The problem of super-resolution deals with recovery of spikes or Dirac masses from low-pass projections in the Fourier domain. This is a standard problem with numerous applications in science and engineering. In this setting, the measurements amount to a stream of smooth pulses where the low-pass nature is due to the pulse shape. This may be an excitation pulse used in time-of-flight imaging such as radar, sonar, lidar and ultrasound. The pulse shape may also be represented as a low-pass filter that is an approximation of,
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point spread function of an optical instrument such as a lens, microscope or a telescope.
transfer function of an electronic sensor such as an antenna or a microphone.
Green’s function of some partial differential equation that represents a physical process (for example, diffusion or fluorescence lifetime imaging).
beampattern of a sensor array.
spectral line shape in spectroscopy (often assumed to be a Cauchy, Gaussian or a Voigt distribution).
, time-delay estimation[7, 8, 9, 10], sparse deconvolution  and, time-of-flight imaging (e.g. optical [12, 13, 14], radar [15, 16, 17] and ultrasound [18, 19]). These applications all address the same challenge: “How can one recover a signal with broadband features (spikes) from a given set of narrowband measurements?”
Our problem can be restated as that of uniform sampling and recovery of spikes with a given bandlimited kernel. Unlike bandlimited or smooth signals which follow a linear recovery principle [20, 21], sparse signals rely on non-linear recovery method. Despite the prevalence of the spike recovery problem across several fields (cf. Table I and Table II in ,  as well as ), the link to sampling theory was established only recently by Vetterli , Blu and co-workers in their study of finite rate of innovation
or FRI signals. These are signals which are described by countable degrees of freedom, per unit time and model a broad class of signals. The FRI sampling has been applied to a number of interesting applications such as channel estimation, radar [15, 16, 17], time-resolved imaging [14, 17, 25], sparse recovery on a sphere, image feature detection [27, 28, 29], medical imaging [30, 18, 19, 31], tomography , astronomy , spectroscopy , unlimited sampling architecture [35, 36] and inverse source problems [37, 38].
In either case, super-resolution or sampling of sparse signals, a common feature is that both problems assume a bandlimited kernel. The choice of Fourier domain for defining bandlimitedness may be restrictive. In practice, many systems and physical phenomena are modeled as linear and time-varying/non-stationary. On the other hand, complex exponentials—that are constituent components of the Fourier transform—are the eigenfunctions of linear time-invariant systems. Polynomial phase models[39, 40, 41] that generalize complex exponentials are often used as an alternative basis for modeling time-varying systems. Such models are specified by basis functions of the form . One notable example is that of quadratic chirps which are specified by . Due to their wide applicability, chirp based transformations , multi-scale orthonormal bases and frames  as well as dictionary based pursuit algorithms  have been derived in the literature. Active imaging systems such as radar  and sonar  use chirps for probing the environment. In , Martone demonstrates the use of polynomial phase basis functions of the fractional Fourier transform for multicarrier communication with time-frequency selective channels. Harms et al.  use chirps for identification of linear time-varying systems. Besides chirps, Fresnel transforms  use polynomial phase representation for digital holography  and diffraction. Other applications of polynomial phase functions include time-frequency representations , DOA estimation , sensor array processing [52, 41], ghost imaging , image encryption  and quantum physics .
Polynomial phase representations were also studied in the context of phase space and mathematical physics. This led to the development of unitary transformations such as the fractional Fourier transform (FrFT)  and the Linear Canonical Transform (LCT) [57, 58]. These transformations generalize the Fourier transform in the same way that polynomial phase functions generalizes the complex exponentials, or .
In the area of signal processing, Almeida first introduced the fractional Fourier transform (FrFT) as a tool for time-frequency representations . Following , a number of papers have extended the Shannon’s sampling theorem to the FrFT domain (cf. and  and references there in). In , Bhandari and Zayed developed the shift-invariant model for the FrFT domain which was later extended in [62, 63]. Sampling of sparse signals in the FrFT domain was studied in . Interestingly, all of the aforementioned transformations and corresponding basis functions are specific cases of the Special Affine Fourier Transform (SAFT).
In this paper, our goal is to extend sampling theory of sparse signals beyond the Fourier domain. We do so by considering the SAFT which is a parametric transformation that subsumes a number of well known unitary transformations used in signal processing and optics. We recently studied sampling theory of bandlimited and smooth signals in the SAFT domain in . By using results developed in , here we derive sampling theorems for sparse signals with three distinct flavors:
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Sampling with arbitrary, bandlimited kernels.
Sampling with smooth, time-limited kernels.
Sparse signal recovery from Gabor transform measurements linked with the SAFT domain.
For this purpose, we introduce two mathematical tools:
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the Special Affine Fourier Series (a generalization of the Fourier Series) for representing time-limited functions.
the Gabor transform associated with the SAFT (a generalization of the usual Gabor transform).
We begin with the definition of the SAFT in Section II. The forward transform, its inverse as well as geometric properties are discussed in the subsections that follow. In order to develop sparse sampling theory for the SAFT, in Section II-D we recall convolution operators for the SAFT domain . This allows us to establish the link between convolution and low-pass orthogonal projection operators. We then recall Shannon’s sampling theorem for SAFT bandlimited functions in Section III. Our main results on sparse sampling theory are presented in Section IV where we discuss three cases. Unlike the Fourier basis functions, the SAFT counterparts are aperiodic. As a workaround, in Section IV-A, we develop the Special Affine Fourier Series (SAFS) for time-limited signals. The SAFS is then used to represent sparse signals and we conclude this work with several future directions in Section V.
Throughout the paper, set of integers, reals and complex numbers is denoted by , and , respectively and denotes a set of positive integers. Continuous-time functions are denoted by while are used for their discrete counterparts. We use script fonts for operators, that is, . For instance, denotes the projection operator and the derivative operator of order is written as . Function/operator composition is denoted by
. We use boldface font for representing vectors and matrices, for exampleand , respectively, and is the matrix transpose. We use is denoted by . Dirac distribution is represented by . All operations linked with are treated in terms of distributions. The Kronecker delta is represented by . The space of square-integrable and absolutely integrable functions is denoted by and , respectively and is the inner-product. We use to denote time-reversal.
|SAFT Parameters||Corresponding Transform|
|Fourier Transform (FT)|
|Offset Fourier Transform|
|Fractional Fourier Transform (FrFT)|
|Offset Fractional Fourier Transform|
|Linear Canonical Transform (LCT)|
|Laplace Transform (LT)|
|Fractional Laplace Transform|
|Bilateral Laplace Transform|
|SAFT Parameters||Corresponding Signal Operation|
|SAFT Parameters||Corresponding Optical Operation|
|Free Space Propagation|
Ii The Special Affine Fourier Transform
The Special Affine Fourier Transform or the SAFT was introduced by Abe and Sheridan  as a generalization of the FrFT. The SAFT can be thought of as a versatile transformation which parametrically generalizes a number of well known unitary and non-unitary transformations as well as mathematical and optical operations. In Table I we list its parameters together with the associated mappings.
Ii-a Forward Transform
Mathematically, the SAFT of a signal is a mapping, which is defined by an integral transformation parameterized by a matrix
When , the matrix is the SAFT parameter matrix,
which is obtained by concatenating the Linear Canonical Transform or the LCT matrix,
(see Table I and ), and, an offset vector, with elements and that represent displacement and modulation, respectively. Let denote the time-frequency co-ordinates. The function in (1) is the parametric SAFT kernel based on a complex exponential of quadratic form,
Both and are parameterized by and hence the SAFT kernel is also parameterized by . The exponential part of the kernel is explicitly written as,
Note that has free parameters and is constrained by . Due to this concatenation of the LCT matrix with a vector, the SAFT is also referred to as the Offset Linear Canonical Transform or the OLCT . The matrix arises naturally in applications involving optics and imaging. We refer the reader to the books [68, 58] for further details on the intuitive meaning of such a matrix representation.
The SAFT of the Dirac distribution is calculated by,
and is non-bandlimited.
Ii-B Inverse Transform
In order to define the inverse-SAFT, we first note that the SAFT satisfies the following composition property,
where is a complex number (phase offset). The elements of the resultant SAFT parameter matrix are specified by,
In the context of phase space, the physical significance of the SAFT parameter matrix is that it maps time-frequency co-ordinates into its affine transformed version,
Hence, the inverse-SAFT is defined by some affine transform that allows for the mapping,
Thanks to the composition property (5), setting,
results in the inverse parameter matrix defining inverse-SAFT which is equivalent to an SAFT with matrix ,
where . Thus, the inverse transform (iSAFT) is defined as an SAFT with matrix in (6),
Ii-C Geometry of the Special Affine Fourier Transform
An intriguing property of the SAFT is its geometrical interpretation in the context of time-frequency representations and the fact that the parameter matrix belongs to a class of area preserving matrices—the ones whose determinant is unity. We elaborate on these aspects starting with the cyclic property of the Fourier transform .
Let be the identity operation that is, which we use to define the Fourier operator composition:
From the last equality, , we conclude that the Fourier operator is periodic with . Due to this periodic structure, the Fourier operator can be represented on a circle as shown in Fig. 1.
Unitary mappings that can be continuously defined on the circle (as opposed to ) were first identified by Condon . This is known as the fractional Fourier transform (FrFT). Qualitatively, the FrFT “fractionalizes” the Fourier transform in the sense that can be defined for an arbitrary point on the circle through by the transformation . We compare the Fourier transform with the FrFT in Fig. 1. As shown in Fig. 2(a), the action of the FrFT on the time-frequency co-ordinates results in rotation of the time-frequency plane  due to —an intrinsic property of the FrFT. This is explained by the co-ordinate transformation matrix—the rotation matrix in case of the FrFT (cf Table I).
The submatrix of may be decomposed in several ways. One interesting decomposition relates to the Fourier transform such that where
are modulation matrices111We refer to and as modulation matrices because whenever in (2), the SAFT in (1) amounts to modulation of the function .. This decomposition implies that the SAFT can be implemented as a Fourier transform using the following sequence of steps,
By simplifying , we observe that it is indeed the SAFT of . In this way, we generalize the previously known result of Zayed  that links the FrFT to the Fourier Transform.
An alternative decomposition relates the SAFT with the FrFT and the Fresnel transform via the elegant Iwasawa Decomposition,
In fact, rotation is a special operation of a class of matrices that belong to the special linear group where,
With the exception of the Laplace, Gauss and Bargmann transforms in Table I, all other operations can be explained by which entails that . Since the basis vectors of form a parallelogram in , its enclosed area must always be unity or the area must be preserved under application of . This aspect has important consequences in ray optics where models paraxial optics [66, 68]. In Figs. 2(b), 2(c) and 2(d) we describe the deformation on due to for the Fresnel transform , the LCT and the SAFT.
Geometrically, the inverse transform relies on specification of which undoes the effect of . For the FrFT, the Fresnel transform and the LCT, the operation is simply the inverse of the matrix, that is (cf. Table I
). The case of the SAFT is unique because it implements an affine transform as opposed to the usual case of a linear transform (cf. compare Fig.2(b,c) and Fig. 2(d)). The presence of an offset in (2) warrants an adjustment by (6) for the SAFT.
Ii-D Convolution Structures in the SAFT Domain
A useful property of the Fourier transform is that the convolution of two functions is equal to the pointwise multiplication of their spectrums. More precisely, . However, this property does not extend to the SAFT domain in that (cf. ). Since convolutions are pivotal to the topic of sampling theory, we will work with a generalized version of the convolution operator, denoted by , which allows for a representation of the form .
Definition 1 (Chirp Modulation).
Let be a matrix. We define the chirp modulation function as,
We also define the –parametrized unitary up and down chirp modulation operation,
respectively. Note that .
Based on the definition of chirp modulated functions, we now define the SAFT convolution operator.
Definition 2 (SAFT Convolution).
Let denote the usual convolution operator. Given functions and , the SAFT convolution operator denoted by , is defined as
where ; the same applies to the function .
In Fig. 3, we explain the SAFT convolution operation defined in (11). Note that the SAFT convolution operation is based on the usual convolution of pre-modulated functions and . This operation, also known as chirping, is a standard procedure in optical information processing  and analog processing where it is implemented via mixing circuits (cf. Fig. 6(a) in ). Similarly, in the field of holography, such operations are used for defining Fresnel transforms (cf. (10) in ). Pre- and post-modulations are critical in our context and enforce the convolution-multiplication property. A formal statement of this result is as follows:
Let and be two given functions and let be defined in (11). Then,
where and denote the SAFT of and , respectively and .
The proof of this theorem is presented in . For further results, we the reader to . The duality principle also holds for the SAFT. Namely, multiplication of functions in the time domain results in convolution in the SAFT domains,
Iii SAFT Domain and Bandlimited Subspaces
In order to set the ground for sampling of sparse signals, we begin by recalling the sampling theorem for SAFT bandlimited signals [2, 63]. The notion of bandlimitedness has a de facto association with the Fourier domain. Below, we consider a more general definition.
Definition 3 (Bandlimited Functions).
Let be a square-integrable function. We say that is –bandlimited and write,
With , we obtain the standard case when is –bandlimited in the Fourier domain.
Shannon’s sampling theorem is restricted to Fourier transforms. In that case, and any can be uniquely recovered from samples provided that . For bandlimited signals in the SAFT sense, the statement of Shannon’s sampling theorem is follows.
Theorem 2 (Shannon’s Sampling Theorem for the SAFT Domain ).
Let be an –bandlimited function in the SAFT domain, that is, . Then, we have,
A detailed proof of this theorem that is based on reproducing kernel Hilbert spaces is given in . Here, we will briefly revisit the key steps. The associated computations will be useful in the context of sampling sparse signals.
It is well known that the sampling theorem for the Fourier domain can be interpreted as an orthogonal projection of onto the subspace of bandlimited functions ,
Thanks to the projection theorem,
In the spirit of the Fourier domain result, in , we derived the subspace of bandlimited functions linked with the SAFT domain which take the form of,
The family is an orthonormal basis for the subspace of bandlimited functions in the SAFT domain. Indeed, with since,
Thanks to the orthonormality and the bandlimitedness properties, the implication of the projection theorem (cf. (13)) is that and by developing this further, we obtain,
As in the classical case, the coefficients are equivalent to low-pass filtering in the SAFT domain followed by uniform sampling. To make this link clear, consider the kernel,
and the expansion coefficients in (15) are indeed the samples.
Iv Sparse Sampling and Super-resolution
When is bandlimited in the Fourier domain, the super-resolution problem boils down to estimating ’s and ’s from measurements of,
This problem can be restated as that of sampling spikes or sparse functions given a bandlimited sampling kernel, . This is because of the equivalence,
where , is the sampling rate and is the sparse signal,
In the previous section, we discussed sampling theory of bandlimited signals in the SAFT domain. By considering sparse signals instead of bandlimited functions, the measurements in context of the SAFT domain amount to,
where denotes the sampling operation or modulation with a -periodic impulse train. As shown in (15), whenever , samples uniquely characterize provided that .
Next, we turn our attention to the problem of recovering a sparse signal from low-pass projections (17). In particular, we will discus three variations on this theme where low-pass projections are attributed to:
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Arbitrary, bandlimited sampling kernels.
Smooth, time-limited sampling kernels.
Gabor functions associated with the SAFT domain.
Since sparse signals are time-limited, their periodic extension allows for a Fourier series representation. That said, the basis functions of the SAFT kernel are aperiodic. As a result, before discussing the recovery of sparse signals, we introduce mathematical tools that allow for Fourier series-like representation of time-limited signals.
Iv-a Special Affine Fourier Series (SAFS)
It is well known that the family of functions , with fundamental harmonic , constitutes an orthonormal basis of . These basis functions are used for representing -periodic functions. Let . Due to the orthonormality and completeness properties, it follows that, for every ,
Inspired by the Fourier series representation, here, we develop a parallel for the SAFT domain which is useful for the task of representing time-limited signals including sparse signals.
In order to determine the basis functions associated with the Special Affine Fourier Series or the SAFS, we first identify the candidate functions and then enforce the orthonormality property. Note that a spike at frequency in the SAFT domain results in the time domain function
With the above as our prototype basis function, we would like to represent a time-limited signal as
which mimics (22), and where the SAFS coefficients are
To enable a representation in the form (25), we enforce orthogonality on the candidate basis functions,
for an appropriate . Computing the inner-product explicitly yields,