Digital computation of the fractional Fourier transform
Summary (3 min read)
I. INTRODUCTION
- Other applications that are currently under study or have been suggested include phase retrieval, signal detection, radar, tomography, and data compression.
- Section I11 reviews some straightforward yet inefficient methods of computing the fractional Fourier transform.
- Fast computational algorithms are presented in Section IV, and simulation examples are given in Section V. Some alternate methods are discussed in Section VI to better situate the suggested algorithm among other possibilities.
B. Relation to the Wigner Distribution and the Concept of Fractional Fourier Domains
- Notice that (7) and ( 8) are special cases of this equation.
- In general, the projection of the Wigner distribution on the ath fractional Fourier domain gives the magnitude squared of the ath fractional Fourier transform of the original function.
- There is actually nothing special about any of the continuum of domains; the privileged status the authors assign to the time and frequency domains can be interpreted as an arbitrary choice of the origin of the parameter a.
- All of the fractional transforms, including the 0th transform (the function itself), are different functional representations of an abstract signal in different domains.
C. Compactness in the Time Domain, the Frequency Domain, and Wigner Space
- It is well known that a function and its Fourier transform cannot be both compact (unless they are identically zero).
- In practice, however, it seems that the authors are always working with a finite time interval and a finite bandwidth.
- The time-bandwidth product can be crudely defined as the product of the temporal extent of the signal and its (double-sided) bandwidth.
- The authors will assume that the time-domain representation of their signal is approximately confined to the interval [-Atla, At121 and that its frequency-domain representation is confined to the interval [-Af/2, Af/Z].
- Ax around the origin, is equivalent to assuming that the Wigner distribution is confined within a circle of diameter Ax, With this, the authors mean that a sufficiently large percentage of the energy of the signal is contained in that circle, For any signal, this assumption can be justified by choosing Ax sufficiently large.
Iv. FAST COMPUTATION OF THE FRACTIONAL FOURIER TRANSFORM
- The fractional Fourier transform is a member of a more general class of transformations that are sometimes called linear canonical transformations or quadratic-phase transforms [20] .
- Members of this class of transformations can be broken down into a succession of simpler operations, such as chirp multiplication, chirp convolution, scaling, and ordinary Fourier transformation.
- Here, the authors will concentrate on two particular decompositions that lead to two distinct algorithms.
A. Method Z
- First, the authors choose to break down the fractional transform into a chirp multiplication followed by a chirp convolution followed by another chirp multiplication [171, [391.
- There are efficient ways of performing the required interpolation [40].
- Overall, the procedure starts with N samples spaced at l/Ax, which uniquely characterize the function f(z), and returns the same for f a (x) .
- A is a diagonal matrix that corresponds to chirp multiplication, and Hz, corresponds to the convolution operation.
- The authors notice that 3': allows us to obtain the samples of the ath transform in terms of the samples of the original function, which is the basic requirement for a definition of the discrete fractional Fourier transform matrix.
B. Method I1
- The authors now turn their attention to an alternative method that does not require Fresnel integrals.
- The authors are again assuming By using ( 25) and ( 24) and changing the order of integration and summation, they obtain.
VI. DISCUSSION OF ALTERNATE METHODS
- The method presented in the previous section is only one of many possible ways to decompose the calculation.
- The overall sequence of steps is of the form convolution-scaling multiplication.
- Yet another decomposition is the following: EQUATION ).
- All these methods result in the same time complexity.
- This is because scaling would require additional interpolations, etc., which will require additional computation.
VII. THE DISCRETE FRACTIONAL FOURIER TRANSFORM
- In Section V, the authors obtained matrices that when multiplied with the samples of a function, gave us the samples of the fractional Fourier transform of the function.
- Thus, it would seem that the authors have found the discrete fractional Fourier transform matrix.
- Remember that the authors have made the assumption that the Wigner distribution of the signal is confined to a circle of diameter Ax around the origin.
- Of course, all of these matrices will yield results that are in increasingly better agreement as the signal energy contained in the circle is increasingly closer to the total energy; therefore, this definition, together with any other definitions, will approach each other in this limit.
A. Discrete Fractional Fourier Transformation
- The relation between the DFT and the continuous ordinary Fourier transform was given in Section 11-D.
- That is, the authors wish a definition that maps the sample vector f of the original function into the sample vector f a of the fractional transform.
- The authors have argued at length that this matrix should not be the simple functional ath power of the ordinary DFT matrix if they are to reach a definition of the discrete fractional Fourier transform relation that directly corresponds to the continuous transform.
- Two candidates for F a that are consistent with their basic requirement are the matrices FY and F:I, which were derived in Section IV.
- Whereas these definitions are technically acceptable, it may be possible to come up with simpler loolung definitions or definitions with other analytically desirable properties.
VIII. CONCLUSIONS
- The fractional Fourier transform is a subclass of a more general class of integral transformations characterized by quadratic complex exponential kernels.
- It is often not possible to evaluate these by direct numerical integration because the fast oscillations of the phase of the complex exponential would imply excessively large sampling rates.
- These methods might also require sampling rates that are significantly higher than the Nyquist rate, depending on the order and particular decomposition employed.
- Since the computation of the fractional transform does not take much longer than the computation of the ordinary Fourier transform, algorithms that can improve performance by employing the fractional transform instead of the ordinary transform can be implemented at no additional cost.
- The authors arrived at two distinct definitions, which of course give results that are equal within the intrinsically necessary approximation of assuming the signals to be limited in both time and frequency.
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Citations
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Cites background from "Digital computation of the fraction..."
...It has been shown that the transform of a continuous function whose time- or space-bandwidth product is can be computed in the order of time [21], just like the ordinary Fourier transform....
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...This improvement comes at no additional cost since computing the fractional Fourier transform is not more expensive than computing the ordinary Fourier transform [21]....
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Cites background or methods from "Digital computation of the fraction..."
..., [57,58]) and the fact that most of these provide a satisfactory approximation to the continuous transform....
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...proposed two innovative approaches for obtaining the DFRFT through sampling of the FRFT [58]....
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...However, both presented cases assumed that the WD of x(t) is zero outside an origin-centered circle of diameter equal to the sampling period [58]....
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...This is a desirable property for a definition of the DFRFT matrix [58]....
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...In order to alleviate some of the problems associated with the DFRFT proposed in [58], a new type of DFRFT, which is unitary, reversible, and flexible, was proposed in [56]....
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Cites background or methods from "Digital computation of the fraction..."
...Recently, a fast digital algorithm has been proposed [ 5 ]....
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...We can find the samples of the th fractional Fourier transform of the input and output processes from the following equations [ 5 ]:...
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...A definition of the discrete fractional Fourier transform is suggested in [ 5 ]....
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...resulting filter can be implemented in time, just like the ordinary Fourier transform [ 5 ], or can be implemented optically with the same kind of hardware as the ordinary Fourier transform [7]‐[12]....
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...Notice that the integrals appearing in (24) are simply fractional Fourier transformations and can be simulated using the procedure given in [ 5 ]....
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References
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