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Optimal ambiguity functions and Weil's exponential sum bound

Complex-valued periodic sequences, u, constructed by Goran Bjorck, are analyzed with regard to the behavior of their discrete periodic narrow-band ambiguity functions A_p(u). The Bjorck sequences, which are defined on Z/pZ for p>2 prime, are unimodular and have zero autocorrelation on (Z/pZ)\{0}. These two properties give rise to the acronym, CAZAC, to refer to constant amplitude zero autocorrelation sequences. The bound proven is |A_p(u)| \leq 2/\sqrt{p} + 4/p outside of (0,0), and this is of optimal magnitude given the constraint that u is a CAZAC sequence. The proof requires the full power of Weil's exponential sum bound, which, in turn, is a consequence of his proof of the Riemann hypothesis for finite fields. Such bounds are not only of mathematical interest, but they have direct applications as sequences in communications and radar, as well as when the sequences are used as coefficients of phase-coded waveforms.

Our setting is almost exclusively limited to the case that N = p is prime. As such, Z/pZ is a field.
We shall use a remarkable construction of CAZAC sequences u p of prime length p to prove optimal behavior of A p (u p ). The construction is due to Göran Björck [8](1985), [9](1990). By optimal behavior, we mean that if p is an odd prime, then (1) |A and therefore the bound (1) above is indeed of optimal order of magnitude. there would be an easier way to prove Weil's bound for Kloosterman sums, which is an essential consequence of [35] and which has withstood the test of time vis a vis evolutionary simplification.
Remark 1.4. Notwithstanding the level of mathematics required to prove the inequality (1), as noted in Remark 1.3, we emphasize that our coding and implementation of Björck's CAZAC sequences is truly elementary. In this regard, see [4], as well as earlier Björck experiments and constructions by one of the authors, e.g., see [7] and references therein, cf. Remark 1.5. In the parlance of waveform design, Theorem 3.8 is an ideal discrete "thumbtack" narrow band ambiguity function which can be used to design ideal phasecoded waveforms devoid of any substantial time or doppler coupling in the continuous narrow band ambiguity function plane. With regard to hardware implementation of these phase-coded waveforms (as well as others stemming from low correlation sequences), the power, bandwidth, and hardware requirements will introduce noise. It is understood that modifications must be made to the formulation of a given low correlation sequence to permit implementation while controlling this noise.

1.2.
Background. The study of CAZAC sequences and of other sequences related to optimal autocorrelation behavior has origins in several important applications, one of the most prominent being in the general area of waveform design associated with radar and communications, see, e.g., according to year of publication [20,21,11,33,34,10,31,25,24,16,2,3,22,12,7,27]. There are hundreds of articles in this area and so this selection may seem arbitrary, although several of these references contain focused lists of contributions and specific applications. Also see Remark 1.5. There are also purely mathematical origins for the construction of CAZAC sequences. One such origin is due to Norbert Wiener, e.g., see new related constructions in [6,5]. Another may be said to have originated in a question by Per Enflo in 1983. This particular mathematical path has been documented and built upon by Bahman Saffari [28]. Enflo's question is the following for a given odd prime p. Is it true that the Gaussian sequences, u : Z/pZ → C, defined by where ζ p = e 2πi/p , r, s ∈ Z and p does not divide r, are the only unimodular sequences of length p, with u[0] = 1, whose Discrete Fourier Transform (DFT) has modulus 1? This is equivalent to asking whether such sequences u with u[0] = 1 are the only bi-unimodular sequences of odd prime length. Enflo was interested in this because of a problem dealing with exponential sums. Enflo's question has a positive answer for p = 3 and p = 5. In 1984, by computer search, Björck discovered counterexamples to the Enflo question for p = 7 and p = 11, see [8]. Later in 1985, Björck saw the role of Legendre symbols in his counterexamples, and this led to his theorems in [9]. It also led to a host of mathematical problems, many still unresolved, about the number of CAZAC sequences for a given length, see, [14,6,7,4], as well as a several valuable oral and email communications by Saffari [29].
Remark 1.5. It is relevant to mention a striking recent application of low correlation sequences to radar in terms of compressed sensing [17]. In this case, the authors use Alltop sequences [1] Theorem 2, cf. [32] Section 2.1.3. It then becomes natural to think in terms of frames generated by Björck sequences for extending the high-resolution radar/compressed sensing setting of the authors of [17].
Another approach to the problem addressed in Section 1.1 is found in [13], cf. [23]. The authors obtain bounds comparable to those found herein, but their class of signals, called the oscillator system, is not necessarily ZAC although excellent cross-correlation criteria are obtained, something we have not pursued. More important, from the point of view of application, the characterization and construction of the oscillator system are decidedly representation theoretic. As such an explicit algorithm associated with the collection of split tori in Sp requires a Bruhat decomposition.
The companion, [4], of this paper not only exhibits the simplicity of implementation stressed in Remark 1.4, but also reflects the combinatorial and geometrical complexity in the ambiguity function domain due to the role of the Legendre symbol in defining Björck sequences. Some of this complexity is characterized by intricate Latin and magic square patterns. Further, the simplicity of implementation gives rise to useful, efficient bounds off of small neighborhoods of (0, 0) in the ambiguity function domain for compactly supported waveforms on R having p lags whose coefficients are the elements of a Björck sequence u p . Also, it is not difficult to see that, as with the oscillator system, there is Fourier invariance of Björck sequences, most simply calculated in the p ≡ 1 (mod 4) case, e.g., [26].
1.3. Outline. We define Björck sequences in Section 2. Properties of Kloosterman sums are proven in Section 3.1. These, in turn, are used along with Weil's results and the proper decomposition formula to express Björck sequences in the way that allows us to prove Theorem 3.8 in Section 3.2. Section 4 provides figures and data which motivated and guided us.

Björck sequences and multiplicative characters
For each prime number p, recall that the Legendre symbol modulo p is the function The preimage of +1 under the Legendre symbol function is the set Q of nonzero quadratic residues modulo p; and the preimage of −1 under the Legendre symbol function is the set Q C of quadratic nonresidues modulo p. Among the many properties of the Legendre symbol, we shall use the fact that it is a character of the multiplicative group (Z/pZ) × . This means that χ, when restricted to (Z/pZ) × , is a group homomorphism into C × ; see [15], Chapters V and VI.
Definition 2.1. The Björck sequence of length p, where p is a prime and p ≡ 1 (mod 4), is defined by for all k ∈ Z/pZ. The Björck sequence of length p, where p is a prime and p ≡ 3 (mod 4), is defined by for all k ∈ Z/pZ.
In the case p ≡ 1 (mod 4), Definition 2.1 is equivalent to the following definition for the Legendre symbol sequence {0, 1, ..., −1, ..., 1} of length p. We replace the first term 0 by 1, every term 1 by and every term −1 by the complex conjugate of η; see [28] for a modest generalization. As proven by Björck and differently in [4], we obtain a CAZAC, and hence bi-unimodular, sequence with three values, viz., 1 at k = 0, and η and η at k ∈ (Z/pZ) × .
In the case p ≡ 3 (mod 4), Definition 2.1 is equivalent to the following definition for the Legendre symbol sequence {0, 1, ..., −1} of length p. Replace the first term 0 by 1, and replace every −1 by As proven by Björck and differently in [4], we obtain a CAZAC, and hence bi-unimodular, sequence with only two values, viz., 1 and ξ. Kloosterman sums are always real-valued, as the following Lemma states. Proof. By the substitution y = −x, we have The following classical description of certain Kloosterman sums was first observed by Hans Salié in equation (52) of [30], using a formula of Ernst Jacobsthal from a footnote on page 239 of [19]. Jacobsthal's footnote refers the reader to his 1906 Ph.D. thesis, but fortunately the proof of his formula is not difficult to derive.
Remark 3.5. In [35], Weil proves his bound for |K[a, b; p]| by first using Lemma 3.3 to rewrite K[a, b; p] as χ[x 2 − 4a]e 2πix/p and then bounding the new sum. Philosophically, then, it would be more direct not to convert the sum to the form exp(2πi(x + ax −1 )/p). Nevertheless, we have applied the transformation in Lemma 3.3 because the latter form of Kloosterman sums is better known than are the details of Weil's proof.

Main bound.
We shall need the following technical lemma, which gives bounds for the ambiguity function of any sequence that is a function of the Legendre symbol.
Lemma 3.6. Fix an odd prime p and complex numbers r, s, t ∈ C. Let χ : Z/pZ → C be the Legendre symbol modulo p, and let U : Z/pZ → C be the function Set R = (r + s)/2, S = (r − s)/2, T = t − R, and ζ p = e 2πi/p . Then for all m, n ∈ Z/pZ {0}, where E 1 [m, n] = RT +RT ζ mn p , and Proof. For any two functions F, G : Z/pZ → C, write Thus, U = Rη + Sχ + T δ, and hence To compute B p (U, U ), we shall compute each of these nine terms separately. Since m = 0, we have B p (δ, δ) = 0. In addition, B p (η, η) = 0, since k∈Z/pZ e −2πikn/p = 0 and n = 0. We also have B p (χ, χ) = A p (χ) by definition. Meanwhile, it is immediate that Combining the nine computations above, and noting that we have B p (U, U ) = |S| 2 A p (χ) + (E 1 + E 2 )/p, where E 1 and E 2 are the quantities in the statement of Lemma 3.6.
The following elementary bound will be needed to prove the p ≡ 3 (mod 4) case of Theorem 3.8.
Lemma 3.7. Let X, Y ∈ R, and let z ∈ C with |z| = 1. Then Proof. Noting that zz = 1, we have We are now ready to state and prove our main result.
Theorem 3.8. Let p be an odd prime, and let u p be the Björck function for p. Then the ambiguity function, A p (u p ), defined on Z/pZ × Z/pZ as  Figure 2, for the case p = 13, illustrates the symmetries inherent in the function A p (u p ) on Z/pZ×Z/pZ. These are fully explained for all p in [4]; and they led to the realization of the complexity involved in proving Theorem 3.8, as well as to a host of geometrical and combinatorial phenomena and problems. Figure 3 illustrates Theorem 3.8 for the case p = 503. Table 1 indicates some of the finer behavior of the quantity (4), over three different ranges of primes. This data suggested to us that 2/ √ p was very nearly the upper bound for |A p (u p )[m, n]|, (m, n) = (0, 0), and it helped lead us to the proof that 2/ √ p + 4/p is an upper bound. In addition, although a number of primes p ≡ 1 (mod 4) require a bound larger than 2/ √ p, we noted that only very few primes p ≡ 3 (mod 4) allowed