现代信号处理：新技术与算法进展

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"现代信号处理" "现代信号处理"是一本基于数学科学研究所研究生课程讲座编写的书籍，专注于探讨信号处理的数学基础及其广泛的应用领域。本书着重于当前的技术挑战，新适应新技术的方法，以及在算法和理论方面的一些近期进展。在信号处理这个无处不在的现代科技领域中，其数学基础和应用是该书的核心。全书基于数学科学研究所的一系列研究生讲座，旨在揭示实际数据中固有对称性的高效分析技术——计算谐波分析，并深入研究图像和一般感知数据的获取、处理和分析所面临的难题。这涵盖了从潜艇的声纳到神经科学家的fMRI研究等一系列应用场景。书中涉及的两大主要领域如下： 1. 计算谐波分析：这是一种技术，目的是利用数据的内在对称性进行有效分析。在现实世界的数据处理中，这通常涉及到复杂信号的分解和特征提取，以便更好地理解数据的结构和模式。例如，在音频处理中，它可以用于识别音乐中的不同乐器或人声；在图像处理中，可以用于检测和识别物体的形状和纹理。 2. 图像和感知数据的处理与分析：这一部分涵盖了从原始数据采集到复杂分析的全过程。无论是遥感图像、医学影像还是其他形式的传感器数据，处理和分析这些数据都需要高级的算法和技术。例如，fMRI（功能性磁共振成像）数据的处理涉及将大脑活动转化为可理解的图像，这对于神经科学研究至关重要。同样，声纳系统在海洋探索或军事应用中，需要对水下环境进行精确的探测和成像。本书不仅介绍了这些领域的基本概念，还讨论了最新进展，如机器学习和深度学习在信号处理中的应用，以及如何利用这些技术改进数据建模和预测能力。此外，它还涵盖了与算法效率相关的优化问题，这对于在大数据时代高效处理和解析信号至关重要。 "现代信号处理"为读者提供了一个深入了解信号处理理论与实践的平台，对于希望在这个快速发展的领域深化理解或从事相关工作的研究生和专业人士来说，是一份宝贵的资源。通过阅读本书，读者可以了解到如何运用数学工具解决实际问题，同时也能把握信号处理领域的最新动态和发展趋势。

8 JEFFERY C. ALLEN AND DENNIS M. HEALY, JR.

generator is transformed to various new reﬂectances in the hyperbolic disk un-

der the action of the possible matching circuits. We look for the closest approach

of this orbit to the load s

with respect to the (pseudo) hyperbolic metric. The

last bound is reducible to a matrix calculation by a hyperbolic version of Ne-

hari’s Theorem [42], a classic result relating analytic approximation to an oper-

ator norm calculation. The resulting Nehari bound gives the circuit designer an

upper limit on the possible performance for any class U ⊆ U

(2) of matching

circuits. For some classes, this bound is tight, telling the circuit designer that

the benchmark is essentially obtainable with matching circuits from the speciﬁed

class. For example, when U is the class of all lumped lossless 2-ports (networks

of discrete inductors and capacitors)

(2, ∞) :=

[

d≥0

(2, d)

and s

= 0, Darlington’s Theorem establishes that

sup{kG

= 0, S, s

−∞

: S ∈ U

(2, ∞)}

= 1 − inf{k∆P (s

, s

∞

: s

∈ BH

∞

provided s

is suﬃciently smooth. In this case, the circuit designer knows that

there are lumped, lossless 2-ports that get arbitrarily close to the Nehari bound.

The limitation of this approach is the requirement that the generator reﬂectance

= 0, which is not always true. Thus, a go od research topic is to relax this

constraint, or to generalize Darlington’s Theorem. Another limitation of the

techniques described in this paper is that the Nehari methods produce only a

bound — they do not supply the matching circuit. However, the techniques do

compute the optimal s

, leading to another excellent research topic — the “uni-

tary dilation” of s

to a scattering matrix with s

= s

. That such substantial

research topics naturally arise shows how an applied problem brings depth to

mathematical investigations.

3. Technical Preliminaries

The real numbers are denoted by R. The complex numbers are denoted by

C. The set of complex M ×N matrices is denoted by C

M×N

. I

and 0

denote

the N ×N identity and zero matrices. Complex frequency is written p = σ + jω.

The open right-half plane is denoted by C

:= {p ∈ C : Re[p] > 0}. The open

unit disk is denoted by D and the unit circle by T.

3.1. Function spaces.

• L

∞

(jR) denotes the class of Lebesgue-measurable functions deﬁned on jR

with norm kφk

∞

:= ess.sup{|φ(jω)| : ω ∈ R}.

• C

(jR) denotes the subspace of those continuous functions on jR that vanish

at ±∞ with sup norm.

10 JEFFERY C. ALLEN AND DENNIS M. HEALY, JR.

Proof. It suﬃces to show closure. If {S

} ⊂ U

(N) converges to S ∈

∞

, C

N×N

), then S

(jω) → S(jω) almost everywhere so that

= lim

m→∞

(jω)

(jω) = S(jω)

S(jω) a.e.

That is, S(jω) is unitary almost everywhere or S ∈ U

(N). ¤

3.4. The weak-∗ topology. We use the weak-∗ top ology on L

∞

(jR) =

(jR)

∗

. A weak-∗ subbasis at 0 ∈ L

∞

(jR) is the collection of weak-∗ open sets

O[w, ε] := {φ ∈ L

∞

(jR) : |hw, φi| < ε},

where ε > 0, w ∈ L

(jR), and

hw , φi :=

∞

−∞

w(jω)φ(jω)dω.

Every weak-∗ open set that contains 0 ∈ L

∞

(jR) is a union of ﬁnite intersections

of these subbasic sets. The Banach–Alaoglu Theorem [47, Theorem 3.15] gives

that the unit ball BL

∞

(jR) is weak-∗ compact. The next lemma shows that the

same holds for a distorted version of the unit ball, a fact that will have signiﬁcant

import for the optimization problems we consider later.

Lemma 3.2. Let c, r ∈ L

∞

(jR) with r ≥ 0 deﬁne the disk

D(c, r) := {φ ∈ L

∞

(jR) : |φ − c| ≤ r a.e.}.

Then D(c, r) a closed, convex subset of L

∞

(jR) that is also weak-∗ compact.

Proof. Closure and convexity follow from pointwise closure and convexity.

To prove weak-∗ compactness, let M

: L

∞

(jR) → L

∞

(jR) be multiplication:

φ := rφ. Observe D(k, r) = k + M

∞

(jR). Assume for now that M

weak-∗ continuous. Then M

∞

(jR) is weak-∗ compact, because BL

∞

(jR)

is weak-∗ compact, and the image of a compact set under a continuous function

is compact. This forces D(k, r) to be weak-∗ compact, provided M

is weak-∗

continuous. To see that M

is weak-∗ continuous, it suﬃces to shows that M

pulls subbasic sets back to subbasic sets. Let ε > 0, w ∈ L

(jR). Then

ψ ∈ M

−1

(O[w, ε])M

ψ ∈ O[w, ε ] ⇐⇒ |hw, rψi| < ε

⇐⇒ |hrw, ψi| < ε ⇐⇒ ψ ∈ O[rw, ε],

noting that rw ∈ L

(jR). ¤

If K is a convex subset L

∞

(jR), then K is closed ⇐⇒ K is weak-∗ closed [17,

page 422]. Because H

∞

) is a closed subspace of L

∞

), is it also weak-∗

closed. Intersecting weak-∗ closed H

∞

) with the weak-∗ compact unit ball

of L

∞

(jR) forces BH

∞

) to be weak-∗ compact.

HYPERBOLIC GEOMETRY, NEHARI’S THEOREM, ELECTRIC CIRCUITS 11

3.5. The Cayley transform. Many computations are more conveniently

placed in function spaces deﬁned on the open unit disk D rather than on the

open right half-plane C

. The notation for the spaces on the disk follows the

preceeding nomenclature with the unit disk D replacing C

and the unit circle

T replacing jR. H

∞

(D) denotes the collection of analytic functions on the

open unit disk with essentially bounded boundary values. C(T) denotes the

continuous functions on the unit circle, A(D) := H

∞

(D)∩C(T) denotes the disk

algebra, and L

∞

(T) denotes the Lebesgue-measurable functions on the unit circle

T with norm determined by the essential bound. A Cayley transform connects

the function spaces on the right half plane to their counterparts on the disk.

Lemma 3.3 ([27, page 99]). Let the Cayley transform c : C

→ D

c(p) :=

p − 1

p + 1

extend to the composition operator c : L

∞

(T) → L

∞

(jR) as

h(p) := H ◦ c(p) (p = jω).

Then c is an isometry mapping











A(D)

∞

(D)

C(T)

∞

(T)











onto











)

∞

)

(jR)

∞

(jR)











3.6. Factoring H

∞

functions. The boundary values and inner-outer factor-

ization of H

∞

functions are notions most conveniently developed on the unit

disk and then transplanted to the right half-plane by the Cayley transform [35].

Let φ ∈ L

(T) have the Fourier expansion in z = exp(jθ)

φ(z) =

∞

n=−∞

φ(n)z

;

φ(n) :=

−π

−jnθ

φ(e

jθ

)

dθ

2π

For 1 ≤ p ≤ ∞, deﬁne H

(D) as the subspace of L

(T) with vanishing negative

Fourier coeﬃcients [27, page 77]:

(D) := {h ∈ L

(T) :

h(n) = 0 for n = −1, −2, . . . }.

Then H

(D) is a closed subspace of L

(T) and as [27, page 3]:

∞

(T) ⊂ H

(T) (1 ≤ p

≤ p

≤ ∞)

Each h ∈ H

(D) admits an analytic extension on the open unit disk [27, p. 77]:

h(z) =

∞

n=0

h(n)z

(z = re

jθ

From the analytic extension, deﬁne h

jθ

) := h(re

jθ

) for 0 ≤ r ≤ 1. For r < 1,

is continuous and analytic. As r increases to 1, h

converges to h in the L

norm, provided 1 ≤ p < ∞. For p = ∞, h

converges to h in the weak-∗ topology

12 JEFFERY C. ALLEN AND DENNIS M. HEALY, JR.

(discussed on page 10). If h

does converge to h in the L

∞

norm, convergence

is uniform and forces h ∈ A(D). Although disk algebra A(D) is a strict subset

of H

∞

(D) in the norm topology, it is a weak-∗ dense subset.

If φ is a positive, measurable function with log(φ) ∈ L

(T) then the analytic

function [48, page 370]:

q(z) = exp

−π

+ z

− z

log |φ(e

2π

(z ∈ D),

is called an outer function. The magnitude of q(z) matches φ [48, page 371]:

lim

r→1

(re

jθ

)| = φ(re

jθ

) (a.e.)

and leads to the equivalence: φ ∈ L

(T) ⇐⇒ q ∈ H

(D). We call q(z) a spectral

factor of φ. Every h ∈ H

∞

(D) admits an inner-outer factorization [48, pages

370-375]:

h(z) = e

jθ

b(z)s(z)q(z),

where the outer function q(z) is a spectral factor of |h| and the inner function

consists of the Blaschke product [48, page 333]

b(z) := z

∞

n=1

− z

1 − ¯z

¯z

6= 0,

(1 − |z

|) < ∞, and the singular inner function

s(z) = exp

−

−π

+ z

− z

dµ(t)

for µ a ﬁnite, positive, Borel measure on T that is singular with respect to

the Lebesgue measure. In the electrical engineering setup, we will see that the

Blaschke products correspond to lumped, lossless circuits while a transmission

line corresponds to a singular inner function.

4. Electric Circuits

The impedance matching problem may be formulated as an optimization of

certain natural ﬁgures of merit over structured sets of candidate electrical match-

ing networks. We begin the formulation in this section, starting with an ex-

amination of the sorts of electrical networks available for impedance matching.

Consideration of various choices of coordinate systems parameterizing the set of

candidate matching circuits leads to the scattering formalism as the most suit-

able choice. Next we consider appropriate objective functions for measuring the

utility of a candidate impedance matching circuit. This leads to description and

characterization of power gain and mismatch functions as natural indicators of

the suitability of our circuits. With the objective function and the parameteriza-

tion of the admissible candidate set, we are in position to formulate impedance

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