the mathematician ramanujan, and digital signal processing
TRANSCRIPT
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P. P. Vaidyanathan California Institute of Technology, Pasadena, CA
The mathematician Ramanujan, and digital signal processing
SAM-2016, Rio, Brazil
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1887 - 1920
Worked with Hardy in 1914 - 1919 1877 - 1947
Self-educated mathematician from India Grew up in poverty in Tamil Nadu His genius was discovered by G. H. Hardy
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Outline • Ramanujan sums (RS): 1918 • Representing periodic signals • New integer bases, integer projections
• Ramanujan periodic dictionaries
• Conclusions • Bounds on period estimation
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Ramanujan-sum representation:
DFT representation:
period
data size N
or a divisor of N period N Example: N = 32; q = 1, 2, 4, 8, 16, 32
period q Every period is represented! Example: N = 32; q = 1, 2, 3, 4, 5, … 32
Very few periods represented
Motivation Consider a periodic signal
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32 point DFT
32 point DFT
(Re part) First, how much can DFT do? Sinusoid example:
(Re part)
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Identifying periods: Not the same as spectrum estimation
DFT, MUSIC, etc., will work, but are not necessarily best …
ω
arbitrary line spectrum
ω fundamental
periodic case
Ramanujan offers something new
fewer parameters of interest
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Hidden periodic components
Does not “look” periodic
Sparse representation? Ramanujan offers something new
period = 12
period = 16
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• Pitch identification acoustics (music, speech, … ) • Time delay estimation in sensor arrays • Medical applications • Genomics and proteomics • Radar • Astronomy
Importance of periodic components
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means k and q are coprime
= gcd of k and q
qth root of unity
Notations
= Euler’s totient function
: d is a divisor (or factor) of q
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The phi-function (Euler’s totient):
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Ramanujan sum (1918)
q = positive integer
has period exactly q
Euler’s totient
sum over coprime frequencies only
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6
Equivalent definition using DFT
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Primitive frequencies, all with SAME period q
φ(q) frequency components: primitive frequencies
Frequency and period in Ramanujan sum
each term has period exactly q
0
DFT grid
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Other ways to write Ramanujan sum:
Orthogonality:
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Theorem: Ramanujan sum is integer valued! Examples: very useful from a
computational perspective
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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
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Ramanujan expanded arithmetic functions (1918)
Sum-of-divisors:
Euler-totient:
von Mangoldt function:
Number-of-divisors:
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Some references from DSP …
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References for this talk
http://systems.caltech.edu/dsp/students/srikanth/Ramanujan/
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Signal duration N = 2^8 = 256
x(n) = x(n) =
An example with
sparse representation
not sparse
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x(n) =
Again, signal duration N = 2^8 = 256
Another example with
x(n) =
sparse representation
not sparse
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PPV, 2014, IEEE SP Trans.
each period is represented by a 1D subspace only
Reason for the problem:
Consider the space spanned by
Every nonzero signal in this space has period EXACTLY q
We call this the Ramanujan subspace
We will use the entire susbpace to represent period q components of x(n)
What to do about it?
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Ramanujan subspace is spanned by:
• This space has dimension exactly!
• This space is also spanned by:
Theorem (PPV, SP Trans. 2014):
• orthogonal for
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Basis for Ramanujan subspace
complex basis
real integer basis cyclic basis
from DFT
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• Integer basis using Ramanujan sum
Any length N signal can be represented as
Theorem:
• Doubly indexed basis
• Orthogonal periodic components
periodic components in
PPV, SP Trans. 2014
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Orthogonal projection onto
Orthogonal Periodic Components
Integer projection operators will find
Periodic components of x(n), period
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= orthogonal projection matrix onto Theorem:
Define ;
Integer Projection Operator
circulant
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Summary of the decomposition
orthogonality
total orthogonality
has period exactly
in
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Example of Ramanujan space projections
Periodic, orthogonal, projections
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Recall signals in have period q (can’t be smaller).
Consider a sum where
This has period (can’t be smaller).
Theorem (LCM property):
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periodic input, period = 64
Example 1
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Method 1:
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Method 2:
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projection = 0
nonzero projections
Period = lcm(1, 2, 4, 8, 16, 32, 64) = 64
Projections onto Ramanujan spaces
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DFT plot
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Ex. 2: Hidden periodic components
period = 12
period = 16
Does not “look” periodic
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(3, 4, 12), (4, 16) union of
periods: 12 and 16
(3, 4, 12, 16) projection indices
index k energy 1.0000 0.0000 2.0000 0.0000 3.0000 24.0000 4.0000 48.0000 6.0000 0.0000 8.0000 0.0000 12.0000 24.0000 16.0000 48.0000 24.0000 0.0000 48.0000 0.0000
Projections onto spaces
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More stuff … • Dictionary methods • 2D versions
• Time-period plane • Ramanujan filter banks • Medical applications
• DNA microsatellites, proteins
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S. Tenneti, P. P. Vaidyanathan
Dictionaries for periodicity
2N
Any period < N can be identified
Ramanujan dictionary
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S. Tenneti, P. P. Vaidyanathan
Hidden periods: 3, 7, 11
DFT does not reveal much
Ramanujan method
periods are clear!
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S. Tenneti, P. P. Vaidyanathan
Hidden periods: 7, 10
DFT Ramanujan Dictionary
integer computations
complex computations
more errors
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Theoretical bound:
Theoretical bound, N hidden periods
Assume period is known to belong to this set:
Min. # of samples for period estimation
S. Tenneti, P. P. Vaidyanathan
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Dictionary method:
Dictionary method, N hidden periods
Assume period is known to belong to this set:
S. Tenneti, P. P. Vaidyanathan
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t
Time-localization
Need a more fundamental approach to time-period plane
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Time
Peri
od
7, 10, 15
S. Tenneti, P. P. Vaidyanathan, ICASSP 2015
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Track period as a function of time
inverse chirp signal
sample spacing
S. Tenneti, P. P. Vaidyanathan, ICASSP 2015
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small l
large l
Ex: q = 9
Can use FIR filter banks in practice:
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Ramanujan filter bank output
chirp signal
l = 5 S. Tenneti, P. P. Vaidyanathan
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Ramanujan filter bank
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STFT, window 256 STFT window 32
Fourier transform does not work
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STFT window size 256
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STFT window size 32
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Protein repeats
Ramanujan filter bank
CS and Bio-Info methods
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Ramanujan filter bank
CS and Bio-Info methods
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Time- period plane
α-helix insertion loop
Pentapeptide
period 5
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5th filter in RFB insertion loops
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2D Ramanujan space decompositions
ICASSP 2015
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original 2
4
1 2
2 3
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History of Number theory in EE …
Coding in digital communications
Acoustics in buildings Array processing
…
Early DSP: NTT, Winograd …
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So, Hardy is wrong!
The ‘real’ mathematics of the ‘real’ mathematicians is almost wholly ‘useless’.
G. H. Hardy 1877 - 1947
Applied mathematics, which is ‘useful’, is trivial.
The ‘real mathematician’ has his conscience clear.
Mathematics is a harmless and innocent occupation.
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Thank you!