Constant Amplitude and Zero Autocorrelation Sequences and Single Pixel Camera Imaging

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1 Constant Amplitude and Zero Autocorrelation Sequences and Single Pixel Camera Imaging Mark Magsino Norbert Wiener Center for Harmonic Analysis and Applications Department of Mathematics University of Maryland, College Park April 4, 2018

2 Frames A finite frame for C N is a set F = {v j } M j=1 constants 0 < A B < where such that there exists A x 2 2 M x, v j 2 B x 2 2 j=1 for any x C N. F is called a tight frame if A = B is possible. Theorem F is a frame for C N if and only if F spans C N.

3 The Frame Operator Let F = {v j } M j=1 be a frame for CN and x C N. (a) The frame operator, S : C N C N, is given by M S(x) = v j, x v j. (b) Given any x C N we can write x in terms of frame elements by M x = x, S 1 v j v j. j=1 j=1 (c) If F is a tight frame with bound A, then S = A Id N.

4 Gabor Frames Definition (a) Let ϕ C N and Λ (Z/NZ) (Z/NZ). The Gabor system, (ϕ, Λ) is defined by (ϕ, Λ) = {e l τ k ϕ : (k, l) Λ}. (b) If (ϕ, Λ) is a frame for C N we call it a Gabor frame.

5 Time-Frequency Transforms Definition Let ϕ, ψ C N. (a) The discrete periodic ambiguity function of ϕ, A p (ϕ), is defined by A p (ϕ)[k, l] = 1 N N 1 j=0 ϕ[j + k]ϕ[j]e 2πijl/N = 1 N τ kϕ, e l ϕ. (b) The short-time Fourier transform of ϕ with window ψ, V ψ (ϕ), is defined by V ψ (ϕ)[k, l] = ϕ, e l τ k ψ.

6 Full Gabor Frames Are Always Tight Theorem Let ϕ C N \ {0}. and Λ = (Z/NZ) (Z/NZ). Then, (ϕ, Λ) is always a tight frame with frame bound N ϕ 2 2.

7 Janssen s Representation Definition Let Λ (Z/NZ) (Z/NZ) be a subgroup. The adjoint subgroup of Λ, Λ (Z/NZ) (Z/NZ), is defined by Λ = {(m, n) : e l τ k e n τ m = e n τ m e l τ k, (k, l) Λ} Theorem Let Λ be a subgroup of (Z/NZ) (Z/NZ) and ϕ C N. Then, the (ϕ, Λ) Gabor frame operator has the form S = Λ N (m,n) Λ ϕ, e n τ m ϕ e n τ m.

8 Λ -sparsity and Tight Frames Theorem Let ϕ C N \ {0} and let Λ (Z/NZ) (Z/NZ) be a subgroup. (ϕ, Λ) is a tight frame if and only if (m, n) Λ, (m, n) 0, A p (ϕ)[m, n] = 0. The frame bound is Λ A p (ϕ)[0, 0].

9 Proof of Λ -sparsity Theorem By Janssen s representation we have S = Λ e n τ m ϕ, ϕ e n τ m = τ m ϕ, e n ϕ e n τ m N (m,n) Λ (m,n) Λ = Λ A p (ϕ)[ m, n]e n τ m = Λ A p (ϕ)[m, n]e n τ m. (m,n) Λ (m,n) Λ If A p (ϕ)[m, n] = 0 for every (m, n) Λ, (m, n) 0, then S is Λ A p (ϕ)[0, 0] times the identity. and so (ϕ, Λ) is a tight frame.

10 Proof of Λ -sparsity Theorem (con t) To show this is a necessary condition, we observe that for S to be tight we need S = Λ A p (ϕ)[m, n]e n τ m = A Id (m,n) Λ which can be rewritten as Λ A p (ϕ)[m, n]e n τ m + ( Λ A p (ϕ)[0, 0] A)Id = 0. (m,n) Λ \{(0,0)}

11 CAZAC Definition Let ϕ C N. ϕ is said to be a constant amplitude zero autocorrelation (CAZAC) sequence if and j (Z/NZ), ϕ j = 1 k (Z/NZ), k 0, 1 N N 1 j=0 ϕ j+k ϕ j = 0. (CA) (ZAC)

12 Examples Quadratic Phase Sequences Let ϕ C N and suppose for each j, ϕ j is of the form ϕ j = e πip(j) where p is a quadratic polynomial. The following quadratic polynomials generate CAZAC sequences: Chu: p(j) = j(j 1) P4: p(j) = j(j N), N is odd Odd-length Wiener: p(j) = sj 2, gcd(s, N) = 1, N is odd Even-length Wiener: p(j) = sj 2 /2, gcd(s, 2N) = 1, N is even

13 Examples Björck Sequences Let p be prime and ϕ C p be of the form ϕ j = e iθ(j). Then ϕ will be CAZAC in the following cases: If p 1 mod 4, then, ( ) ( ) j 1 p θ(j) = arccos p 1 + p If p 3 mod 4, then, { ( ) ( ) arccos 1 p 1+p, if j p = 1 0, otherwise

14 Connection to Hadamard Matrices Theorem Let ϕ C N and let H be the circulant matrix given by H = ϕ τ 1 ϕ τ 2 ϕ τ N 1 ϕ Then, ϕ is a CAZAC sequence if and only if H is Hadamard. In particular there is a one-to-one correspondence between CAZAC sequences and circulant Hadamard matrices.

15 Connection to Cyclic N-roots Definition x C N is a cyclic N-root if it satisfies x 0 + x x N 1 = 0 x 0 x 1 + x 1 x x N 1 x 0 = 0 x 0 x 1 x 2 x N 1 = 1

16 Connection to Cyclic N-roots Theorem (a) If ϕ C N is a CAZAC sequence then, ( ϕ1, ϕ 2,, ϕ 0 ϕ 1 is a cyclic N-root. (b) If x C N is a cyclic N-root then, is a CAZAC sequence. ϕ 0 ϕ N 1 ϕ 0 = x 0, ϕ j = ϕ j 1 x j (c) There is a one-to-one correspondence between CAZAC sequences which start with 1 and cyclic N-roots. )

17 5-Operation Equivalence Proposition Let ϕ C N be a CAZAC sequence. Then, the following are also CAZAC sequences: (a) c C, c = 1, cϕ (b) k Z/NZ, τ k ϕ (c) l (Z/NZ), e l ϕ (d) m Z/NZ, gcd(m, N) = 1, π m ϕ[j] (e) n = 0, 1, c n ϕ The operation π m is defined by π m ϕ[j] = ϕ[mj] and c 0, c 1 is defined by c 0 ϕ = ϕ and c 1 ϕ = ϕ.

18 5-Operation Group Action Let G be the set given by {(a, b, c, d, f ) : a {0, 1}, b, c, d, f Z/pZ, c 0} and define the operation : G G G by (a, b, c, d, f ) (h, j, k, l, m) = (a + h, cj + b, ck, l + ( 1) h kd, m + ( 1) h (f jc)). Then, (G, ) is a group of size 2p 3 (p 1).

19 5-Operation Group Action (cont d) To each element (a, b, c, d, f ) G we associate the operator ω f e d π c τ b c a, which form a group under composition, where ω = e 2πi/p. The composition is computed by the operation for (G, ) and obtaining the operator associated with the computed result. The group of operators under composition forms a group action for U p p, the set of p-length vectors comprised entirely of p-roots of unity. There are p(p 1) many CAZACs in U p p which start with 1. Adding in any scalar multiples of roots of unity, there are p 2 (p 1) many.

20 5-Operation Orbits Theorem Let p be an odd prime and let ϕ U p p be the Wiener sequence ϕ[n] = e 2πisn2 /p, where s Z/pZ. Denote the stabilizer of ϕ under the group (G, ) as G ϕ. If p 1 mod 4, then G ϕ = 4p. If p 3 mod 4, then G ϕ = 2p. In particular, the orbit of ϕ has size p 2 (p 1)/2 if p 1 mod 4 and size p 2 (p 1) if p = 3 mod 4.

21 Sketch of Proof Take a linear operator ω f e d π c τ b c a and write the system of equations that would describe fixing each term of ϕ[n]. Use the n = 0, 1 cases to get expressions for f and d in terms of the other variables. Use any other n > 1 and substitute the expressions for f and d to obtain the condition c 2 ( 1) a mod p.

22 Sketch of Proof (con t) In the case a = 0, there are always the solutions c ±1 mod p. If a = 1, then it depends if 1 is a quadratic residue modulo p. It is if p 1 mod 4 and is not if p 3 mod 4. All variables are solved for except b and the solutions leave it as a free parameter. Thus there are 4p and 2p stabilizers for the p 1 mod 4 and p 3 mod 4 cases respectively.

23 DPAF of Chu Sequence DPAF of length 15 Chu sequence A p (ϕ Chu )[k, l] : { e πi(k2 k)/n, k l mod N 0, otherwise Doppler Shift Magnitude Time Shift 0.0 Figure: DPAF of length 15 Chu sequence.

24 Example: Chu/P4 Seqeunce Proposition Let N = abn where gcd (a, b) = 1 and ϕ C N be the Chu or P4 sequence. Define K = a, L = b and Λ = K L. (a) Λ = N a N b. (b) (ϕ, Λ) is a tight Gabor frame bound NN.

25 DPAF of Even Length Wiener Sequence DPAF of a 16 length Wiener sequence A p (ϕ Wiener )[k, l] : { e πisk2 /N, sk l mod N 0, otherwise Doppler Shift Magnitude Time Shift 0.0 Figure: DPAF of length 16 P4 sequence.

26 DPAF of Björck Sequence DPAF of length 13 Bjorck sequence Doppler Shift Magnitude Time Shift 0.0 Figure: DPAF of length 13 Björck sequence.

27 DPAF of a Kronecker Product Sequence 27 DPAF of Bjorck(7) ( P4(4) Kronecker Product: Let u C M, v C N. (u v)[am + b] = u[a]v[b] Dopp er Shift Magnitude Time Shift 0.0 Figure: DPAF of Kroneker product of length 7 Bjorck and length 4 P4.

28 Example: Kronecker Product Sequence Proposition Let u C M be CAZAC, v C N be CA, and ϕ C MN be defined by the Kronecker product: ϕ = u v. If gcd (M, N) = 1 and Λ = M N, then (ϕ, Λ) is a tight frame with frame bound MN.

29 Gram Matrices and Discrete Periodic Ambiguity Functions Definition Let F = {v i } M i=1 be a frame for CN. The Gram matrix, G, is defined by G i,j = v i, v j. In the case of Gabor frames F = {e lm τ km ϕ : m 0,, M 1}, we can write the Gram matrix in terms of the discrete periodic ambiguity function of ϕ: G m,n = Ne 2πikn(ln lm)/n A p (ϕ)[k n k m, l n l m ]

30 Gram Matrix of Chu and P4 Sequences Lemma Let ϕ C N be the Chu or P4 sequence and let N = abn where gcd(a, b) = 1. Suppose G is the Gram matrix generated by the Gabor system (ϕ, K L) where K = a and L = b. Then, (a) The support of the rows (or columns) of G either completely conincide or are completely disjoint. (b) If two rows (or columns) have coinciding supports, they are scalar multiples of each other.

31 Example: P4 Gram Matrix

32 Tight Frames from Gram Matrix Theorem Let ϕ C N be the Chu or P4 sequence and let N = abn where gcd(a, b) = 1. Suppose G is the Gram matrix generated by the Gabor system (ϕ, K L) where K = a and L = b. Then, (a) rank(g) = N. (b) G has exactly one nonzero eigenvalue, NN. In particular (a) and (b) together imply that the Gabor system (ϕ, K L) is a tight frame with frame bound NN.

33 Proof K L = ab. G mn 0 if and only if (l n l m ) (k n k m ) mod N. This can only occur at the intersection of K and L, i.e., j (Z/N Z), (l n l m ) (k n k m ) jab mod N Fix an m, we can write k n as k n = a(j m + jb) for some j By the column ordering of G we can write the n-th column by n = k n N + l n /b.

34 Proof (con t) We want to look at the first N columns so we want n < N and thus require k n < ab. Thus, (j m + jb) < b. There is exactly one such j (Z/N Z) and it is j = j m /b. Consequently, for each row m, there is exactly 1 column n N where G mn 0 and the first N columns of G are linearly independent. rank(g) = N.

35 Proof (con t) Let g n be the n-th column of G, n < N. The goal is to show that Gg n = NN g n. Gg n [m] is given by the inner product of row m and column n. The n-th column is the conjugate of the n-th row. If Gg n [m] 0, then G mn 0 and G nn 0.

36 Proof (con t) Lemma implies row m and n have coinciding supports and are constant multiples of each other thus, G m( ) = C m g n where C m = 1. Therefore, Gg n [m] = C m g n 2 2 = N2 N C m. G nn = N, so g n [m] = NC m. Finally, Gg n [m] = (NN )(NC m ) and we conclude the first N columns of G are eigenvectors of G with eigenvalue NN.

37 Future work: Continuous CAZAC Property Is it possible to generalize CAZAC to the real line? The immediate problem is the natural inner product to use for autocorrelation is the L 2 (R) inner product, but if f (x) = 1 for every x R, then clearly f L 2 (R).

38 Future work: Continuous CAZAC Property Alternatives and ideas: Define a continuous autocorrelation on L 2 (T) and push torus bounds to infinity. Distribution theory, esp. using tempered distributions. Wiener s Generalized Harmonic Analysis, which includes a theory of mean autocorrelation on R.

39 Future work: Single Pixel Camera My work on this is with John Benedetto and Alfredo Nava-Tudela and is an ongoing project. The original concept is due to Richard Baraniuk. The idea is to construct a camera using only a single light receptor or sensor. This is accomplished by filtering through a pixel grid that either admits or blocks light. Baraniuk s original design does this with digital micromirror devices. Several collections with different pixel grids are required but compressed sensing theory allows this to be done efficiently.

40 Questions?

Constructing Tight Gabor Frames using CAZAC Sequences

Constructing Tight Gabor Frames using CAZAC Sequences Mark Magsino mmagsino@math.umd.edu Norbert Wiener Center for Harmonic Analysis and Applications Department of Mathematics University of Maryland, College