniformly distributed on
the unit sphere) in a high-dimensional space $\mathbb{R}^{n}$ and have high
confidence that they are all almost orthogonal to one another.

Consider the $d\times n$ matrix $\bm{W}$ with
$\bm{w}_{1}^{\top},\dots,\bm{w}_{d}^{\top}$ as its rows, which we shall call
the _projection_ matrix for brevity. Given a arbitrary vector
$\bm{u}\in\mathbb{R}^{n}$ in this space (which we will for concreteness call
the _input_ vector), the _projected_ vector $\bm{W}\bm{u}$, i.e., the vector
whose entries are the inner products of the random vectors
$\bm{w}_{1},\dots,\bm{w}_{d}$ with the input vector $\bm{u}$, comprises the
“coordinates” of the input vector $\bm{u}$ along these “almost-orthogonal”
directions.

Thus, it is intuitively plausible that the entries of the projected vector can
serve as _features_ for further processing by an ML model that has been fed
with this input vector, i.e., that these entries serve as a _representation_
of the input vector that may be more useful to an ML training algorithm than
the original input when it comes to training an ML model for some task. When
the dimension $d$ of the projected vector is lower than the dimension $n$ of
the input vector, which corresponds to the selection of only a small subset of
$d$ almost-orthogonal random vectors to compose the rows of the projection
matrix, out of the possible exponentially many (in $n$) rows possible, the
above intuition is supported by an approximate distance-preserving property
called the Johnson-Lindenstrauss lemma. This is the traditional method of
extracting features via random projection.

###  3.3 Expansion and sparsification

What happens if $d$, the number of rows in the projection matrix (each row
being a randomly-selected vector, and the rows all being almost-orthogonal as
discussed above), is larger than $n$, the dimension of the input vector? The
projected vector now has dimension $d>n$, so the input dimensionality has been
_expanded_ by this random projection operation.

Now, any attempt to develop an ML model whose decisions are understandable,
and ideally explainable, to humans will necessarily involve a reduction in the
dimensionality of the representation from that of the input vector, $n$. The
traditional method of getting a representation achieves this by restricting
$d<n$. However, if $d>n$, then the reduction of the dimensionality of the
representation has to be done separately, in a step that we call
_sparsification_.

Sparsification of the projected vector when $d>n$ may be carried out either
deterministically or stochastically. In the deterministic approach, we may
simply retain a certain fixed number $k$ ($k<n<d$) out of the $d$ entries in
the projected vector, say the $k$ largest values, and drop all the other
entries in the projected vector, thereby obtaining a sparse representation
with just $k$ significant entries. This appears to be the sparsification
method employed in biological neural networks like the olfactory system of the
fruit fly, as modeled by Dasgupta et al. in [16].

In the stochastic approach, the number of significant entries in the sparse
representation constructed out of the projected vector is not fixed to be some
$k$, but may vary depending on the representation (and therefore the input)
vector. For example, we may retain only the top, say, 5% of the entries in the
projected vector. More generally, we may compare the projected vector term by
term against a (deterministic or random) _threshold_ vector selected in some
way, and retain only the entries in the projected vector that are larger than
the corresponding entries in the threshold vector. Note that the choice of
threshold vector for comparison will determine the number of significant
entries in the resulting sparse representation vector.333For example, the
threshold entries may be selected such that on average, $k$ of the $d$ entries
of the projected vector exceed their corresponding thresholds [1]. This number
of significant entries in the sparse representation vector is a random
variable that could even be significant compared to the dimension $n$ of the
input vector.

###  3.4 ReLU MLP layer as expand-and-sparsify operator

Consider an $n$-dimensional input $\bm{u}\in\mathbb{R}^{n}$ to a single layer
of a feedforward neural network (MLP) comprising $d>n$ neurons, each being a
_Rectified Linear Unit_ (ReLU) with the following activation function:

$\text{ReLU}(x)=\begin{cases}x,&x\geq 0,\\\ 0,&x<0.\end{cases}$ (1)

Suppose the $i$th neuron has $n$ weights represented by the vector
$\bm{w}_{i}=[w_{i,1},\dots,w_{i,n}]^{\top}\in\mathbb{R}^{n}$ and threshold
parameter $\tau_{i}\in\mathbb{R}$ so that its output when given input $\bm{u}$
is $v_{i}=\text{ReLU}(\bm{w}_{i}^{\top}\bm{u}-\tau_{i})$, $i=1,\dots,d$. Thus
the output of the MLP layer of $d$ such neurons is the vector

$\bm{v}=\begin{bmatrix}v_{1}\\\ \vdots\\\
v_{d}\end{bmatrix}=\begin{bmatrix}\text{ReLU}(\bm{w}_{1}^{\top}\bm{u}-\tau_{1})\\\
\vdots\\\
\text{ReLU}(\bm{w}_{d}^{\top}\bm{u}-\tau_{d})\end{bmatrix}=\text{ReLU}\left(\begin{bmatrix}\bm{w}_{1}^{\top}\bm{u}-\tau_{1}\\\
\vdots\\\
\bm{w}_{d}^{\top}\bm{u}-\tau_{d}\end{bmatrix}\right)=\text{ReLU}(\bm{W}\bm{u}-\bm{\tau}),$
(2)

where $\bm{W}$ is the $d\times n$ matrix whose rows are
$\bm{w}_{1}^{\top},\dots,\bm{w}_{d}^{\top}$, and
$\bm{\tau}=[\tau_{1},\dots,\tau_{d}]^{\top}$ is the vector of neuron
thresholds.

From (2), it is clear that if $\bm{w}_{1},\dots,\bm{w}_{d}$ are i.i.d. (say,
uniformly on the unit sphere in $\mathbb{R}^{n}$), then $\bm{W}\bm{u}$ is the
projected input corresponding to the expansion operation on the input $\bm{u}$
with the random projection matrix $\bm{W}$, and the output $\bm{v}$ of the MLP
layer is the result of the expand-and-sparsify operation after stochastic
sparsification by comparing every entry of the projected input $\bm{W}\bm{u}$
against a threshold value for that entry and retaining only those entries that
are greater than the corresponding threshold. Of course, in practice
$\bm{w}_{1},\dots,\bm{w}_{d}$ are only i.i.d. at the time they are initialized
before gradient descent training begins [21, 22], but this observation will
nevertheless prove useful, as shown later.

With these preliminaries, let us now review the body of known results about
such expand-and-sparsify operations.

##  4 Function approximation with single hidden layer

###  4.1 Definitions and notation

We follow the analysis of Dasgupta and Tosh in [1]. They assume that the
sparsification leaves a binary vector, i.e., the activation function of the
$d$ neurons in the MLP layer is not ReLU (1) but the unit step function. It
follows that the output of the MLP layer is not given by (2) but by the binary
$d$-dimensional vector

$\bm{z}=[z_{1},\dots,z_{d}]^{\top}\in\\{0,1\\}^{d},$ (3)

where for $j=1,\dots,d$,

$z_{j}=\begin{cases}1,&\bm{w}_{j}^{\top}\bm{u}\geq\tau_{j},\\\
0,&\text{otherwise}.\end{cases}$ (4)

Although we write $\tau_{j}$ for brevity, each entry $\tau_{j}$ of the
threshold vector $\bm{\tau}$ is actually a function of $\bm{w}_{j}$, defined
as the $100(1-k/d)$th percentile of the random variable
$\bm{w}_{j}^{\top}\bm{u}$. It follows that the vector $\bm{z}$ in (3) is
$k$-sparse _in expectation_ , i.e., the mean number of $1$s in $\bm{z}$ is
$k$.

###  4.2 Mathematical Results for Expand-and-Sparsify

Consider now the problem of approximating a scalar function $f(\cdot)$ of the
input $\bm{u}$. Specifically, we want to approximate $f$ using a two-layer MLP
where the single hidden layer of $d$ neurons has output (3), and the output
“layer” of the MLP is a single linear node. To be precise, the output of the
MLP is a weighted average of the hidden-layer outputs $\bm{z}$:

$\hat{f}(\bm{u})=\frac{w_{1}^{\prime}z_{1}+\cdots+w_{d}^{\prime}z_{d}}{z_{1}+\cdots+z_{d}},$
(5)

whenever the denominator is positive, where
$w_{1}^{\prime},\dots,w_{d}^{\prime}$ are the weights connecting the single
output neuron to the $d$ neurons of the hidden layer. In particular, the input
space $\mathcal{U}$ is divided into regions
$\mathcal{U}_{j}=\\{\bm{u}\in\mathcal{U}:\bm{w}_{j}^{\top}\bm{u}\geq\tau_{j}\\}$
where $z_{j}=1$, $j=1,\dots,d$, and $w_{j}^{\prime}$ is defined as the
expected value of $f(\cdot)$ over the region $\mathcal{U}_{j}$.

If all weights of the hidden-layer weight matrix $\bm{W}$ are i.i.d. zero-mean
Gaussian and the distribution of the input $\bm{u}$ to the MLP is supported on
an $m$-dimensional ($m<n$) submanifold of $\mathcal{U}$ with some other mild
conditions, then for $k\gg\log d$ the function approximation error made by (5)
is upper bounded by $O((k/d)^{1/m})$ [1, Thm. 8].

Note that although this is a remarkable universal approximation result holding
for an untrained neural network with random hidden-layer weights selected
independent of the input, it does require the sparsity $k$ of the hidden-layer
output to be $\gg\log d$.444In [1], Dasgupta and Tosh say that this is to
ensure that every input in the submanifold excites at least one neuron in the
hidden layer.

Recall that $d$, the number of random vectors selected as rows of the random
projection matrix $\bm{W}$, can be exponential in the dimension $n$ of the
input $\bm{u}$, so $\log d\sim n$, which means that $k$, the sparsity of the
output of the hidden layer of the MLP, could be a significant fraction of $n$,
although it is unlikely that we need such a large $k$ for approximation
accuracy.

Dasgupta and Tosh also derive a result (see [1, Thm. 9]) that imposes no lower
bound on sparsity $k$ of the projected vector $\bm{W}\bm{u}$ when the
distribution of the projection weights $\bm{W}$ is mildly dependent on the
distribution of the input $\bm{u}$, but that result is for deterministic
sparsification, i.e., only the $k$ highest values of $\bm{W}\bm{u}$ are
retained. These assumptions do not describe an MLP, but they do give hope that
similar results may be available for the stochastic sparsification that occurs
in an MLP.

We will now try to explain several unusual or puzzling phenomena that have
been empirically observed in MLPs, some of which have later been observed in
other ML models as well. The canonical use case to keep in mind during our
ensuing discussion is that of using a “deep” MLP for a classification task,
say classifying the handwritten digits in the MNIST dataset [23].

##  5 Explanations of Observed Phenomena

Let us now go over the phenomena listed in the Introduction.

  1. 1.

Why neural networks perform better than the classical kernel machine ML models
like support vector machines and random forests on the same problems: We
believe the reason is that the kernel machines deterministically restrict the
dimension of the representation to be much smaller than that of the input.
While this may work for a majority of the inputs, a statistically significant
fraction of the input space will be better served by a representation with
more dimensions, which is possible with the stochastic sparsification inherent
in an MLP layer. In particular, certain input values (especially noisy inputs)
may require a significantly denser (less sparse) representation than most of
the other input values, in which case the larger number of significant entries
in the output of this MLP layer may also be serving to distribute the noise
over many more entries and thereby improve the reliability of the
representation. Some indications of this have been observed in the literature
(see [24, Secs. 4–6]).

  2. 2.

Why neural networks are curiously vulnerable to adversarially crafted inputs
that are imperceptibly different (to a human) from inputs which do not result
in erroneous output: In [18, Suppl. Info.], Dasgupta et al. define and study
the probability that two inputs $\bm{u}$ and $\bm{u}^{\prime}$ that are close
in $\mathbb{R}^{n}$ will yield expand-and-sparsify binary representations
$\bm{z}$ and $\bm{z}^{\prime}$ respectively whose entries are highly
correlated. Although this correlation is high, i.e.,
$\bm{z}^{\top}\bm{z}^{\prime}$ is tightly concentrated (see [18, Lem. 4,
Suppl. Info.]), it is intuitively clear that it should be possible to craft an
input $\bm{u^{\prime}}$ that is very close to a known observed input $\bm{u}$
but whose sparse binary representation $\bm{z^{\prime}}$ differs from
$\bm{z}$, the representation of $\bm{u}$. An _adversarial input_
$\bm{u^{\prime}}$ is one such input, chosen such that the difference between
$\bm{z^{\prime}}$ and $\bm{z}$ cascades progressively up the higher layers of
the MLP until the final output of the MLP, typically a classification of
$\bm{u^{\prime}}$, is different from that of $\bm{u}$. Our intuition would say
that such vulnerability to adversarial inputs increases with the number of
layers of the MLP.

  3. 3.

Why randomly-initialized neural networks, even without training, can perform
fairly well: this is explained as a direct consequence of the high
approximation accuracy of (5) as discussed above.

  4. 4.

Why vanilla stochastic gradient descent without regularization is usually
sufficient to train a neural network: As remarked in [9], often the weights of
the MLP change by very little from their random initial values even after
training. This is readily explainable in light of the above phenomenon which
shows that the randomly initialized weights are often “good enough.” If these
initial weights require hardly any fine-tuning for even better performance,
then it also follows that we need not be concerned with issues such as
gradient decay prevention or the curious effectiveness of such a simple
unregularized gradient descent optimization method – they are mostly
unnecessary because the expand-and-sparsify operation with a large-enough
expansion will deliver good to excellent results even before any numerical
optimization is attempted. The gradient descent iterations during training
serve only to perturb these initial values of the parameters555These
perturbations have been analyzed asymptotically by Chizat and Bach in [9] and
more realistically in great detail in the book-length treatment of Roberts and
Yaida [10]., and therefore deliver only a first-order impact on model
performance, in contrast to the zeroth-order impact of architecture and
parameter initialization.

  5. 5.

Why a trained neural network can be drastically pruned without loss of
performance: if the expansion of the expand-and-sparsify stage is to a large
enough dimension, then there are many possible choices for the sparsification
factor that should yield good performance. It is plausible that aggressive
sparsification will lead to many hidden-layer nodes not being activated by any
input in the training and/or test set, which allows for their deletion without
any effect on performance.666The observation that a trained neural network can
be pruned aggressively without significant impact on performance has been
previously noted in the literature and described as part of the Lottery Ticket
Hypothesis [2]. However, this intuition also indicates that aggressive pruning
will render the MLP more susceptible to adversarial attack. Moreover, it also
implies that the model may underperform on selected test sets comprising
natural (unmodified) inputs from regions of the input space that require
milder sparsification for optimum processing. This conjecture should be
tested.

  6. 6.

Why training a neural network with dropout leads to better performance of the
trained model: the explanation for this is the same as that of the previous
phenomenon, namely that there is a range of choices for sparsification that
will all yield trained models that perform well. Dropout [4] simply
corresponds to a further level of stochasticization of the sparsification
step, and so it is not surprising that it can do well in scenarios calling for
high sparsification in the expand-and-sparsify operation, although another
effect of dropout may be to create stochastic ensembles of MLP models with
little interaction between them, thereby yielding the performance boost from
ensembles [25]. Our intuition implies that the effect of dropout is comparable
to increasing the sparsification factor, and this is also a conjecture that
should be tested.

  7. 7.

Why a neural network so overparameterized that it can memorize a training set
with arbitrary labels can nevertheless generalize well if trained on a
training set with correct labeling, even though it still learns the training
set with zero error: The most significant consequence of the expand-and-
sparsify operation is that “overparameterization” is a red herring, since a
wide range of values of the expansion factor and sparsification factor will
yield models with good performance, and a judicious selection of these two
factors may even obviate the need for much training of the weights in the
corresponding hidden layers. In other words, it is misleading to simply count
the number of weights in the hidden layers and compare them to the number of
dimensions in the input, or the number of training inputs, and then declare
the MLP as a whole to be overparameterized or not as the case may be. For a
training set (think of MNIST) whose examples have been assigned random labels,
we know that the expand-and-sparsify operation can approximate even this
scrambled label assignment accurately, although the estimation of the linear
weights $w_{j}^{\prime}$ via backpropagation may require a large number of
training steps because of the complexity of the regions $\mathcal{U}_{j}$ in
this case compared to the case with correctly labeled training data. This
increase in required training time over the scrambled-labels dataset has been
reported in the literature [5].

##  6 Next steps

Much work remains to be done, both in theory and in practice, to close the gap
between the tantalizing preliminary results reported in [1] and a rigorous
body of knowledge that not only explains observed phenomena in neural
networks, but can also be used to design better-performing neural networks.

We have made two testable conjectures in connection with our discussion of
phenomena 5 and 6 in the previous section. Our next step should be to test
these conjectures.

Another inference that follows from the expand-and-sparsify methodology is
that a wide first hidden layer should be used for better performance of an
MLP. However, the interplay between this wide first hidden layer and the folk
wisdom to go “deep rather than wide” when designing such neural network
architectures needs to be examined. The theoretical reason for going deep was
the reduction in complexity required for function approximation, but if, as we
have argued above, the expand-and-sparsify operation renders conventional
notions of complexity irrelevant, then the wisdom of going deep needs to be
scrutinized in depth. The optimum depth-to-width results derived in [10]
should be compared to empirical observations for different expansion and
sparsification factors for MLPs of different numbers of layers.

From a theoretical perspective, we suggest as “best-practice” the choice of
expansion and sparsification factors in the expand-and-sparsify model for MLPs
with different numbers of total layers. The result [1, Thm. 3] on the size $d$
of the MLP hidden layer required for a given accuracy of function
approximation with a $k$-sparse representation after expand-and-sparsify is
only applicable to deterministic sparsification (i.e., retaining the $k$
largest values). Thus, it needs to be generalized both to stochastic
sparsification and to adapt to inputs that live on a lower-dimensional
subdomain of the $n$-dimensional input space.

## References

  * [1] S. Dasgupta, C. Tosh, “Expressivity of Expand-and-Sparsify Representations,” https://arxiv.org/abs/2006.03741, Jun. 2020.
  * [2] J. Frankle, M. Carbin, “The Lottery Ticket Hypothesis – Finding Sparse, Trainable Neural Networks,” ICLR 2019, https://openreview.net/forum?id=rJl-b3RcF7.
  * [3] J. Frankle, D.J. Schwab, A.S. Morcos, “Training batchnorm and only batchnorm: On the expressive power of random features in CNNs,” https://arxiv.org/abs/2003.00152, Mar. 2020.
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# Time Index Modulation-Driven Standalone RIS Mechanism for Symbiotic Radio

M. Ertug Pihtili , Mehmet C. Ilter , Ertugrul Basar

###### Abstract

The rising demand for energy and spectrum resources in next-generation
Internet-of-things (IoT) systems accounts for innovative modes of information
and power transfer. One potential solution is to harness the active
transmission capability of devices to facilitate data transmission and
wireless energy harvesting (WEH) for backscatter communication so as to form a
symbiotic radio (SR) environment in a mutualistic manner. Additionally,
incorporating reconfigurable intelligent surfaces (RISs) into the SR
environment can provide an additional link and enhance the reliability of
backscatter communication, thereby reinforcing the symbiotic relationships
between active and passive devices. This paper proposes a novel SR system
where a standalone RIS sustains its functions through WEH based on a low-power
RIS structure and establishes mutualistic symbiosis by utilizing a signal
conveyed by the primary transmitter (PTx) to assist ongoing transmissions and
convey information to the primary receiver (PRx). The PTx employs time index
modulation (TIM) to transmit information to the PRx and power to the RIS and
energy harvester (EH). A log-likelihood ratio (LLR)-based detector is
presented to address challenges in the TIM scheme. Finally, the performance of
the proposed scheme is investigated in terms of harvested direct current (DC)
power at the RIS and EH, as well as the bit error rate (BER) at the PRx.

###### Index Terms:

Wireless energy harvesting, reconfigurable intelligent surfaces, symbiotic
radio, index modulation.

##  I Introduction

The integration of wireless energy harvesting (WEH) into future Internet-of-
things (IoT) architectures is increasingly vital as the coexistence between
cellular and IoT devices demands additional energy resources to support the
growing number of connected devices. In order to address rising demand,
symbiotic relationships among cellular and IoT devices have been discovered,
whereby the passive transmission capabilities of IoT devices leverage the
power and spectrum resources of active transmission technology used by
cellular devices. The concept, known as symbiotic radio (SR), has been
introduced to enable energy- and spectrum-efficient communications within the
network [1].

Similar to SR, IoT devices can utilize ambient backscatter communications
(AmBC) by leveraging existing ambient RF signals in the environment. However,
AmBC suffers from direct link interference from surrounding ambient RF
signals, which limits detection performance at the target receiver [2].
Contrary to AmBC, the SR concept forms a novel architecture in which active
and passive systems rely on mutualistic communication through joint decoding
at the receiver and an additional link provided by a backscatter device (BD)
rather than parasitic communication, which is a drawback of AmBC [1]. In the
SR framework, the primary transmitter (PTx) shares its energy and spectrum
resources with the secondary transmitter (STx). Meanwhile, STx operates as a
passive device that also transmits its information to primary receiver (PRx)
by manipulating the resources of PTx. To establish mutualistic symbiosis, the
PTx maintains a higher transmission rate than the PRx, which means the symbol
duration of the STx is longer than that of PTx [2]. In [3], the authors reveal
that the mutualistic condition depends on the average strength of direct and
backscatter links, and [1] mentions that the backscatter link can be improved
by utilizing reconfigurable intelligent surfaces (RISs). The RIS-aided SR
system has recently introduced in [4], where the RIS performs two functions
simultaneously: assisting PTx transmission and transmitting its information by
orchestrating the phase shifts on the incident signals.

Specifically, RISs are passive devices consisting of $\displaystyle N$ unit
cells (UCs) that operate as reflectors to manipulate the wireless channel by
tuning the phase of the reflected signal [5]. RIS UCs are tunable devices
equipped with PIN diodes, varactors, and RF switches, that control the
wireless medium and enable the reflection function. Nevertheless, UCs can also
be operated as absorbers for WEH on the RIS. In [6], a novel RIS controller
architecture, called integrated architecture, is discussed to form a low-power
RIS setup. Based on this integrated architecture, WEH protocols for self-
sustainable operations are explored in [7], assuming perfect absorbers with no
reflected signals caused by the RIS. To achieve a self-sustainable
architecture, the harvested direct current (DC) power on the RIS should be at
least equal to its total power consumption. Although the integrated
architecture provides lower power consumption than conventional RIS
controllers, the type of UCs can make self-sustainable operations infeasible
if varactor-based UCs are employed, as they are driven by power-hungry
digital-to-analog converters (DACs). The authors examined the power
consumption of various UCs and claimed that it is possible to design an RF
switch-based RIS without DACs [8]. An RF switch-based RIS prototype was
designed, demonstrating that by connecting one port of the RF switch to the
absorber, WEH can be achieved on RIS [9].

00footnotetext: Notations: Italic letters $\displaystyle x$ denote scalar
values, while lowercase $\displaystyle\mathbf{x}$ and uppercase
$\displaystyle\mathbf{X}$ in boldface represent vectors and matrices,
respectively. The superscripts $\displaystyle(\cdot)^{T}$ and
$\displaystyle(\cdot)^{H}$ denote the transpose and the Hermitian transpose
operations, respectively. $\displaystyle\angle\cdot$,
$\displaystyle\lvert\cdot\rvert$, and $\displaystyle\lVert\cdot\rVert$ denote
the angle, absolute value, and the norm of a vector/matrix, respectively.
$\displaystyle\log_{a}(\cdot)$ represents the logarithm with base
$\displaystyle a$. The operator $\displaystyle\mathbb{E}[\cdot]$ indicates
averaging/statistical expectation. $\displaystyle\Pr(\cdot)$ represents the
probability of an event. $\displaystyle\lceil\cdot\rceil$ and
$\displaystyle\lfloor\cdot\rfloor$ denote the ceil and floor operations,
respectively. The symbol $\displaystyle j$ represents the imaginary unit
$\displaystyle\sqrt{-1}$, while $\displaystyle I$ and $\displaystyle Q$ denote
a complex symbol’s in-phase and quadrature parts, respectively.
$\displaystyle\mathcal{CN}(\mu,\sigma^{2})$ represents a circularly symmetric
complex Gaussian (CSCG) random variable with mean $\displaystyle\mu$ and
variance $\displaystyle\sigma^{2}$. The set of complex matrices of dimensions
$\displaystyle m\times n$ is denoted by $\displaystyle\mathbb{C}^{m\times n}$,
the set of non-negative integers of dimensions $\displaystyle m\times n$ is
denoted by $\displaystyle\mathbb{N}^{m\times n}$, and
$\displaystyle\mathbb{Z}^{+}$ denotes the set of integers greater than zero.
The binomial coefficient is denoted by $\displaystyle C(n,k)$. The
$\displaystyle\mathbf{0}_{n\times m}$ denotes the all-zero matrix of size
$\displaystyle n\times m$.

To address the increasing power demand, simultaneous wireless information and
power transfer (SWIPT) emerges as a candidate for next-generation green
communication systems, wherein power and information transmission are handled
separately across various domains: power (power splitting), time (time
switching), and space (antenna splitting) [10]. An alternative approach to
SWIPT, called Information Harvesting (IH), has been proposed to
synergistically transfer information and power to IoT devices by exploiting a
far-field RF power transfer mechanism for data transmission through index
modulation (IM) while avoiding signal splitting into different domains [11].

IM can map information bits by altering the on/off status of transmission
entities in a communication system, such as transmit antennas and subcarriers
[12]. This distinctive feature enables the integrated transfer of power and
information by means of IM. To facilitate data transfer and wireless energy
harvesting (WEH) efficiently, a time index modulation (TIM) technique that
employs time slots to convey additional information is presented in [13]. In
[14], an orthogonal frequency division multiplexing (OFDM)-IM scheme is
proposed by inactivating some subcarriers, and its performance under the low-
complexity log-likelihood ratio (LLR) detector is investigated. Furthermore,
[15] presents a dual-mode OFDM-IM system by replacing inactive subcarriers
with a different constellation set from those used in active ones. In [16],
the error performance of several OFDM-IM-based schemes is compared using an
LLR-based detector. Building upon the concept introduced in [11], a novel
OFDM-based information harvesting concept is proposed in [17], where OFDM-IM
is applied to an ongoing wireless power transfer mechanism. [18] proposes an
RIS-based IM scheme in which the indices of the activated group of RIS UCs are
utilized to convey environmental data collected by RIS. To prevent power loss
caused by inactivating a group of RIS UCs, partitioning the RIS UCs into two
orthogonal groups is proposed in [19].

In this paper, we present a novel TIM-driven SR environment empowered by a
standalone RIS architecture. This architecture utilizes an integrated
controller to reduce the power consumption of the RIS and employs 2-bit
resolution RF switches as UCs to eliminate power-hungry DACs, thereby
promoting sustained functionality through WEH. Consequently, the RIS operates
as a BD in the SR environment by initially applying WEH, subsequently
assisting PTx transmission strategy and conveying RIS information by inducing
phase shifts selected from a discrete set. Additionally, PTx implements the
TIM scheme to power the RIS and EH while ensuring reliable communication with
the PRx. Moreover, we propose an LLR-based detector tailored for the TIM-based
SR scheme within our framework to address decoding challenges.

##  II Symbiotic Radio Model

In the proposed scheme, the primary system, which consists of a single antenna
PTx and an $\displaystyle M_{\text{R}}$ antenna PRx, relies on active RF
transmission technology. The secondary system comprises a standalone RIS with
$\displaystyle N=N_{1}+N_{2}+N_{3}$ UCs and a single antenna EH, both of which
are passive IoT devices. These passive IoT devices harness the power and
spectrum resources of the primary system to sustain their operations such that
the primary and secondary systems rely on symbiotic relationships [1]. In Fig.
1, PTx leverages the TIM technique to send information to the PRx and power to
the EH through the direct link and the link empowered by RIS. Therein, the RIS
is partitioned into three groups: the first group, with $\displaystyle N_{1}$
UCs, assists the PTx transmission; the second group uses $\displaystyle N_{2}$
UCs as absorbers to ensure the RIS’s self-sustainability through WEH by
exploiting the PTx signal; and $\displaystyle N_{3}$ UCs in the third group
are dedicated to conveying the RIS information by inducing phase shifts in the
reflected signal. The PTx-RIS and RIS-PRx channels are defined as
$\displaystyle\mathbf{h}_{r}\triangleq\left[{\mathbf{h}}_{r,1},{\mathbf{h}}_{r,2},{\mathbf{h}}_{r,3}\right]^{T}\in\mathbb{C}^{N\times
1}$ and
$\displaystyle\mathbf{G}_{d}\triangleq\left[{\mathbf{G}}_{d,1},{\mathbf{G}}_{d,2},{\mathbf{G}}_{d,3}\right]\in\mathbb{C}^{N_{\text{R}}\times
N}$, respectively, where subscripts $\displaystyle l=\\{1,2,3\\}$ associated
with the RIS groups with $\displaystyle\\{N_{1},N_{2},N_{3}\\}$ UCs. All links
follow the Rician channel model, and $\displaystyle v$-th entry of PTx-PRx,
and $\displaystyle(v,n)$-th entry of RIS-PRx link can be written as:

$\displaystyle\displaystyle h_{d}^{v}$
$\displaystyle\displaystyle=\sqrt{\nu_{p}}\left(\sqrt{\frac{\kappa_{p}}{\kappa_{p}+1}}{h}_{d,\mathrm{LoS}}^{n}+\sqrt{\frac{1}{\kappa_{p}+1}}h_{d,\mathrm{NLoS}}^{n}\right),$
(1) $\displaystyle\displaystyle g_{d}^{v,n}$
$\displaystyle\displaystyle=\sqrt{\nu_{p}}\left(\sqrt{\frac{\kappa_{p}}{\kappa_{p}+1}}g_{d,\mathrm{LoS}}^{n,v}+\sqrt{\frac{1}{\kappa_{p}+1}}g_{d,\mathrm{NLoS}}^{n,v}\right),$
(2)

where $\displaystyle v\in\\{1,2,\ldots,M_{R}\\}$ and $\displaystyle
n\in\\{1,2,\ldots,N\\}$. Herein, $\displaystyle\nu_{p}$ and
$\displaystyle\kappa_{p}\geq 0$ for $\displaystyle{p}\in\\{h_{d},g_{d}\\}$
represent the path loss and the Rician factors, respectively. The terms
$\displaystyle{h}_{d,\mathrm{LoS}}$, and $\displaystyle
g_{d,\mathrm{LoS}}^{v,n}$ are the deterministic line-of-sight (LoS) components
with $\displaystyle\lvert{h}^{v}_{d,\mathrm{LoS}}\rvert=\lvert
g_{d,\mathrm{LoS}}^{v,n}\rvert=1$ and
$\displaystyle{h}^{v}_{d,\mathrm{NLoS}}$, and $\displaystyle
g_{d,\mathrm{NLoS}}^{v,n}$ represent non-LoS components in which each entry is
independent and identically distributed (i.i.d.) according to
$\displaystyle\mathcal{CN}\left(0,1\right)$. The definitions of the remaining
channels follow similar principles and can be derived analogously.

Figure 1: System model of TIM-aided RIS-SR scheme.

###  II-A Primary transmission model

The PTx utilizes the TIM scheme to convey information to the PRx and power to
the EH over direct and RIS links. Hence, the PTx also sends bits by indexing
information and power transmission stages into distinct time slots, allowing
deterministic power signals to participate in information transmission. Thus,
there is an ongoing information transmission even when the PTx switches to a
power stage.

####  II-A1 Information Stage

In the information stage, $\displaystyle L$ information signals $\displaystyle
x_{i}\in\mathcal{M}$ with a power of $\displaystyle P_{L}$ are drawn from
$\displaystyle M$-PSK/QAM constellation points, where
$\displaystyle\mathcal{M}=\\{x_{i}:1\leq i\leq M\\}$. Therefore,
$\displaystyle\eta_{m}=L\log_{2}(M)$ bits can be sent over the total block
length. By leveraging TIM to alter the transmission ordering of the
$\displaystyle L$ information signals, $\displaystyle r=C(K,L)$ distinct
mappings of $\displaystyle L$ information signals can be achieved. Thus,
additional $\displaystyle\eta_{r}=\left\lfloor\log_{2}(r)\right\rfloor$ bits
are conveyed, such that $\displaystyle\eta=\eta_{m}+\eta_{r}$ bits are
transmitted over one block duration that spans $\displaystyle K$ transmission
stages.

According to $\displaystyle\eta_{r}$ bits, the time indices of $\displaystyle
L$ information signals are chosen from the set for each block duration, which
is given by

$\mathcal{T}=\begin{Bmatrix}\mathbf{t}_{1}&\mathbf{t}_{2}&\ldots&\mathbf{t}_{2^{\eta_{r}}}\end{Bmatrix},$
(3)

in which there are $\displaystyle 2^{\eta_{r}}$ possible legitimate time-index
selections out of $\displaystyle r$. After deciding on
$\displaystyle\mathbf{t}_{\alpha}\in\mathbb{Z}^{+}$, where
$\displaystyle\alpha\in\\{1,2,\ldots,2^{\eta_{r}}\\}$, the time-index vector
is defined as
$\displaystyle\boldsymbol{\tau}_{\mathbf{t}_{\alpha}}=\begin{bmatrix}\tau_{1}&\tau_{2}&\ldots&\tau_{K}\end{bmatrix}^{T}\in\mathbb{N}^{K\times
1}$, and $\displaystyle\tau_{k}$ is set to 1 at every information transmission
stage, where $\displaystyle k=1,2,\ldots,K$.

####  II-A2 Power Stage

The remaining $\displaystyle(K-L)$ time slots are used for deterministic power
signal $\displaystyle\omega=\omega_{I}+j\omega_{Q}$ with a power of
$\displaystyle\lvert\omega\rvert^{2}=P_{H}$, where $\displaystyle P_{H}\geq
P_{L}$, and $\displaystyle\tau_{k}$ is set to 0 at each power transmission
stage.

By following the construction of
$\displaystyle\boldsymbol{\tau}_{\mathbf{t}_{\alpha}}$, the
$\displaystyle\eta_{m}$ bits are mapped to the $\displaystyle M$-PSK/QAM
symbols, where each symbol carries $\displaystyle\eta_{m}/L$ bits. The mapping
is done considering the legitimate signal set $\displaystyle\mathcal{S}$,
which has a length of $\displaystyle M^{L}$ and is defined as follows:

$\mathcal{S}=\begin{Bmatrix}\mathbf{x}_{1}&\mathbf{x}_{2}&\mathbf{x}_{3}&\cdots&\mathbf{x}_{M^{L}}\end{Bmatrix}.$
(4)

Figure 2: PTx transmission strategy.

Therein, $\displaystyle\mathbf{x}_{n}\in\mathcal{S}$, where $\displaystyle
n\in\\{1,2,\ldots,M^{L}\\}$, represents a vector of length $\displaystyle L$,
$\displaystyle\mathbf{x}_{n}=\begin{bmatrix}{x}_{i,1}^{n}&{x}_{i,2}^{n}&\cdots&{x}_{i,L}^{n}\end{bmatrix}^{T}$
in which, $\displaystyle P_{L}=\mathbb{E}[|x_{i}|^{2}]$ is normalized to unit
power. Consequently, the elements of $\displaystyle\mathbf{x}_{n}$, namely the
symbols, are mapped into the time slots based on
$\displaystyle\boldsymbol{\tau}_{\mathbf{t}_{\alpha}}$ to initialize the
transmission stage. Accordingly, the signal $\displaystyle
s_{k}\in\\{\mathcal{S},\omega\\}$ is transmitted at each time instant
$\displaystyle k$.

In Fig. 2, the transmission strategy of PTx is illustrated for $\displaystyle
K=6$ and $\displaystyle L=3$. The total block duration is assumed to be less
than the channel coherence time, ensuring that the channel coefficients remain
unchanged for a block of length $\displaystyle K$, though they might change
from one block to another. Herein, PTx alternates between information and
power stages according to
$\displaystyle\boldsymbol{\tau}_{\mathbf{t}_{\alpha}}=\begin{bmatrix}1&0&1&0&0&1\end{bmatrix}$,
where $\displaystyle\mathbf{t}_{\alpha}=\begin{bmatrix}1&3&6\end{bmatrix}$,
indicating that the locations of the information and power stages in time
domain carry additional information bits.

###  II-B RIS Absorption Function and WEH

The absorbing function of UCs enables RIS operations through WEH without
requiring an external battery or power grid. Hence, RF energy harvesting on
the RIS can be achieved through RF switches since they can act as both
reflectors and absorbers [8]. Accordingly, RF switches are utilized in RIS as
UCs, with $\displaystyle N_{2}$ UCs serving as perfect absorbers and the
remaining $\displaystyle N-N_{2}$ UCs functioning as perfect reflectors. The
RF switch allocates one output port for an impedance-matching component, which
absorbs the energy of incoming signals instead of reflecting them. The
absorbed signals are fed to the RF combining network, which combines the
signals in the RF domain [20] and outputs them to the nonlinear rectenna as
input for obtaining harvested DC power such that the rectenna’s input at time
instant $\displaystyle k$ is mathematically given by

$\mathrm{Q}_{k}=\left|\sum_{n_{2}=N_{1}+1}^{N_{2}}{h}_{r,2}^{n_{2}}s_{k}\right|^{2}.$
(5)

The constant-linear-constant (CLC) model is employed to capture the practical
characteristics of a rectenna, including turn-on sensitivity and saturation.
[21]. In the CLC rectenna model, the harvested DC power,
$\displaystyle\mathcal{P}_{\text{DC}}(\mathrm{Q})$ is expressed as follows:

$\mathcal{P}_{\text{DC}}(\mathrm{Q}_{k})=\begin{cases}0,&\mathrm{Q}_{k}\in[0,\mathcal{P}_{\text{on}})\\\
\rho(\mathrm{Q}_{k}-\mathcal{P}_{\text{on}}),&\mathrm{Q}_{k}\in[\mathcal{P}_{\text{on}},\mathcal{P}_{\text{sat}})\\\
\rho(\mathcal{P}_{\text{sat}}-\mathcal{P}_{\text{on}}),&\mathrm{Q}_{k}\in[\mathcal{P}_{\text{sat}},\infty)\end{cases}.$
(6)

Here, $\displaystyle\rho$ denotes the WEH efficiency, and
$\displaystyle\mathrm{Q}_{k}$ represents the input power to the rectifier at
time instant $\displaystyle k$. The turn-on sensitivity
$\displaystyle\mathcal{P}_{\text{on}}$ is the minimum received DC power
required to activate the rectenna for RF-to-DC conversion. Saturation occurs
when the rectenna reaches its maximum power harvesting capacity. The maximum
harvested DC power is expressed as
$\displaystyle\mathcal{P}_{\text{max}}=\eta(\mathcal{P}_{\text{sat}}-\mathcal{P}_{\text{on}})$.
Since the input power of the rectifier circuit varies depending on the
specific application, $\displaystyle\mathcal{P}_{\text{sat}}$ and
$\displaystyle\mathcal{P}_{\text{on}}$ should be adjusted according to the
input power level. When the input power to the rectifier circuit is high,
increasing $\displaystyle\mathcal{P}_{\text{sat}}$ can optimize the benefits
of the high input power.

Due to the fact that the RIS relies on WEH through its $\displaystyle N_{2}$
absorbers to serve its functions, the harvested DC power
$\displaystyle\mathcal{P}_{\text{DC}}(\mathrm{Q}^{\text{RIS}}_{k})$ should be
sufficient to maintain the total power consumption of the RIS. The total power
consumption required to operate the RIS consists of two parts: the static
power consumption caused by the RIS controller and the dynamic power
consumption generated by the RIS UCs. Therefore, the total power consumption
modeling of RIS can be expressed as follows:

$P_{\text{RIS}}=P_{\text{stc}}\,+\,P_{\text{dyn}}$ (7)

where $\displaystyle P_{\text{stc}}$, and $\displaystyle P_{\text{dyn}}$ refer
to the static and dynamic power consumption of RIS, respectively.
$\displaystyle P_{\text{stc}}$ consists of the power consumption of the
control board, $\displaystyle P_{\text{cb}}$, which has a constant value, and
the power consumption of the drive circuits, $\displaystyle P_{\text{drv}}$,
which varies with the type of UC driving mechanism. Therefore, various UCs,
such as varactors and RF switches, exhibit distinct power consumption
characteristics [6]. Since varactor-based UCs require power-hungry DACs to
achieve continuous phase shifts, they lead to increased static power
consumption. As $\displaystyle N$ increases, $\displaystyle P_{\text{stc}}$
tends to dominate $\displaystyle P_{\text{dyn}}$, making the self-sustainable
condition challenging [7]. To accomplish a standalone RIS architecture, one
can utilize RF switches directly driven by RIS controllers without DACs [8].
Additionally, an integrated architecture for the RIS controller is a favorable
approach to constructing a standalone RIS, as it consumes very low power
during static conditions compared to conventional RIS controllers [6]. Thus,
RF switches combined with an integrated architecture could offer a potential
solution to meet the standalone RIS condition despite the discrete phase shift
capability of RIS compared to the continuous phase shifts provided by
varactor-based UCs. The power consumption of the integrated architecture for
varactors and RF switches is given by, respectively,

${P}_{\text{RIS}}=\begin{cases}\left\lceil N/N_{\text{cb}}\right\rceil
P_{\text{cb}}+N\left(2P_{\text{drv}}+P_{\text{varactor}}\right),&\text{for
varactor}\\\ \left\lceil N/N_{\text{cb}}\right\rceil
P_{\text{cb}}+NP_{\text{switch}},&\text{for RF switch}\end{cases},$ (8)

where $\displaystyle N_{\text{cb}}$ is the number of UCs controlled by one RIS
controller, and $\displaystyle
P_{\text{dyn}}\in\\{P_{\text{switch}},P_{\text{varactor}}\\}$. The condition
that verifies the standalone RIS operation can be given as follows:

$\frac{1}{K}\sum_{k=1}^{K}\mathcal{P}_{\text{DC}}(\mathrm{Q}^{\text{RIS}}_{k})\geq
P_{\text{RIS}},$ (9)

which implies that the average harvested DC power over a block of length
$\displaystyle K$ should be at least equal to or greater than the power
consumption of the RIS.

###  II-C RIS-enabled passive transmission model

During the information stage, the RIS performs two functions: it assists PTx
transmission using $\displaystyle N_{1}$ UCs and transmits RIS information
through $\displaystyle N_{3}$ UCs by applying a phase shift to the signal.
Since RF switches are utilized, they can induce discrete phase shifts based on
the $\displaystyle b$-bit resolution from a discrete set, i.e.,
$\displaystyle\phi_{l}\in$
$\displaystyle\mathcal{J}=\left\\{0,{2\pi}/{B},\ldots,{(B-1)\pi}/{B}\right\\}$,
where $\displaystyle\phi_{l}$ represents the common phase shift within the
$\displaystyle l$-th group. The RIS can generate $\displaystyle B=2^{b}-1$
different phase shift levels, as one output port of the RF switches is
dedicated to an absorber. Once designed, these phase shifts remain invariant
across all channel realizations in the RF switch-based RIS architecture. If a
2-bit resolution RF switch is employed, the possible phase shifts can be
expressed as follows:

$\phi_{l}\in\begin{Bmatrix}\phi_{1}&\phi_{2}&\phi_{p}\end{Bmatrix}.$ (10)

Considering the discrete set in (10), in the power stage, the RIS UCs
configure the phases of $\displaystyle N-N_{2}=N_{1}+N_{2}$ UCs to
$\displaystyle\phi_{p}$ for power transmission. In the information stage, the
RIS selects phase shifts from the set $\displaystyle\mathcal{J}^{\prime}$,
where $\displaystyle\mathcal{J}^{\prime}=\begin{Bmatrix}\theta_{c}:1\leq c\leq
J\end{Bmatrix}$, to represent one bit of information based on the two possible
phase options $\displaystyle\theta_{c}\in\\{\phi_{1},\phi_{2}\\}$; therefore,
the modulation order of RIS information is $\displaystyle J=2$. The first
group of $\displaystyle N_{1}$ UCs can alter their phase shifts at each
information transmission slot by selecting the closest phase from the set
$\displaystyle\mathcal{J}^{\prime}$ to assist PTx transmission. Meanwhile, to
achieve mutualistic SR [1], the RIS aligns the third group of $\displaystyle
N_{3}$ UCs to an invariant phase shift across $\displaystyle L$ information
slots during a block of length $\displaystyle K$. Therefore, the symbol period
of RIS information is larger than that of PTx, provided that $\displaystyle
L>1$, and the parameter $\displaystyle\delta=L$ represents the spreading gain
that affects the mutualistic symbiosis, and increasing $\displaystyle\delta$
improves the detection performance of RIS information at PRx [3].

Let $\displaystyle\psi_{l}=\beta_{l}e^{-j\phi_{l}}$ denote the reflect
beamforming coefficient at the $\displaystyle l$-th RIS group in which
$\displaystyle\beta_{l}\in[0,1]$ is the common reflection amplitude within the
$\displaystyle l$-th group and
$\displaystyle\boldsymbol{\Psi}_{k}=\begin{bmatrix}\psi_{1}&\psi_{2}&\psi_{3}\end{bmatrix}^{T}$
denote the reflection vector. The received signal at PRx at time instant
$\displaystyle k$ is written as:

$\displaystyle\displaystyle\mathbf{y}_{k}=$
$\displaystyle\displaystyle\mathbf{h}_{d}s_{k}+{\sum_{n_{1}=1}^{N_{1}}\mathbf{g}_{d,1}^{n_{1}}e^{-j\phi_{l}}{h}_{r,1}^{n_{1}}s_{k}}$
$\displaystyle\displaystyle+\sum_{n_{3}=N_{2}+1}^{N_{3}}\mathbf{g}_{d,3}^{n_{3}}e^{-j\phi_{l}}{h}_{r,3}^{n_{3}}s_{k}+\mathbf{z}_{k}.$
(11)

Herein, the reflection amplitudes are fixed to $\displaystyle\beta_{l}=1$ for
the first and third groups to maximize the reflected power, while
$\displaystyle\beta_{2}=0$ since the second group absorbs the signal such that
$\displaystyle\psi_{2}=0$. Additionally,
$\displaystyle\mathbf{z}_{k}\in\mathbb{C}^{M_{\text{R}}\times 1}$ denotes the
additive white Gaussian noise (AWGN) sample vector, where each element
$\displaystyle{z}_{k}\sim\mathcal{CN}\left(0,\sigma^{2}\right)$ has a zero
mean and variance $\displaystyle\sigma^{2}$, added to the signal at each
information stage. Defining
$\displaystyle\underline{\mathbf{f}}\triangleq\begin{bmatrix}\mathbf{G}_{r,1}\mathbf{h}_{r,1}&\mathbf{G}_{r,2}\mathbf{h}_{r,2}&\mathbf{G}_{r,3}\mathbf{h}_{r,3}\end{bmatrix}\in\mathbb{C}^{N_{\text{R}}\times
3}$ as PTx-RIS-PRx cascaded channel matrix, the (11) can be rewritten as:

$\mathbf{y}_{k}=\mathbf{h}_{d}s_{k}+\underline{\mathbf{f}}\boldsymbol{\Psi}_{k}s_{k}+\mathbf{z}_{k}.$
(12)

Similarly, the received signal at EH is given by

${\varepsilon}_{k}=h_{e}s_{k}+\underline{\mathbf{v}}\boldsymbol{\Psi}_{k}s_{k},$
(13)

where
$\displaystyle\underline{\mathbf{v}}\triangleq\begin{bmatrix}\mathbf{h}_{r,1}\mathbf{g}_{e,1}&\mathbf{h}_{r,2}\mathbf{g}_{e,2}&\mathbf{h}_{r,3}\mathbf{g}_{e,3}\end{bmatrix}\in\mathbb{C}^{1\times
3}$ denotes the PTx-RIS-EH cascaded channel matrix and AWGN term is omitted
since it is too low to be harvested. Therein,
$\displaystyle\mathrm{Q}_{\text{EH}}=|\varepsilon_{k}|^{2}$ is input for the
CLC rectenna defined in (6) and the average harvested DC power can be
calculated by putting $\displaystyle\mathrm{Q}^{\text{EH}}_{k}$ into (9).

The cascaded channel of the PTx-RIS-PRx (EH) links can be estimated using the
method in [22]. Subsequently, we assume that PTx perfectly knows the cascaded
channel and direct link channel. Therefore, PTx can transmit the phase
knowledge and the corresponding transmission stage to the RIS prior to each
transmission by determining the phase shift of the first group through
$\displaystyle\psi_{1}=\arg\min_{\theta_{c}\in\mathcal{J}^{\prime}}\lvert\theta_{c}-\mathbb{E}[(\angle\mathbf{G}_{r,1}\mathbf{h}_{r,1}+\angle
h_{d})]\rvert^{2}$. Accordingly, the RIS sets the phase of $\displaystyle
N_{1}$ UCs and arranges the phase of $\displaystyle N_{3}$ UCs at time instant
$\displaystyle k$.

##  III SR-based Decoder

The PRx task involves jointly estimating the time-index vector
$\displaystyle\boldsymbol{\tau}_{\mathbf{t}_{\alpha}}$ as well as the
information of PTx and RIS by considering all possible time-index selections
and corresponding symbols from the sets $\displaystyle\mathcal{S}$ and
$\displaystyle\mathcal{J}^{\prime}$, respectively. To do so, PRx employs a
maximum likelihood (ML) detector and minimizes the following metric:

$\left(\boldsymbol{\hat{\tau}_{\mathbf{t}_{\alpha}}},\mathbf{\hat{x}}_{n},\hat{\psi}_{3}\right)=\arg\min_{\begin{subarray}{c}{\mathbf{x}_{n}\in\mathcal{S}}\\\
{{\mathbf{t}_{\alpha}}\in\mathcal{T}}\\\
{\psi_{3}\in\mathcal{J^{\prime}}}\end{subarray}}\sum_{k=1}^{L}\lVert\mathbf{y}_{k}-\mathbf{h}_{d}s_{k}+\underline{\mathbf{f}}\boldsymbol{\Psi}_{k}s_{k}\rVert^{2}.$
(14)

Require : $\displaystyle\mathbf{y}_{k}$, $\displaystyle\mathbf{h}_{d}$,
$\displaystyle\underline{\mathbf{f}}$,
$\displaystyle\boldsymbol{\Lambda}_{c}$, $\displaystyle\boldsymbol{\Omega}$,
$\displaystyle\mathcal{J}^{\prime}$, $\displaystyle\mathcal{M}$,
$\displaystyle\sigma^{2}$, $\displaystyle K$, $\displaystyle L$

Ensure : $\displaystyle\vartheta_{k}$ is the LLR of $\displaystyle k$-th time
instant

1 for  _$\displaystyle k=1$ to $\displaystyle K$_ do

2 $\displaystyle\Delta_{i}$ =
$\displaystyle-\frac{1}{\sigma^{2}}\lVert\mathbf{y}_{k}-\mathbf{h}_{d}x_{1}+\underline{\mathbf{f}}\boldsymbol{\Lambda}_{1}x_{1}\rVert^{2}$

3 $\displaystyle\Delta_{p}$ =
$\displaystyle-\lVert\mathbf{y}_{k}-\mathbf{h}_{d}\omega+\underline{\mathbf{f}}\boldsymbol{\Omega}\omega\rVert^{2}$

4 for  _$\displaystyle c=1$ to $\displaystyle J$_ do

5 for  _$\displaystyle i=2$ to $\displaystyle M$_ do

6
$\displaystyle\xi_{1}=-\frac{1}{\sigma^{2}}\lVert\mathbf{y}_{k}-\mathbf{h}_{d}x_{i}+\underline{\mathbf{f}}\boldsymbol{\Lambda}_{c}x_{i}\rVert^{2}$

7
$\displaystyle\xi_{2}=\max\left\\{\Delta_{i},\xi_{1}\right\\}+\ln(1+\exp(-\lvert\xi_{1}-\Delta_{i}\rvert))$

8 $\displaystyle\Delta_{i}=\xi_{2}$

9

10 end for

11

12 end for

13
$\displaystyle\vartheta_{k}=\ln(L^{2})-\ln((K-L)^{2})+\Delta_{i}-\Delta_{p}$

14

15 end for

return $\displaystyle\boldsymbol{\vartheta}$

Algorithm 1 LLR-Based Detector for TIM

Require : $\displaystyle\boldsymbol{\vartheta}$, $\displaystyle\mathcal{T}$,
$\displaystyle\eta_{r}$, $\displaystyle K$, $\displaystyle L$

Ensure : $\displaystyle\mathbf{t}_{\alpha}$ belongs to the legitimate set
$\displaystyle\mathcal{T}$ and $\displaystyle{\mathbf{t}_{\alpha}^{\iota}}$ is
the $\displaystyle\iota$-th element of $\displaystyle\mathbf{t}_{\alpha}$.

1

2$\displaystyle\vartheta=\boldsymbol{\vartheta}_{k}$

3 $\displaystyle\mathbf{b}=\mathbf{0}_{2^{\eta_{r}}\times 1}$

4 for  _$\displaystyle{\alpha}=1$ to $\displaystyle 2^{\eta_{r}}$_ do

5 if  _$\displaystyle L >1$_ then

6 for  _$\displaystyle{\iota}=1$ to $\displaystyle L$_ do

7 $\displaystyle a$ = $\displaystyle{\mathbf{t}_{\alpha}^{\iota}}$

8 $\displaystyle\mathbf{b}_{\alpha}\mathrel{+}=\boldsymbol{\vartheta}_{a}$

9

10 end for

11

12 end if

13 else if  _L = 1_ then

14 $\displaystyle a$ = $\displaystyle{\mathbf{t}_{\alpha}}$

15 $\displaystyle\mathbf{b}_{\alpha}=\boldsymbol{\vartheta}_{a}$

16

17 end if

18

19 end for

20$\displaystyle\mathbf{t}_{\alpha}=\arg\max_{\alpha}\left\\{\mathbf{b}_{\alpha}\right\\}$

return $\displaystyle\boldsymbol{\hat{\tau}_{\mathbf{t}_{\alpha}}}$

Algorithm 2 Mapping Rule-aided Info. Slot Selection

Since the ML detector searches for all possible mapping rules and the
corresponding symbols, the decoding complexity increases as $\displaystyle K$,
$\displaystyle L$, and $\displaystyle M$ grow. The computational complexity of
the ML detector is approximately
$\displaystyle\mathcal{O}(2^{\eta_{r}}JM^{L})$ in terms of complex
multiplications. Therefore, we propose an LLR-based detector for our scheme to
reduce the decoding complexity while maintaining the reliability of decoding
at PRx.

Let us define two vectors for the information and power stages from
$\displaystyle\boldsymbol{\Psi}_{k}$ to ease the exposition. To describe the
power stage, we can represent $\displaystyle\boldsymbol{\Psi}_{k}$ as
$\displaystyle\boldsymbol{\Omega}=\begin{bmatrix}e^{-j\phi_{p}}&0&e^{-j\phi_{p}}\end{bmatrix}$,
and for the information stage,
$\displaystyle\Lambda_{c}=\begin{bmatrix}\psi_{1}&0&\theta_{c}\end{bmatrix}$.
Now, we can write the expression representing the logarithm ratio of the a
posteriori probabilities of the symbols, taking into account that their values
may indicate the selection of either information or power signal. The
expression is as follows:

$\vartheta_{k}=\ln\left(\frac{\sum_{j=1}^{{J}}\sum_{i=1}^{M}\Pr\left(s_{k}=x_{i},\psi_{3}=\theta_{c}\mid\mathbf{y}_{k}\right)}{\Pr\left(s_{k}=\omega,\psi_{3}=\phi_{p}\mid\mathbf{y}_{k}\right)}\right).$
(15)

A higher value of $\displaystyle\vartheta_{k}$ in (15) indicates a higher
likelihood of information being transmitted at time instant $\displaystyle k$.
Applying Bayes’ formula, with
$\displaystyle\Sigma_{i=1}^{M}\Pr\left(s_{k}=x_{i}\right)=\Sigma_{c=1}^{J}\Pr\left(s_{k}=\theta_{c}\right)=L/K$
and
$\displaystyle\Pr(s_{k}=\omega)=\Pr\left(\psi_{3}=\phi_{p}\right)=(K-L)/K$,
(15) can be expanded as:

$\displaystyle\displaystyle\vartheta_{k}=$
$\displaystyle\displaystyle\ln\left(\sum_{j=1}^{{J}}\sum_{i=1}^{M}\exp\left(-\frac{1}{\sigma^{2}}\lVert\mathbf{y}_{k}-\mathbf{h}_{d}x_{i}+\underline{\mathbf{f}}\boldsymbol{\Lambda}_{c}x_{i}\rVert^{2}\right)\right)$
$\displaystyle\displaystyle-\ln\left(\exp\left(-\lVert\mathbf{y}_{k}-\mathbf{h}_{d}\omega+\underline{\mathbf{f}}\boldsymbol{\Omega}\omega\rVert^{2}\right)\right)$
(16)
$\displaystyle\displaystyle+\ln\left({L^{2}}\right)-\ln\left({(K-L)^{2}}\right).$

TABLE I: MAPPING RULE of
$\displaystyle\boldsymbol{{\tau}_{\mathbf{t}_{\alpha}}}$ for $\displaystyle
K=4$ and $\displaystyle L=2$.

Bits | $\displaystyle\boldsymbol{{\tau}_{\mathbf{t}_{\alpha}}}$ | $\displaystyle\mathcal{T}$
---|---|---
00 | $\displaystyle\begin{bmatrix}1&0&1&0\end{bmatrix}$ | $\displaystyle\mathbf{t}_{1}=\begin{bmatrix}1&3\end{bmatrix}$
01 | $\displaystyle\begin{bmatrix}1&0&0&1\end{bmatrix}$ | $\displaystyle\mathbf{t}_{2}=\begin{bmatrix}1&4\end{bmatrix}$
10 | $\displaystyle\begin{bmatrix}0&1&0&1\end{bmatrix}$ | $\displaystyle\mathbf{t}_{3}=\begin{bmatrix}2&4\end{bmatrix}$
00 | $\displaystyle\begin{bmatrix}0&1&1&0\end{bmatrix}$ | $\displaystyle\mathbf{t}_{4}=\begin{bmatrix}2&3\end{bmatrix}$
X | $\displaystyle\begin{bmatrix}1&1&0&0\end{bmatrix}$ | -
X | $\displaystyle\begin{bmatrix}0&0&1&1\end{bmatrix}$ | -

(a)

(b)

Figure 3: Average harvested DC power (5) with respect to $\displaystyle N_{2}$
UCs for (a) standalone RIS and (b) EH.

Then, recursion initializes with $\displaystyle\Delta_{i}$ =
$\displaystyle-\frac{1}{\sigma^{2}}\lVert\mathbf{y}_{k}-\mathbf{h}_{d}x_{1}+\underline{\mathbf{f}}\boldsymbol{\Lambda}_{1}x_{1}\rVert^{2}$.
To avoid numerical overflow, we utilize the following identity:
$\displaystyle\ln\left(e^{\xi_{1}}+e^{\xi_{2}}+\cdots+e^{\xi_{\chi}}\right)=\ln\left(e^{\Delta_{i}}+e^{\xi_{\chi}}\right)=\max\left\\{\Delta_{i},\xi_{\chi}\right\\}+\ln\left(1+e^{-\left|\xi_{\chi}-\Delta_{i}\right|}\right)$
[14] at each iteration in which
$\displaystyle\xi_{\chi}=\lVert\mathbf{y}_{k}-\mathbf{h}_{d}x_{i}+\underline{\mathbf{f}}\boldsymbol{\Lambda}_{j}x_{i}\rVert^{2}$
by setting $\displaystyle c=1$ to $\displaystyle J$ and $\displaystyle i=2$ to
$\displaystyle M$. Since there is one possible power signal once
$\displaystyle\Delta_{p}$ =
$\displaystyle-\lVert\mathbf{y}_{k}-\mathbf{h}_{d}\omega+\underline{\mathbf{f}}\boldsymbol{\Omega}\omega\rVert^{2}$
is evaluated and keep the same for each iteration. Afterward, LLR value is
obtained by Algorithm 1 for each time instant $\displaystyle k$ and they are
collected in a vector which is given by

$\boldsymbol{\vartheta}=\begin{bmatrix}\vartheta_{1}&\vartheta_{2}&\ldots&\vartheta_{k}\end{bmatrix}.$
(17)

(a)

(b)

(c)

Figure 4: BER of the proposed scheme for various TIM setups: (a) PTx, (b) RIS,
and (c) PTx and RIS for varying $\displaystyle P_{H}$ values.

Finally, the receiver decides on the time indices of $\displaystyle L$
information slots $\displaystyle\mathbf{t}_{\alpha}$ based on maximum LLR
values with the aid of $\displaystyle\boldsymbol{\vartheta}$ and estimates
$\displaystyle\boldsymbol{\hat{\tau}_{\mathbf{t}_{\alpha}}}$. It can be seen
that the proposed LLR detector simplifies decoding complexity to around
$\displaystyle\mathcal{O}(K(JM+1))$ in terms of complex multiplications.

The mapping rule for $\displaystyle(K=4,L=2)$ is demonstrated in Table I. When
selecting $\displaystyle(4,2)$, there are $\displaystyle 6$ possible mappings.
Nevertheless, only $\displaystyle 2^{\left\lfloor\log_{2}(6)\right\rfloor}=4$
legitimate mappings can be chosen, as there are four possible bit
combinations. Therefore, it is likely to decide on an illegitimate set with
Algorithm 1.

Assuming that $\displaystyle\hat{L}$ is the number of estimated information
slots in $\displaystyle\boldsymbol{\hat{\tau}_{\mathbf{t}_{\alpha}}}$, in
cases where $\displaystyle\hat{L}>L$ or $\displaystyle\hat{L}<L$, or even when
$\displaystyle\hat{L}=L$, the receiver can detect
$\displaystyle\boldsymbol{{\tau}_{\mathbf{t}\alpha}}$, which is not specified
in the mapping rule, such that the detection process will fail. Thus, we
present an LLR detector that operates in cooperation with the mapping rule to
prevent the detection of illegitimate sets. Hence, regarding the potential
scenario where the PRx might choose
$\displaystyle\boldsymbol{{\tau}_{\mathbf{t}\alpha}}$ that does not belong to
the mapping rule, Algorithm 2 is provided to compute the maximum LLR sum for
determining time-indices $\displaystyle\mathbf{t}_{\alpha}$ from legitimate
set $\displaystyle\mathcal{T}$ [14]. Ultimately, to estimate
$\displaystyle(\mathbf{\hat{x}}_{n},\hat{\psi}_{3})$ for selected time
instants (information slots), the receiver applies an ML detector, which can
be expressed as:

$\left(\mathbf{\hat{x}}_{n},\hat{\psi}_{3}\right)=\arg\min_{\begin{subarray}{c}{\mathbf{x}_{n}\in\mathcal{S}}\\\
{\psi_{3}\in\mathcal{J^{\prime}}}\end{subarray}}\sum_{\alpha=1}^{L}\lVert\mathbf{y}_{\alpha}-\mathbf{h}_{d}s_{\alpha}+\underline{\mathbf{f}}\boldsymbol{\Psi}_{\alpha}s_{\alpha}\rVert^{2}.$
(18)

where $\displaystyle\mathbf{y}_{\alpha},s_{\alpha},\boldsymbol{\Psi}_{\alpha}$
corresponds to the estimated time indices in
$\displaystyle\boldsymbol{\hat{\tau}_{\mathbf{t}_{\alpha}}}$ computed by
Algorithm 2.

##  IV Numerical Results

In this section, we evaluate the performance of our proposed scheme in terms
of average harvested DC power and bit error rate (BER). For all channels, a
Rician factor $\displaystyle\kappa_{p}=5$ is considered, and the 3GPP Indoor
hotspot (InH) path loss model is employed, which is expressed as follows [23]:

$\nu_{p}(d)=32.8+16.9\log_{10}(d)+20\log_{10}(f),$ (19)

where $\displaystyle f$ is the carrier frequency, and $\displaystyle d$ refers
to the distance, with $\displaystyle d\in\\{5,10,14\\}$ for the PTx-RIS, RIS-
PRx (EH), and PTx-RIS-PRx (EH) links, respectively. Unless otherwise
specified, $\displaystyle 4$-QAM is employed at PTx, and $\displaystyle
M_{R}=4$, and $\displaystyle N_{1}=60$ are used in the simulations. The
simulation parameters are presented in Table II.

TABLE II: SIMULATION PARAMETERS

| Parameter | Value
---|---|---
|  |  |
PTx | $\displaystyle P_{L},P_{H}$ |  $\displaystyle 30$ dBm, $\displaystyle 34$ dBm
$\displaystyle f$ |  $\displaystyle 2$ GHz
|  |  |
RIS |  $\displaystyle N$, $\displaystyle N_{\text{cb}}$ | $\displaystyle 256,4$
$\displaystyle P_{\text{cb}}$, $\displaystyle P_{\text{dyn}}$ |  $\displaystyle 50$ $\displaystyle\mu$W , $\displaystyle 40$ $\displaystyle\mu$W [6]
$\displaystyle P_{\text{switch}}$, $\displaystyle P_{\text{varactor}}$ |  $\displaystyle 1\mu$W [9], $\displaystyle 0$ W [8]
$\displaystyle\rho$, $\displaystyle\mathcal{P}_{\text{on}}$, $\displaystyle\mathcal{P}_{\text{sat}}$ |  $\displaystyle 0.75$, $\displaystyle 150$ $\displaystyle\mu$W, $\displaystyle 70$ mW
|  |  |
EH |  $\displaystyle\rho$, $\displaystyle\mathcal{P}_{\text{on}}$, $\displaystyle\mathcal{P}_{\text{sat}}$ |  $\displaystyle 0.75$, $\displaystyle 50$ $\displaystyle\mu$W, $\displaystyle 0.1$ mW

We aim to demonstrate the proposed standalone RIS architecture in terms of
average harvested DC power and the number of $\displaystyle N_{2}$ UCs
required to meet the condition (9). Fig. LABEL:fig:RIS_Harvested indicates
that the minimum number of $\displaystyle N_{2}$ UCs required for a standalone
RIS architecture is approximately two times fewer for RF switches compared to
varactors. This difference arises as varactors are driven by power-hungry DACs
that make accomplishing a standalone operation challenging, resulting in
roughly 8 dB more power consumption than RF switches. As expected, reserving
more time slots for the power stage in the TIM scheme increases the average
harvested DC power, as illustrated in Figs. LABEL:fig:RIS_Harvested and
LABEL:fig:EH_Harvested. Importantly, around $\displaystyle N_{2}=140$, the CLC
rectenna reaches saturation region for all TIM schemes. Herein, it is crucial
to mention the parameters of the CLC rectenna model can be reconfigured
according to $\displaystyle N_{2}$ UCs to leverage the benefits effectively.
Furthermore, allocating more $\displaystyle N_{2}$ UCs at the RIS reduces the
average harvested DC power at the EH, as the total number of reflectors, which
is $\displaystyle N-N_{2}$ UCs, decreases. Since $\displaystyle N_{2}=35$
satisfies the standalone condition (9) for all schemes in Fig.
LABEL:fig:RIS_Harvested, we exploit this in our simulations to ensure that RIS
performs its functions by WEH in the SR environment.

In Figs. LABEL:fig:BER_PTx and LABEL:fig:BER_RIS, we demonstrate the BER
performance of our scheme for ML and LLR detectors with respect to direct link
(DL) signal-to-noise ratio (SNR),
$\displaystyle\gamma_{\text{DL}}=\nu_{h_{d}}(d)/\sigma^{2}$. It can be
observed from Figs. LABEL:fig:BER_PTx and LABEL:fig:BER_RIS that BER
performances of the ML and LLR detectors align with one another. However, the
LLR detector reduces the complexity by approximately $\displaystyle 86\%$
compared to the ML detector in terms of complex multiplications, considering
the TIM $\displaystyle(8,2)$ scheme. Additionally, based on the setup in [4],
we compare our scheme with benchmarks. We assume that the benchmark schemes
maintain RIS information unchanged for $\displaystyle L$ information slots for
a block size of $\displaystyle K=8$, without utilizing TIM at the PTx. Thus,
the benchmark schemes convey $\displaystyle\eta_{m}=L\log_{2}(M)$ bits and
$\displaystyle(K-L)$ power signals in a given block. In Fig.
LABEL:fig:BER_PTx, the benchmark scheme B.2 outperforms our proposed scheme.
Nevertheless, when considering the $\displaystyle(8,2)$ case, since our scheme
exploits six time slots for power transfer while the benchmark B.2 uses four
time slots, the average harvested DC power in a block size of $\displaystyle
K$ should be higher than that of B.2. If the benchmark utilizes six time slots
(B.1) for power transfer, all TIM schemes show their superiority against it.
Thus, modulating bits in the time domain rather than the signal domain
enhances the BER performance of the SR system. In Fig. LABEL:fig:BER_RIS, the
proposed scheme outperforms benchmark B.1 in terms of BER of RIS information.
Despite B.3 having a spreading gain four times that of the TIM
$\displaystyle(8,1)$ case, their performances are close due to the fact that
PTx and RIS information are coupled. Therefore, increasing the modulation
order $\displaystyle M$ of PTx has a detrimental impact on RIS information.
Additionally, as the DL SNR increases, the BER of B.2 and TIM
$\displaystyle(8,4)$ cases become indistinguishable. Moreover, the BER
performance of the RIS improves with increasing spreading gain
$\displaystyle\delta$, which also enhances the BER of PTx in the high SNR
region, as observed in Fig. LABEL:fig:BER_PTx.

To emphasize the effect of $\displaystyle P_{H}$ on decoding performance in
the LRR detector, the BER for PTx and RIS information is plotted in Fig.
LABEL:fig:PH_BER with $\displaystyle\gamma_{\text{DL}}=10$ dB. As
$\displaystyle P_{H}$ increases, the distinction between information and power
stages becomes more accurate, leading to a better estimation of time-index
bits in the LLR detector. Interestingly, the impact of $\displaystyle P_{H}$
on the BER of RIS information is highly coupled with the spreading factor
$\displaystyle\delta$. Therefore, as $\displaystyle\delta$ improves,
increasing $\displaystyle P_{H}$ becomes more meaningful for achieving better
BER performance. Furthermore, the BER of PTx and RIS information saturates
after a specific SNR value in all TIM schemes, indicating that further
increasing $\displaystyle P_{H}$ will not affect decoding performance.

##  V Concluding Remarks

The coexistence of cellular and IoT devices in next-generation communication
frameworks becomes more power and bandwidth-hungry as the number of connected
devices increases. Therefore, delivering power and information in a
mutualistic manner is of paramount importance for designing next-generation
networks. This paper presents a novel low-power, standalone RIS architecture
that serves as a BD in an SR environment. In this setup, the PTx utilizes a
TIM scheme to transfer information to the PRx and power to the RIS and EH,
respectively. We propose an LLR-based detector to reduce computational
complexity at the PRx and investigate the proposed scheme in terms of average
harvested DC power and BER performance while ensuring the necessary conditions
for being a standalone RIS. Our results demonstrate that a standalone RIS can
establish a mutualistic symbiosis with the PTx, and the proposed system can
achieve reliable communication while incorporating power transfer into its
transmission strategy without demanding additional power resources.

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