novelty over optimization updates, where a generated molecule is
considered“novel” if there does not exist any existing molecule in ChEMBL with
a Tanimoto similarity score greater than $0.8$. In panels (b,c,d), we estimate
substructure similarity using the Tversky metric between TextGrad molecules
and clinically approved drugs for each target. While there are a wide range of
similarity scores, we observe that similarity with known drugs has little
correlation with molecule performance, suggesting that TextGrad’s generation
process is weakly influenced by “memorized” knowledge of clinically approved
structures.

One of the primary concerns with using LLM models in scientific discovery is
their capacity to simply memorize and regurgitate their training sets instead
of performing logical reasoning [105]. Since recent models like gpt-4o are
trained on massive, opaque datasets, a key concern is that the molecules
TextGrad generates may simply be duplicates of clinically approved molecules
for their respective targets. To quantify and test the extent or existence of
this “memorization” effect, we compare the molecules that TextGrad generates
with approved drugs for their respective protein targets, as well as unrelated
but existing druglike chemical compounds.

We first compare TextGrad generated molecules for a particular protein target
with all the clinically approved small molecules found in Drugbank for the
same target (see E.3 for filtering criteria) using the Tversky similarity
score on RDKit daylight chemical fingerprints. The Tversky similarity score is
not symmetric, and compares substructures between a variant and reference
molecule [106]. In our application, this is preferred as it assigns a high
score to generated (variant) molecules that contain most or all of the
substructures in the Drugbank (reference) molecule, even if there exist other
extraneous substructures in the generated molecule that reduce the symmetric
overlap between the two molecules. A Tversky score of $1.0$ indicates that the
TextGrad molecule is a complete subset of the Drugbank molecule, while $0.0$
indicates no overlap.

In order to analyze the most relevant chemical structures, we restricted our
analysis to a set of high performing TextGrad molecules, rather than all
molecules from all iterations. In particular, using the overall score
described in Equation 19, we selected the best molecule for each protein
target and for each initial fragment, for a total of $87$ generated molecules
($29$ druggable targets x $3$ initial fragments) and $118$ clinically approved
drugs across $29$ targets. For each target, we compute all pairwise Tversky
scores between the $3$ TextGrad molecules and the Drugbank molecules approved
for that target, setting the Drugbank molecule as the reference, and the
TextGrad molecule as the variant.

We observed that the distribution of Tversky similarity scores was quite
broad, with a median of $0.42$ but ranging from $0.14$ to $0.90$ (Figure 2
(b)). However, we observe that Tversky similarity between TextGrad and
DrugBnak molecules is actually slightly anti-correlated with molecule
performance as measured by the overall score, suggesting that TextGrad ’s
optimization procedure is at best weakly influenced by prior knowledge of
approved drugs. In fact, for a variety of generated molecules along a gradient
of similarity scores, the generated molecules exhibit QED and Vina scores that
match or even exceed their DrugBank counterparts (Figure 2 (d)).

Beyond the similarity to known drugs, we are also interested in observing if
TextGrad is generating novel molecules, or discovering previously unknown
properties in existing compounds, for example a strong binding affinity to a
protein target in a compound that has not been associated with the protein. To
answer this question, we perform a Tanimoto similarity search across all of
ChEMBL, a manually curated database of $2.4$ million bioactive molecules with
drug-like properties [107]. Unlike the Tversky score, the Tanimoto metric
measures the symetric overlap between two chemical structures, where $1.0$
indicates an exact match, and $0.0$ no overlap. We classify a TextGrad
compound as “novel” if and only if there does not exist any molecule in ChEMBL
with a with a Tanimoto similarity score over 0.80.

While the starting fragments are known molecules, we observe that as the
number of iterations increases, TextGrad generates molecules that are
progressively less likely to be previously known compounds. By the $6$th
iteration, 95% of all the molecules generated by TextGrad across all $58$
targets and $3$ starting fragments are novel using the criteria above (Figure
2 (a)). By observing the trajectories of generated molecules and analyzing the
textual gradients, we hypothesize that the feedback provided by the Vina and
QED scores encourages TextGrad to explore chemical space beyond known
molecules by progressively adding and removing combinations of functional
groups. Since these chemical updates can form a combinatorial number of unique
structures, it is reasonable that TextGrad would reach previously unexplored
regions of chemical space in a relatively small number of iterations.

###  E.6 Implicit Objectives

Supplementary Figure 3: Safety Properties We evaluate the predicted
harmfullness of TextGrad molecules using the ADMET-AI model and compare them
to clinically approved drugs. For both mutagenicity and clinical toxicity,
$1.0$ indicates a highly likelihood for harm and $0.0$ a low likelihood. We
observe that despite the fact that neither of these characteristics are
directly encoded into TextGrad’s objective function, TextGrad implicitly
avoids proposing harmful molecules.

Another key concern in applying LLMs in science is their propensity for
hallucinations, where models generate factually incorrect or illogical
responses while attempting to satisfy user requests [105]. In our setting,
this hallucinations could manifest by TextGrad proposing invalid, toxic, or
otherwise undesirable molecules in order to optimize its objective function.
We control for severe hallucinations by preprocessing molecules using RDKit
sanitization, ensuring at the bare minimum that TextGrad generates chemically
valid molecules. However, this simple preprocessing step does not completely
specify desirable chemical behavior. A direct strategy would be to
exhaustively encode all possible metrics for desirability in drug molecules
beyond druglikeness and binding affinity into the objective function, and
extend it to include sythesizability, toxicity, among other criteria.
Unfortunately, this approach is not realistically feasible as not all criteria
for desirability have mature computational metrics. Thus, a key question is
whether TextGrad obeys so-called implicit objectives during its optimization
process that curtails illogical or undesirable behavior.

To evaluate the extent or existence of undesirable molecules, we can
characterize the harmfulness of the generated molecules, focusing on
mutagenisis and clinical toxicity. Mutagenicity refers to the ability of a
drug to induce genetic alterations, which may lead to DNA damage and harmful
long-term affects. Clinical toxicity refers to a broad range of adverse short
or long-term side effects. Importantly, neither druglikeness nor binding
affinity are strongly correlated with these criteria, and thus these
desirability metrics are not directly optimized or otherwise explicitly
encoded in TextGrad’s objectives.

To evaluate the propensity for mutagenisis and clinical toxicity, we employ
the ADMET-AI model, that predicts these scores from the chemical structures of
molecules [108]. ADMET-AI employs a deep learning model trained on multiple
relevant datasets. In particular, for mutagenicity, ADMET-AI is trained on
$7,255$ drugs from the Ames dataset, a bacterial reverse mutation assay for
rapidly screening large numbers of compounds for can induced genetic damage
and frameshift mutations. A predicted label of $1.0$ indicates a high
likelihood that the drug will induce mutagenesis, while a label of $0.0$
indicates a low likelihood. For clinical toxicity, ADMET-AI is trained on the
ClinTox dataset, a dataset of $1,484$ drugs consisting of molecules that have
failed clinical trials for toxicity reasons and also drugs that are associated
with successful trials. Similarly, a predicted label of $1.0$ indicates a high
likelihood of clinical toxicity, while a label of $0.0$ indicates a low
likelihood.

Once again, we restrict our analysis to the best performing generated
molecules with druggable targets as measured by the overall score, and select
the best molecule for each protein target and for each initial fragment, for a
total of $87$ generated molecules. We then compare their predicted mutagenisis
and clinical toxicity to the the $118$ clinically approved molecules from
DrugBank. We observe that for both Mutagenicity and Clinical Toxicity, the
molecules generated by TextGrad have predicted distributions that indicate a
low likelihood of harmful effects, and closely match the distributions of the
clinically approved molecules. Together, these results suggest that TextGrad
implicitly avoids proposing harmful molecules, even though these criteria are
not directly encoded in its loss function.

##  Appendix F Treatment Plan Optimization

###  F.1 Prompts

Radiotherapy treatment plan evaluation can based on various dimensions,
therefore there is no single score that can indicate the quality of plans. We
adopt LLM to compute the “loss” by prompting it to assess the plan quality
with clinical protocols. Specifically, LLM is used to compare each protocol
with the current plan and produce the final assessment.

Treatment Plan Loss Prompt Please act as an impartial and objective
professional radiation oncologist. Your job is to evaluate the quality of
radiation therapy treatment plans based on their dose volume histogram. You
should consider the following protocols: <Clinical Protocols> Note that the
order above does not indicate priority; always prioritize the regions that
have protocols that are more significantly violated. Here is the dose-volume
histograms of the candidate plans for evaluation; each entry in the dose-
volume histograms (DVH) table indicates the percentage of volume receiving a
dose higher than a certain Gy (specified in the first column). <DVH table> We
also provide the statistics of the above DVH table for ease of evaluation.
<DVH statistics> Now, based on the protocols, and the DVH, please evaluate the
plans. Avoid any positional biases and ensure that the order in which the
responses are presented does not influence your decision. Be as objective as
possible. Your answer must: 1. (Evidence) Extract the corresponding entries
from the DVH table based on the factors to consider. 2. (Interpretation) Based
on the extracted entries from DVH table, interpret them as: {percentage}% of
volume encompassing the dose more than {dose} Gy for {organ}. 3. (Analysis)
Combine the interpretation and the factors, analyze whether each of the
numerical factors are met. Based on the analysis on DVH tables, you need to
produce a final evaluation. When all factors are satisfied, you should always
answer no improvement is needed. If improvements are required, please provide
suggestions on where to improve. Please ensure to follow the format below to
return the evaluation results: <FINAL> Decision: [The plan does/doesn’t need
to be improved.] Reasons: [list all factors that are not satisfied with
detailed reasons, e.g. protocol X on (PTV/rectum/bladder/fh/body) is not
satisfied because …] </FINAL> The final answer in the end must strictly follow
the format above.

###  F.2 Inner-loop optimization for treatment planning

We employ a two-loop optimization approach [109], which includes (i) an inner
loop for inverse planning and (ii) an outer loop for optimizing the
hyperparameters of the inner loop. The inner loop focuses on traditional
fluence map optimization, seeking to determine the optimal fluence map $x$ by
minimizing a cost function that combines multiple weighted objectives for
various targets and organs at risk. This cost function is defined as:

$\displaystyle\min_{x}$
$\displaystyle\sum_{t=1}^{N_{t}}w_{t}([Kx]_{t}-d_{t})^{2}+\sum_{s=1}^{N_{s}}w_{s}\Theta([Kx]_{s}-d_{s})([Kx]_{s}-d_{s})$
(20) s.t. $\displaystyle D_{95}([Kx]_{t})=d_{t},$ $\displaystyle x\geq 0,$

Here, $\\{w_{t}\\}_{t=1}^{N_{t}}$ and $\\{w_{s}\\}_{s=1}^{N_{s}}$ are the
importance weights (the hyperparameters optimized by TextGrad) that balance
the various objectives for $N_{t}$ PTV targets and $N_{s}$ OARs, respectively.
$K$ denotes the dose influence matrix, which specifies the dose per fluence
unit delivered to each voxel in the volume by each beamlet.
$\\{d_{t}\\}_{t=1}^{N_{t}}$ and $\\{d_{s}\\}_{s=1}^{N_{s}}$ are the scalar
objective doses for each structure. The cost function essentially penalizes
squared deviations from the target objective doses for the PTV targets and
penalizes squared overdosing for the OARs only when doses exceed $d_{s}$. The
Heaviside function $\Theta$ is used to ensure the objective considers only
positive values. The minimization is constrained to positive $x$ values and
ensures that $D_{95}$ — the minimum dose received by 95% of the structure
volume — matches the prescribed dose for the clinical goal. We use matRad [65]
with interior point algorithms to solve the inner-loop optimization.

###  F.3 Additional Experimental Details

#### Dataset

The dataset used in this study comprised imaging and treatment plans for 5
prostate cancer patients who underwent intensity-modulated radiation therapy
(IMRT). Available data for each patient includes CT scans, delineated
anatomical structures, and clinically approved treatment plans obtained via
Eclipse®.

#### Method

As we mentioned in 3.5, TextGrad is used to optimize the hyperparameters
(e.g., importance weights for PTV and OARs) of the inner-loop numerical
optimizer that generates the treatment plan. This optimization is done using a
variation of vanilla TextGrad, i.e. “projected gradient descent with momentum
updates”.In particular, three prostate cancer treatment plans optimized by
clinicians, along with their corresponding hyperparameters, are provided.
These examples guide the updates of the hyperparameters. This procedure can be
viewed as an analogy to projection, as the updated hyperparameters are “softly
projected” onto a feasible set defined by the three in-context examples.
Moreover, the historical hyperparameters and the textual gradients from past
iterations, as an analogy to momentum, are also included in the prompts for
updating the hyperparameters. This additional context helps refine the
optimization process. The optimization will be stopped if the loss suggests
all protocols meet, other wise, it will be stopped if the maximum number of
iterations (we set it to 10) is reached.

#### Initialization

The hyperparameters i.e. the importance weights are all initialized at 100 for
different organs. The dose objectives are set to 70.20 for PTV, 0.00 for
bladder and rectum, and 30.00 for femoral heads and body, and fixed during
optimization.

###  F.4 Additional Results

In Supplementary Table 1 and 2, we show additional results on comparing
TextGrad optimized plan with clinicians optimized plans.

Supplementary Table 1: PTV dose metrics. Several dose metrics of the PTV target are displayed for all the clinical and TextGrad optimized plans, including the mean and minimum doses, as well as the $D_{95}$. For all the metrics, we include the average deviations from the clinical goal across 5 plans and the standard deviation in brackets. Values in bold represent the best for each PTV target. Target | Method | Mean dose [Gy] | Min dose [Gy] | Max dose [Gy] |  $\textbf{D}_{95}$ [Gy]
---|---|---|---|---|---
PTV | Clinical Goal | 70.20 | $\approx 70.20$ | $\approx 70.20$ | 70.20
Radiation Oncologist | +1.97 (0.36) | -8.88 (2.31) | +4.66 (0.82 ) | -0.10 (0.15)
TextGrad |  +0.51 (0.09) |  -8.48 (2.38) |  +3.63 (0.87) |  +0.00 (0.00)
Supplementary Table 2: Organs at Risk (OARs) dose metrics. We show mean dose capturing OAR sparing. Lower values demonstrate better OAR sparing which is desirable, as this number indicates organs at risk, which should not get more than dosage than what is listed in the clinical guidelines. For all the metrics, we include the average mean dose across 5 plans and the standard deviation in brackets. Organ | Method | Mean dose [Gy] $\downarrow$ |  $\textbf{D}_{5}$$\downarrow$ |  $\textbf{D}_{50}$$\downarrow$
---|---|---|---|---
Rectum | Radiation Oncologist | 23.88 (6.45) | 64.26 (10.00) | 20.04 (5.50)
TextGrad |  17.18 (4.2) |  58.82 (18.81) |  9.54 (0.70)
Bladder | Radiation Oncologist | 22.39 (5.55) | 67.81 (6.44) | 14.78 (8.42)
TextGrad |  20.92 (0.79) |  65.96 (6.96) |  14.11 (3.17)


fpga short=FPGA, long=Field Programmable Gate Array, soc short=SoC,
long=System on Chip, phy short=PHY, long=physical layer, dl short=DL,
long=deep learning, dnn short=DNN, long=deep neural network, lsdnn
short=LSDNN, long=least square deep neural network, cnn short=CNN,
long=Convolutional Neural Network, csi short=CSI, long=channel state
information, ls short=LS, long=least squares, mmse short=MMSE, long=minimum
mean square error, mse short=MSE, long=mean square error, ofdm short=OFDM,
long=Orthogonal Frequency Division Multiplexing, srcnn short=SRCNN,
long=Super-Resolution CNN, dncnn short=DnCNN, long=Denoising CNN, snr
short=SNR, long=signal-to-noise ratio, iresnet short=iResNet,
long=Interpolation ResNet, resnet short=ResNet, long=Residual Neural Network,
fcnns short=FCNNs, long=Fully Connected Neural Networks, fcnn short=FCNN,
long=Fully Connected Neural Network, sta short=STA, long=Spectral Temporal
Averaging, ifft short=IFFT, long=Inverse Fast Fourier Transform, lts
short=LTS, long=Long Training Sequence, trfi short=TRFI, long=Time domain
Reliability test Frequency domain Interpolation, rnns short=RNNS,
long=Recurrent Neural Networks, rnn short=RNN, long=Recurrent Neural Network,
lstm short=LSTM, long=Long Short-Term Memory, gru short=GRU, long=Gated
Recurrent Unit, awgn short=AWGN, long=Additive White Gaussian Noise, ps
short=PS, long=processing system, pl short=PL, long=programmable logic, pe
short=PE, long=processing element, mac short=MAC, long=Multiply-Accumulate,
tdl short=TDL, long=tapped delay line, vtv short=VTV, long=vehicle-to-vehicle,
rtv short=RTV, long=roadside-to-vehicle, nmse short=NMSE, long=normalized mean
squared error, ber short=BER, long=bit error rate, relu short=ReLU,
long=Rectified Linear Unit, bpsk short=BPSK, long=Binary Phase Shift Keying,
sbs short=SBS, long=Symbol by Symbol,

# Low Complexity High Speed Deep Neural Network Augmented Wireless Channel
Estimation

Syed Asrar ul haq, Varun Singh, Bhanu Teja Tanaji and Sumit Darak  Electronics
and Communications Department, IIIT-Delhi, India-110020
e-mail: {syedh, varun22189, bhanu22155<EMAIL_ADDRESS>
###### Abstract

The channel estimation (CE) in wireless receivers is one of the most critical
and computationally complex signal processing operations. Recently, various
works have shown that the deep learning (DL) based CE outperforms conventional
minimum mean square error (MMSE) based CE, and it is hardware-friendly.
However, DL-based CE has higher complexity and latency than popularly used
least square (LS) based CE. In this work, we propose a novel low complexity
high-speed Deep Neural Network-Augmented Least Square (LC-LSDNN) algorithm for
IEEE 802.11p wireless physical layer and efficiently implement it on Zynq
system on chip (ZSoC). The novelty of the LC-LSDNN is to use different DNNs
for real and imaginary values of received complex symbols. This helps reduce
the size of DL by 59% and optimize the critical path, allowing it to operate
at 60% higher clock frequency. We also explore three different architectures
for MMSE-based CE. We show that LC-LSDNN significantly outperforms MMSE and
state-of-the-art DL-based CE for a wide range of signal-to-noise ratios (SNR)
and different wireless channels. Also, it is computationally efficient, with
around 50% lower resources than existing DL-based CE.

##  I Introduction

The wireless physical layer (PHY) enables efficient transmission of digital
information from transmitter to receiver in the presence of fading channels
and hardware impairments. With the evolution of dl, various works have
explored its usefulness for wireless PHY [1, 2, 3, 4, 5, 6, 7, 8, 9]. In [1],
wireless PHY is replaced with single DL model and has shown to offer better
performance and adaptability in unknown channel environment. However, a single
DL-based PHY is incompatible with existing standards due to the need for
intermediate control signal communications between transmitters and receivers
to allow multiple users to coexist in wireless networks. The alternative
standard-compatible approach is to replace or augment one or more signal
processing operations with DL model [6, 5, 7].

Channel estimation (CE) is a critical and computationally intensive task. It
estimates csi at the receiver using known transmitted data. Wireless receivers
utilize csi for reliable data reception, while transmitters employ it to
select appropriate channel and PHY parameters. Additionally, csi enhances PHY
security in wireless networks and improves the accuracy of indoor localization
techniques. The widely used least square (LS) based CE is simple and easy to
implement but performs poorly at low signal-to-noise ratio (SNR). Another
statistical method is minimum mean square error (MMSE) based CE, which offers
improved performance but needs computationally intensive matrix inverse
operation.

The CE approach depends on wireless PHY. In cellular PHY, CE uses pilots
embedded in the data frame, while in IEEE 802.11 PHY, CE is done using a
preamble. In this work, we focus on preamble-based CE for IEEE 802.11p PHY.
Without loss of generality, the proposed approach can be extended to cellular
PHY with additional interpolation operations [2, 3]. For preamble-based PHY,
dl is employed to enhance the estimation of each ofdm symbol either as pre-
processing [8], or as postprocessing [9]. While dl-based CE has demonstrated
improved functional accuracy, limited research has been conducted on mapping
of dl-based CE on soc [9]. This is critical since wireless PHY must be
deployed on edge platforms, and hence, constraints of such platforms must be
taken into account in algorithm design [10].

In this work, we aim to design low-complexity, high-speed DL-based CE and
efficiently implement it on SoC. The contributions of the paper are summarized
as follows:

  1. 1.

We propose a novel low complexity high-speed Deep Neural Network-Augmented
Least Square (LC-LSDNN) algorithm for the IEEE 802.11p wireless PHY. The
novelty of the LC-LSDNN is to use distinct DNNs for real and imaginary
components of received complex symbols, reducing DNN size and optimizing the
critical path for operation at higher clock frequencies.

  2. 2.

We assess CE performance in end-to-end PHY across various SNRs and wireless
channels. We show that the proposed LC-LSDNN outperforms existing approaches.
Furthermore, we demonstrate that the DL models can be trained using a high SNR
dataset instead of an ideal channel model, making it practically useful.

  3. 3.

We efficiently map the LC-LSDNN on Zynq ZC706 SoC. We explore three different
architectures for MMSE-based CE and optimize each architecture for the lowest
word length (WL) and maximum feasible clock frequency. We show that the LC-
LSDNN requires fewer resources, lower power, and lower latency than state-of-
the-art CE approaches.

The paper is organized as follows: Section II discusses the system model and
reviews existing CE approaches. In Section III, we discussed the proposed LC-
LSDNN algorithm, simulation, and complexity results, followed by architecture
details in Section IV and implementation results in Section V. Section VI
concludes the paper.

##  II System Model and Review of Wireless Channel Estimation Approaches

We consider an OFDM-based IEEE 802.11p standard wireless PHY as shown in Fig.
1. It consists of a data frame comprising a preamble followed by the data
field as shown in Fig. 2. The preamble consists of ten short training symbols
(STS) for signal detection and timing synchronization and two long training
symbols (LTS) for CE. The preamble is OFDM modulated, employing 64 sub-
carriers. Among these, 52 are allocated to LTS, while the remaining are null
sub-carriers. The information bits to be transmitted are subjected to data
modulation using quadrature amplitude modulation (QAM) followed by OFDM
modulation. Of 64 sub-carriers, 48 are used for data, 12 are null subcarriers,
and 4 are pilots for carrier offset estimation at the receiver.

At the receiver, OFDM demodulation is performed, followed by CE using LTS. The
estimated channel is used to equalize the received data subcarriers, followed
by data demodulation. We use bit error rate (BER) as an end-to-end performance
metric and normalized mean square error (NMSE) as a performance metric for
estimation performance.

Figure 1: Building blocks of IEEE 802.11p OFDM transceiver PHY. Figure 2:
Frame structure of IEEE 802.11p OFDM transceiver PHY.

When a transmitted signal traverses a wireless channel, it undergoes fading
due to multipath propagation and experiences a Doppler shift resulting from
mobility between transmitter and receiver. This work assumes a low mobility
scenario where the channel estimated using the preamble remains unchanged for
the entire data frame. However, the proposed work can be extended to high
mobility scenarios using data-pilot aided (DPA) approaches where channel
estimated using a preamble can be updated further using real and virtual
pilots [8]. Next, we review various CE approaches.

###  II-A Statistical Channel Estimation Approaches

If $X\in\mathbb{C}^{K}$ represents the transmitted OFDM symbol with $K$
subcarriers, $H\in\mathbb{C}^{K}$ denotes the channel gain vector for the
current ofdm symbol, and $Z\in\mathbb{C}^{K}$ represents the awgn, then the
received signal $Y\in\mathbb{C}^{K}$ can be expressed as follows:

$Y=H\odot X+Z$ (1)

where $\odot$ is the element-wise multiplication operator. The LS estimate
aims to minimize the squared error between the received signal and actual
channel response. In an ofdm PHY with circular convolution due to CP,
$\hat{H}_{LS}[i,k]$ for subcarrier $k$ of symbol $i$ is simplified as follows:

$\hat{H}_{LS}[i,k]=\frac{Y[i,k]}{X[i,k]}$ (2)

The channel estimates are calculated for both LTS, and averaged to get the
final estimate. LS estimation is simple and easy to implement, but it performs
poorly at low SNRs due to its lack of consideration for the influence of
noise. MMSE leverages prior knowledge of channel and noise statistics to
improve the LS. It is given as

$\hat{H}_{MMSE}=R_{HH}\left(R_{HH}+\frac{K\mathcal{N}_{0}}{E_{p}}I\right)^{-1}\hat{H}_{LS}$
(3)

As discussed in Section V, MMSE offers improved performance but
computationally complex.

###  II-B DL Augmented Channel Estimation

DL is currently being explored for CE due to its exceptional feature
extraction capabilities from raw data, enabling it to comprehend and generate
meaningful representations [11]. DL architectures for inference are hardware-
friendly due to simple arithmetic operations. As discussed in Section I, we
focus on low-complexity DL-augmented CE compared to existing approaches where
CE is replaced with large-size DL, resulting in a high area, power, and delay
penalty. In [9], a dnn-augmented CE is discussed, followed by its realization
on ZSoC. Though it outperforms LS and MMSE, it has high computation complexity
and latency compared to LS. We aim to address these challenges by novel
contributions at the algorithm and architecture levels.

##  III Proposed DL Augmented Channel Estimation

In this section, we present the proposed low complexity least square augmented
DNN-based CE (LC-LSDNN) followed by performance analysis using floating-point
arithmetic and theoretical complexity analysis with existing DL-based CE.

Figure 3: DL-based CE in (a) [9], and (b) Proposed LC-LSDNN.

###  III-A Proposed LC-LSDNN

In 802.11p PHY, LS-based CE produces a complex output,
$\hat{H}_{LS}\in\mathbb{C}^{k}$. Existing methods [9, 6] concatenate real and
imaginary components for DNN processing, as shown in Fig. 3(a). However, this
approach has high memory and compute complexity. Recent work [7] explores a
more efficient strategy using separate neural networks for real and imaginary
components. Though this approach seems promising, an in-depth analysis of its
functionality and complexity on floating and fixed-point architectures has not
been done yet.

The proposed approach employs two distinct DNNs to process the real and
imaginary components independently, as illustrated in Fig. 3(b). The LS
estimate, denoted as $\hat{H}_{LS}\in\mathbb{C}^{k_{on}}$, is initially
decomposed into its real $\Re\\{\hat{H}_{LS}\\}\in\mathbb{R}^{k_{on}}$ and
imaginary component $\Im\\{\hat{H}_{LS}\\}\in\mathbb{R}^{k_{on}}$, where
$k_{on}$ are the active subcarriers. These components are individually fed
into their respective trained DNNs, producing outputs that correspond to the
real and imaginary components of the channel estimates. The outputs of these
two DNNs are combined, resulting in complex-valued channel estimates
$\hat{H}_{LC-LSDNN}\in\mathbb{C}^{k_{on}}$.

The LC-LSDNN consists of a fcnn with a single hidden layer utilizing ReLU
activation function and an output layer with a linear activation function. The
input and output layer is of size $k_{on}$ while the hidden layer size is
$k_{on}/2$.

TABLE I: DNN parameters.

Parameters | Values
---|---
[9] Architecture | $2\times k_{on};k_{on};2\times k_{on}$
LC-LSDNN Architecture | $k_{on};k_{on}/2;k_{on}$
Hidden layer activation function | ReLU
Loss Function | MSE
Optimizer | ADAM
Epochs | 500
Number of training samples | 20000
Number of testing samples | 2000

The DNN is trained using a MATLAB-simulated dataset. Input comprises LS-
estimated channel values, while the corresponding channel frequency response
(CFR) can be used as labels. Since CFR is not available in real-world systems,
we used received signals with very high SNR as training labels [6]. This can
be easily obtained by placing the transmitter and receivers close to each
other. 20,000 training samples were generated, with 16,000 samples allocated
for the training and 4,000 samples designated for validation. Independent
testing sets comprising 2,000 samples for each testing SNR were also generated
to evaluate the system performance. The training dataset was generated for a
fixed SNR of 10dB. The rest of the DNN parameters are given in Table I.

###  III-B Performance of Floating Point Architectures

We have validated the functional correctness of the LC-LSDNN over a wide range
of SNR under realistic multipath fading environments with zero-doppler
conditions. We use well-known channel models [12] commonly associated with
IEEE 802.11p systems, replicating vtv and rtv environments. As shown in Fig.
4, the NMSE and BER performance improves with SNR for all approaches. The LC-
LSDNN offers nearly the same NMSE and BER as that of [9] for all SNRs and
three different channels. Both DL-based approaches outperform the LS and MMSE.
Furthermore, DL-based CE offers BER, which is nearly the same as perfect CE,
thereby validating their functional superiority.

Figure 4: NMSE and BER performance comparison of LC-LSDNN with LS, MMSE and
[9] approaches for (a) VTV- urban expressway oncoming, (b)VTV- urban canyon,
and (c) RTV- urban canyon channel models.

###  III-C Theoretical Complexity analysis

In Table II, we compare the LC-LSDNN and [9] in terms of a number of
parameters and mac operations. [9] utilizes a single dnn with an input and
output layer of size $2\times k_{on}$ and a hidden layer with a size half of
that of the input layer. In contrast, the LC-LSDNN employs two DNNs with
identical architecture as in [9], featuring an input and output layer size of
$k_{on}$ and a hidden layer size again half that of the input layer. As shown
in Table II, LC-LSDNN offers significant savings of 49% and 50% in the number
of parameters and MAC operations, respectively, over [9].

TABLE II: Complexity Comparisong of LC-LSDNN and [9].

| Hidden Layer | Output Layer |  Total Parameters |  Total MACs
---|---|---|---|---
# Weights | #Bias | # Weights | #Bias
[9] | $104\times 52$ | $52$ | $52\times 104$ | $104$ | 10972 | 10816
LC-LSDNN | $2(52\times 26)$ | $2(26)$ | $2(26\times 52)$ | $2(52)$ | 5564 | 5408

##  IV Hardware Architectures for CE on ZSoC

The algorithms to architecture mapping of MMSE and LC-LSDNN on the ZSoC are
discussed in this section. Other algorithms, such as LS and [9], are already
realized on ZSoC, and their source codes are publicly available.

###  IV-A LC-LSDNN Architecture

The LC-LSDNN algorithm comprises four operations: LS estimation,
normalization, DNN inference, and denormalization. Fig. 5 illustrates the
proposed FPGA architecture for LC-LSDNN realization. The received complex LTS
undergo LS estimation in the receiver, which, on hardware, involves six real
multiplications, three additions, and two real divisions. Depending on FPGA
resources, these operations can be parallelized for subcarriers to optimize
the latency. For IEEE 802.11p, BPSK modulated LTS enables simplifying LS to
multiplexer operations, as LS output remains unchanged or negated based on the
reference LTS being +1 or -1.

Figure 5: The architecture of LC-LSDNN.

The LS output is normalized by subtracting the mean and dividing it by the
standard deviation. The mean and standard deviation values are pre-calculated
during the DNN training and are pre-stored in the internal memory. The DNN
processes the normalized LS estimates. In LC-LSDNN, we use different DNNs for
real and imaginary components of the input samples. The fundamental
computational unit within the DNN is a neuron or pe. Each pe executes a mac
operation on inputs, employing their associated weights. These inputs are
sourced from the previous layer of the network. After the mac operation, the
bias is added, and the resulting output is processed by the relu activation
function. The relu decides between assigning zero or retaining the PE’s
output, depending on whether the output is negative or positive respectively.
The schematic depiction of a pe’s architectural design is provided in Fig. 5.

All the pes within a layer are implemented in parallel. At the same time, the
operations within each pe are pipelined to achieve a balanced trade-off
between resource utilization and latency. The output of the dnn is
denormalized to obtain the final output. The outputs from both dnns are
combined to obtain complex samples, which are then used for channel
equalization.

###  IV-B MMSE Architecture

As discussed in [9], the MMSE is computationally complex and from Eq. 3, we
can observe that the arithmetic operations in MMSE are not hardware-friendly.
The architecture of the MMSE involves the LS estimation followed by
multiplication with weight matrix as shown in Fig. 6. The calculation of the
MMSE weight matrix involves the addition of the inverse of the SNR to the
diagonal elements of the channel auto-correlation matrix $R_{HH}$, followed by
matrix inverse operation of size $k_{on}\times k_{on}$. The resulting inverse
matrix is then multiplied with the channel auto-correlation matrix to obtain
the MMSE weight matrix.

Figure 6: mmse Architecture.

Compared to existing works, we focus on comparing the complexity of MMSE using
different matrix inversion approaches. The matrix inversion through the method
of cofactors is unfeasible due to its excessively high computation time. In
this work, we have designed three mmse architectures using three matrix
inversion approaches on ZSoC: 1) Gauss-Jordan [13], 2) QR[14], and 3) LU
decomposition [15]. The hardware architectures of these MMSE algorithms are
designed using reference matrix inversion examples provided by AMD-Xilinx. The
functional and complexity performance of these architectures is discussed in
Section V.

TABLE III: Complexity comparison of various CE approaches for SPFL WL and SDMA
with fixed 50 MHz clock frequency.

Architectures | Interface | Execution time(us) | LUT | FF | BRAM | DSP | Total Power(W) | PL Power (W)
---|---|---|---|---|---|---|---|---
LS | SDMA | 13.3 | 14721 | 15094 | 14 | 96 | 1.969 | 0.438
MMSE | GJ | SDMA | 25859 | 32621 | 24917 | 124 | 362 | 2.479 | 0.948
QR | SDMA | 23041 | 37497 | 29019 | 149 | 383 | 2.193 | 0.663
LU | SDMA | 21359 | 43356 | 32870 | 181 | 522 | 2.711 | 1.18
[9] | SDMA | 27.99 | 41224 | 37956 | 79.5 | 208 | 2.593 | 1.063
MM | 26.84 | 39285 | 35622 | 79.5 | 208 | 2.204 | 0.674
LC-LSDNN | SDMA | 17.46 | 38589 | 37779 | 35 | 184 | 2.128 | 0.598
MM | 17.07 | 38670 | 37618 | 38 | 184 | 2.152 | 0.622

Figure 7: Hardware-software co-design architecture of CE on ZSoC.

###  IV-C System Level Architecture for CE on ZSoC

We have realized various CE approaches on ZSoC comprising dual core Cortex-A9
processor from ARM as processing system (PS) and ultra-scale FPGA from AMD-
Xilinx as programmable logic (PL). The corresponding architecture obtained via
hardware-software co-design is shown in Fig. 7. All CE hardware IPs are
designed with an advanced extensible interface (AXI) stream interface. These
IPs can be configured by the processor using AXI-Lite interface, and data
communication with memory is done using AXI direct memory access (AXI DMA). In
LC-LSDNN, AXI broadcaster IP is used to split the complex estimates into their
real and imaginary components before feeding them to their respective DNNs,
and the AXI combiner is employed to merge their outputs before transmitting
them to the PS for further processing. The remainder of the wireless PHY is
implemented in the PS, along with the DMA controller, interrupt controller,
and scheduler.

##  V Performance and Complexity Analysis

In this section, we present functional performance and complexity results on
the ZC706 platform, comparing them with state-of-the-art CE approaches. We
evaluate three matrix inversion algorithms for MMSE, as detailed in Section
IV-B. Figure 8(a) illustrates that MMSE outperforms LS across all SNRs for
DPFL WL. However, the performance of MMSE-QR degrades slightly at higher SNR
due to erroneous matrix inverse operations. In the case of SPFL WL in Fig.
8(b), the performance of all MMSE hardware IPs degrades at high SNR. Hence, we
do not explore fixed-point architectures for MMSE and use DPFL WL for the
complexity analysis.

Figure 8: Double precision and Single precision floating point implementation
of MMSE channel estimation schemes.

Since the data to be processed by the receiver PHY is buffered in memory, the
interface and type of communication also impact the latency performance. We
have considered three different types of interfaces: 1) Simple direct memory
access (SDMA), 2) Scatter Gather DMA (SGDMA), and 3) Memory-mapped (MM). All
these approaches are based on the AXI protocol. In SDMA and SGDMA, the DMA is
used to read and write the data between memory and hardware IP, while MM
allows the hardware IP to directly read and write the data from memory. Among
DMAs, SDMA can access contiguous data, while SGDMA can access non-contiguous
data. Thus, SDMA needs frequent configuration for non-contiguous data, making
SGDMA a preferred choice.

TABLE IV: Complexity comparison of various CE approaches using SDMA and SGDMA
with architecture operating at the maximum clock frequency.

Architectures | Max. clock frequency | Interface | Execution time(us) | LUT | FF | BRAM | DSP | Total Power(W) | PL Power (W)
---|---|---|---|---|---|---|---|---|---
MMSE | LU | 83.3 | SDMA | 13366 | 43797 | 40332 | 183 | 515 | 3.082 | 1.557
SGDMA | 13376 | 44148 | 41845 | 183 | 515 | 3.196 | 1.671
GJ | 111.11 | SDMA | 13811 | 32505 | 34471 | 126 | 358 | 3.356 | 1.833
SGDMA | 13796 | 34203 | 37198 | 126 | 358 | 3.344 | 1.821
QR | 111.11 | SDMA | 12045 | 37795 | 38426 | 151 | 395 | 2.746 | 1.225
SGDMA | 12006 | 39435 | 41153 | 151 | 395 | 2.758 | 1.237
[9] | 125 | SDMA | 14.67 | 35193 | 38945 | 83 | 138 | 2.164 | 0.634
SGDMA | 12.5 | 36786 | 41672 | 83 | 138 | 2.593 | 1.072
LC-LSDNN | 125 | SDMA | 11.144 | 36827 | 47371 | 35 | 116 | 2.569 | 1.048
200 | SDMA | 7.41 | 58387 | 89137 | 35 | 112 | 4.217 | 2.681
SGDMA | 5.36 | 60016 | 91896 | 35 | 112 | 4.286 | 2.75

TABLE V: Complexity comparison of fixed-point LC-LSDNN and [9] with SGDMA at
maximum clock frequency.

Architectures | Word Length | Max. clock frequency | Execution time(us) | LUT | FF | BRAM | DSP | Total Power(W) | PL Power (W)
---|---|---|---|---|---|---|---|---|---
[9] | <24,8>, <18,2> | 125 | 6.844 | 28801 | 30170 | 31 | 169 | 2.548 | 1.027
LC-LSDNN | <24,8>, <18,2> | 200 | 2.52 | 86165 | 89244 | 9 | 172 | 5.764 | 4.228

In Table III, we compare four CE approaches in terms of FPGA resource
utilization, execution time, and power consumption. We assume the input data
is in contiguous memory and, hence, limit our discussion to SDMA and MM
interfaces. Also, all architectures are designed to work at a fixed clock
frequency of 50 MHz. Among conventional statistical approaches, LS is
efficient but performs poorly compared to MMSE in CE functionality. In the
case of MMSE, MMSE-LU demonstrates the fastest execution time with a slight
increase in resources and power consumption. On the contrary, the MMSE-GJ
offers the highest execution time with lower resources and power consumption,
among other MMSE approaches. Though MMSE offers better performance than LS,
there is a need for alternative approaches that can offer performance that is
the same or better than MMSE with comparable complexity as that of LS.

Next, we compare the LS with [9] and proposed LC-LSDNN. Please note that the
results for LC-LSDNN are for both the real and imaginary DNNs. For SDMA, LC-
LSDNN offers a 37.6% increase in speed, accompanied by a substantial 55.97%
reduction in BRAM utilization, an 11.5% reduction in DSP utilization, and
17.93% lower power consumption than [9]. Both DL-based approaches
significantly outperform MMSE in functional and complexity comparison. This is
mainly due to replacing complex MMSE signal processing operations with
hardware-friendly DL arithmetic operations. By removing DMA and using the MM
interface, the execution time of LC-LSDNN is reduced by 2.2% compared to the
SDMA and by 36.4% compared to the [9] with further reduction in resource
utilization due to the absence of DMA, as LC-LSDNN can use two parallel
interfaces for memory read and write compared to a single DNN in [9]. The LC-
LSDNN has only 25% higher execution time than LS compared to 100% higher
execution time in LSDNN.

Next, we optimized the critical path and increased the clock frequency at
which the architecture operates without losing functional accuracy. As shown
in Table V, the maximum clock frequency is different for each architecture,
and LC-LSDNN can work at highest frequency due to a smaller critical path.

Among the MMSE variants, MMSE-LU had the longest critical path, resulting in
the smallest attainable clock frequency of 83.3 MHz. In contrast, MMSE-GJ and
MMSE-QR achieved higher frequencies of 111.11 MHz. These adjustments
significantly reduced execution times, with decreases of 46%, 47.7%, and 37.4%
for MMSE-GJ, MMSE-QR, and MMSE-LU, respectively. Notably, with the higher
clock periods, MMSE-QR achieved the shortest execution time.

Compared to 125 MHz in [9], LC-LSDNN can operate at a much higher clock
frequency of 200 MHz. At this frequency, LC-LSDNN’s execution time is reduced
to $5.36\mu s$, which is 57% faster than [9] at 125 MHz and 59.7% faster than
LS at 50 MHz. However, this improvement in latency is accompanied by an
increase in power dissipation, with LC-LSDNN consuming 39.5% more power than
[9]. Even when operated at the same clock frequency of 125 MHz, LC-LSDNN is
13% faster than [9], highlighting the low complexity and high-speed
characteristics of the LC-LSDNN.

We optimized the architectures further using fixed-point WL. Our detailed
study identified a minimum WL of 24 bits, with 8 bits for the integer part and
16 bits for the fractional part, denoted as $<24,8>$, ensuring computational
accuracy. Additionally, a WL of $<18,2>$ proved sufficient for DNN parameters.
With the reduced WL, the execution time of LC-LSDNN was further reduced by
51%, and BRAM utilization decreased by 74.2%. However, DSP utilization
increased by 34.8%, due to further possibility of parallel operations.
Depending on the desired execution time, we can reduce resource utilization by
serializing the architecture.

##  VI Conclusion

We developed a novel low complexity high speed deep neural network augmented
least square (LC-LSDNN) based channel estimation (CE) and efficiently mapped
it on zynq system on chip (ZSoC). The novelty of the LC-LSDNN is to use
different DNNs for real and imaginary values of received complex symbols. This
helps reduce the size of DL by around 50% and optimize the critical path,
allowing it to operate at 60% higher clock frequency. In addition, we offer
significant savings in resource utilization and power consumption over state-
of-the-art DL-based CE without compromising on functional accuracy.

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# Constrained quantization for a uniform distribution with respect to a family
of constraints

1Megha Pandey  and  2Mrinal K. Roychowdhury  1Department of Mathematical
Sciences
Indian Institute of Technology (Banaras Hindu University)
Varanasi, 221005, India.  2School of Mathematical and Statistical Sciences
University of Texas Rio Grande Valley
1201 West University Drive
Edinburg, TX 78539-2999, USA<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

In this paper, with respect to a family of constraints for a uniform
probability distribution we determine the optimal sets of $n$-points and the
$n$th constrained quantization errors for all positive integers $n$. We also
calculate the constrained quantization dimension and the constrained
quantization coefficient. The work in this paper shows that the constrained
quantization dimension of an absolutely continuous probability measure depends
on the family of constraints and is not always equal to the Euclidean
dimension of the underlying space where the support of the probability measure
is defined.

###### Key words and phrases:

Probability measure, constrained quantization error, optimal sets of
$n$-points, constrained quantization dimension, constrained quantization
coefficient

###### 2010 Mathematics Subject Classification:

60Exx, 94A34.

##  1\. Introduction

Let $P$ be a Borel probability measure on $\mathbb{R}^{k}$ equipped with a
metric $d$ induced by a norm $\|\cdot\|$ on $\mathbb{R}^{k}$, and
$r\in(0,\infty)$. Then, for $n\in\mathbb{N}$, the $n$th constrained
quantization error for $P$, of order $r$ with respect to a family of
constraints $\\{S_{j}:j\in\mathbb{N}\\}$ with $S_{1}$ nonempty, is defined as

$V_{n,r}:=V_{n,r}(P)=\inf\Big{\\{}\int\mathop{\min}\limits_{a\in\alpha}d(x,a)^{r}dP(x):\alpha\subseteq\bigcup_{j=1}^{n}S_{j},~{}1\leq\text{card}(\alpha)\leq
n\Big{\\}},$ (1)

where $\text{card}(A)$ represents the cardinality of the set $A$. The number

$V_{r}(P;\alpha):=\int\mathop{\min}\limits_{a\in\alpha}d(x,a)^{r}dP(x)$

is called the distortion error for $P$, of order $r$, with respect to a set
$\alpha\subseteq\mathbb{R}^{k}$. Write
$V_{\infty,r}(P):=\mathop{\lim}\limits_{n\to\infty}V_{n,r}(P)$. Then, the
number $D_{r}(P)$ defined by

$D_{r}(P):=\mathop{\lim}\limits_{n\to\infty}\frac{r\log
n}{-\log(V_{n,r}(P)-V_{\infty,r}(P))},$

if it exists, is called the constrained quantization dimension of $P$ of order
$r$ and is denoted by $D_{r}(P)$. For any $\kappa>0$, the number

$\lim_{n\to\infty}n^{\frac{r}{\kappa}}(V_{n,r}(P)-V_{\infty,r}(P)),$ (2)

if it exists, is called the $\kappa$-dimensional constrained quantization
coefficient for $P$ of order $r$. Constrained quantization has recently been
introduced, see [PR1, PR2]. Unconstrained quantization, which traditionally in
the literature is known as quantization, is a special case of constrained
quantization. For unconstrained quantization, one can see [DFG, DR, GG, GL,
GL1, GL2, GL3, GN, KNZ, P, P1, R1, R2, R3, Z1, Z2]. If $\int
d(x,0)^{r}dP(x)<\infty$ is satisfied, then the infimum in (1) exists (see
[PR1]). A set $\alpha\subseteq\mathop{\bigcup}\limits_{j=1}^{n}S_{j}$ for
which the infimum in (1) exists and does not contain more than $n$ elements is
called an optimal set of $n$-points for $P$. In unconstrained quantization,
the optimal sets of $n$-points are referred to as optimal sets of $n$-means.
This paper deals with the Euclidean metric induced by the Euclidean norm
$\|\cdot\|$. Thus, instead of writing $V_{r}(P;\alpha)$ and
$V_{n,r}:=V_{n,r}(P)$ we will write them as $V(P;\alpha)$ and
$V_{n}:=V_{n}(P)$. Let us take the family $\\{S_{j}:j\in\mathbb{N}\\}$ of
constraints, that occurs in (1) as follows:

$S_{j}=\\{(x,y):-\frac{1}{j}\leq x\leq 1\text{ and }y=x+\frac{1}{j}\\}$ (3)

for all $j\in\mathbb{N}$. Let $P$ be a Borel probability measure on
$\mathbb{R}^{2}$ which has support the closed interval $\\{(x,y):0\leq x\leq
1,\,y=0\\}$. Moreover, $P$ is uniform on its support. In this paper we
determine the optimal sets $\alpha_{n}$ of $n$-points for $P$ such that

$\alpha_{n}\subseteq\bigcup_{j=1}^{n}S_{j}$

for all $n\in\mathbb{N}$. We also calculate the constrained quantization
dimension and the constrained quantization coefficient for $P$. As mentioned
in Remark 4.3, unlike the unconstrained quantization dimension of an
absolutely continuous probability measure, the constrained quantization
dimension of an absolutely continuous probability measure depends on the
family of constraints and is not always equal to the Euclidean dimension of
the space where the support of the probability measure is defined (for some
more details, one can see [PR1]).

##  2\. Preliminaries

In this section, we give some basic notations and definitions which we have
used throughout this paper. Notice that for any elements
$p,q\in\mathbb{R}^{2}$, if $e$ an element on the boundary of their Voronoi
regions, then

$\rho(p,e)-\rho(q,e)=0,$

where for any two elements $(a,b)$ and $(c,d)$ in $\mathbb{R}^{2}$,
$\rho((a,b),(c,d))$ represents the squared Euclidean distance between the two
elements. Such an equation is known as a canonical equation. Let $P$ be a
Borel probability measure on $\mathbb{R}^{2}$ which is uniform on its support
the line segment given by

$J:=\\{(x,y):0\leq x\leq 1\text{ and }y=0\\}.$

Let $P_{1},P_{2}$ be the marginal distributions of $P$, i.e.,
$P_{1}(A)=P(A\times\mathbb{R})$ for all $A\in\mathfrak{B}$, and
$P_{2}(B)=P(\mathbb{R}\times B)$ for all $B\in\mathfrak{B}$, where
$\mathfrak{B}$ is the Borel $\sigma$-algebra on $\mathbb{R}$. Then,
$P=P_{1}\times P_{2}$. Thus, for any $A\in\mathfrak{B}$ and
$B\in\mathfrak{B}$, $P(A\times B)=P_{1}(A)P_{2}(B)$. Since $P$ has support
$[0,1]\times\\{0\\}$, we have

$1=P([0,1]\times\\{0\\})=P_{1}([0,1])P_{2}(\\{0\\}),$

i.e., $P_{1}([0,1])=1$ and $P_{2}(\\{0\\})=1$, i.e., $P_{1}$ and $P_{2}$ have
supports $[0,1]$ and $\\{0\\}$, respectively. The probability density function
$f(x,y)$ for $P$ is given by

$f(x,y)=\left\\{\begin{array}[]{cc}1&\text{ if }0\leq x\leq 1\text{ and
}y=0,\\\ 0&\text{ otherwise}.\end{array}\right.$

Moreover,

$dP(x,0)=P(dx\times\\{0\\})=P_{1}(dx)=dP_{1}(x)=f(x,0)dx.$

Thus, we see that the probability distribution $P$ is uniform on its support
$J$ is equivalent to say that $P_{1}$ is a uniform distribution on the closed
interval $[0,1]$. Let $\textbf{X}=(X,Y)$ be a random vector with probability
distribution $P$. Then, notice that for any $0\leq a<b\leq 1$, we have

$E(\textbf{X}:\textbf{X}\in[a,b]\times\\{0\\})=\int_{[a,b]\times\\{0\\}}(x,0)dP(x,0)=\Big{(}\int_{a}^{b}x\,dP_{1}(x),0\Big{)}=\Big{(}\frac{a+b}{2},0\Big{)}.$
(4)

With respect to a finite set $\alpha\subset\mathbb{R}^{2}$, by the Voronoi
region of an element $a\in\alpha$, it is meant the set of all elements in
$\mathbb{R}^{2}$ which are nearest to $a$ among all the elements in $\alpha$,
and is denoted by $M(a|\alpha)$.

In this paper, we investigate the constrained quantization for the probability
measure $P$ with respect to the family of constraints given by

$S_{j}=\\{(x,y):-\frac{1}{j}\leq x\leq 1\text{ and }y=x+\frac{1}{j}\\}\text{
for all }j\in\mathbb{N},$ (5)

i.e., the constraints $S_{j}$ are the line segments joining the points
$(-\frac{1}{j},0)$ and $(1,1+\frac{1}{j})$ which are parallel to the line
$y=x$. The perpendicular on a constraint $S_{j}$ passing through a point
$(x,x+\frac{1}{j})\in S_{j}$ intersects the real line at the point
$(2x+\frac{1}{j},0)$ where $-\frac{1}{j}\leq x\leq 1$; and it intersects $J$
if $0\leq 2x+\frac{1}{j}\leq 1$, i.e., if

$-\frac{1}{2j}\leq x\leq\frac{1}{2}-\frac{1}{2j}.$ (6)

Thus, for all $j\in\mathbb{N}$, there exists a one-one correspondence between
the elements $(x,x+\frac{1}{j})$ on $S_{j}$ and the elements
$(2x+\frac{1}{j},0)$ on the real line if $-\frac{1}{j}\leq x\leq 1$. Thus, for
all $j\in\mathbb{N}$, there exist bijective functions $U_{j}$ such that

$U_{j}(x,x+\frac{1}{j})=(2x+\frac{1}{j},0)\text{ and
}U_{j}^{-1}(x,0)=\Big{(}\frac{1}{2}(x-\frac{1}{j}),\frac{1}{2}(x-\frac{1}{j})+\frac{1}{j}\Big{)},$
(7)

where $-\frac{1}{j}\leq x\leq 1$.

In the following sections we give the main results of the paper.

##  3\. Optimal sets of $n$-points and the $n$th constrained quantization
errors

In this section, we calculate the optimal sets of $n$-points and the $n$th
constrained quantization errors for all $n\in\mathbb{N}$. Let us first give
the following lemma.

######  Lemma 3.1.

Let $\alpha_{n}\subseteq\mathop{\cup}\limits_{j=1}^{n}S_{j}$ be an optimal set
of $n$-points for $P$ such that

$\alpha_{n}:=\\{(a_{j},b_{j}):1\leq j\leq n\\},$

where $a_{1}<a_{2}<a_{3}<\cdots<a_{n}$. Then, $\alpha_{n}\subseteq S_{n}$ and
$(a_{j},b_{j})=U_{n}^{-1}(E(\textbf{X}:\textbf{X}\in
M((a_{j},b_{j})|\alpha_{n})))$, where $M((a_{j},b_{j})|\alpha_{n})$ are the
Voronoi regions of the elements $(a_{j},b_{j})$ with respect to the set
$\alpha_{n}$ for $1\leq j\leq n$.

###### Proof.

Let $\alpha_{n}:=\\{(a_{j},b_{j}):1\leq j\leq n\\}$, as given in the statement
of the lemma, be an optimal set of $n$-points. Take any
$(a_{q},b_{q})\in\alpha_{n}$. Since
$\alpha_{n}\subseteq\mathop{\cup}\limits_{j=1}^{n}S_{j}$, we can assume that
$(a_{q},b_{q})\in S_{t}$, i.e., $b_{q}=a_{q}+\frac{1}{t}$ for some $1\leq
t\leq n$. Since the Voronoi region of $(a_{q},b_{q})$, i.e.,
$M((a_{q},b_{q})|\alpha_{n})$ has positive probability, we can assume that
$M((a_{q},b_{q})|\alpha_{n})$ intersects the support of $P$ at the points
$(a,0)$ and $(b,0)$, where $0\leq a<b\leq 1$. Hence, the distortion error
contributed by $(a_{q},b_{q})$ in its Voronoi region
$M((a_{q},b_{q})|\alpha_{n})$ is given by

$\displaystyle\int_{M((a_{q},b_{q})|\alpha_{n})}\rho((x,0),(a_{q},b_{q}))\,dP=\int_{a}^{b}\|(x,0)-(a_{q},b_{q})\|^{2}\,dx$
$\displaystyle=\frac{1}{3}(b-a)\Big{(}a^{2}-3(a+b)a_{q}+ab+3a_{q}^{2}+b^{2}+3b_{q}^{2}\Big{)}$
$\displaystyle=\frac{1}{3}(b-a)\Big{(}a^{2}-3(a+b)a_{q}+ab+3\Big{(}a_{q}+\frac{1}{t}\Big{)}^{2}+3a_{q}^{2}+b^{2}\Big{)}.$

The above expression is minimum if $a_{q}=\frac{at+bt-2}{4t}$. Now, putting
$a_{q}=\frac{at+bt-2}{4t}$, we have the above distortion error as

$\frac{(b-a)\left(t^{2}\left(5a^{2}+2ab+5b^{2}\right)+12t(a+b)+12\right)}{24t^{2}}.$

Since $1\leq t\leq n$, the above distortion error is minimum if $t=n$. Thus,
for $t=n$, we see that $(a_{q},b_{q})\in S_{n}$, and

$a_{q}=\frac{1}{2}(\frac{a+b}{2}-\frac{1}{n}),\text{ and
}b_{q}=a_{q}+\frac{1}{n}=\frac{1}{2}(\frac{a+b}{2}-\frac{1}{n})+\frac{1}{n},$

which implies

$(a_{q},b_{q})=U_{n}^{-1}\Big{(}\Big{(}\frac{a+b}{2},0\Big{)}\Big{)},$

which by (4) yields that

$(a_{q},b_{q})=U_{n}^{-1}(E(\textbf{X}:\textbf{X}\in
M((a_{j},b_{j})|\alpha_{n}))).$

Since $(a_{q},b_{q})\in\alpha_{n}$ is chosen arbitrarily, the proof of the
lemma is complete. ∎

######  Remark 3.2.

By (6) and (7), and Lemma 3.1, we can conclude that all the elements in an
optimal set of $n$-points must lie on $S_{n}$ between the two elements
$U_{n}^{-1}(0,0)$ and $U_{n}^{-1}(1,0)$, i.e., between the two elements
$(-\frac{1}{2n},\frac{1}{2n})$ and $(\frac{n-1}{2n},\frac{n+1}{2n})$. If this
fact is not true, then the constrained quantization error can be strictly
reduced by moving the elements in the optimal set between the elements
$(-\frac{1}{2n},\frac{1}{2n})$ and $(\frac{n-1}{2n},\frac{n+1}{2n})$ on
$S_{n}$, in other words, the $x$-coordinates of all the elements in an optimal
set of $n$-points must lie between the two numbers $-\frac{1}{2n}$ and
$\frac{n-1}{2n}$ (see Figure 1).

######  Lemma 3.3.

Let $\alpha_{n}$ be an optimal set of $n$-points for $P$. Then,
$U_{n}(\alpha_{n})$ is an optimal set of $n$-means for $P$.

###### Proof.

By Lemma 3.1, $\alpha_{n}\subseteq S_{n}$ for all $n\in\mathbb{N}$. Let
$\alpha_{n}:=\\{(a_{j},b_{j}):1\leq j\leq n\\}$ be an optimal set of
$n$-points for $P$ such that $a_{1}<a_{2}<\cdots<a_{n}$. Then, by Remark 3.2,
we have $-\frac{1}{2n}\leq a_{1}<a_{2}<\cdots<a_{n}\leq\frac{n-1}{2n}$.
Moreover, as $(a_{j},b_{j})\in S_{n}$, we have $b_{j}=a_{j}+\frac{1}{n}$ for
all $1\leq j\leq n$.

Notice that the boundary of the Voronoi region of the element $(a_{1},b_{1})$
intersects the support of $P$ at the elements $(0,0)$ and
$(a_{1}+a_{2}+\frac{1}{n},0)$, the boundaries of the Voronoi regions of
$(a_{j},b_{j})$ for $2\leq j\leq n-1$ intersect the support of $P$ at the
elements $(a_{j-1}+a_{j}+\frac{1}{n},0)$ and $(a_{j}+a_{j+1}+\frac{1}{n},0)$,
and the boundary of the Voronoi region of $(a_{n},b_{n})$ intersects the
support of $P$ at the elements $(a_{n-1}+a_{n}+\frac{1}{n},0)$ and $(1,0)$.
Thus, the distortion error due to the set $\alpha_{n}$ is given by

$\displaystyle
V(P;\alpha_{n})=\int_{\mathbb{R}}\min_{a\in\alpha_{n}}\|(x,0)-a\|^{2}dP(x)$
$\displaystyle=\int_{0}^{a_{1}+a_{2}+\frac{1}{n}}\rho((x,0),(a_{1},a_{1}+\frac{1}{n}))\,dx+\sum_{i=2}^{n-1}\int_{a_{i-1}+a_{i}+\frac{1}{n}}^{a_{i}+a_{i+1}+\frac{1}{n}}\rho((x,0),(a_{i},a_{i}+\frac{1}{n}))\,dx$
$\displaystyle\qquad+\int_{a_{n-1}+a_{n}+\frac{1}{n}}^{1}\rho((x,0),(a_{n},a_{n}+\frac{1}{n}))\,dx.$

Since $V(P;\alpha_{n})$ gives the optimal error and is differentiable with
respect to $a_{i}$ for all $1\leq i\leq n$, we have $\frac{\partial}{\partial
a_{i}}V(P;\alpha_{n})=0$ implying

$\frac{1}{n}+2a_{1}=a_{2}-a_{1}=a_{3}-a_{2}=\cdots=a_{n}-a_{n-1}=1-\frac{1}{n}-2a_{n}.$

Then, we can assume that there is a constant $d$ depending on $n$, such that

$\frac{1}{n}+2a_{1}=a_{2}-a_{1}=a_{3}-a_{2}=\cdots=a_{n}-a_{n-1}=1-\frac{1}{n}-2a_{n}=d$
(8)

yielding

$a_{2}=d+a_{1},\,a_{3}=2d+a_{1},\,a_{4}=3d+a_{1},\,\cdots,\,a_{n}=(n-1)d+a_{1}$

i.e.,

$a_{j}=(j-1)d+a_{1}\text{ for }2\leq j\leq n.$ (9)

Again, by (8), we have

$a_{1}=\frac{1}{2}(d-\frac{1}{n})\text{ and
}a_{n}=\frac{1}{2}(1-\frac{1}{n}-d).$ (10)

Putting the above values of $a_{1}$ and $a_{n}$ in the expression
$a_{n}=(n-1)d+a_{1}$, and then upon simplification, we have $d=\frac{1}{2n}$.
Putting the values of $d$ by (9) and (10), we have

$a_{j}=\frac{2j-3}{4n}\text{ for }1\leq j\leq n.$ (11)

Then, we see that the boundary of the Voronoi region of the element
$(a_{1},b_{1})$ intersects the support of $P$ at the elements $(0,0)$ and
$(a_{1}+a_{2}+\frac{1}{n},0)$, i.e., at the elements

$(0,0)\text{ and }(\frac{1}{n},0),$

the boundaries of the Voronoi regions of $(a_{j},b_{j})$ for $2\leq j\leq n-1$
intersect the support of $P$ at the elements $(a_{j-1}+a_{j}+\frac{1}{n},0)$
and $(a_{j}+a_{j+1}+\frac{1}{n},0)$, i.e., at the elements

$\Big{(}\frac{j-1}{n},0\Big{)}\text{ and }\Big{(}\frac{j}{n},0\Big{)},$

and the boundary of the Voronoi region of $(a_{n},b_{n})$ intersects the
support of $P$ at the elements $(a_{n-1}+a_{n}+\frac{1}{n},0)$ and $(1,0)$,
i.e., at the elements

$\Big{(}\frac{n-1}{n},0\Big{)}\text{ and }(1,0).$

Thus, we deduce that for $1\leq j\leq n$, the boundaries of the elements
$(a_{j},b_{j})$ in the optimal set $\alpha_{n}$ of $n$-points intersect the
support of $P$ at the elements $(\frac{j-1}{n},0)$ and $(\frac{j}{n},0)$.
Hence, by Lemma 3.1, we have

$(a_{j},b_{j})=U_{n}^{-1}(E(\textbf{X}:\textbf{X}\in[\frac{j-1}{n},\frac{j}{n}]\times\\{0\\}))=U_{n}^{-1}((\frac{2j-1}{2n},0))\text{
for }1\leq j\leq n.$

We know that for the uniform distribution $P$, the optimal set of $n$-means
(see [RR]) is given by

$\left\\{\Big{(}\frac{2j-1}{2n},0\Big{)}:1\leq j\leq n\right\\}.$

Since

$U_{n}(\alpha_{n})=\\{U_{n}(a_{j},b_{j}):1\leq j\leq
n\\}=\left\\{\Big{(}\frac{2j-1}{2n},0\Big{)}:1\leq j\leq n\right\\},$

the proof of the lemma is complete. ∎

Figure 1. Points in the optimal sets of $n$-points for $1\leq n\leq 4$.

Let us now give the following theorem, which is the main theorem of the paper.

######  Proposition 3.4.

A set $\alpha_{n}\subseteq S_{n}$ is an optimal set of $n$-points for $P$ if
and only if $U_{n}(\alpha_{n})$ is an optimal set of $n$-means for $P$.

###### Proof.

Let $\alpha_{n}\subseteq S_{n}$ be an optimal set of $n$-points for $P$. Then,
by Lemma 3.3, the set $U_{n}(\alpha_{n})$ is an optimal set of $n$-means for
$P$. Next assume that for a set $\beta_{n}\subseteq S_{n}$, the set
$U_{n}(\beta_{n})$ is an optimal set of $n$-means for $P$, we need to show
that $\beta_{n}$ is an optimal set of $n$-points for $P$. For the sake of
contradiction, assume that there exists a set $\gamma_{n}\subseteq S_{n}$ such
that $\gamma_{n}$ is an optimal set of $n$-points for $P$ and
$\gamma_{n}\neq\beta_{n}$. Then, by Lemma 3.3, the set $U_{n}(\gamma_{n})$ is
an optimal set of $n$-means for $P$. Since, the optimal set of $n$-means for
$P$ is unique, we must have $U_{n}(\gamma_{n})=U_{n}(\alpha_{n})$. Since
$U_{n}$ is an injective function, then we have $\gamma_{n}=\beta_{n}$, which
is a contradiction. Thus, the proof of the proposition is complete. ∎

Let us now give the following theorem, which is the main theorem of the paper.

######  Theorem 3.5.

An optimal set of $n$-points for the probability distribution $P$ is given by

$\left\\{\Big{(}\frac{1}{2}\left(\frac{2j-1}{2n}-\frac{1}{n}\right),\frac{1}{2}\left(\frac{2j-1}{2n}-\frac{1}{n}\right)+\frac{1}{n}\Big{)}:1\leq
j\leq n\right\\},$

with $n$th constrained quantization error

$V_{n}=\frac{4n^{2}+12n+13}{24n^{2}}.$

###### Proof.

Let $\alpha_{n}:=\\{(a_{j},b_{j}):1\leq j\leq n\\}$ be an optimal set of
$n$-points for $P$ such that $a_{1}<a_{2}<\cdots<a_{n}$. By Proposition 3.4,
we know that $U_{n}(\alpha_{n})$ is an optimal set of $n$-means for $P$, i.e.,

$U_{n}(\alpha_{n})=\left\\{\Big{(}\frac{2j-1}{2n},0\Big{)}:1\leq j\leq
n\right\\}.$

Since $U_{n}$ is an injective function, we have

$\alpha_{n}=U_{n}^{-1}\Big{\\{}\Big{(}\frac{2j-1}{2n},0\Big{)}:1\leq j\leq
n\Big{\\}}=\Big{\\{}U_{n}^{-1}\Big{(}\frac{2j-1}{2n},0\Big{)}:1\leq j\leq
n\Big{\\}}$

i.e.,

$\alpha_{n}=\left\\{\Big{(}\frac{1}{2}\left(\frac{2j-1}{2n}-\frac{1}{n}\right),\frac{1}{2}\left(\frac{2j-1}{2n}-\frac{1}{n}\right)+\frac{1}{n}\Big{)}:1\leq
j\leq n\right\\}.$

Writing

$(a_{j},b_{j})=\Big{(}\frac{1}{2}\left(\frac{2j-1}{2n}-\frac{1}{n}\right),\frac{1}{2}\left(\frac{2j-1}{2n}-\frac{1}{n}\right)+\frac{1}{n}\Big{)},$

for $1\leq j\leq n$, we have the $n$th constrained quantization for $n$-points
as

$\displaystyle
V_{n}=\int_{\mathbb{R}}\min_{a\in\alpha_{n}}\|(x,0)-a\|^{2}dP(x)$
$\displaystyle=\int_{0}^{a_{1}+a_{2}+\frac{1}{n}}\rho((x,0),(a_{1},a_{1}+\frac{1}{n}))\,dx+\sum_{i=2}^{n-1}\int_{a_{i-1}+a_{i}+\frac{1}{n}}^{a_{i}+a_{i+1}+\frac{1}{n}}\rho((x,0),(a_{i},a_{i}+\frac{1}{n}))\,dx$
$\displaystyle\qquad+\int_{a_{n-1}+a_{n}+\frac{1}{n}}^{1}\rho((x,0),(a_{n},a_{n}+\frac{1}{n}))\,dx,$

which upon simplification yields

$V_{n}=\frac{4n^{2}+12n+13}{24n^{2}}.$

Thus, the proof of the theorem is complete (see Figure 1). ∎

##  4\. Constrained quantization dimension and constrained quantization
coefficient

In this section, we show that the constrained quantization dimension $D(P)$
exists and equals two. We further show that the $D(P)$-dimensional constrained
quantization coefficient for $P$ exists as a finite positive number.

######  Theorem 4.1.

The constrained quantization dimension $D(P)$ of the probability measure $P$
exists, and $D(P)=2$.

###### Proof.

By Theorem 3.5, the $n$th constrained quantization error is given by

$V_{n}=\frac{4n^{2}+12n+13}{24n^{2}}.$

Notice that $V_{\infty}=\mathop{\lim}\limits_{n\to\infty}V_{n}=\frac{1}{6}$.
Hence, the constrained quantization dimension is given by

$D(P)=\lim_{n\to\infty}\frac{2\log
n}{-\log(V_{n}-V_{\infty})}=\lim_{n\to\infty}\frac{2\log(n)}{-\log\left(\frac{4n^{2}+12n+13}{24n^{2}}-\frac{1}{6}\right)}=2,$

which is the theorem. ∎

######  Theorem 4.2.

The $D(P)$-dimensional constrained quantization coefficient for $P$ exists,
and equals $\frac{1}{2}$.

###### Proof.

We have

$V_{n}=\frac{4n^{2}+12n+13}{24n^{2}}\text{ and
}V_{\infty}=\mathop{\lim}\limits_{n\to\infty}V_{n}=\frac{1}{6},$

and hence, using (2), we have the $D(P)$-dimensional constrained quantization
coefficient as

$\lim_{n\to\infty}n(V_{n}-V_{\infty})=\lim_{n\to\infty}n\left(\frac{4n^{2}+12n+13}{24n^{2}}-\frac{1}{6}\right)=\frac{1}{2}.$

Thus, the proof of the theorem is complete. ∎

######  Remark 4.3.

For the absolutely continuous probability measure, considered in this paper,
we have obtained that the constrained quantization dimension is two which is
not equal to the Euclidean dimension of the underlying space where the support
of the probability measure is defined. On the other hand, it is well-known
that the unconstrained quantization dimension of an absolutely continuous
probability measure always equals the Euclidean dimension of the space where
the support of the probability measure is defined (see [BW]).

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