Designing unimodular waveforms with an impulse-like autocorrelation is central in the general area of waveform design, and it is particularly relevant in several applications in the areas of radar and communications. In the former, the waveforms can play a role in effective target recognition, e.g., [1-8]; and in the latter they are used to address synchronization issues in cellular (phone) access technologies, especially code division multiple access (CDMA), e.g., [9,10]. The radar and communications methods combine in recent advanced multifunction RF systems (AMRFS). In radar there are two main reasons that the waveforms should be unimodular, that is, have constant amplitude. First, a transmitter can operate at peak power if the signal has constant peak amplitude—the system does not have to deal with the surprise of greater than expected amplitudes. Second, amplitude variations during transmission due to additive noise can be theoretically eliminated. The zero autocorrelation property ensures minimum interference between signals sharing the same channel.

Constructing unimodular waveforms with zero autocorrelation can be related to fundamental questions in harmonic analysis as follows. Let be the real numbers, the integers, the complex numbers, and set. The aperiodic autocorrelation of a waveform, is defined as

A general problem is to characterize the family of positive bounded Radon measures F, whose inverse Fourier transforms are the autocorrelations of bounded waveforms X. A special case is when on and X is unimodular on. This is the same as when the autocorrelation of X vanishes except at 0, where it takes the value 1. In this case, X is said to have perfect autocorrelation. An extensive discussion on the construction of different classes of deterministic waveforms with perfect autocorrelation can be found in [11]. Instead of aperiodic waveforms that are defined on, in some applications, it might be useful to construct periodic waveforms with similar vanishing properties of the autocorrelation function. Let be an integer and be the finite group with addition modulo n. The periodic autocorrelation of a waveform is defined as

It is said that is a constant amplitude zero autocorrelation (CAZAC) waveform if each and

The literature on CAZACs is overwhelming. A good reference on this topic is [3], among many others. Literature on the general area of waveform design include [12-14]. Comparison between periodic and aperiodic autocorrelation can be found in [15].

Here the focus is on the construction of stochastic aperiodic waveforms. Henceforth, the reference to waveforms shall imply aperiodic waveforms unless stated otherwise. These waveforms are stochastic in nature and are constructed from certain random variables. Due to the stochastic nature of the construction, the expected value of the corresponding autocorrelation function is analyzed. It is desired that everywhere away from zero, the expectation of the autocorrelation can be made arbitrarily small. Such waveforms will be said to have almost perfect autocorrelation and will be called zero autocorrelation stochastic waveforms. First discrete waveforms, , are constructed such that X has almost perfect autocorrelation and for all This approach is extended to the construction of continuous waveforms, , with similar spike like behavior of the expected autocorrelation and for all Thus, these waveforms are unimodular. The stochastic and non-repetitive nature of these waveforms means that they cannot be easily intercepted or detected by an adversary. Previous work on the use of stochastic waveforms in radar can be found in [16-18], where the waveforms are only real-valued and not unimodular. In comparison, the waveforms constructed here are complex valued and unimodular. In addition, frame properties of frames constructed from these stochastic waveforms are discussed. This is motivated by the fact that frames have become a standard tool in signal processing. Previously, a mathematical characterization of CAZACs in terms of finite unit-normed tight frames (FUNTFs) has been done in [2].

Let be a random variable with probability density function Assuming to be absolutely continuous, the expectation of denoted by is

The Gaussian random variable has probability density function given by The mean or expectation of this random variable is and the variance, is In this case it is also said that follows a normal distribution and is written as The characteristic function of at , is denoted by. For further properties of expectation and characteristic function of a random variable the reader is referred to [19].

Let be a Hilbert space and let where is some index set, be a collection of vectors in. Then is said to be a frame for if there exist constants and such that for any

The constants A and B are called the frame bounds. Thus a frame can be thought of as a redundant basis. In fact, for a finite dimensional vector space, a frame is the same as a spanning set. If the frame is said to be tight. Orthonormal bases are special cases of tight frames and for these,

If is a frame for then the map given by is called the analysis operator. The synthesis operator is the adjoint map given by

The frame operator is given by For a tight frame, the frame operator is just a constant multiple of the identity, i.e., where is the identity map. Every can be represented as

Here is also a frame and is called the dual frame. For a tight frame, is just Tight frames are thus highly desirable since they offer a computationally simple reconstruction formula that does not involve inverting the frame operator. The minimum and maximum eigenvalues of are the optimal lower and upper frame bounds respectively [20]. Thus, for a tight frame all the eigenvalues of the frame operator are equal to each other. For the general theory on frames one can refer to [20,21].

The construction of discrete unimodular stochastic waveforms, , with almost perfect autocorrelation is done in Section 2. This is first done with the Gaussian random variable and then generalized to other random variables. The variance of the autocorrelation is also estimated. The section also addresses the construction of stochastic waveforms in higher dimensions, i.e., construction of, that have almost perfect autocorrelation and are unit-normed, considering the usual norm in In Section 3 the construction of unimodular continuous waveforms with almost perfect autocorrelation is done using Brownian motion.

As mentioned in Section 1.2, frames are now a standard tool in signal processing due to their effectiveness in robust signal transmission and reconstruction. In Section 4, frames in are constructed from the discrete waveforms of Section 2 and the nature of these frames is analyzed. In particular, the maximum and minimum eigenvalues of the frame operator are estimated. This helps one to understand how close these frames are to being tight. Besides, it follows, from the eigenvalue estimates, that the matrix of the analysis operator, F, for such frames, can be used as a sensing matrix in compressed sensing.

Let be independent identically distributed (i.i.d.)

random variables following a Gaussian or normal distribution with mean 0 and variance i.e., Define by

where i is. Thus, for each, and X is unimodular. The autocorrelation of X at is

where the limit is in the sense of probability. Theorem 2.1 shows that the waveform given by (3) has autocorrelation whose expectation can be made arbitrarily small for all integers

Theorem 2.1. Given the waveform defined in (3) has autocorrelation such that

Proof. 1) When

and so

2) Let One would like to calculate

Let Then Let Then for each

and. Thus, by the Dominated Convergence Theorem [19], which justifies the interchange of limit and integration below, one obtains

where the last line uses the fact that the s are i.i.d. random variables. Here is the characteristic function at of which is the same as that for any other due to their identical distribution. The characteristic function at of a Gaussian random variable with mean 0 and variance is Thus

3) When a similar calculation for gives

Together, this shows that given and any

which indicates that the expectation of the autocorrelation at any integer can be made arbitrarily small depending on the choice of.                    □

As shown in Theorem 2.1 the expectation of the autocorrelation can be made arbitrarily small but this is not useful unless one can estimate the variance of the autocorrelation. Denoting the variance of by

one has

First consider

By applying the Lebesgue Dominated Convergence Theorem one can bring the expectation inside the double sum to get

The sum

may have cancelations among terms involving n with terms involving m. Suppose that for a fixed n and m there are indices that cancel in each of the four sums in (4). Due to symmetry, the same number i.e., of terms will cancel in each sum. Depending on n and m, lies between 0 and k, i.e., For the sake of making the notation less cumbersome, will from now on be written as. When If or then Each sum in (4) has k terms and of these get cancelled leaving terms. One can re-index the variables in (4) and write it as

where the sign depends on whether is less than or greater than Thus

.

Due to the independence of the s, this means

The minimum is attained for and the maximum at Thus

and

This gives

A similar calculation can be done for Thus for

So far the construction of discrete unimodular zero autocorrelation stochastic waveforms has been based on Gaussian random variables. This construction can be generalized to many other random variables. The unimodularity of the waveforms is not affected by using a different random variable. The following theorem characterizes the class of random variables that can be used to get the desired autocorrelation.

Theorem 2.2. Let be a sequence of i.i.d. random variables with characteristic function Suppose that the probability density function of the s is even and that goes to 0 as t goes to infinity. Then, given the waveform given by

has almost perfect autocorrelation.

Proof. Since the density function of each is even this means that the characteristic function is real valued [19]. Following the calculation in the proof of Theorem 2.1, the expected autocorrelation of for is

and this goes to zero with by the hypothesis.      □

Example 2.3. Suppose the s follow a bilateral distribution that has density with and characteristic function. Then for,

and this can be made arbitrarily small with.

In the same way as was done in the Gaussian case, for

and

Thus

Example 2.4. Suppose that the s follow the Cauchy distribution with density function Note thatdisregarding the constant this is the characteristic function of the random variable considered in Example 2.3. The characteristic function of the s is now the same as the distribution function in Example 2.3. For

which can be made arbitrarily small with Also,

Here one is interested in constructing waveforms , It is desired that has unit norm and the expectation of its autocorrelation can be made arbitrarily small. One way to construct is based on the construction of the one dimensional example given in Section 2.1. This is motivated by the higher dimensional construction in the deterministic case [2]. As before, is a sequence of i.i.d. Gaussian random variables with mean zero and variance. Next, one defines. The waveform is then defined as

In this case, the autocorrelation is given by

where is the usual inner product in. The length or norm of any is thus given by

From (5),

Thus the s are unit-normed. The following Theorem 2.5 shows that the expected autocorrelation of v can be made arbitrarily small everywhere except at the origin.

Theorem 2.5. Given the waveform defined in (5) has autocorrelation such that

Proof. As defined in (6),

When

Thus,

For due to (5),

Consider

Similarly, for, one gets

              □

Thus the waveform as defined in this section is unit-normed and has autocorrelation that can be made arbitrarily small.

Remark 2.6. As in the one dimensional construction, it is easy to see that here too the construction can be done with random variables other than the Gaussian. In fact, all random variables that can be used in the one dimensional case, i.e., ones satisfying the properties of Theorem 2.2, can also be used for the higher dimensional construction.

It can be shown that the periodic case follows the same nature as the aperiodic case. The sequence is defined in the same way as in Section 2.1, i.e.,

where Following the definition given in (2), when

When the expectation of the autocorrelation is

For

where one uses the fact that the s are i.i.d.. A similar calculation for negative values of k suggests that the autocorrelation can be made arbitrarily small, depending on for all non-zero values of k. Also, as in the aperiodic case, this result can be obtained for random variables other than the Gaussian.

This section discusses the construction of sets of vectors in as given by (8). The frame properties of such sets are analyzed. In fact, it is shown that the expectation of the eigenvalues of the frame operator are close to each other, the closeness increasing with the size of the set. The bounds on the probability of deviation of the eigenvalues from the expected value is also derived. The related inequalities arise from an application of Theorem 4.1 [22] below.

Theorem 4.1. (Azuma’s Inequality) Suppose that is a martingale and

almost surely. Then for all positive integers and all positive reals

Consider vectors in i.e., in (8). Then and

Considering the set the frame operator of V is

Theorem 4.2. 1) Consider the set where the vectors are given by (9). The minimum eigenvalue, and the maximum eigenvalue, of the frame operator of V satisfy

where

2) The deviation of the minimum and maximum eigenvalue of from their expected value is given, for all positive reals by

Proof. 1) The frame operator of

is given in (10). The eigenvalues of are and where

Let

so that

Note that for and are independent and so Also, since the s are i.i.d. and the characteristic function of the s is symmetric,

and therefore

Thus

The above estimate on implies that

Since and, (12) implies

Noting that and one finally gets, after setting

2) To prove 2) we use the Doob martingale and Azuma’s inequality [22]. For let Here the Doob martingale is the sequence where

and

Note that and Also,

So by Azuma’s Inequality (see Theorem 4.1)

Since this means

and

.

Going back to the actual frame operator, whose eigenvalues are and the following estimates hold.

and

Corollary 4.3. The eigenvalues of the frame operator considered in Theorem 4.2 satisfy, for all positive reals r,

where

Proof. Due to part 1) of Theorem 4.2

This implies, as a consequence of part 2) of Theorem 4.2, that

In a similar way, from part 1) of Theorem 4.2,

which implies, as a consequence of part 2) of Theorem 4.2, that

Remark 4.4. In Theorem 4.2, as M tends to infinity, the value of in (11) can be made arbitrarily small based on the choice of This in turn implies that the two eigenvalues can be made arbitrarily close to each other, with On the other hand, for a fixed M, as tends to zero, (11) becomes

For general d and M, in order to use existing results on the concentration of eigenvalues of random matrices [23, 24], a slightly different construction of the frame needs to be considered. Let be i.i.d. random variables following a Gaussian distribution with mean zero and variance It can be shown that

and the variance

One can define the following two dimensional sequence. For

Consider the mapping given by

As before, let and consider the set of unit vectors in. The frame operator of this set is

.

Let

so that The matrix A has entries with mean zero and variance According to results in [23], if as , then the smallest and largest eigenvalues of converge almost surely to and, respectively.

Theorem 4.5. Let be the singular values of the matrix A given by (14). Then the following hold.

1) Given there is a large enough d such that

2)

where and are universal positive constants.

Proof. Let be the mapping that associates to a matrix it largest singular value. Equip with the Frobenius norm

Then the mapping is convex and 1-Lipschitz in the sense that

for all pairs of d by M matrices [24].

We think of A as a random vector in The real and imaginary parts of the entries of are supported in Let P be a product measure on. Then as a consequence of the concentration inequality (Corollary 4.10, [24]) we have

where is the median of. It is known that the minimum and maximum singular values of A converge almost surely to and, respectively, as d, M tend to infinity and. As a consequence, for each and M sufficiently large, one can show that the medians belong to the fixed interval

which gives

For the smallest singular value we cannot use the concentration inequality as used for since the smallest singular value is not convex. However, following results in [25] (Theorem 3.1) that have been used in [26] in a similar situation as here, one can say that whenever, where is greater than a small constant,

where and are positive universal constants.   □

Remark 4.6. Note that the square of the singular values of A are the eigenvalues of and so the estimates given in (15) and (16) give insight into the corresponding deviation of the eigenvalues of the frame operator.

Remark 4.7. (Connection to compressed sensing) The theory of compressed sensing [27-29] states that it is possible to recover a sparse signal from a small number of measurements. A signal is k-sparse in a basis

if x is a weighted superposition of at most k elements of. Compressed sensing broadly refers to the inverse problem of reconstructing such a signal x from linear measurements with, ideally with. In the general setting, one has, where is a sensing matrix having the measurement vectors as its columns, x is a length-M signal and y is a length-d measurement.

The standard compressed sensing technique guarantees exact recovery of the original signal with very high probability if the sensing matrix satisfies the Restricted Isometry Property (RIP). This means that for a fixed k, there exists a small number, such that

for any k-sparse signal x. By imitating the work done in [26] (Lemmas 4.1 and 4.2), it can be shown, due to Theorem 4.5, that matrices A of the type given in (14) satisfy

the RIP condition and can therefore be used as measurement matrices in compressed sensing. These matrices are different from the traditional random matrices used in compressed sensing in that their entries are complexvalued and unimodular instead of being real-valued and not unimodular.

Example 4.8. This example illustrates the ideas in this subsection. First consider M = 5 and d = 3 so that there are 5 vectors in Taking from a normal distribution with mean 0 and variance a realization of the matrix is

.

Then taking is

The condition number, ratio of the maximum and minimum eigenvalues, of As the number of vectors M is increased, the condition number gets closer to 1. Figure 1 shows the behavior of the condition number with the increase in the number of vectors.