利用信号相关性的分布式压缩感知

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"这篇论文探讨了压缩感知(Compressive Sensing)原理在信号处理中的应用，特别是分布式压缩感知(Distributed Compressive Sensing, DCS)的理论与算法。压缩感知是一种信号采集框架，它指出稀疏信号的小部分线性投影就足以恢复信号的关键信息。论文提出的新理论——分布式压缩感知，旨在利用多信号集合内部和之间的相关结构来设计新的编码算法。作者们提出了‘信号集合的联合稀疏性’这一新概念，并对分布式压缩感知的恢复性能极限进行了理论分析。" 正文: 压缩感知(Compressive Sensing)是现代信号处理领域的一个重要概念，它颠覆了传统的采样理论，指出对于稀疏信号，可以通过远低于奈奎斯特定理所规定的速率进行采样，仍能准确重构原始信号。这一理论的核心在于，信号的大部分信息集中在少数非零成分上，而这些非零成分可以通过精心设计的测量矩阵捕获。在本文中，作者们将关注点转向了分布式压缩感知(DCS)，这是一个扩展了压缩感知思想的应用场景，特别适用于处理多个相关的信号集合。在分布式系统中，多个传感器或节点可能同时收集到相关联的数据，这些数据共享某些结构特征。DCS的目标就是利用这些内在的相关性，提高数据采集和处理的效率。论文引入了“信号集合的联合稀疏性”这一概念，它指的是一个信号集合中所有信号共有的稀疏结构。通过利用这种联合稀疏性，可以设计出新的编码策略，使得在分布式环境下，各节点能够协同工作，共同恢复信号。这种方法不仅考虑了单个信号的稀疏性，还考虑了信号间的相关性，从而优化了整体的恢复性能。作者们对DCS的理论贡献在于，他们分析了分布式压缩感知的恢复性能极限。这包括了在给定的测量条件下，如何设计最优的测量矩阵，以及如何在保证恢复精度的同时，最大限度地减少所需的测量次数。这种理论分析对于实际应用具有重要的指导意义，例如在无线通信、图像处理、医疗成像等领域，可以有效降低数据传输和存储的需求，提高系统的效率。论文进一步讨论了如何实现这些分布式编码算法，以及这些算法在处理大规模数据集时的性能表现。此外，作者们也强调了这个理论是向已故的Hyeokho Choi致敬，他是他们的同事、导师和朋友，这显示了研究者们对他的深深敬意和怀念。这篇论文深入探讨了压缩感知的分布式应用，通过引入信号集合的联合稀疏性，为多信号环境下的数据采集和处理提供了新的理论基础和技术手段，这对于提升信号处理系统的效率和可靠性具有重大意义。

2.6 Properties of random measurements

In addition to oﬀering substantially reduced measurement rates, CS has many attractive and in-

triguing properties, particularly when we employ random projections at the sensors. Random

measurements are universal in the sense that any sparse basis can be used, allowing the same en-

co ding strategy to be app lied in diﬀerent sens ing environments. Random measurements are also

future -proof: if a better sparsity-inducing basis is found for the s ignals, then the same measurements

can be used to recover a more accurate view of the environment. Random cod ing is also robust: the

measurements coming from each sens or have equal priority, unlike Fourier or wavelet coeﬃcients in

current coders. Finally, random measurements allow a progressively better recovery of the data as

more measurements are obtained; one or more measurements can also be lost without corrupting

the entire recovery.

2.7 Related work

Several researchers have formulated joint measurement settings for CS in sensor networks that

exploit inter-signal correlations [28–32]. In their approaches, each sen sor n ∈ {1, 2, . . . , N} simul-

taneously records a single reading x(n) of some spatial ﬁeld (temperature at a certain time, for

example).

Each of the sensors generates a p s eu dorandom sequence r

(m), m = 1, 2, . . . , M, and

modulates the reading as x(n)r

(m). Each sensor n then transmits its M numbers in sequ en ce to

the collection point where the measurements are aggregated, obtaining M measurements y(m) =

n=1

x(n)r

(m). T hus, deﬁning x = [x(1), x(2), . . . , x(N)]

and φ

= [r

(m), r

(m), . . . , r

(m)],

the collection point automatically receives the measurement vector y = [y(1), y(2), . . . , y(M)]

Φx after M trans mission s teps. The samples x(n) of the spatial ﬁeld can then be recovered using

CS provided that x has a sparse representation in a known b asis. These methods have a major

limitation: since they operate at a single time instant, they exploit only inter-signal and not intra-

signal correlations; that is, they essentially assume that the sensor ﬁeld is i.i.d. from time instant

to time instant. In contrast, we will develop signal mod els and algorithms that are agnostic to the

spatial sampling structure and that exploit both inter- and intra-signal correlations.

Recent work has adapted DCS to the ﬁnite rate of innovation signal acquisition framework [47]

and to the continuous-time setting [48]. Since the original submission of this paper, additional work

has focus ed on the analysis and proposal of recovery algorithms for jointly sparse signals [49, 50].

3 Joint Sparsity Signal Models

In this section, we generalize the notion of a signal being sparse in some basis to the notion of an

ensemble of signals being jointly sparse.

3.1 Notation

We will use the following notation for signal ensembles and our measurement model. Let Λ :=

{1, 2, . . . , J} denote the set of indices for the J signals in the ensemble. Denote the signals in the

ensemble by x

, with j ∈ Λ and assume that each signal x

∈ R

. We use x

(n) to denote sample

n in signal j, and assume for the sake of illustration — but without loss of generality — that these

signals are sparse in the canonical basis, i.e., Ψ = I. The entries of the signal can take arbitrary

real values.

We denote by Φ

the measurement matrix for signal j; Φ

is M

× N and, in general, the

entries of Φ

are diﬀerent for each j. Thus, y

= Φ

consists of M

< N random measurements

ote t hat in Section 2.7 only, N refers t o the number of sensors, since each sensor acquires a signal sample.

伪随机序列

调制

融合

of x

. We w ill emphasize random i.i.d. Gaussian matrices Φ

in the following, but other schemes

are possible, including random ±1 Bernoulli/Rademacher matrices, and so on.

To compactly rep resent the signal and measurement ensembles, we denote M =

j∈Λ

and

deﬁne X ∈ R

, Y ∈ R

, and Φ ∈ R

M×JN

X =













, Y =













, and Φ =







0 . . . 0

0 Φ

. . . 0

0 0 . . . Φ







, (4)

with 0 denoting a m atrix of appropriate size with all entries equ al to 0. We th en have Y =

ΦX. Equation (4) shows that sep arate measurement matrices have a characteristic block-diagonal

structure when the entries of the sparse vector are grouped by signal.

Below we propose a general framework for joint sparsity models (JSMs) and three example

JSMs that apply in diﬀerent situations.

3.2 General framework for joint sparsity

We now propose a general framework to quantify the sparsity of an ensemble of correlated signals

, x

, . . . , x

, which allows us to compare the complexities of diﬀerent signal ens embles and to

quantify their measurement requirements. The framework is based on a factored repr esentation of

the signal ensemble that decouples its location and value information.

To motivate this factored representation, we begin by examining the structure of a single sparse

signal, where x ∈ R

with K ≪ N nonzero entries. As an alternative to the notation used in (1),

we can decouple the location and value information in x by writing x = P θ, where θ ∈ R

contains

only the nonzero entries of x, and P is an identity submatrix, i.e., P contains K columns of the

N × N identity matrix I. Any K-sparse signal can be written in similar fashion. To model the

set of all possible sparse signals, we can then let P be the set of all identity submatrices of all

possible sizes N × K

′

, with 1 ≤ K

′

≤ N. We refer to P as a sparsity model. Whether a signal

is su ﬃciently sparse is deﬁned in the context of this model: given a signal x, one can consider all

possible factorizations x = P θ with P ∈ P. Among these factorizations, the unique representation

with smallest dimensionality for θ equals the sparsity level of the signal x under the model P.

In the signal ensemble case, we consider factorizations of the form X = P Θ where X ∈ R

as above, P ∈ R

JN×δ

, and Θ ∈ R

for various integers δ. We refer to P and Θ as the location

matrix and value vector, respectively. A joint sparsity model (JSM) is deﬁned in terms of a set P

of admissible location matrices P with varying numbers of columns; we specify below additional

conditions that the matrices P must satisfy for each model. For a given ensemble X, we let

(X) ⊆ P d en ote the set of feasible location matrices P ∈ P for which a factorization X = P Θ

exists. We deﬁne the joint sparsity lev el of the signal ensemble as follows.

Deﬁnition 1 The joint sparsity level D of the signal ensemble X is the numbe r of columns of the

smallest matrix P ∈ P

(X).

In contrast to th e single-signal case, there are several natural choices for what matrices P

should be members of a joint sparsity model P. We restrict our attention in the sequel to what

we call common/innovation component JSMs. In these models each signal x

is generated as a

combination of two comp on ents: (i) a common component z

, which is present in all signals, and

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