SAMPLING OF CORRELATED BANDLIMITED CONTINUOUS SIGNALS BY JOINT TIME-VERTEX GRAPH FOURIER TRANSFORM Zhongyi Ni Feng Jiy Hang Sheng Hui Feng Bo Hu

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SAMPLING OF CORRELATED BANDLIMITED CONTINUOUS SIGNALS BY JOINT

TIME-VERTEX GRAPH FOURIER TRANSFORM

Zhongyi Ni∗, Feng Ji†, Hang Sheng∗, Hui Feng∗, Bo Hu∗

∗School of Information Science and Engineering, Fudan University, Shanghai 200433, China

†School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore

∗{20300720007, 20110720036, hfeng, bohu}@fudan.edu.cn, †jifeng@ntu.edu.sg

ABSTRACT

When sampling multiple signals, the correlation between

the signals can be exploited to reduce the overall number of

samples. In this paper, we study the sampling theory of mul-

tiple correlated signals, using correlation to sample them at

the lowest sampling rate. Based on the correlation between

signal sources, we model multiple continuous-time signals as

continuous time-vertex graph signals. The graph signals are

projected onto orthogonal bases to remove spatial correlation

and reduce dimensions by graph Fourier transform. When

the bandwidths of the original signals and the reduced dimen-

sion signals are given, we prove the minimum sampling rate

required for recovery of the original signals, and propose a

feasible sampling scheme.

Index Terms—Time-vertex graph signal, sampling the-

ory, graph signal processing

1. INTRODUCTION

Sampling of continuous-time signals not only plays a crucial

role in digital signal processing systems but also provides sig-

niﬁcant savings in the cost of signal storage and processing.

The theory of perfect sampling and recovery for a bandlimited

signal has been extensively studied [1, 2]. In many scenar-

ios, we need to observe multiple signal sources at the same

time, such as biological research, sensor networks, and so-

cial networks. In these cases, the observed continuous-time

signals may be correlated with each other, which introduces

additional information. That is, the observations from sources

at a moment constitute high-dimensional and redundant data.

It would be wasteful to still sample each signal according to

its bandwidth.

For the sampling theory of multiple signals, [3] studied

the problem of the exact recovery of multidimensional signals

from projections. [4,5] relate the bandwidth and correlation to

the sampling rate required for signal reconstruction. In gen-

eral, we hope to further reduce the data dimension by correla-

tion. Principal Component Analysis (PCA) is one of the most

common ways to reduce the dimension of data by project-

ing high-dimensional data into an orthogonal space [6]. This

dimension reduction process is consistent with graph signal

processing which adopts the adjacency matrix as its funda-

mental framework [7]. The covariance matrix is the adjacency

matrix, which is used to characterize the correlation between

signal sources. Eigenvectors of the adjacency matrix are de-

ﬁned as the graph Fourier bases, which constitute the orthog-

onal space. We can apply graph Fourier transform (GFT) to

the data at each moment of the signal sources and remove the

redundant spatial information [8].

Therefore, we use the time-vertex graph signal (TVGS)

processing framework to study the sampling of correlated sig-

nals [9, 10]. We model the correlated signal sources as a

graph, whose each vertex relates to a continuous function

of time. In this way, multiple continuous-time signals are

modeled as a continuous time-vertex graph signals (CTVGS).

There are some research results about the sampling of time-

point diagram signals. Based on joint time-vertex Fourier

transform (JFT) [11], Yu et al. [12] proposed a joint sampling

scheme, which further reduce the amount of samples of the

ﬁnite time-vertex graph signal. Ji et al. [10] considered the

sampling of TVGS with inﬁnite time components and gave

an extension of the separate sampling scheme. However, the

above works only consider sampling based on the spectrum of

the dimension-reducted signals and thus fail to give the mini-

mum sampling rate required for recovery of the CTVGS.

In this paper, the bandlimited signals are modeled as

CTVGS, whose low-dimensional representations are ob-

tained by GFT. When the bandwidths of the signals before

and after dimension reduction are given, we prove the low-

est sampling rate required for signal recovery and propose a

speciﬁc method to obtain the minimum sample set. Finally,

we veriﬁed the feasibility of sampling and recovery through

simulation experiments.

2. MODEL

An undirected graph can be represented as G= (V,E,A),

where V={v1, . . . , vN}is the set of Nvertices, Eis the

set of edges, and Ais an N×Nsymmetric weighted adja-

cency matrix. The weight a(i, j)of the edge (vi, vj)∈ V re-

arXiv:2210.04504v1 [eess.SP] 10 Oct 2022

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SAMPLINGOFCORRELATEDBANDLIMITEDCONTINUOUSSIGNALSBYJOINTTIME-VERTEXGRAPHFOURIERTRANSFORMZhongyiNi,FengJiy,HangSheng,HuiFeng,BoHuSchoolofInformationScienceandEngineering,FudanUniversity,Shanghai200433,ChinaySchoolofElectricalandElectronicEngineering,NanyangTechnologicalUniversity,Singaporef20300...

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SAMPLING OF CORRELATED BANDLIMITED CONTINUOUS SIGNALS BY JOINT TIME-VERTEX GRAPH FOURIER TRANSFORM Zhongyi Ni Feng Jiy Hang Sheng Hui Feng Bo Hu

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