36e8ceaInteractive explorer for compressibility of quantized filtered Gaussian time seriesJeremy Magland 1import { useMemo } from 'react'
2import katex from 'katex'
89a240bRewrite the math section as the formula actually computed, with every term definedJeremy Magland 4function Tex({ tex, display }: { tex: string; display?: boolean }) {
5 const html = useMemo(
6 () => katex.renderToString(tex, { displayMode: !!display, throwOnError: false }),
7 [tex, display],
8 )
9 return <span dangerouslySetInnerHTML={{ __html: html }} />
10}
12/** One term of the formula: the symbol, then what it is. */
13function Def({ tex, children }: { tex: string; children: React.ReactNode }) {
14 return (
15 <>
16 <dt>
17 <Tex tex={tex} />
18 </dt>
19 <dd>{children}</dd>
20 </>
21 )
36e8ceaInteractive explorer for compressibility of quantized filtered Gaussian time seriesJeremy Magland 22}
24/**
89a240bRewrite the math section as the formula actually computed, with every term definedJeremy Magland 25 * The theoretical rate R exactly as `model/theory.ts` computes it, with every
26 * symbol defined. The derivation that justifies it is still to be written.
36e8ceaInteractive explorer for compressibility of quantized filtered Gaussian time seriesJeremy Magland 27 */
28export default function MathSection() {
29 return (
30 <div className="math-section">
31 <p>
89a240bRewrite the math section as the formula actually computed, with every term definedJeremy Magland 32 The dashed line on the compression chart is R, the predicted bits per sample. It is
33 computed in three steps: the spectrum of the stored signal, the residual an ideal
34 predictor leaves, and the entropy of that residual on the integer grid.
36e8ceaInteractive explorer for compressibility of quantized filtered Gaussian time seriesJeremy Magland 35 </p>
89a240bRewrite the math section as the formula actually computed, with every term definedJeremy Magland 36
37 <Tex display tex="S_z(f) \;=\; \sigma^2\,\big|H(f)\big|^2 \;+\; \sigma_q^2, \qquad H(f) \;=\; \sum_{n=0}^{L-1} h_n\, e^{-2\pi i f n}" />
38 <Tex display tex="\sigma_e \;=\; 2^{\,\int_0^{1/2} \log_2 S_z(f)\,df}" />
39 <Tex display tex="R \;=\; -\sum_{z \in \mathbb{Z}} p_z \log_2 p_z, \qquad p_z \;=\; \Phi\!\left(\frac{z + \tfrac12}{\sigma_e}\right) - \Phi\!\left(\frac{z - \tfrac12}{\sigma_e}\right)" />
41 <dl className="defs">
42 <Def tex="\sigma">
43 standard deviation of the i.i.d. Gaussian input, in quantization steps (the step is the
44 unit, so rounding is to the nearest integer)
45 </Def>
46 <Def tex="h_0,\dots,h_{L-1}">
47 the FIR kernel the input is convolved with — the taps drawn in the kernel plot, L of
48 them
49 </Def>
50 <Def tex="H(f)">
51 the kernel's frequency response, the quantity plotted in dB as |H(f)|
52 </Def>
53 <Def tex="f">
54 frequency in cycles per sample, running from 0 to ½ (Nyquist); the plots label the same
55 axis in Hz, as f times the sample rate
56 </Def>
57 <Def tex="\sigma_q^2">
58 variance charged to rounding, treated as additive white noise: 1/12 for the roundoff
59 alone, 1/6 when dither is on (the dither is stored in the integers, so its 1/12 adds)
60 </Def>
61 <Def tex="S_z(f)">
62 power spectrum of the stored integer signal, in steps² per unit frequency
63 </Def>
64 <Def tex="\sigma_e">
65 standard deviation of the innovation — what an ideal linear predictor still cannot
66 predict from all earlier samples. The exponent is the Szegő–Kolmogorov formula for the
67 one-step prediction error, the geometric mean of the spectrum.
68 </Def>
69 <Def tex="\Phi">standard normal cumulative distribution function</Def>
70 <Def tex="p_z">
71 probability that the innovation, rounded to the integer grid, lands on z
72 </Def>
73 <Def tex="R">
74 bits per sample; the compression ratio the chart marks is 16/R, against 16-bit integer
75 storage
76 </Def>
77 </dl>
79 <p className="card-note">
80 The integral is evaluated by the midpoint rule on 8192 points and the sum over z is taken
81 out to where the remaining mass is negligible. A derivation — and an account of where
82 modeling the roundoff as white noise stops being fair — is still to be written.
36e8ceaInteractive explorer for compressibility of quantized filtered Gaussian time seriesJeremy Magland 83 </p>
84 </div>
85 )
86}