Xampling：突破奈奎斯特定理的降采样技术

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"降采样理论——Xampling框架与实践" 降采样理论是信号处理领域的一个重要概念，特别是在信号压缩感知理论中具有广泛的应用。这一理论允许我们在低于奈奎斯特定理所规定的速率下对连续时间模拟信号进行采样，从而在保留信号信息的同时降低数据量，节省存储和计算资源。 Eldar提出的Xampling（交叉采样）是一种设计方法论，它结合了经典采样理论与压缩感知领域的最新发展，旨在克服奈奎斯特采样限制。Xampling框架主要由以下四个原则构成： 1. 广泛的信号模型：Xampling能够捕获不同类型的信号，包括宽频带、非线性或非高斯分布的信号。 2. 低采样率：通过有效利用信号结构的稀疏性，Xampling能够在远低于奈奎斯特速率的情况下进行采样。 3. 高效的模拟和数字实现：Xampling设计考虑了硬件实施的效率，力求在减少硬件复杂性的同时保持性能。 4. 低速率基带处理：在数字域中进行低速率处理，降低了计算复杂度，同时也简化了系统设计。 Xampling框架对多种次奈奎斯特采样方法进行了评估，揭示了这些方法存在的问题。例如，许多传统的方法在模拟实现上可能遇到困难，数字域内的计算负担过重，且缺乏基带处理能力。其中，调制宽带转换器（Modulated Wideband Converter, MWC）是一个例外，它满足了模型、速率和实现标准，但在基带处理方面仍有待加强。降采样理论的应用不仅限于通信系统，还扩展到了图像处理、医学成像、雷达和遥感等领域。在这些应用中，降采样能够显著减少数据采集和传输的成本，同时保持信号的可恢复性。通过有效的压缩和恢复算法，如迭代阈值算法或基于凸优化的方法，可以从低速率采样数据中重构原始信号。降采样理论和Xampling方法论提供了一种创新的思路，挑战了传统的采样理论，并在实际应用中展示了巨大的潜力。它们对于设计更高效、更节能的信号处理系统具有重要意义，是未来通信和信号处理技术发展的重要方向。

4 XAMPLING—PART I: PRACTICE

Neglecting this mathematical issue, a realtime system should

react gracefully to spectral changes.

Examples. Before proceeding, we examine two straightfor-

ward sampling solutions in light of the Xampling criteria.

Uniform sampling at f

NYQ

obviously contradicts (X2) and

cannot be considered as a sub-Nyquist system. Furthermore,

in wideband settings, the implementation may be impractical

since the rates of conventional Nyquist ADC devices are still

far below the wideband regime [32], contradicting (X3). Base-

band processing (X4) is also not possible, since by deﬁnition

the samples arrive at the high Nyquist rate, thus extracting

the information signals I(t), Q(t) must involve computational

complexity proportional to f

NYQ

The second approach utilizes the scheme of Fig. 1, by

searching for the carrier frequencies f

prior to sampling,

namely using analog components. The motivation for this

solution is two-fold. Standard receivers often compensate for

slight deviations from the prespeciﬁed carrier values f

, by

performing ﬁne analog tunings to the local oscillator until its

frequency is locked to the actual carrier value [33], [34]. The

idea is therefore to use the same locking mechanism to search

for f

over the entire wideband regime. The fact that Fig. 1

satisﬁes all the four Xampling criteria, once the carriers f

are

known, is the second motivation.

Unfortunately, this solution is practically infeasible when

the carrier frequencies f

are unknown apriori and can lie

anywhere below f

max

. Locking over the entire wideband

spectrum is time consuming; during this time the signal

cannot be acquired. To shorten this period, the sampling rate

must be increased much above the minimal, contradicting

(X2). Furthermore, such a locking stage is both hardware and

software excessive. A standard tunable oscillator can cover

only a narrow range of frequencies [35], which may require

hundreds of devices to cover the range until f

max

. In addition,

when initializing the mechanism far away from the true carrier,

it may lock to a spurious frequency. Only high-level data-

aided algorithms can identify this situation and re-initialize

the hardware. This severely burdens the DSP, contradicting

also (X3). Furthermore, whenever the band positions change,

the locking needs to be reinitiated, and again the signal cannot

be acquired until this task is completed.

B. Xampling = Compressed sensing for analog signals

We now brieﬂy describe the CS framework [14], [15], and

distinguish between CS and Xampling.

The majority of compressed sensing publications study

variants of the underdetermined sparse recovery problem. The

signal model assumes a vector x in the ﬁnite space R

or C

which has only a few nonzero entries. Sampling, referred to

as sensing in this framework, is carried out by computing the

linear projection

y = Ax, (4)

with A having far fewer rows than columns. Results from

this ﬁeld [14], [15] show that under suitable conditions, the

linear sensing is stably invertible, even when the length of y

is proportional to the number of nonzeros in x, rather than the

ambient dimension n.

Sensing of sparse vectors is the discrete counterpart of

the sub-Nyquist problem illustrated in Fig. 1. However, it is

not straightforward to generalize the discrete CS formulation

to analog signals. The difﬁculty can be noticed immediately

in the signal model. Sparsity is deﬁned in CS by count-

ing the number of nonzeros in x, while analog sparsity of

x(t) involves an uncountable number of zeros and nonzeros.

Another difference relates to the sensing matrix A. In the

analog context, A corresponds to generalized sampling, where

measurements are inner products with the input x(t) [17],

[36]–[38]. This leads to a structured and deterministic matrix

A, as it needs to be implemented in hardware. In contrast,

mainstream CS results are stated for random unstructured

sensing matrices. A ﬁnal important difference is the issue of

recovery complexity. Na

ıve extensions of CS-type algorithms

to the inﬁnite dimensions, such as `

linear programming

min

kxk

s.t. ky − Axk ≤ , (5)

or greedy techniques, lead to undeﬁned or difﬁcult problems.

For example, an optimization over the continuous signal x(t)

[39]

min

x(t)

obj(x(t)) s.t. ky[n] − A(x(t))k

≤ , (6)

where the objective is a sparsity-promoting function L

(R) →

R and the constraint involves the inﬁnite sample set y[n], the

sampling operator A : L

(R) → l

(R), and the continuous

signal. A program such as (6) is not a well-deﬁned optimiza-

tion structure [39]. In turn, discretization methods result in

very large scale CS systems, which impose a severe burden

on the digital processing units. In contrast, continuous recon-

struction in sampling theory, though hypothetically involving

inﬁnite sequences, is practically performed in realtime over a

well localized set of samples.

The baseband processing is another distinct aspect. Lately,

there has been growing interest in processing in the com-

pressed domain, e.g. machine learning and classiﬁcations [40],

or manifold learning [41], [42]. These works exploit the fact

that certain properties are approximately invariant under a

linear transformation. This allows, to some extent, learning and

classiﬁcation tasks to be carried out directly on the short-length

vector y of (4). The methods [40]–[42] are speciﬁc to discrete

vectors. In contrast, sub-Nyquist methods, both classic [3]–

[6] and recent [7]–[12], are all based on spreading or aliasing

techniques which result in a mixture of the signal content. In

particular, the information contents I(t), Q(t) are not invariant

under the mixture, and a DSP algorithm cannot be carried out

directly on the samples. To emphasize, baseband processing

in Xampling means the ability to translate the seemingly-

corrupted samples to the original information bits or analog

message. Learning, classiﬁcation, and other DSP algorithms

can then follow so that there is no need to approximate their

results in the compressed domain. The crucial requirement

of (X4) is that the information is to be extracted without

interpolating to the Nyquist grid.

The CS paradigm aims at avoiding high-rate redundant

sampling. The discrete CS framework [14], [15] initiated a

long line of highly inﬂuential works. However, it still remains

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