Some results on convolution idempotents May 28, 2020 1 IIT - - PowerPoint PPT Presentation

Some results on convolution idempotents

P.Charantej Reddy 1 Aditya Siripuram 1 Brad Osgood 2 May 28, 2020

1IIT Hyderabad, India 2Stanford University

1

Problem Statement

Convolution idempotent (Defjnition) A mapping h : ZN → CN is a convolution idempotent if h ∗ h = h. Where ZN are integers modulo N and ∗ is circular convolution.

• The DFT coeffjcients are either
• r
• We have

, where an indicator, is support set

• For example:
• If

then

• If

then

Zero-set problem Given a positive integer and a set , fjnd all idempotents that vanish on . Motivation comes from sampling and Fuglede’s conjecture.

2

Problem Statement

Convolution idempotent (Defjnition) A mapping h : ZN → CN is a convolution idempotent if h ∗ h = h. Where ZN are integers modulo N and ∗ is circular convolution.

• The DFT coeffjcients are either 0 or 1

h ∗ h = h = ⇒ (Fh)2 = Fh

• We have

, where an indicator, is support set

• For example:
• If

then

• If

then

Zero-set problem Given a positive integer and a set , fjnd all idempotents that vanish on . Motivation comes from sampling and Fuglede’s conjecture.

2

Problem Statement

Convolution idempotent (Defjnition) A mapping h : ZN → CN is a convolution idempotent if h ∗ h = h. Where ZN are integers modulo N and ∗ is circular convolution.

• The DFT coeffjcients are either 0 or 1

h ∗ h = h = ⇒ (Fh)2 = Fh

• We have hJ = F−11J , where 1J an indicator, J is support set
• For example:
• If

then

• If

then

Zero-set problem Given a positive integer and a set , fjnd all idempotents that vanish on . Motivation comes from sampling and Fuglede’s conjecture.

2

Problem Statement

Convolution idempotent (Defjnition) A mapping h : ZN → CN is a convolution idempotent if h ∗ h = h. Where ZN are integers modulo N and ∗ is circular convolution.

• The DFT coeffjcients are either 0 or 1

h ∗ h = h = ⇒ (Fh)2 = Fh

• We have hJ = F−11J , where 1J an indicator, J is support set
• For example:
• If J = ZN then h = δ
• If J = {} then h = 0

Zero-set problem Given a positive integer and a set , fjnd all idempotents that vanish on . Motivation comes from sampling and Fuglede’s conjecture.

2

Problem Statement

Convolution idempotent (Defjnition) A mapping h : ZN → CN is a convolution idempotent if h ∗ h = h. Where ZN are integers modulo N and ∗ is circular convolution.

• The DFT coeffjcients are either 0 or 1

h ∗ h = h = ⇒ (Fh)2 = Fh

• We have hJ = F−11J , where 1J an indicator, J is support set
• For example:
• If J = ZN then h = δ
• If J = {} then h = 0

Zero-set problem Given a positive integer N and a set Z ⊆ ZN, fjnd all idempotents h : ZN → CN that vanish on Z. Motivation comes from sampling and Fuglede’s conjecture.

2

Problem Statement

Convolution idempotent (Defjnition) A mapping h : ZN → CN is a convolution idempotent if h ∗ h = h. Where ZN are integers modulo N and ∗ is circular convolution.

• The DFT coeffjcients are either 0 or 1

h ∗ h = h = ⇒ (Fh)2 = Fh

• We have hJ = F−11J , where 1J an indicator, J is support set
• For example:
• If J = ZN then h = δ
• If J = {} then h = 0

Zero-set problem Given a positive integer N and a set Z ⊆ ZN, fjnd all idempotents h : ZN → CN that vanish on Z. Motivation comes from sampling and Fuglede’s conjecture.

2

Motivation I: Sampling

• Sampling is a well studied problem in Signal Processing
• Traditional sampling with uniformly spaced samples can be

ineffjcient while sampling signals with fragmented spectra.

Figure 1: Example signal with two fragments, for .

• In traditional setting average number of samples taken per second is

at-least for the example signal in Figure 1.

• Can we do better than this by using the frequency support

information?

3

Motivation I: Sampling

• Sampling is a well studied problem in Signal Processing
• Traditional sampling with uniformly spaced samples can be

ineffjcient while sampling signals with fragmented spectra.

Ff(s) s 1 3 2 Figure 1: Example signal with two fragments, for F = {0, 2}.

• In traditional setting average number of samples taken per second is

at-least for the example signal in Figure 1.

• Can we do better than this by using the frequency support

information?

3

Motivation I: Sampling

• Sampling is a well studied problem in Signal Processing
• Traditional sampling with uniformly spaced samples can be

ineffjcient while sampling signals with fragmented spectra.

Ff(s) s 1 3 2 Figure 1: Example signal with two fragments, for F = {0, 2}.

• In traditional setting average number of samples taken per second is

at-least 3 for the example signal in Figure 1.

• Can we do better than this by using the frequency support

information?

3

Motivation I: Sampling

• Sampling is a well studied problem in Signal Processing
• Traditional sampling with uniformly spaced samples can be

ineffjcient while sampling signals with fragmented spectra.

Ff(s) s 1 3 2 Figure 1: Example signal with two fragments, for F = {0, 2}.

• In traditional setting average number of samples taken per second is

at-least 3 for the example signal in Figure 1.

• Can we do better than this by using the frequency support

information?

3

Motivation I: Sampling (contd)

Multi-coset sampling

• Two samples every second
• At 0s, 0.25s, 1s, 1.25s, 2s, 2.25s, . . .

t 1 2

sampled

Figure 2: Top-left: Example sampling pattern, Top-right: Spectrum of signal and Bottom: Spectrum of sampled signal

4

Motivation I: Sampling (contd)

Multi-coset sampling

• Two samples every second
• At 0s, 0.25s, 1s, 1.25s, 2s, 2.25s, . . .

t 1 2 Ff(s) s 1 3 2

sampled

Figure 2: Top-left: Example sampling pattern, Top-right: Spectrum of signal and Bottom: Spectrum of sampled signal

4

Motivation I: Sampling (contd)

Multi-coset sampling

• Two samples every second
• At 0s, 0.25s, 1s, 1.25s, 2s, 2.25s, . . .

t 1 2 Ff(s) s 1 3 2 |Ffsampled(s)| s 1 3 2 Figure 2: Top-left: Example sampling pattern, Top-right: Spectrum of signal and Bottom: Spectrum of sampled signal

4

Multi-coset sampling

• Non-uniform deterministic sampling techniques1
• Frequency support is known
• With out loss of generality we make following assumptions
• Spectrum is non zero only in +ve frequency
• Location of spectrum is known
• Spectrum is non zero at integer intervals

Figure 3: Example signal with two fragments, for .

• For any signal

in the space, the Fourier transform is non zero only when the frequency is in .

1Arthur Kohlenberg. “Exact interpolation of band-limited functions”. In: Journal of

Applied Physics 24.12 (1953), pp. 1432–1436.

5

Multi-coset sampling

• Non-uniform deterministic sampling techniques1
• Frequency support is known
• With out loss of generality we make following assumptions
• Spectrum is non zero only in +ve frequency
• Location of spectrum is known
• Spectrum is non zero at integer intervals

Figure 3: Example signal with two fragments, for .

• For any signal

in the space, the Fourier transform is non zero only when the frequency is in .

1Kohlenberg, “Exact interpolation of band-limited functions”.

5

Multi-coset sampling

• Non-uniform deterministic sampling techniques1
• Frequency support is known
• With out loss of generality we make following assumptions
• Spectrum is non zero only in +ve frequency
• Location of spectrum is known
• Spectrum is non zero at integer intervals

Ff(s) s 1 3 2 Figure 3: Example signal with two fragments, for F = {0, 2}.

• For any signal f in the space, the Fourier transform Ff(s) is non

zero only when the frequency s is in ∪n∈F[n, n + 1].

1Kohlenberg, “Exact interpolation of band-limited functions”.

5

Multi-coset sampling (contd)

• Let us cosider the sampling pattern

∞

• k=−∞

pJ (t − kN), where pJ (t) =

• m∈J

δ(t − m/N)

t

1 N

. . . 1 . . . 2 . . . Figure 4: Sampling pattern: Samples are taken at every second, and at a 1/N second ofgset

• And J ⊆ [0, N − 1] and N (> maxF + 1) are parameters that need

to be picked.

6

Multi-coset sampling (contd)

• From elementary Fourier analysis, the sampled signal has spectrum

Ffsampled(s) =

∞

• k=−∞

hJ (k)Ff(s − k)

• here hJ (n) is discrete Fourier transform of 1J

Figure 5: Shifted spectrum for

7

Multi-coset sampling (contd)

• From elementary Fourier analysis, the sampled signal has spectrum

Ffsampled(s) =

∞

• k=−∞

hJ (k)Ff(s − k)

• here hJ (n) is discrete Fourier transform of 1J

Ff(s − k) s 1 2 3 4 5 6 Figure 5: Shifted spectrum for k = 0

7

Multi-coset sampling (contd)

• From elementary Fourier analysis, the sampled signal has spectrum

Ffsampled(s) =

∞

• k=−∞

hJ (k)Ff(s − k)

• here hJ (n) is discrete Fourier transform of 1J

Ff(s − k) s 1 2 3 4 5 6 Figure 5: Shifted spectrum for k = 0, 1

7

Multi-coset sampling (contd)

• From elementary Fourier analysis, the sampled signal has spectrum

Ffsampled(s) =

∞

• k=−∞

hJ (k)Ff(s − k)

• here hJ (n) is discrete Fourier transform of 1J

Ff(s − k) s 1 2 3 4 5 6 Figure 5: Shifted spectrum for k = 0, 1, 2.

7

Multi-coset sampling: connection to the proposed problem

Proposition If hJ satisfjes hJ (k1 − k2) = 0 whenever k1, k2 ∈ F, k1 = k2, then f can be recovered from the sampled spectrum. Proof

• sampled
• Note that
• The

fragment overlaps with the fragment in the aliasing terms at a shift of

• These aliasing terms are scaled by

, which is 0

• For

not of this form the shift does not overlap with any fragment

8

Multi-coset sampling: connection to the proposed problem

Proposition If hJ satisfjes hJ (k1 − k2) = 0 whenever k1, k2 ∈ F, k1 = k2, then f can be recovered from the sampled spectrum. Proof

• Ffsampled(s) = hJ (0)Ff(s) +
• k=0

hJ (k)Ff(s − k)

• Note that hJ (0) = |J | = 0
• The kth

2 fragment overlaps with the kth 1 fragment in the aliasing

terms at a shift of k1 − k2

• These aliasing terms are scaled by hJ (k1 − k2), which is 0
• For k not of this form the shift Ff(s − k) does not overlap with any

fragment

8

Motivation I: Sampling (conclusion)

Observations

• The average number of samples taken per second is |J |
• If hJ satisfjes hJ (k1 − k2) = 0 whenever k1, k2 ∈ F, k1 = k2, then

f can be recovered from the sampled spectrum

• Starting from the fragment set we can construct the difgerence set

with elements as k1 − k2 and try to build an idempotent that vanishes on difgerence set

Figure 6: Example signal with two fragments, for .

• Let

then

9

Motivation I: Sampling (conclusion)

Observations

• The average number of samples taken per second is |J |
• If hJ satisfjes hJ (k1 − k2) = 0 whenever k1, k2 ∈ F, k1 = k2, then

f can be recovered from the sampled spectrum

• Starting from the fragment set we can construct the difgerence set

with elements as k1 − k2 and try to build an idempotent that vanishes on difgerence set

Ff(s) s 1 3 2 Figure 6: Example signal with two fragments, for F = {0, 2}.

• Let J = {0, 1}, N = 4 then hJ =
• 2

1 + i 1 − i

• 9

Motivation II: Fuglede’s conjecture

Spectral sets A set J ⊆ ZN is spectral if there exists a square unitary (or unitary up to scaling) submatrix of F with columns indexed by J

Figure 7: is a primitive root of unity.

10

Motivation II: Fuglede’s conjecture

Spectral sets A set J ⊆ ZN is spectral if there exists a square unitary (or unitary up to scaling) submatrix of F with columns indexed by J 1 1 1 1 1 ζ ζ2 ζ3 1 ζ2 ζ4 ζ6 1 ζ3 ζ6 ζ9              

Figure 7: ζ = e−2πi/4 is a primitive 4th root of unity.

10

Tiling sets

Tiling sets A set J ⊆ ZN tiles ZN if there exists a set K ⊂ ZN such that every i ∈ ZN can be written uniquely as i = j + k mod N, with j ∈ J , k ∈ K.

• For example,

tiles :

• From the defjnition we can write:
• r

11

Tiling sets

Tiling sets A set J ⊆ ZN tiles ZN if there exists a set K ⊂ ZN such that every i ∈ ZN can be written uniquely as i = j + k mod N, with j ∈ J , k ∈ K.

• For example, {0, 3} tiles Z4:

{0, 3} ⊕ {0, 2} = Z4

• From the defjnition we can write:
• r

11

Tiling sets

Tiling sets A set J ⊆ ZN tiles ZN if there exists a set K ⊂ ZN such that every i ∈ ZN can be written uniquely as i = j + k mod N, with j ∈ J , k ∈ K.

• For example, {0, 3} tiles Z4:

{0, 3} ⊕ {0, 2} = Z4

0 1 2 3

• From the defjnition we can write:
• r

11

Tiling sets

Tiling sets A set J ⊆ ZN tiles ZN if there exists a set K ⊂ ZN such that every i ∈ ZN can be written uniquely as i = j + k mod N, with j ∈ J , k ∈ K.

• For example, {0, 3} tiles Z4:

{0, 3} ⊕ {0, 2} = Z4

0 1 2 3

• From the defjnition we can write:

1J ∗ 1K = 1ZN , or hJ hK = δ,

11

Fuglede’s conjecture

Fuglede’s conjecture,2 spectral-tile direction If J ⊆ ZN is spectral then J tiles ZN. known to be true in the case when N is a prime power and when N is pmq.

2Bent Fuglede. “Commuting self-adjoint partial difgerential operators and a group

theoretic problem”. In: Journal of Functional Analysis 16.1 (1974), pp. 101–121.

12

Fuglede’s conjecture: connection to the proposed problem

• Starting with the spectral set J to prove that it is tiling set we need

to fjnd K such that the following holds: hJ hK = δ

• Here K is the set of translates
• We need to fjnd

such that vanishes on whenever does not: for any such that

• This reduces to the problem of an idempotent

that vanishes on a given set

13

Fuglede’s conjecture: connection to the proposed problem

• Starting with the spectral set J to prove that it is tiling set we need

to fjnd K such that the following holds: hJ hK = δ

• Here K is the set of translates
• We need to fjnd K such that hK vanishes on ZN \ {0} whenever hJ

does not: hK(n) = 0 for any n = 0 such that hJ (n) = 0

• This reduces to the problem of an idempotent hK that vanishes on a

given set

13

Back to the zero-set problem

Zero-set problem Given a positive integer N and a set Z ⊆ ZN, fjnd all idempotents h : ZN → CN that vanish on Z.

• Let us defjne zero-set:

Z(h) = {n ∈ ZN : h(n) = 0}

• These are zeroes of polynomials with 0 or 1 coeffjcients
• Hence are very structured

14

The structure of zero-sets

0 1 2 3 4 5

Figure 8: GCD sets A6, A1, A2, A3 for N = 6

• Consider the indices {0, 1, 2, 3, 4, 5} partitioned according to their

GCD with 6

• This creates the GCD sets Ak for k = 1, 2, 3, 6.

Lemma The zero set of an idempotent is a disjoint union of GCD sets.

• For e.g. if

then .

• Thus

are potential zero sets

15

The structure of zero-sets

0 1 2 3 4 5

Figure 8: GCD sets A6, A1, A2, A3 for N = 6

• Consider the indices {0, 1, 2, 3, 4, 5} partitioned according to their

GCD with 6

• This creates the GCD sets Ak for k = 1, 2, 3, 6.

Lemma The zero set of an idempotent is a disjoint union of GCD sets.

• For e.g. if h(2) = 0 then h(4) = 0.
• Thus A2, A1 ∪ A2, A3 ∪ A2 are potential zero sets

15

Zero-set problem (reformulated)

Problem iN(D) Given a positive integer N and a set of divisors D ⊆ DN let Z = {i ∈ ZN : gcd(i, N) ∈ D} Find all index sets J such that the idempotent hJ = F−11J vanishes

• n Z.

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Solutions to iN(D) for N a prime power3

• Assume N = pM
• Solutions can be constructed from base−p expansions of elements of

J

• Consider the case N = 25, D = {21, 23}

20 21 22 23 24 17 1 1 25 1 1 1 3 1 1 27 1 1 1 1

Figure 9: Example solution to i32({2, 8}).

3Aditya Siripuram and Brad Osgood. Convolution Idempotents with a given Zero-set.

• 2020. arXiv: 2001.00739 [cs.IT].

17

Sampling two dimensional signal with fragmented spectrum

1 2 1 2 3 s1 s2

Figure 10: Example fragmented spectrum of a 2-D signal. The Fourier transform Ff(s1, s2) is non zero only in the shaded regions.

• In two dimensional sampling each fragment occupies a single cell of

an integer lattice. Ff(s1, s2) = 0 when (s1, s2) / ∈

• (m,n)∈F

[m, m + 1] × [n, n + 1].

18

Sampling two dimensional signal with fragmented spectrum

1 2 1 2 3 s1 s2

Figure 11: Example fragmented spectrum of a 2-D signal. The Fourier transform Ff(s1, s2) is non zero only in the shaded regions.

• An analysis similar to the 1-D sampling leads to two dimensional

idempotents h ∈ CN×M.

• When N and M are coprime, the entries in h can be re-indexed to

an NM− length vector that is an idempotent in CNM 4

4Irving John Good. “The interaction algorithm and practical Fourier analysis”. In:

Journal of the Royal Statistical Society. Series B (Methodological) (1958),

• pp. 361–372.

19

Solutions to iN(D) for N a product of two primes

• Assume N = pq
• We can have D = {1}, {p}, {q}, {p, q}, {1, p}, or {1, q}

Main result If N = pq,

• Any solution to ipq({1}) is either a (disjoint) union of translates of

{0, p, 2p, . . . , (q − 1)p} or a (disjoint) union of translates of {0, q, 2q, . . . , (p − 1)q}

• The only solution to ipq({p, q}) is ZN.
• As a corollary, any solution to

is a solution to either

• r
• In particular, there is no idempotent that vanishes only on
• Proof technique using properties of Ramanujan sums5

5PP Vaidyanathan. “Ramanujan sums in the context of signal processing—Part I:

Fundamentals”. In: IEEE transactions on signal processing 62.16 (2014),

• pp. 4145–4157.

20

Solutions to iN(D) for N a product of two primes

• Assume N = pq
• We can have D = {1}, {p}, {q}, {p, q}, {1, p}, or {1, q}

Main result If N = pq,

• Any solution to ipq({1}) is either a (disjoint) union of translates of

{0, p, 2p, . . . , (q − 1)p} or a (disjoint) union of translates of {0, q, 2q, . . . , (p − 1)q}

• The only solution to ipq({p, q}) is ZN.
• As a corollary, any solution to ipq({1}) is a solution to either

ipq({1, p}) or ipq({1, q})

• In particular, there is no idempotent that vanishes only on Apq(1)
• Proof technique using properties of Ramanujan sums5

5Vaidyanathan, “Ramanujan sums in the context of signal processing—Part I:

Fundamentals”.

20

Open problems and future work

• Investigate generalizations to the product of two prime powers
• Understand the solution space of i(D) when D corresponds to

spectral or tiling sets, and explore the connection to Fuglede’s conjecture better

Thank you !!

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References i

Coven, Ethan M and Aaron Meyerowitz. “Tiling the integers with translates of one fjnite set”. In: Journal of Algebra 212.1 (1999),

• pp. 161–174.

Dutkay, Dorin Ervin and CHUN-KIT LAI. “Some reductions of the spectral set conjecture to integers”. In: Mathematical Proceedings of the Cambridge Philosophical Society. Vol. 156. 01. Cambridge Univ

• Press. 2014, pp. 123–135.

Fuglede, Bent. “Commuting self-adjoint partial difgerential operators and a group theoretic problem”. In: Journal of Functional Analysis 16.1 (1974), pp. 101–121. Good, Irving John. “The interaction algorithm and practical Fourier analysis”. In: Journal of the Royal Statistical Society. Series B (Methodological) (1958), pp. 361–372.

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References ii

Herley, Cormac and Ping Wah Wong. “Minimum rate sampling and reconstruction of signals with arbitrary frequency support”. In: IEEE Transactions on Information Theory 45.5 (1999), pp. 1555–1564. Iosevich, Alex, Nets Katz, and Terence Tao. “The Fuglede spectral conjecture holds for convex planar domains”. In: Mathematical Research Letters 10.5 (2003), pp. 559–569. Kohlenberg, Arthur. “Exact interpolation of band-limited functions”. In: Journal of Applied Physics 24.12 (1953), pp. 1432–1436. Laba, Izabella. “The spectral set conjecture and multiplicative properties

• f roots of polynomials”. In: Journal of the London Mathematical

Society 65.03 (2002), pp. 661–671. Lam, T.Y. and K.H. Leung. “On vanishing sums of roots of unity”. In: Journal of algebra 224.1 (2000), pp. 91–109.

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References iii

Lin, Yuan-Pei and PP Vaidyanathan. “Periodically nonuniform sampling

• f bandpass signals”. In: IEEE Transactions on Circuits and Systems II:

Analog and Digital Signal Processing 45.3 (1998), pp. 340–351. Osgood, Brad. Lectures on the Fourier Transform and Its Applications. American Mathematical Society, 2018. Ramanujan, Srinivasa. “On certain trigonometrical sums and their applications in the theory of numbers”. In: Trans. Cambridge Philos. Soc 22.13 (1918), pp. 259–276. Siripuram, Aditya. “Sampling and Interpolation of Discrete Signals: Orthogonality, Universality and Uncertainty”. PhD thesis. Stanford University, 2014. Siripuram, Aditya and Brad Osgood. Convolution Idempotents with a given Zero-set. 2020. arXiv: 2001.00739 [cs.IT]. – .“LP relaxations and Fuglede’s conjecture”. In: 2018 IEEE International Symposium on Information Theory (ISIT). IEEE. 2018, pp. 2525–2529.

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References iv

Siripuram, Aditya, William Wu, and Brad Osgood. “Discrete Sampling: A graph theoretic approach to Orthogonal Interpolation”. In: IEEE Transactions on Information Theory (2019). Tao, Terence. “Fuglede’s conjecture is false in 5 and higher dimensions”. In: arXiv preprint math/0306134 (2003). Vaidyanathan, PP. “Ramanujan sums in the context of signal processing—Part I: Fundamentals”. In: IEEE transactions on signal processing 62.16 (2014), pp. 4145–4157. – .“Ramanujan sums in the context of signal processing—Part II: FIR representations and applications”. In: IEEE Transactions on Signal Processing 62.16 (2014), pp. 4158–4172. Venkataramani, Raman and Yoram Bresler. “Perfect reconstruction formulas and bounds on aliasing error in sub-Nyquist nonuniform sampling of multiband signals”. In: IEEE Transactions on Information Theory 46.6 (2000), pp. 2173–2183.

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