The Fractional Fourier Transform (FrFT) has a wide range of applications in fields such as optics, quantum mechanics, image processing, and communications. The FrFT of a function ƒ(x) of order a is defined asFa[ƒ(x)]=∫−∞∞Ba(x,x′)ƒ(x′)dx′  (1)
where the kernel Ba(x, x′) is defined as
                                          B            a                    ⁡                      (                          x              ,                              x                '                                      )                          =                                            e                              i                ⁡                                  (                                                            π                      ⁢                                                                        ϕ                          ^                                                /                        4                                                              -                                          ϕ                      /                      2                                                        )                                                                    ❘                                                sin                  ⁡                                      (                    ϕ                    )                                                  ⁢                                  ❘                                      1                    /                    2                                                                                ×                      e                          iπ              ⁡                              (                                                                            x                      2                                        ⁢                                          cot                      ⁡                                              (                        ϕ                        )                                                                              -                                      2                    ⁢                                          xx                      ′                                        ⁢                                          csc                      ⁡                                              (                        ϕ                        )                                                                              +                                                            x                      ′2                                        ⁢                                          cot                      ⁡                                              (                        ϕ                        )                                                                                            )                                                                        (        2        )            
where ϕ=ax/2 and {circumflex over (ϕ)}=sgn [sin(ϕ)]. This applies to the range 0<|ϕ|<π, or 0<|a|<2. In discrete time, the N×1 FrFT of an N×1 vector can be modeled asXa=Fax  (3)
where Fa is an N×N matrix whose elements are given by
                                          F            a                    ⁡                      [                          m              ,              n                        ]                          =                              ∑                                          k                =                0                            ,                              k                ≠                                  (                                      N                    -                    1                    +                                                                  (                        N                        )                                                                    2                        )                                                                              )                                                      N                    ⁢                                          ⁢                                                    u                k                            ⁡                              [                m                ]                                      ⁢                          e                                                -                  j                                ⁢                                  π                  2                                ⁢                ka                                      ⁢                          u                              k                ⁡                                  [                  n                  ]                                                                                        (        4        )            
and where uk[m] and uk[n] are the eigenvectors of the matrix S defined by
                    S        =                              (                                                                                                                                                                                    C                            0                                                                                                                                                1                                                                                                                                                          0                                                                                        ⋮                                                                                        1                                                              ⁢                                                                                                                                            1                                                                                                                                                  C                            1                                                                                                                                                                                    1                                                                                        ⋮                                                                                        0                                                              ⁢                                                                                                                                            0                                                                                                                      1                                                                                                                                                                                C                      2                                                                                                            ⋮                                                                                        0                                                              ⁢                                                                                                                                            …                                                                                                                      …                                                                                                                                                          …                                                                                        ⋱                                                                                        ⋯                                                              ⁢                                                                                                                                            1                                                                                                                      0                                                                                                                                                          0                                                                                        ⋮                                                                                                              C                                              N                        -                        1                                                                                                                  )                    ⁢                                          ⁢          and                                    (        5        )                                          C          n                =                              2            ⁢                                                  ⁢            cos            ⁢                                                  ⁢                          (                                                                    2                    ⁢                    π                                    N                                ⁢                n                            )                                -          4                                    (        6        )            
The FrFT is a useful approach for separating a signal-of-interest (SOI) from interference and/or noise when the statistics of either are non-stationary (i.e., at least one device is moving, Doppler shift occurs, time-varying signals exist, there are drifting frequencies, etc.). The FrFT enables translation of the received signal to an axis in the time-frequency plane where the SOI and interference/noise are not separable in the frequency domain, as produced by a conventional Fast Fourier Transform (FFT), or in the time domain. Various algorithms for estimating the best rotational parameter a and applications using FrFTs have been developed.
Signal separation using the FrFT can be visualized using the concept of a Wigner Distribution (WD). The WD is a time-frequency representation of a signal. The WD may be viewed as a generalization of the Fourier Transform, which is solely the frequency representation. The WD of a signal x(t) can be written asWx(t,ƒ)=∫−∞∞x(t+τ/2)x*(t−τ/2)e−2πjτƒdτ  (7)
The projection of the WD of a signal x(t) onto an axis ta gives the energy of the signal in the FrFT domain a, |Xa(t)|2. Letting α=aπ/2, this may be written as|Xα(t)|2=∫−∞∞Wx(t cos(α)−ƒ sin(α),t sin(α)+ƒ cos(α))dƒ  (8)
In discrete time, the WD of a signal x[n] can be written as
                                          W            x                    ⁡                      [                                          n                                  2                  ⁢                                      f                    s                                                              ,                                                kf                  s                                                  2                  ⁢                  N                                                      ]                          =                              e                          j              ⁢                              π                N                            ⁢              kn                                ⁢                                    ∑                              m                =                                  l                  1                                                            l                2                                      ⁢                                                  ⁢                                          x                ⁡                                  [                  m                  ]                                            ⁢                                                x                  *                                ⁡                                  [                                      n                    -                    m                                    ]                                            ⁢                              e                                  j                  ⁢                                                            2                      ⁢                      π                                        N                                    ⁢                  km                                                                                        (        9        )            
where l1=max(0, n−(N−1)) and l2=min(n, N−1). This particular implementation of the discrete WD is valid for non-periodic signals. Aliasing is avoided by oversampling the signal x[n] using a sampling rate ƒs. (samples per second) that is at least twice the Nyquist rate.
The WD of an SOI and interference illustrates the benefit of filtering with the FrFT. See graph 100 of FIG. 1. In non-stationary environments, the SOI x(t) and interference xI(t) vary as a function of time and frequency. The channel, too, will vary similarly. The two signals overlap in the time domain (ta=0) and in the frequency domain (ta=1), so they cannot be completely separated using either of these axes. This means that conventional time-based and frequency-based techniques, such as MMSE filtering and FFT analysis, respectively, will not achieve good signal separation. The FrFT can be used to seek the axes where the signals overlap the least, and hence can best be separated.
Repeated FrFT domain filtering can improve separation capability even further. In FIG. 1, two rotations are required to completely filter out the interference. First, rotate to a1, 0<a1<2, to filter out the upper portion of the interfering signal xI(t), and then rotate to a2, 0<a2<2, to filter out the remaining lower part of xI(t). In Sud, S., “Interference Cancellation by Repeated Filtering in the Fractional Fourier Transform Domain Using Mean-Square Error Convergence”, Int. Journal of Engineering Research and Applications (IJERA), Vol. 6, No. 5, Part-2, pp. 43-47 (May 2016), the optimum rotational axes are found by searching, using an MMSE criterion, which resulted in improvement in mean-square error (MSE) over the single stage filter. However, further improvement in performance is still possible. Accordingly, an improved MMSE-FrFT approach may be beneficial.