Linear signal synthesis using the Radon-Wigner transform

Wood, J.C.; Barry, D. T.

doi:10.1109/78.301845

Cited by 119 publications

(58 citation statements)

References 4 publications

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“…The WD is reconstructed from the RWT using the inverse Radon transform. A good agreement between the RWTs and the WDs obtained experimentally [see Measuring the RWT allows the identification of the quadratic chirp components of signals [1], which is particularly relevant for the characterization of optical systems. The angle α corresponding to the intensity peak in the RWT indicates that the chirp exp½iπx 2 =ðλf c Þ has parameter f c ¼ −s 2 λ −1 tan α.…”

mentioning

confidence: 87%

“…The set of the WD projections at an arbitrary angle in phase-space, known as a Radon-Wigner transform (RWT) [1], contains the same information about the signal as the WD. In contrast to the WD, the RWT takes only nonnegative values and, therefore, can be directly measured [2].…”

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confidence: 99%

“…In the case of one-dimensional (1D) signals, the RWT is a two-dimensional (2D) function defined as Pðα; xÞ ¼ Z dqW ðx cos α − s 2 q sin α; q cos α þ s −2 x sin αÞ; ð1Þ where W ðx; qÞ ¼ R dx 0 Γðx þ x 0 =2; x − x 0 =2Þ expð−i2πx 0 qÞ is the WD of the signal, ðx; qÞ are the space and frequency coordinates, Γðx 1 ; x 2 Þ ¼ hf ðx 1 Þf Ã ðx 2 Þi is the mutual intensity of the signal f ð·Þ, and s is a normalization parameter with dimension of length that depends on the system implementation. The transformation angle α varies in a π interval.…”

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Phase-space tomography with a programmable Radon–Wigner display

et al. 2011

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We show the adaptation of a multifunctional optical system consisting of two spatial light modulators for the optimal measurement of the Radon-Wigner transform of one-dimensional signals. The proposed Radon-Wigner display allows reconstructing the Wigner distribution and the phase or the mutual intensity of fully or partially coherent fields, respectively. It is also suitable for the analysis of two-dimensional rotationally symmetric or separable in Cartesian coordinates optical fields. 0 qÞ is the WD of the signal, ðx; qÞ are the space and frequency coordinates, Γðx 1 ; x 2 Þ ¼ hf ðx 1 Þf Ã ðx 2 Þi is the mutual intensity of the signal f ð·Þ, and s is a normalization parameter with dimension of length that depends on the system implementation. The transformation angle α varies in a π interval. The kernel of the fractional FT needed for the acquisition of the corresponding projections, Pðα; xÞ ¼ hj R duf ðuÞK α ðx; uÞj 2 i, is given bySeveral schemes for the RWT visualization-RadonWigner displays (RWDs)-have been proposed. They are based on the use of either combinations of Fresnel zone plates [3] or varifocal glass lenses [4,5]. In spite of important achievements, these setups have not found wide application in signal processing due to their complex design and the involvement of nonconventional optical elements. Moreover, in most of the schemes, each projection has a different magnification depending on the angle α.The use of spatial light modulators (SLMs) allows constructing an RWD that avoids the previous inconveniences and provides flexibility to the system. Here, we demonstrate how a multifunctional optical scheme consisting of two SLMs can be easily converted into an RWD. Its performance is verified experimentally. We also show that the RWD, primarily created for the description of 1D signals, can also be applied for the analysis of 2D optical fields.

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confidence: 87%

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Phase-space tomography with a programmable Radon–Wigner display

et al. 2011

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“…2,34 This approach led to the development of a chart that contains a continuous representation of the FRT of a signal as a function of its fractional order. 35 This representation may also be useful in optics, since it explicitly shows the propagation of a signal inside a GRIN medium.…”

Section: Radon-wigner-based Phase Spaces a (X P) Chartmentioning

confidence: 99%

Wigner-related phase spaces for signal processing and their optical implementation

2000

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Phase spaces are different ways to represent signals. Owing to their properties, they are often used for signal compression and recognition with high discrimination abilities. We present several recently introduced Wigner-related sets of representations that have improved signal processing performance, and we introduce an optical implementation. This study deals with the generalized Wigner spaces, the fractional Fourier transform, and the x -p and the r -p representations. The optical implementations are demonstrated and discussed.

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“…Recently, a new time-frequency analyzing tool, the Radon -Wigner transform, was suggested 2,3 and used for the time-frequency representation of signals. 4,5 Without being named, this approach led exactly to a chart that contains a continuous representation of the FRT of a signal as a function of the fractional Fourier order. As we mention below, we call this representation the ͑x, p͒ chart, and it may also be useful in optics because it explicitly shows the propagation of a signal inside a gradedindex (GRIN) medium.…”

Section: Introductionmentioning

confidence: 99%

New signal representation based on the fractional Fourier transform: definitions

et al. 1995

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The fractional Fourier transform is a mathematical operation that generalizes the well-known Fourier transform. This operation has been shown to have physical and optical fundamental meanings, and it has been experimentally implemented by relatively simple optical setups. Based on the fractional Fourier-transform operation, a new space-frequency chart definition is introduced. By the application of various geometric operations on this new chart, such as radial and angular shearing and rotation, optical systems may be designed or analyzed. The field distribution, as well as full information about the spectrum and the space -bandwidth product, can be easily obtained in all the stages of the optical system.

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Linear signal synthesis using the Radon-Wigner transform

Cited by 119 publications

References 4 publications

Phase-space tomography with a programmable Radon–Wigner display

Phase-space tomography with a programmable Radon–Wigner display

Wigner-related phase spaces for signal processing and their optical implementation

New signal representation based on the fractional Fourier transform: definitions

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