Distributed Noise-Shaping Quantization: II. Classical Frames

Abstract

This chapter constitutes the second part in a series of papers on distributed noise-shaping quantization. In the first part, the main concept of distributed noise shaping was introduced and the performance of distributed beta encoding coupled with reconstruction via beta duals was analyzed for random frames (Chou and Güntürk, Constr Approx 44(1):1–22, 2016). In this second part, the performance of the same method is analyzed for several classical examples of deterministic frames. Particular consideration is given to Fourier frames and frames used in analog-to-digital conversion. It is shown in all these examples that entropic rate-distortion performance is achievable.

Appendix: Greedy Quantizer for Complex Measurements

In this section we will provide a generalization of the complex-valued ΣΔ quantization algorithm given in [3, Proposition 3.1] and the greedy noise-shaping quantization algorithm given in [6, Theorem  2.1]. The result, which is applicable to both real and complex quantization alphabets, offers nontrivial improvements in the complex case, thanks to the use of general semi-norms to measure closeness.

Lemma 2

Let $\mathscr{A}$ be a quantization alphabet in $\mathbb{C}$ , B _∗ be the closed unit ball of a semi-norm | ⋅ |_∗ on $\mathbb{C}$ treated as a vector space over $\mathbb{R}$ , and H: = (H _n, m ) _{n, m ∈ [N]} be an N × N real-valued lower-triangular matrix with unit diagonal. Suppose there exist positive real numbers μ, δ, γ such that 

$$\displaystyle{ \delta B_{{\ast}} +\mathscr{ A} \supset \gamma B_{{\ast}} }$$

(27)

 and 

$$\displaystyle{ \mu +\delta \max _{n\in [N]}\sum _{m<n}\vert H_{n,m}\vert \leq \gamma. }$$

(28)

 Then for any $y \in \mathbb{C}^{N}$ such that | y _n |_∗ ≤ μ for all n ∈ [N], there exist $q \in \mathscr{ A}^{N}$ and $u \in \mathbb{C}^{N}$ such that 

$$\displaystyle{y - q = Hu}$$

 where | u _n |_∗ ≤ δ for all n ∈ [N].

Proof

The proof of this result is yet another adaptation of a well-known induction argument. By our assumption on H, we are seeking to satisfy the equations

$$\displaystyle{ u_{n} = \left (y_{n} -\sum _{m<n}H_{n,m}u_{m}\right ) - q_{n} }$$

(29)

for all n ∈ [N].

Since | y ₀ |_∗ ≤ μ ≤ γ, (27) implies that there exist $q_{0} \in \mathscr{ A}$ and u ₀ ∈ δB _∗ such that u ₀ + q ₀ = y ₀. Hence (29) is satisfied for n = 0 and | u ₀ |_∗ ≤ δ.

For the induction step, assume that | u _m |_∗ ≤ δ for all m < n, and let

$$\displaystyle{w_{n}:= y_{n} -\sum _{m<n}H_{n,m}u_{m}.}$$

Using sub-additivity and homogeneity of | ⋅ |_∗ followed by the condition given in (28), we get

$$\displaystyle{\vert w_{n}\vert _{{\ast}}\leq \mu +\delta \sum _{m<n}\vert H_{n,m}\vert \leq \gamma;}$$

hence, because of (27) again, there exist $q_{n} \in \mathscr{ A}$ and u _n ∈ δB _∗ such that u _n + q _n = w _n , i.e. (29) holds. □

Special known cases. There are certainly many ways to choose $\mathscr{A}$ and | ⋅ |_∗. We first note two important special cases of practical importance. Here L denotes $\vert \mathscr{A}\vert$.

• $(\mathbb{R})$ Real arithmetic progression

  This quantizer uses $\mathscr{A}:=\mathscr{ A}_{L,\delta }:=\{ (-L + 2l - 1)\delta: 1 \leq l \leq L\} \subset \mathbb{R}$, i.e. the origin-symmetric arithmetic progression of length L and spacing 2δ along with $\vert z\vert _{{\ast}}:= \vert \mathfrak{R}(z)\vert$. Then B _∗ is the infinite vertical strip $\{z: \vert \mathfrak{R}(z)\vert \leq 1\}$ and (27) holds for γ: = Lδ. Using the algorithm in Lemma 2, $y \in \mathbb{R}^{N}$ results in $u \in \mathbb{R}^{N}$, and ∥y∥ _∞ ≤ μ implies ∥u∥ _∞ ≤ δ so that the setup becomes identical to that of [6].

• $(\mathbb{C})$ Complex square lattice quantizer

  This quantizer assumes L = K ² for some positive integer K and sets $\mathscr{A}:=\mathscr{ A}_{K,\delta } + i\mathscr{A}_{K,\delta } \subset \mathbb{C}$ along with $\vert z\vert _{{\ast}}:=\max (\vert \mathfrak{R}(z)\vert,\vert \mathfrak{I}(z)\vert )$. B _∗ can be identified with [−1, 1]² (as a subset $\mathbb{R}^{2}$) so that (27) is valid for γ: = Kδ. Since $\vert z\vert _{{\ast}}\leq \vert z\vert \leq \sqrt{2}\vert z\vert _{{\ast}}$ for any $z \in \mathbb{C}$, ∥y∥ _∞ ≤ μ implies | y _n |_∗ ≤ μ for all n and Lemma 2 then yields $\|u\|_{\infty }\leq \sqrt{2}\delta$.

  When K is even, the resulting $\mathscr{A}$ has no real points and it may be desirable to require that $y \in \mathbb{R}^{N}$ always yields $q \in \mathbb{R}^{N}$. In this case, we may instead use the slightly larger alphabet $\mathscr{A}:=\mathscr{ A}_{K,\delta } + i\mathscr{A}_{K+1,\delta }$ for which L = K(K + 1). This choice indeed corresponds to the one made in [3]. Another natural possibility in this case is to use the 1-norm in $\mathbb{R}^{2}$ coupled with the diamond lattice as shown in Figure 1 for K = 2.

  Fig. 1

  

  Lattice covering for the 1-norm in $\mathbb{R}^{2}$ where L = 4 and γ = 2δ.

  Note that for the square (or diamond) lattice quantizer of K ² levels, using the Euclidean norm | ⋅ | on $\mathbb{C}$ would be sub-optimal. Indeed, the largest value of γ that can be used in (27) is $\gamma = \frac{K} {\sqrt{2}}\delta$.

Hexagonal norm for a tri-level complex alphabet. It is natural to ask if a complex quantization alphabet $\mathscr{A}$ with fewer than 4 levels can be used in connection with the noise-shaping quantization algorithm of Lemma 2. For L = 3, we may set $\mathscr{A}$ to be the vertices of an equilateral triangle in $\mathbb{C}$ centered at the origin. If the Euclidean norm is used, then it is not difficult to prove that the largest value of γ that can be used in (27) is $\gamma = \frac{2} {\sqrt{3}}\delta$ (see Figure 2 for a demonstration of this covering). In this case, ∥y∥ _∞ ≤ μ yields ∥u∥ _∞ ≤ δ.

Fig. 2

Three identical hexagons at scale δ covering a larger hexagon at scale $\gamma = \frac{4} {3}\delta$ compared to three identical circular discs at scale δ covering a larger circular disc at scale $\rho = \frac{2} {\sqrt{3}}\delta$.

An alternative we have found useful is to employ the norm | ⋅ |_∗ induced by a regular hexagonal body whose sides are aligned with the sides of the triangle. Then, as shown in Figure 2, we can attain $\gamma = \frac{4} {3}\delta$. By choosing the scale of the hexagonal body suitably, we can ensure $\vert z\vert _{{\ast}}\leq \vert z\vert \leq \frac{2} {\sqrt{3}}\vert z\vert _{{\ast}}$ so that ∥y∥ _∞ ≤ μ implies | y _n |_∗ ≤ μ for all n, and therefore Lemma 2 yields $\|u\|_{\infty }\leq \frac{2} {\sqrt{3}}\delta$. Despite the increase in the bound for ∥u∥ _∞ , there is a sizable gain in the “expansion factor” γ∕δ from $\frac{2} {\sqrt{3}}$ to $\frac{4} {3}$. This gain is crucial for beta encoding because any β up to this expansion factor is admissible for stability via Lemma 2 provided $\mathscr{A}$, γ, and δ are suitably scaled to meet (27) and (28) simultaneously.