The finite harmonic oscillator and its associated sequences

Abstract

A system of functions (signals) on the finite line, called the oscillator system, is described and studied. Applications of this system for discrete radar and digital communication theory are explained.

One-dimensional analog signals are complex valued functions on the real line ℝ. In the same spirit, one-dimensional digital signals, also called sequences, might be considered as complex valued functions on the finite line 𝔽_p, i.e., the finite field with p elements. In both situations the parameter of the line is denoted by t and is referred to as time. In this work, we will consider digital signals only, which will be simply referred to as signals. The space of signals ℋ = ℂ(𝔽_p) is a Hilbert space with the Hermitian product given by

A central problem is to construct interesting and useful systems of signals. Given a system 𝔖, there are various desired properties that appear in the engineering wish list. For example, in various situations (1, 2), one requires that the signals will be weakly correlated, i.e., that for every φ≠ϕ∈𝔖

This property is trivially satisfied if 𝔖 is an orthonormal basis. Such a system cannot consist of more than dim(ℋ) signals; however, for certain applications, e.g., code division multiple access (CDMA) (3) a larger number of signals is desired; in that case, the orthogonality condition is relaxed.

During the transmission process, a signal ϕ might be distorted in various ways. Two basic types of distortions are time shift ϕ(t) ↦ L_τϕ(t) = ϕ(t + τ) and phase shift ϕ(t) ↦ M_wϕ(t) = e ϕ(t), where τ, w ∈ 𝔽_p. The first type appears in asynchronous communication and the second type is a Doppler effect due to relative velocity between the transmitting and receiving antennas. In conclusion, a general distortion is of the type ϕ ↦ M_wL_τϕ, suggesting that for every ϕ ≠ φ ∈𝔖, it is natural to require (1) the following stronger condition

Because of technical restrictions in the transmission process, signals are sometimes required to admit low peak-to-average power ratio (4), i.e., that for every ϕ ∈ 𝔖 with ‖ϕ‖₂ = 1

Finally, several schemes for digital communication require that the above properties will continue to hold also if we replace signals from 𝔖 by their Fourier transform.

In this article we construct a system of (unit) signals 𝔖_O, consisting of an order of p³ signals, where p is an odd prime, called the oscillator system. These signals constitute, in an appropriate formal sense, a finite analogue for the eigenfunctions of the harmonic oscillator in the real setting and, in accordance, they share many of the nice properties of the latter class. In particular, the system 𝔖_O satisfies the following properties

1. Autocorrelation (ambiguity function). For every ϕ ∈ 𝔖_O we have

   
2. Cross-correlation (cross-ambiguity function). For every φ≠ϕ∈𝔖_O we have

   

   for every τ, w ∈ 𝔽_p.
3. Supremum. For every signal ϕ ∈ 𝔖_O we have

   

4. Fourier invariance. For every signal ϕ ∈ 𝔖_O its Fourier transform ϕ̂ is (up to multiplication by a unitary scalar) also in 𝔖_O.

In Figs. 1, 2, and 3, the ambiguity function of a signal from the oscillator system is compared with that of random signal and a typical chirp.

Fig. 1.

Ambiguity function of an “oscillator” signal.

Fig. 2.

Ambiguity function of a random signal.

Fig. 3.

Ambiguity function of a chirp.

Remark 1.

Explicit algorithm that generates the oscillator system is given in supporting information (SI) Appendix.

The oscillator system can be extended to a much larger system 𝔖_E, consisting of an order of p⁵ signals if one is willing to compromise Properties 1 and 2 for a weaker condition. The extended system consists of all signals of the form M_wL_τϕ for τ, w ∈ 𝔽_p, and ϕ ∈ 𝔖_O. It is not hard to show that # (𝔖_E) = p²·# (𝔖_O) ≈ p⁵. As a consequence of Eqs. 1 and 2 for every ϕ ≠ φ ∈ 𝔖_E we have

The characterization and construction of the oscillator system is representation theoretic and we devote the rest of the article to an intuitive explanation of the main underlying ideas. As a suggestive model example we explain first the construction of the well known system of chirp (Heisenberg) signals, deliberately taking a representation theoretic point of view (see refs. 2 and 5 for a more comprehensive treatment).

Model Example (Heisenberg System)

Let us denote by ψ : 𝔽_p → ℂ^× the character ψ(t) = e. We consider the pair of orthonormal bases Δ = {δ_a : a ∈ 𝔽_p} and Δ^⋁ = {ψ_a : a ∈ 𝔽_p}, where ${\psi }_a(t)=1/\sqrt{p}\psi (at)$, and δ_a is the Kronecker delta function, δ_a(t) = 1, if t = a and δ_a(t) = 0 if t ≠ a.

Characterization of the Bases Δ and Δ^v.

Let L : ℋ → ℋ be the time shift operator Lϕ(t) = ϕ(t + 1). This operator is unitary and it induces a homomorphism of groups L : 𝔽_p → U(H) given by L_τϕ(t) = ϕ(t + τ) for any τ ∈ 𝔽_p.

Elements of the basis Δ^⋁ are character vectors with respect to the action L, i.e., L_τψ_a = ψ(aτ)ψ_a for any τ ∈ 𝔽_p. In the same fashion, the basis Δ consists of character vectors with respect to the homomorphism M : 𝔽_p → U(H) given by the phase shift operators M_wϕ(t) = ψ(t)ϕ(t).

The Heisenberg Representation.

The homomorphisms L and M can be combined into a single map π̃ : 𝔽_p × 𝔽_p → U(ℋ) which sends a pair (τ, w) to the unitary operator π̃(τ, w) = ψ (−1/2τw) M_w ○ L_τ. The plane 𝔽_p × 𝔽_p is called the time-frequency plane and will be denoted by V. The map π̃ is not an homomorphism since, in general, the operators L_τ and M_w do not commute. This deficiency can be corrected if we consider the group H = V × 𝔽_p with multiplication given by

The map π̃ extends to a homomorphism π : H → U(ℋ) given by

The group H is called the Heisenberg group and the homomorphism π is called the Heisenberg representation.

Maximal Commutative Subgroups.

The Heisenberg group is no longer commutative; however, it contains various commutative subgroups which can be easily described. To every line L ⊂ V that passes through the origin, one can associate a maximal commutative subgroup A_L = {(l, 0) ∈ V × 𝔽_p : l ∈ L}. It will be convenient to identify the subgroup A_L with the line L.

Bases Associated with Lines.

Restricting the Heisenberg representation π to a subgroup L yields a decomposition of the Hilbert space ℋ into a direct sum of one-dimensional subspaces

where χ runs in the set L^⋁ of (complex valued) characters of the group L. The subspace ℋ_χ consists of vectors ϕ ∈ ℋ such that π(l)ϕ = χ(l)ϕ. In other words, the space ℋ_χ consists of common eigenvectors with respect to the commutative system of unitary operators {π(l)}_l∈L such that the operator π (l) has eigenvalue χ (l).

Choosing a unit vector ϕ_χ ∈ ℋ_χ for every χ ∈ L^⋁ we obtain an orthonormal basis ℬ_L = {ϕ_χ : χ ∈ L^⋁}. In particular, Δ^⋁ and Δ are recovered as the bases associated with the lines T = {(τ, 0) : τ ∈ 𝔽_p} and W = {(0, w) : w ∈ 𝔽_p}, respectively. For a general L the signals in ℬ_L are certain kind of chirps. Concluding, we associated with every line L ⊂ V an orthonormal basis ℬ_L, and overall we constructed a system of signals consisting of a union of orthonormal bases

For obvious reasons, the system 𝔖_H will be called the Heisenberg system.

Properties of the Heisenberg System.

It will be convenient to introduce the following general notion. Given two signals φ, ϕ ∈ ℋ, their matrix coefficient is the function m_φ,ϕ : H → ℂ given by m_φ,ϕ(h) = 〈φ, π(h)ϕ〉. In coordinates, if we write h = (τ, w, z), then m_φ,ϕ(h) = ψ(−(1/2)τw + z) 〈φ, M_w ○ L _τϕ〉. When φ = ϕ the function m_ϕ,ϕ is called the ambiguity function of the vector ϕ and is denoted by A_ϕ = m_ϕ,ϕ.

The system 𝔖_H consists of p + 1 orthonormal bases,^‖ altogether p (p + 1) signals and it satisfies the following properties (2, 5):

1. Autocorrelation. For every signal ϕ ∈ ℬ_L the function-A_ϕ- is the characteristic function of the line L, i.e.,

   
2. Crosscorrelation. For every φ ∈ ℬ_L and ϕ ∈ ℬ_M, where L ≠ M, we have

   

   for every v ∈ V. If L = M, then m_ϕ,φ is the characteristic function of some translation of the line L.
3. Supremum. A signal ϕ ∈ 𝔖_H is a unimodular function, i.e., $\left|\varphi (t)\right|=1/\sqrt{p}$ for every t ∈ 𝔽_p; in particular, we have

   

Remark 2.

Note the main differences between the Heisenberg and the oscillator systems. The oscillator system consists of an order of p³ signals, whereas the Heisenberg system consists of an order of p² signals. Signals in the oscillator system admit an ambiguity function concentrated at 0 ∈ V (thumbtack pattern, see Fig. 1) whereas signals in the Heisenberg system admit ambiguity function concentrated on a line (see Fig. 3).

The Oscillator System

Reflecting back on the Heisenberg system we see that each vector ϕ ∈ 𝔖_H is characterized in terms of action of the additive group G_a = 𝔽_p. Roughly, in comparison, each vector in the oscillator system is characterized in terms of action of the multiplicative group G_m = 𝔽_p^×. Our next goal is to explain the last assertion. We begin by giving a model example.

Given a multiplicative character** χ : G_m → ℂ^×, we define a vector χ ∈ ℋ by

We consider the system ℬ_std = {χ : χ ∈ G_m^⋁, χ ≠ 1}, where G_m^⋁ is the dual group of characters.

Characterizing the System ℬ_std.

For each element a ∈ G_m let ρ_a : ℋ → ℋ be the unitary operator acting by scaling ρ_aϕ(t) = ϕ(at). This collection of operators form a homomorphism ρ : G_m → U(ℋ).

Elements of B_std are character vectors with respect to ρ, i.e., the vector χ satisfies ρ_a (χ) = χ(a)χ for every a ∈ G_m. In more conceptual terms, the action ρ yields a decomposition of the Hilbert space ℋ into character spaces ℋ = ⊕ℋ_χ, where χ runs in the group G_m^⋁. The system ℬ_std consists of a representative unit vector for each space ℋ_χ, χ ≠ 1.

The Weil Representation.

We would like to generalize the system ℬ_std in a similar fashion as we generalized the bases Δ and Δ^⋁ in the Heisenberg setting. To do this we need to introduce several auxiliary operators.

Let ρ_a : ℋ → ℋ, a ∈ 𝔽_p^×, be the operators acting by ρ_aϕ(t) = σ(a)ϕ(a⁻¹t) (scaling), where σ is the unique quadratic character of 𝔽_p^×; let ρ_T : ℋ → ℋ be the operator acting by ρ_Tϕ(t) = ψ(t²)ϕ(t) (quadratic modulation); and finally, let ρ_S : ℋ → ℋ be the operator of Fourier transform

where ν is a normalization constant (6). The operators ρ_a, ρ_T and ρ_S are unitary. Let us consider the subgroup of unitary operators generated by ρ_a, ρ_S, and ρ_T. This group turns out to be isomorphic to the finite group Sp = SL₂(𝔽_p); therefore, we obtained a homomorphism ρ : Sp → U(ℋ). The representation ρ is called the Weil representation (7) and it will play a prominent role in this article.

Systems Associated with Maximal (Split) Tori.

The group Sp consists of various types of commutative subgroups. We will be interested in maximal diagonalizable commutative subgroups. A subgroup of this type is called maximal split torus. The standard example is the subgroup consisting of all diagonal matrices

which is called the standard torus. The restriction of the Weil representation to a split torus T ⊂ Sp yields a decomposition of the Hilbert space ℋ into a direct sum of character spaces ℋ = ⊕ ℋ_χ, where χ runs in the set of characters T ^⋁. Choosing a unit vector ϕ_χ ∈ ℋ_χ for every χ we obtain a collection of orthonormal vectors ℬ_T = {ϕ_χ : χ ∈ T^⋁, χ ≠ σ}. Overall, we constructed a system

which will be referred to as the split oscillator system. We note that our initial system ℬ_std is recovered as ℬ_std = ℬ_A.

Systems Associated with Maximal (Nonsplit) Tori.

From the point of view of this article, the most interesting maximal commutative subgroups in Sp are those that are diagonalizable over an extension field rather than over the base field 𝔽_p. A subgroup of this type is called maximal nonsplit torus. It might be suggestive to first explain the analogue notion in the more familiar setting of the field ℝ. Here, the standard example of a maximal nonsplit torus is the circle group SO(2) ⊂ SL₂ (ℝ). Indeed, it is a maximal commutative subgroup that becomes diagonalizable when considered over the extension field ℂ of complex numbers.

The analogy above suggests a way to construct examples of maximal nonsplit tori in the finite field setting as well. Let us assume for simplicity that −1 does not admit a square root in 𝔽_p. The group Sp acts naturally on the plane V = 𝔽_p × 𝔽_p. Consider the symmetric bilinear form B on V given by

An example of maximal nonsplit torus is the subgroup T_ns ⊂ Sp consisting of all elements g ∈ Sp preserving the form B, i.e., g ∈ T_ns, if and only if B(gu, gv) = B(u, v) for every u, v ∈ V. In the same fashion, as in the split case, restricting the Weil representation to a nonsplit torus T yields a decomposition into character spaces ℋ = ⊕ ℋ_χ. Choosing a unit vector ϕ_χ ∈ ℋ_χ for every χ ∈ T^⋁ we obtain an orthonormal basis ℬ_T. Overall, we constructed a system of signals

The system 𝔖_O^ns will be referred to as the nonsplit oscillator system. The construction of the system 𝔖_O = 𝔖_O^s ∪ 𝔖_O^ns, together with the formulation of some of its properties, is the main contribution of this article.

Behavior Under Fourier Transform.

The oscillator system is closed under the operation of Fourier transform, i.e., for every ϕ ∈ 𝔖_O we have (up to a multiplication by a unitary scalar) that ϕ̂ ∈ 𝔖_O. Indeed, the Fourier transform on the space ℂ (𝔽_p) appears as a specific operator ρ (w) in the Weil representation, where

Given a signal ϕ ∈ ℬ_T ⊂ 𝔖_O, its Fourier transform ϕ̂ = ρ(w)ϕ is, up to a unitary scalar, a signal in ℬ_T′ where T′ = wTw⁻¹. In fact, 𝔖_O is closed under all the operators in the Weil representation! Given an element g ∈ Sp and a signal ϕ ∈ ℬ_T we have, up to a unitary scalar, that ρ(g)ϕ ∈ ℬ_T′, where T′ = gTg⁻¹.

In addition, the Weyl element w is an element in some maximal torus T_w (the split type of T_w depends on the characteristic p of the field) and as a result signals ϕ ∈ ℬ_Tw are, in particular, eigenvectors of the Fourier transform. As a consequence, a signal ϕ ∈ ℬ_Tw and its Fourier transform ϕ̂ differ by a unitary constant, and therefore are practically the “same” for all essential matters.

These properties might be relevant for applications to orthogonal frequency division multiplexing (OFDM) (8) where one requires good properties both from the signal and its Fourier transform.

Relation to the Harmonic Oscillator.

Here, we give the explanation why functions in the nonsplit oscillator system 𝔖_O^ns constitute a finite analogue of the eigenfunctions of the harmonic oscillator in the real setting. The Weil representation establishes the dictionary between these two, seemingly, unrelated objects. The argument works as follows.

The one-dimensional harmonic oscillator is given by the differential operator D = ∂² − t². The operator D can be exponentiated to give a unitary representation of the circle group ρ : SO (2, ℝ) → U (L²(ℝ)), where ρ(θ) = e^iθD. Eigenfunctions of D are naturally identified with character vectors with respect to ρ. The crucial point is that ρ is the restriction of the Weil representation of SL₂(ℝ) to the maximal nonsplit torus SO (2, ℝ) ⊂ SL₂ (ℝ).

Summarizing, the eigenfunctions of the harmonic oscillator and functions in 𝔖_O^ns are governed by the same mechanism, namely, both are character vectors with respect to the restriction of the Weil representation to a maximal nonsplit torus in SL₂. The only difference appears to be the field of definition, which for the harmonic oscillator is the reals and for the oscillator functions is the finite field.

Applications

Two applications of the oscillator system will be described. The first application is to the theory of discrete radar. The second application is to CDMA systems. We will give a brief explanation of these problems, while emphasizing the relation to the Heisenberg representation.

Discrete Radar.

The theory of discrete radar is closely related (2) to the finite Heisenberg group H. A radar sends a signal ϕ(t) and obtains an echo e(t). The goal (9) is to reconstruct, in maximal accuracy, the target range and velocity. The signal ϕ(t) and the echo e(t) are, in principal, related by the transformation

where the time shift τ encodes the distance of the target from the radar and the phase shift encodes the velocity of the target. Equivalently saying, the transmitted signal ϕ and the received echo e are related by an action of an element h₀ ∈ H, i.e., e = π(h₀)ϕ. The problem of discrete radar can be described as follows. Given a signal ϕ and an echo e = π(h₀)ϕ extract the value of h₀.

It is easy to show that |m_ϕ,e (h)| = |A_ϕ (h · h₀)| and it obtains its maximum at h₀⁻¹. This suggests that a desired signal ϕ for discrete radar should admit an ambiguity function A_ϕ, which is highly concentrated around 0 ∈ H, which is a property satisfied by signals in the oscillator system (Property 2).

Remark 2.

It should be noted that the system 𝔖_O is “large” consisting of approximately p³ signals. This property becomes important in a jamming scenario.

Code Division Multiple Access (CDMA).

We are considering the following setting.

• There exists a collection of users i ∈ I, each holding a bit of information b_i ∈ ℂ (usually, b_i is taken to be an Nth root of unity).

• Each user transmits his bit of information, say, to a central antenna. In order to do that, he multiplies his bit b_i by a private signal ϕ_i ∈ ℋ and forms a message u_i = b_iϕ_i.

• The transmission is carried through a single channel (for example, in the case of cellular communication the channel is the atmosphere), therefore the message received by the antenna is the sum

  

The main problem (3) is to extract the individual bits b_i from the message u. The bit b_i can be estimated by calculating the inner product

The last expression above should be considered as a sum of the information bit b_i and an additional noise caused by the interference of the other messages. This is the standard scenario also called the synchronous scenario. In practice, more complicated scenarios appear, e.g., asynchronous scenario, in which each message u_i is allowed to acquire an arbitrary time shift u_i(t) ↦ u_i(t + τ_i); phase shift scenario, in which each message u_i is allowed to acquire an arbitrary phase shift u_i(t) ↦ eu_i(t) and probably also a combination of the two where each message u_i is allowed to acquire an arbitrary distortion of the form u_i(t) ↦ eu_i(t + τ_i).

The previous discussion suggests that what we are seeking is a large system 𝔖 of signals that will enable a reliable extraction of each bit b_i for as many users transmitting through the channel simultaneously.

Definition 3 (stability conditions).

Two unit signals φ ≠ ϕ are called stably cross-correlated if |m_ϕ,ψ (v)| ≪ 1 for every v ∈ V. A unit signal ϕ is called stably autocorrelated if |A_ϕ (v)| ≪ 1, for every v ≠ 0. A system 𝔖 of signals is called a stable system if every signal ϕ ∈ 𝔖 is stably autocorrelated and any two different signals φ, ϕ ∈ 𝔖 are stably cross-correlated.

Formally what we require for CDMA is a stable system 𝔖. Let us explain why this corresponds to a reasonable solution to our problem. At a certain time t the antenna receives a message

which is transmitted from a subset of users J ⊂ I. Each message u_i, i ∈ J, is of the form u_i = b_ieϕ_i(t+τ_i) = b_iπ(h_i)ϕ_i, where h_i ∈ H. In order to extract the bit b_i we compute the matrix coefficient

where R_hi is the operator of right translation R_hiA_ϕi(h) = A_ϕi(hh_i).

If the cardinality of the set J is not too big then by evaluating m_ϕi,u at h = h_i⁻¹, we can reconstruct the bit b_i. It follows from Eqs. 1 and 2 that the oscillator system 𝔖_O can support an order of p³ users, enabling reliable reconstruction when an order of $\sqrt{p}$ users are transmitting simultaneously.

Remark about field extensions.

All the results in this article were stated for the basic finite field 𝔽_p for the reason of making the terminology more accessible. However, they are valid for any field extension of the form F_q with q = pⁿ. Complete proofs appear in ref. 6.