The mathematician Ramanujan, and digital signal processing

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1 The mathematician Ramanujan, and digital signal processing P. P. Vaidyanathan California Institute of Technology, Pasadena, CA SAM-2016, Rio, Brazil

1887-1920 Self-educated mathematician from India Grew up in poverty in Tamil Nadu

2 Self-educated mathematician from India Grew up in poverty in Tamil Nadu His genius was discovered by G. H. Hardy Worked with Hardy in

3 Outline Ramanujan sums (RS): 1918 Representing periodic signals New integer bases, integer projections Ramanujan periodic dictionaries Bounds on period estimation Conclusions

4 Motivation Consider a periodic signal period data size N DFT representation: Example: N = 32; q = 1, 2, 4, 8, 16, 32 Very few periods represented period N or a divisor of N Ramanujan-sum representation: Example: N = 32; q = 1, 2, 3, 4, 5, 32 Every period is represented! period q

5 First, how much can DFT do? Sinusoid example: (Re part) (Re part) 32 point DFT 32 point DFT

6 Identifying periods: Not the same as spectrum estimation arbitrary line spectrum ω DFT, MUSIC, etc., will work, but are not necessarily best Ramanujan offers something new periodic case fundamental fewer parameters of interest ω

7 Hidden periodic components period = 16 period = 12 Does not look periodic Sparse representation? Ramanujan offers something new

8 Importance of periodic components Pitch identification acoustics (music, speech, ) Time delay estimation in sensor arrays Medical applications Genomics and proteomics Radar Astronomy

9 Notations = gcd of k and q means k and q are coprime qth root of unity = Euler s totient function : d is a divisor (or factor) of q

10 The phi-function (Euler s totient):

11 Ramanujan sum (1918) sum over coprime frequencies only q = positive integer has period exactly q Euler s totient

12 Equivalent definition using DFT 6

13 Frequency and period in Ramanujan sum each term has period exactly q φ(q) frequency components: primitive frequencies DFT grid 0 Primitive frequencies, all with SAME period q

14 Other ways to write Ramanujan sum: Orthogonality:

15 Theorem: Ramanujan sum is integer valued! Examples: very useful from a computational perspective

16 Ramanujan-sum representation: Ramanujan sum What did Ramanujan do with Ramanujan sums? What did DSP people do with them? What did PP do with them? Does it solve the limitations of DFT? New directions

17 Ramanujan expanded arithmetic functions (1918) Number-of-divisors: Sum-of-divisors: Euler-totient: von Mangoldt function:

18 Some references from DSP

19 References for this talk

20 An example with Signal duration N = 2^8 = 256 x(n) = x(n) = sparse representation not sparse

21 Another example with Again, signal duration N = 2^8 = 256 x(n) = sparse representation x(n) = not sparse

22 Reason for the problem: What to do about it? each period is represented by a 1D subspace only Consider the space spanned by We call this the Ramanujan subspace Every nonzero signal in this space has period EXACTLY q We will use the entire susbpace to represent period q components of x(n) PPV, 2014, IEEE SP Trans.

23 Ramanujan subspace is spanned by: Theorem (PPV, SP Trans. 2014): This space has dimension exactly! This space is also spanned by: orthogonal for

24 Basis for Ramanujan subspace real integer basis cyclic basis complex basis from DFT

25 Theorem: Any length N signal can be represented as Orthogonal periodic components Integer basis using Ramanujan sum Doubly indexed basis periodic components in PPV, SP Trans. 2014

26 Orthogonal Periodic Components Periodic components of x(n), period Orthogonal projection onto Integer projection operators will find

27 Integer Projection Operator circulant Define ; Theorem: = orthogonal projection matrix onto

28 Summary of the decomposition in orthogonality total orthogonality has period exactly

29 Example of Ramanujan space projections Periodic, orthogonal, projections

30 Theorem (LCM property): Consider a sum where This has period (can t be smaller). Recall signals in have period q (can t be smaller).

31 Example 1 periodic input, period = 64

32 Method 1:

33 Method 2:

34 Projections onto Ramanujan spaces nonzero projections Period = lcm(1, 2, 4, 8, 16, 32, 64) = 64 projection = 0

35 DFT plot

36 Ex. 2: Hidden periodic components period = 16 period = 12 Does not look periodic

37 Projections onto spaces projection indices (3, 4, 12, 16) index k energy union of (3, 4, 12), (4, 16) periods: 12 and 16

38 More stuff Dictionary methods 2D versions Time-period plane Ramanujan filter banks Medical applications DNA microsatellites, proteins

39 Dictionaries for periodicity Ramanujan dictionary 2N S. Tenneti, P. P. Vaidyanathan Any period < N can be identified

40 Hidden periods: 3, 7, 11 DFT does not reveal much Ramanujan method periods are clear! S. Tenneti, P. P. Vaidyanathan

41 Hidden periods: 7, 10 Ramanujan Dictionary integer computations DFT complex computations more errors S. Tenneti, P. P. Vaidyanathan

42 Min. # of samples for period estimation Assume period is known to belong to this set: Theoretical bound: Theoretical bound, N hidden periods S. Tenneti, P. P. Vaidyanathan

43 Assume period is known to belong to this set: Dictionary method: Dictionary method, N hidden periods S. Tenneti, P. P. Vaidyanathan

44 Time-localization t Need a more fundamental approach to time-period plane

45 7, 10, 15 Period Time S. Tenneti, P. P. Vaidyanathan, ICASSP 2015

46 inverse chirp signal Track period as a function of time sample spacing S. Tenneti, P. P. Vaidyanathan, ICASSP 2015

47 Can use FIR filter banks in practice: Ex: q = 9 small l large l

48 chirp signal Ramanujan filter bank output l = 5 S. Tenneti, P. P. Vaidyanathan

49 Ramanujan filter bank

50 Fourier transform does not work STFT, window 256 STFT window 32

51 STFT window size 256

52 STFT window size 32

53 CS and Bio-Info methods Ramanujan filter bank Protein repeats

54 CS and Bio-Info methods Ramanujan filter bank

55 Pentapeptide α-helix insertion loop Time- period plane period 5

56 5 th filter in RFB insertion loops

57 2D Ramanujan space decompositions ICASSP 2015

58 2 original

59 History of Number theory in EE Coding in digital communications Early DSP: NTT, Winograd Acoustics in buildings Array processing

60 So, Hardy is wrong! G. H. Hardy The real mathematics of the real mathematicians is almost wholly useless. The real mathematician has his conscience clear. Mathematics is a harmless and innocent occupation. Applied mathematics, which is useful, is trivial.

61 Thank you!

PROPERTIES OF RAMANUJAN FILTER BANKS. P. P. Vaidyanathan and S. Tenneti

PROPERTIES OF RAMANUJAN FILTER BANKS. P. P. Vaidyanathan and S. Tenneti ROERTIES OF RAMANUJAN FILTER BANKS.. Vaidyanathan and S. Tenneti Dept. of Electrical Engineering, MC 136-93 California Institute of Technology, asadena, CA 91125, USA E-mail: ppvnath@systems.caltech.edu,