Lattices as Multidimensional Sampling Structures: Part I

Vishal Monga [email protected]

September 20, 2007

1 Lattices

Definition: A lattice Λ in the N-dimensional real space RN , is the set of all linear combinations of N linearly independent real vectors v1,..., vN of size N × 1, which are known as the basis vectors of Λ, with integer coefficients. Λ can be written as the set

Λ = {Vn | n ∈ ZN }, (1) · ¸ . . where Z denotes the set of all integers and V = v1 . ··· . vN is a N × N real-valued matrix, with the T basis vectors v1,..., vN as its columns. For any point Vn in the lattice, the vector n = [n1, . . . , nN ] is the representation of the point in the lattice with respect to V, which is often called the generating or the sampling matrix.

For any lattice basis vectors are not unique. In fact, if EN×N is a matrix of integer numbers and det(E) = ±1, then Vˆ = EV provides another set of basis vectors for Λ.

Definition: Given a lattice Λ, one can find a unit cell U(Λ) such that the entire RN space can be tiled by U(Λ) and its translations to all lattice points.

The unit cell associated with a lattice is not unique. However, two unit cells-namely Fundamental Parallelepiped and Voronoi Cell, are often encountered in the literature.

1. Fundamental parallelepiped: represents set of all points in RN within the region enclosed by the basis vectors and it can be expressed as the set ( ) XN N P(Λ) = x ∈ R | x = αivi, ∀ 0 ≤ αi ≤ 1 . (2) i=1

1 y y

1

1 2 4 / w2 2 v2

x x v1 3 6 w1 1/3 2/3

3 3 2 1 3 0 V =  /  W =  /   0 2   −1 4 1 2     / /  (a) Basic lattice Λ (b) Reciprocal lattice Λ∗

Figure 1: Example of a lattice in two dimensions.

2. Voronoi cell: represents the set of all points in RN closer to origin 0 than to any of the lattice points and it can be expressed as

© ª V(Λ) = x ∈ RN | d(x, 0) ≤ d(x, p), ∀ p ∈ Λ , (3) where d(a, b) is the Euclidean distance between the points a and b.

Examples of these unit cells for a lattice whose generator matrix is V is shown in Fig. 1(a).

y y

P(Λ)

V(Λ)

x x

(a) Fundamental parallelepiped (b) Voronoi cell

Figure 2: Unit cells for the lattice of Fig 1(a).

Although a unit cell for a lattice is not unique, hyper-volume of the unit cell of a lattice denoted by d(Λ) = det(V) is unique. Another useful term associated with the lattices is Sampling density, which is computed as 1 Sampling density = . (4) d(Λ)

2 Definition: Let Λ and Γ be two lattices. Λ is a sublattice of Γ, if every point in Λ is also a point in Γ. The ratio d(Λ)/d(Γ) is called the index of Λ in Γ.

The set c + Λ = {c + x | x ∈ Λ} for any c ∈ Γ is called a coset or class of Λ in Γ. Two cosets c + Λ and d + Λ are identical if and only if c − d ∈ Λ.

Definition: Given a lattice Λ, the set of all vectors y such that yT x ∈ Z; ∀x ∈ Λ is called the reciprocal (polar) lattice of Λ and it can be represented as the set: © ª Λ∗ = Wn = V−T n | n ∈ ZN . (5)

T It can be seen that W V = IN×N and 1 d (Λ∗) = . (6) d (Λ) Thus, denser Λ ensures a sparser Λ∗ and vice versa. An example for a reciprocal lattice is shown in Fig. 1(b) for the lattice shown in Fig. 1(a).

Theorem: Let Λ1 and Λ2 be two lattices of dimension N. Intersection of these lattices Λ1 ∩ Λ2 is also a lattice, however its dimension could be less than or equal to N. A necessary and sufficient condition for Λ1 ∩Λ2 −1 N×N to be N-dimensional, is that V1 V2 ∈ Q , where Q is the set of rational numbers.

Λ1 ∩ Λ2 is the largest lattice that is contained in both Λ1 and Λ2. Similarly,

Λ1 + Λ2 = {x + y | ∀ x ∈ Λ1, ∀ y ∈ Λ2} (7) is the smallest lattice that contains both Λ1 and Λ2. Furthermore,

∗ ∗ ∗ (Λ1 + Λ2) = Λ1 ∩ Λ2 and (8) ∗ ∗ ∗ (Λ1 ∩ Λ2) = Λ1 + Λ2. (9)

Thus, if an algorithm is present to find the intersection of lattices; it can be used to find sum of lattices as well.

3 2 Sampling Over Lattices

Definition: A function ψ(x) is periodic with a non-singular periodicity matrix V if ψ(x) = ψ(x + Vn) ∀n ∈ ZN .

When a lattice is used to describe the periodicity of a function, it is usually referred as the periodicity lattice and a unit cell of the lattice is referred as the fundamental period of the function.

N Definition: Given a continuous signal sc(x), x ∈ R , a sampled signal over a lattice Λ with a generating matrix V is defined as: X N s(x) = sc(x)δ (x − Vn) , x ∈ R . (10) n∈ZN

2.1 Fourier Transform of Signals Sampled on Lattices

X X S(u) = s(x) exp(−2πjuT x) = s(Vn) exp(2πjuT Vn), (11) x∈Λ n∈ZN T where u = [u1, u2, . . . , uN ] represents the frequency coordinates.

Let r be a point in Λ∗. S(u + r) can be written as: X S(u + r) = S(Vn) exp(−2πj(u + r)T Vn) (12) n∈ZN X = S(Vn) exp(−2πjuT Vn) exp(−2πjrT Vn). (13) n∈ZN From the definition of the reciprocal lattice rT x is an integer ∀ x ∈ Λ, r ∈ Λ∗. Thus, Fourier transform is periodic with W = V−T the generator matrix for the reciprocal lattice of Λ.

Similarly, the inverse Fourier transform can be computed as: Z s(x) = d(Λ) S(u) exp(2πjuT x)du), x ∈ Λ. (14) U(Λ∗)

2.2 Sampling on Cosets

Let Ψ be discrete set of points in RN such that it is a union of distinct cosets of a sublattice Λ in a lattice Γ. Thus, we have [P Ψ = (ci + Λ), (15) i=1

4 where ci is a vector in Γ. Sampled signal can be written as X T s(u) = sc(x) exp(−2πju x) (16) x∈Ψ and its Fourier transform can accordingly be written as

XP X T S(u) = s(ci + x) exp(−2πju (ci + x)) (17) i x∈Λ XP X T T = exp(−2πju ci) s(ci + x) exp(−2πju x) (18) i=1 x∈Λ XP T = exp(−2πju ci)Si(u), (19) i=1 whose periodicity is determined by the lattice Γ∗.

2.3 Generalized Nyquist Sampling Theorem

1 X S(u) = S (u − r), (20) d(Λ) c r∈Λ∗ where Sc(u) is the continuous Fourier transform of sc(x).

It is possible to recover the original continuous signal from the sampled signal perfectly if and only if the support ∗ region of Sc(u) is limited within the fundamental period of the reciprocal lattice Λ . Use the reconstruction filter ½ d(Λ); u ∈ U(Λ∗) H (u) = (21) r 0; o.w. to interpolate between the samples. In order to avoid aliasing, use the filter ½ 1; u ∈ U(Λ∗) H (u) = (22) AA 0; o.w.

∗ and to recover the original signal perfectly, U(Λ ) should be at least 2 × BW(Sc(u)), where BW(Sc(u)) is the support region of the continuous signal.

2.4 Sampling Efficiency

By using very high sampling density, it can be ensured U(Λ∗) covers the support of the continuous time signal. However, at the same time, it is desired that the number of samples that is enough to reconstruct the original signal is the smallest. This can be achieved if U(Λ∗) covers the support of the continuous signal as tightly as possible.

5 Definition: For a circularly bandlimited signal, the ratio of the hyper-volume of the hyper-sphere that defines the bandwidth of the signal to the hyper-volume of the smallest U(Λ∗) that covers this hyper-sphere is called the sampling efficiency of the sampling structure.

V (1) ρ(Λ) = N , (23) d(Λ∗) where VN (1) is the hyper-volume of the unit hyper-sphere in N dimensions.

In Fig. 2.4, sampling efficiencies of 3 different geometries is shown. As a matter of fact, it can be shown for a

r =1 r =1 r =1

Λ Λ Λ d( ∗)=4 d( ∗)=4 d( ∗)=2√3

ρ(Λ)= π 0.785 ρ(Λ)= π 0.785 ρ(Λ)= π 0.907 4 ≈ 4 ≈ 2√3 ≈

Figure 3: Sampling efficiencies of different geometries. circularly bandlimited signal hexagonal sampling lattice has the highest efficiency among the others.

In Table. 1, the sampling efficiency of a rectangular lattice is compared against a hexagonal lattice.

M η = ρ(ΛR) ρ(ΛH ) 1 1.000 2 0.866 3 0.705 5 0.353 8 0.062

Table 1: Rectangular vs. Hexagonal sampling efficiency

It can be seen that for 1-D signals, there is no difference between the structures. However, as N increases the efficacy of hexagonal sampling is obvious. For example, for a continuous 2-D signal, hexagonal lattice can reconstruct the signal with approximately 13.4% less samples than a rectangular lattice requires.

6 3 Sampling Rate Conversion

Let s(x) is the sampled version of a continuous signal sc(x) on a lattice Λ1, we would like to obtain s(y) which is the sampled version of sc(x) on a lattice Λ2 by using s(x).

3.1 Upsampling

If Λ1 ⊂ Λ2, then the upsampling operation can be defined by padding zeros ½ s(x), x ∈ Λ1 sU (x) = U(s(x)) = (24) 0, x ∈ Λ2 \ Λ1, where Λ2 \ Λ1 represents the set of points in Λ2 but not in Λ1.

In order to fill the zero-padded samples, the following interpolation filter is applied ( d(Λ1) ; u ∈ U(Λ∗) H (u) = d(Λ2) 1 (25) U ∗ ∗ 0; u ∈ U(Λ2) \U(Λ1).

An example of upsampling is shown in Fig. 4.

y ∗ v Λ1 Λ1

2 1

1

∗ U(Λ1) −2 −1 1 2 x −1 1 u

−1

−2 −1

y ∗ v Λ2 Λ2

2 1

1 H (u, v) = 0 U ∗ U(Λ2) ∗ U(Λ1) −2 −1 1 2 x −1 1 u

−1 HU (u, v) = 2

−2 −1

Figure 4: Example of upsampling

7 3.2 Downsampling

If Λ1 ⊃ Λ2, then the downsampling operation can be defined as

sD(x) = D(s(x)) = s(x), for x ∈ Λ2. (26)

In order to prevent aliasing apply the filter ( d(Λ1) , u ∈ U(Λ∗) H (u) = d(Λ2) 2 (27) D ∗ ∗ 0, u ∈ U(Λ1) \U(Λ2) before the downsampling operation. An example of upsampling is shown in Fig. 4.

y ∗ v Λ1 Λ1

2 1

1 H (u, v) = 0 D ∗ U(Λ1)

∗ U(Λ2) −2 −1 1 2 x −1 1 u

1 −1 HD(u, v) = 4

−2 −1

y ∗ v Λ2 Λ2

2 1

1

∗ U(Λ2) −2 −1 1 2 x−1 1 u

−1

−2 −1

Figure 5: Example of downsampling

3.3 Conversion Between Arbitrary Lattices

If non of the lattices is a sublattice of each other, then the conversion can be achieved by upsampling to Λ3 = Λ1 + Λ2, then downsampling to Λ2. Instead applying HU and HD separately, two filters can be combined into a single filter H(u) as ( d(Λ1) , u ∈ U(Λ∗ ∩ Λ∗) H(u) = d(Λ2) 1 2 (28) ∗ ∗ ∗ 0, u ∈ U(Λ3) \U(Λ1 ∩ Λ2)

8 and applied after the upsampling operation. An example for conversion between non-inclusive lattices is shown in Fig. 6.

y ∗ v Λ1 Λ1

2 1

1

∗ U(Λ1) −2 −1 21 x −1 1 u

−1

−2 −1

y ∗ v Λ2 Λ2

2 1

1

∗ U(Λ2) −2 −1 1 x −1 12 u

−1

−2 −1

y ∗ v Λ3 Λ3

2

1 H(u, v) = 0 ∗ U(Λ3)

∗ U(Λ2) −2 −1 1 2 x −1 u

1 −1 H(u, v) = 2

−2 −1

Figure 6: Example of conversion between non-inclusive lattices

9