# Linear Canonical Transforms

The Fourier transform rotates the time-frequency content of a signal over 90 degrees from the time axis to the frequency axis. Already in the 1970s it was observed that certain optical systems rotated the signal over an arbitrary angle, which became known as the fractional Fourier transform since it it acts like a fractional (i.e., a real) power of the Fourier operator. The fractional Fourier transform is thus depending on this rotation angle forms a one-parameter family of transforms.

Around the same time, quantum physicists realized that certain systems could transform the position and momentum as a vector into their new values leaving the quantum mechanics invariant. All such transforms were obtained by multiplying the position-momentum vector with a unimodular matrix. In the scalar case such a matrix depends on 3 free parameters. In a general N-dimensional space we have a 2 by 2 block matrix that form a real symplectic group Sp(2N). These transforms are the essence of the family of linear canonical transforms (LCT).

When in 2D optics the vector of position and momentum of the optical ray is considered, the same kind of transformation can be used, and the fractional Fourier transform like many other (fractional) transforms can be seen as a special case of the LCT. The reader less familiar with the subject should be warned that these fractional transforms are not directly related to the equally important and equally flourishing domain known as fractional calculus which studies fractional derivatives and fractional integrals.

Since the early days many papers appeared on all kinds of fractional transforms, and even several books, among which a basic one on *The Fractional Fourier Transform* by Haldun M. Ozaktas, Zeev Zalevsky, and M. Alper Kutay in 2001. There the emphasis was on definitions, mathematical properties and computation and their applications in optics and signal processing. The LCT is already there but it is not the main focus. Two of the authors of that book are now also editor of the present one. One could think of it as the LCT analog of their fractional Fourier transform book, but less extensive, and it is not a monograph. Several experts are contributing to the present book. It gives an up-to-date overview of the many aspects of the LCT. As it appears as a volume of the *Springer Series in Optical Sciences*, there is an understandable bias towards the optical viewpoint and applications, with less emphasis on the quantum physics.

The fifteen chapters are subdivided into three parts: (1) Fundamentals, (2) Discretization and computation, (3) Applications. All the aspects are covered: operator theory, theoretical physics, analysis, and group theory in the early chapters, discrete approximations and digital implementation in the second part, and it ends with some applications in the last part.

In the first part one gets the basics in some 100 pages: some history and of course the definition and properties, the kernels when written as an integral transform, all the types and special cases, and the effects of the transform in phase-space. The eigenmodes are as important as the Gauss-Hermite eigenfunctions of the Fourier transform. So there is a separate chapter dealing with the eigenfunctions. Also uncertainty principles play an important role when it comes to sampling theory for these transforms. The first part is completed with two chapters covering extensively the optical aspects of the LCT.

The second part deals with the computational aspects. While the fractional Fourier transform resulted in a rotation in the time-frequency plane, the LCT will result in oblique transforms. It is then important to obtain some analogs of bandwidth, sampling theorems, and degrees of freedom in the signal when one wants to come to an implementation of a fast discrete LCT transform. Analyzing these effects is related to a decomposition of the LCT in a sequence of elementary transforms which boils down to a sequence of chirp multiplications and Fourier transforms. Several possibilities are proposed to come to a fast digital LCT implementation, although no software is provided. The approach taken is from a signal processing viewpoint. One chapter gives an alternative that is based on optical interpretation. That alternative approach relies on coherent self-imaging, known as Talbot effect which describes Fresnel diffraction of a strictly periodic wavefront. Knowing that Fresnel diffraction is a special case of LCT. Therefore various generalizations of self-imaging in the wider LCT context can also lead to a practical implementation of discrete LCT.

The application part illustrates how LCT can be used to solve certain problems like deterministic phase retrieval, analyzing holographic systems, double random phase encoding, speckle metrology and quantum states of light.

This book is a most welcome addition to the literature. The subjects discussed appear in quite diverse contexts and it is therefore difficult to get the same overview as it is presented here. The general approach is the same as was used in the fractional Fourier book that I mentioned above, but it is less thorough. Moreover, since the different chapters are written by different authors, and because they highlight different aspects, the notation is not always strictly uniform, but that should not be hindering too much. The part on the algorithmic implementation is rather detailed but no software is provided, so it is a challenge for computer scientists to design an optimal implementation so that it can become standard software in signal processing packages.

**Submitted by Adhemar Bultheel |

**9 / Feb / 2016