DETERMINATION OF IDLE TONES IN SD MODULATION BY ERGODIC THEORY

1. DETERMINATION OF IDLE TONES IN SD MODULATION BY ERGODIC THEORY Nguyen T. Thao City University of New York thao@ee.ccny.cuny.edu http://www-ee.engr.ccny.cuny.edu/www/web/thao/nguyen.html Sampta’09, Lumini, France, May 21

2. QUANTIZING FRAME EXPANSIONS where coarsely quantized error:

3. SD MODULATION where Nth order SD coarsely quantized [1] I.Daubechies and R.DeVore, "Reconstructing a bandlimited function from very coarsely quantized data: A family of stable sigma-delta modulators of arbitrary order", Annals of Mathematics, vol. 158, no. 2, pp. 643-674, Sept. 2003. error:

4. COMMON KNOWLEDGE • quantization = nonlinear system • sinusoidal input ==> harmonics • several sinusoids ==> intermodulation

5. COMMON KNOWLEDGE • quantization = nonlinear system • sinusoidal input ==> harmonics • several sinusoids ==> intermodulation

6. SPECTRAL ANALYSIS autocorrelation spectral density

7. SPECTRAL ANALYSIS autocorrelation spectral density

8. SCALAR QUANTIZATION

9. SCALAR QUANTIZATION

10. SCALAR QUANTIZATION where and where (1-periodic) where (1-periodic) If is irrational, has a uniform distribution in and where

11. SCALAR QUANTIZATION Fourier expansion: where and (1-periodic) (1-periodic) If is irrational, has a uniform distribution in and where

12. SCALAR QUANTIZATION where , and

13. INTERMODULATION If the vectors have a uniform distribution in , then where intermodulation product

14. NON-OVERLOADED SD MODULATION but and

15. NON-OVERLOADED SD MODULATION Nth order integrator

16. NON-OVERLOADED SD MODULATION Nth order integrator where linear system state model nonlinear memoryless (1-periodic) where

17. IDEAL SD MODULATION nonlinear memoryless (1-periodic) where

18. CONSTANT INPUT CASE where (1-periodic) where (1-periodic) If the vectors have a uniform distribution in , then more general than

19. TIME-VARYING INPUT CASE where where , and where where If the vectors have a uniform distribution in , then

20. QUESTIONS TO ADDRESS • Assuming that is true, will we have with or • How do we know that the vectors have a • What if the vectors do not have a uniform distribution in ? • What happens if the SD modulator is overloaded ? uniform distribution in ?

21. HILBERT SPACE SETTING where inner-product in (Hilbert space) Suppose that g satisfies In this case, g is an eigenfunction ofUof eigenvaluel where

22. UNITARY OPERATOR • is measure preserving (for every ) • U is a unitary operator • The space Espanned by the eigenfunctions of U yields an orthonormal basis of eigenfunctions with eigenvalues of the form discrete spectrum “continuous” spectrum

23. APPLICATION TO SD MODULATION , where define eigenvalues of U are where intermodulation product extra tone from DC discrete spectrum “continuous” spectrum

24. QUESTIONS TO ADDRESS • Assuming that is true, will we have with or • How do we know that the vectors have a • What if the vectors do not have a uniform distribution in ? • What happens if the SD modulator is overloaded ? uniform distribution in ?

25. where is a transformation of Additional assumption: preserves Lebesgue measure The vectors have a uniform distribution in if and only if T is ergodic with respect to Lebesgue measure ERGODIC THEORY Uniform distribution of where Additional assumption : A such that

26. ERGODIC THEORY Let T be a transformation of and m a probability measure on Definition 1: Tis m-preserving if for every Definition 2: A m-preservingtransformationT is m-ergodic if (up to a set of measure 0) can be achieved only when or Theorem (Birkhoff): If T is m-preserving and m-ergodic, then for any and almost every

27. ERGODIC THEORY Assume that Tis m-preserving. Proposition: The operator is unitary with respect to the inner product The eigenvalues of in the space are of the form Theorem: T is m-ergodic if and only if there exists a unique such that

28. APPLICATION TO SD MODULATION m = Lebesgue measure , eigenvalues of U are where intermodulation product extra tone from DC Theorem: T is m-ergodic if and only if there exists a unique such that if and only if are rationally independent must be irrational cannot be zero

29. RATIONAL DC COMPONENT Assume that (irreducible fraction) Typically is confined to a subset Sd of of the type where and depends on the initial conditions Define m the uniform probability measure in this subset preserves the measure m

30. The eigenvalues of in the space are where intermodulation product extra tone from initial condition RATIONAL DC COMPONENT where and depends on the initial conditions Define m the uniform probability measure in Sd preserves the measure m

31. IDEAL SD MODULATION not overloaded where nonlinear memoryless (1-periodic) where

32. OVERLOADED SD MODULATION 1-bit where nonlinear memoryless (1-periodic) where

33. OVERLOADED SD MODULATION 1-bit where nonlinear memoryless (1-periodic) where

34. OVERLOADED SD MODULATION 1-bit where where (1-periodic)

35. OVERLOADED SD MODULATION same tone frequencies where where (1-periodic)