The Golden Ratio and Signal Quantization

Transcription

1 The Golden Ratio and Signal Quantization Tom Hejda, based on the work of Ingrid Daubechies et al. Doppler Institute & Department of Mathematics, FNSPE, Czech Technical University in Prague Combinatorics on Words, Frysava 2011 Tom Hejda (CTU) Golden Ratio Encoder Frysava '11 1 / 13

2 Abstract Optický nebo zvukový signál je obecn funkce z podmnoºiny R n do R. V p ípad fotograe je deni ní obor celá plocha scény a obor hodnot je jas v daném míst (pro ernobílý obrázek). V p ípad zvukového signálu je deni ní obor as a obor hodnot je amplituda. Pokud chceme signál p evést na digitální data, nutn musíme ztratit n jakou informaci, nebo kone ný objem dat pojme pouze kone né mnoºství hodnot. Omezení deni ního oboru se provádí tzv. samplováním, kdy se vezmou data pouze z kone ného po tu bod, obvykle z m íºky. Omezení oboru hodnot se provádí tzv. kvantizací, kdy z hodnoty nap tí x na výstupu m ícího p ístroje vygenerujeme kone nou posloupnost bit (nul a jedni ek), která p edepsaným zp sobem reprezentuje danou hodnotu nap tí. Práv kvantizací se zabývá tento p ísp vek. B ºn se pouºívá binární kvantizace, kdy se postupn tou jednotlivé bity binárního rozvoje x. Problém v²ak je, ºe jeden chybn p e tený bit nutn znamená chybný výsledek. Toto se pokusíme odstranit pouºítím soustavy o základu zlatý ez τ 1.618, který e²í rovnici τ 2 = τ + 1. S úsp chem vyuºijeme n kolika jeho vlastností: zmín ná rovnice má koecienty pouze 0 a ±1; platí 1 < τ < 2; soustava o základu τ je redundantní ( ísla mají více vyjád ení). Tom Hejda (CTU) Golden Ratio Encoder Frysava '11 2 / 13

3 Motivation Signal: function f : (R n ) (R) Sampling: restriction f : ˆM n (R), where M N Quantization: restriction f : ˆM n nite set Tom Hejda (CTU) Golden Ratio Encoder Frysava '11 3 / 13

4 Motivation Signal: function f : (R n ) (R) Sampling: restriction f : ˆM n (R), where M N Quantization: restriction f : ˆM n nite set Tom Hejda (CTU) Golden Ratio Encoder Frysava '11 3 / 13

5 Motivation Signal: function f : (R n ) (R) Sampling: restriction f : ˆM n (R), where M N Quantization: restriction f : ˆM n nite set Tom Hejda (CTU) Golden Ratio Encoder Frysava '11 3 / 13

6 Encoder Denition A/D Encoder is a device that takes a signal (voltage) and gives its (some) representation. Theory: Encoder : R + A N Reality: Encoder : X = interval A N Tom Hejda (CTU) Golden Ratio Encoder Frysava '11 4 / 13

7 Encoder Denition A/D Encoder is a device that takes a signal (voltage) and gives its (some) representation. Theory: Encoder : R + A N Reality: Encoder : X = interval A N Tom Hejda (CTU) Golden Ratio Encoder Frysava '11 4 / 13

8 Encoder Denition A/D Encoder is a device that takes a signal (voltage) and gives its (some) representation. Theory: Encoder : R + A N Reality: Encoder : X = interval A N Tom Hejda (CTU) Golden Ratio Encoder Frysava '11 4 / 13

9 Technology q τ (u) := { 1 u τ 0 u < τ Tom Hejda (CTU) Golden Ratio Encoder Frysava '11 5 / 13

10 Technology q τ (u) := { 1 u τ 0 u τ Reality: no this component is accurate Tom Hejda (CTU) Golden Ratio Encoder Frysava '11 5 / 13

11 Common Encoders Binary encoder x = lim N b n 2 n N n=0 + Exponential accuracy, error = O ( 2 N) Not robust (with respect to parameters) Recursive process: x X = [0, 2); u 0 := x; b n := q(u n ); u n+1 := 2(u n b n ); Tom Hejda (CTU) Golden Ratio Encoder Frysava '11 6 / 13

12 Common Encoders Σ -encoder x = lim N 1 N N n=0 b n + Robust (with respect to multiplication / quantization errors) Low accuracy, error = O (1/N) Recursive process: x X = [0, 1); u 0 (0, 1) arbitrary; b n := q(u n + x); u n+1 := u n + x b n ; Tom Hejda (CTU) Golden Ratio Encoder Frysava '11 7 / 13

13 GRE Golden ratio encoder Golden ratio encoder: Golden mean φ = 1+ 5 satises φ 2 = φ N x = lim b n φ n N k=0 + Robust (with respect to multiplication / quantization errors) + Exponential accuracy: error = O ( φ N) Recursive process: T Q : [ un x X = [0, 2); u 0 := x, u 1 := 0; u n+2 := u n+1 + u n b n ; b n := Q(u n, u n+1 ), Q : R 2 A = {0, 1}. u n+1 ] [ un+1 u n+2 ] = [ ] [ ] 0 1 un Q(u n, u n+1 ) 1 1 u n+1 [ ] 0 1 Tom Hejda (CTU) Golden Ratio Encoder Frysava '11 8 / 13

14 GRE Golden ratio encoder Golden ratio encoder: Golden mean φ = 1+ 5 satises φ 2 = φ N x = lim b n φ n N k=0 + Robust (with respect to multiplication / quantization errors) + Exponential accuracy: error = O ( φ N) Recursive process: T Q : [ un x X = [0, 2); u 0 := x, u 1 := 0; u n+2 := u n+1 + u n b n ; b n := Q(u n, u n+1 ), Q : R 2 A = {0, 1}. u n+1 ] [ un+1 u n+2 ] = [ ] [ ] 0 1 un Q(u n, u n+1 ) 1 1 u n+1 [ ] 0 1 Tom Hejda (CTU) Golden Ratio Encoder Frysava '11 8 / 13

15 GRE Golden ratio encoder x X = [0, 2); u 0 := x, u 1 := 0; u n+2 := u n+1 + u n + b n ; b n := Q(u n, u n+1 ) Proposition Let x [0, 2) and dene b n and u n as above. Then x = b n φ n if and only if the seqence (u n ) is bounded. What properties we needed for the proof? β satises β d = a d 1 β d a 1 β + a 0 with a i { 1, 0, 1} β (1, 2) n=0 Tom Hejda (CTU) Golden Ratio Encoder Frysava '11 9 / 13

16 GRE Golden ratio encoder x X = [0, 2); u 0 := x, u 1 := 0; u n+2 := u n+1 + u n + b n ; b n := Q(u n, u n+1 ) Proposition Let x [0, 2) and dene b n and u n as above. Then x = b n φ n if and only if the seqence (u n ) is bounded. What properties we needed for the proof? β satises β d = a d 1 β d a 1 β + a 0 with a i { 1, 0, 1} β (1, 2) n=0 Tom Hejda (CTU) Golden Ratio Encoder Frysava '11 9 / 13

17 Quantizer Q Parameter α: Q α (u n, u n+1 ) := q(u n + αu n+1 ) Choice α = 1: no multipliers Proposition Let T Q1 as above and R Q1 := [0, 1] 2. Then T Q1 (R Q1 ) = R Q1. Accuracy of multiplier: α α Accuracy of quantizer: aky bit-quantizer q ν 1,ν 2, ν 1 < ν 2 : 0, if u < ν 1, q ν 1,ν 2 (u) := 1, if u > ν 2, 0 or 1, if u [ν 1, ν 2 ] Q 1 is not robust to aky q ν 1,ν 2, because (u n ) can get unbounded no matter how close ν 1,2 are to 1. Tom Hejda (CTU) Golden Ratio Encoder Frysava '11 10 / 13

18 Quantizer Q Parameter α: Q α (u n, u n+1 ) := q(u n + αu n+1 ) Choice α = 1: no multipliers Proposition Let T Q1 as above and R Q1 := [0, 1] 2. Then T Q1 (R Q1 ) = R Q1. Accuracy of multiplier: α α Accuracy of quantizer: aky bit-quantizer q ν 1,ν 2, ν 1 < ν 2 : 0, if u < ν 1, q ν 1,ν 2 (u) := 1, if u > ν 2, 0 or 1, if u [ν 1, ν 2 ] Q 1 is not robust to aky q ν 1,ν 2, because (u n ) can get unbounded no matter how close ν 1,2 are to 1. Tom Hejda (CTU) Golden Ratio Encoder Frysava '11 10 / 13

19 Quantizer Q Parameter α: Q α (u n, u n+1 ) := q(u n + αu n+1 ) Choice α = 1: no multipliers Proposition Let T Q1 as above and R Q1 := [0, 1] 2. Then T Q1 (R Q1 ) = R Q1. Accuracy of multiplier: α α Accuracy of quantizer: aky bit-quantizer q ν 1,ν 2, ν 1 < ν 2 : 0, if u < ν 1, q ν 1,ν 2 (u) := 1, if u > ν 2, 0 or 1, if u [ν 1, ν 2 ] Q 1 is not robust to aky q ν 1,ν 2, because (u n ) can get unbounded no matter how close ν 1,2 are to 1. Tom Hejda (CTU) Golden Ratio Encoder Frysava '11 10 / 13

20 Quantizer Q Parameter α: Q α (u n, u n+1 ) := q(u n + αu n+1 ) Choice α = 1: no multipliers Proposition Let T Q1 as above and R Q1 := [0, 1] 2. Then T Q1 (R Q1 ) = R Q1. Accuracy of multiplier: α α Accuracy of quantizer: aky bit-quantizer q ν 1,ν 2, ν 1 < ν 2 : 0, if u < ν 1, q ν 1,ν 2 (u) := 1, if u > ν 2, 0 or 1, if u [ν 1, ν 2 ] Q 1 is not robust to aky q ν 1,ν 2, because (u n ) can get unbounded no matter how close ν 1,2 are to 1. Tom Hejda (CTU) Golden Ratio Encoder Frysava '11 10 / 13

21 Quantizer Q: Choices α > 1 Proposition Let 0 µ (2φ 2 (φ + 2)) Then exists set R µ and ranges of choices of α, ν 1, ν 2 such that T ν Q 1,ν 2 (R α µ ) R µ + B µ. Imagine ugly inequalities for α min,max, ν min,max (µ) here. Additive noise: u n+2 := u n + u n+1 b n + ɛ n Theorem Let µ, α min,max (µ) as above. For every α (α min (µ), α max (µ)) there exists ν 1 < ν 2 and η > 0 such that GRE with Q ν 1,ν 2 α is stable for α α < η and ɛ n < µ. Stable encoder = the encoding converges, but the string (b n ) need not to represent x. Tom Hejda (CTU) Golden Ratio Encoder Frysava '11 11 / 13

22 Quantizer Q: Choices α > 1 Proposition Let 0 µ (2φ 2 (φ + 2)) Then exists set R µ and ranges of choices of α, ν 1, ν 2 such that T ν Q 1,ν 2 (R α µ ) R µ + B µ. Imagine ugly inequalities for α min,max, ν min,max (µ) here. Additive noise: u n+2 := u n + u n+1 b n + ɛ n Theorem Let µ, α min,max (µ) as above. For every α (α min (µ), α max (µ)) there exists ν 1 < ν 2 and η > 0 such that GRE with Q ν 1,ν 2 α is stable for α α < η and ɛ n < µ. Stable encoder = the encoding converges, but the string (b n ) need not to represent x. Tom Hejda (CTU) Golden Ratio Encoder Frysava '11 11 / 13

23 Quantizer Q: Choices α > 1 Proposition Let 0 µ (2φ 2 (φ + 2)) Then exists set R µ and ranges of choices of α, ν 1, ν 2 such that T ν Q 1,ν 2 (R α µ ) R µ + B µ. Imagine ugly inequalities for α min,max, ν min,max (µ) here. Additive noise: u n+2 := u n + u n+1 b n + ɛ n Theorem Let µ, α min,max (µ) as above. For every α (α min (µ), α max (µ)) there exists ν 1 < ν 2 and η > 0 such that GRE with Q ν 1,ν 2 α is stable for α α < η and ɛ n < µ. Stable encoder = the encoding converges, but the string (b n ) need not to represent x. Tom Hejda (CTU) Golden Ratio Encoder Frysava '11 11 / 13

24 Conclusion We constructed a A/D encoder with: 1 exponential accuracy 2 robustness with respect to multiplication and quantization 3 the usage of a Golden ratio :--) Tom Hejda (CTU) Golden Ratio Encoder Frysava '11 12 / 13

25 References 1 I. Daubechies; Ö. Ylmaz. Robust and practical analog-to-digital conversion with exponential precision. IEEE Trans. Inform. Theory 52 (2006), no. 8, I. Daubechies, C.S. Güntürk, Y. Wang, Ö. Ylmaz. The Golden Ratio Encoder. arxiv: , 7 Sep 2008 Tom Hejda (CTU) Golden Ratio Encoder Frysava '11 13 / 13