5. THE CLASSES OF FOURIER TRANSFORMS
        There are four classes of Fourier transform, which are represented in the following table. So
        far, we have concentrated on the discrete Fourier transform.
                             Table 1. The classes of Fourier transforms*
                                            Periodic                 Aperiodic
                    Continuous          Discrete aperiodic      Continuous aperiodic
                                          Fourier series           Fourier integral
                    Discrete            Discrete periodic        Continuous periodic
                                           Discrete FT            Discrete-time FT
            * The class of the Fourier transform depends upon the nature of the function which
            is transformed. This function may be discrete or continuous and it may be periodic
            or aperiodic. The names and the natures of corresponding transforms are shown in
            the cells of the table.
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                                  D.S.G. POLLOCK: Lectures in Lodz
      (i) The Fourier integral:
                              1  ∞         iωt                       ∞         −iωt
                     x(t)=2π −∞ξ(ω)e dω              ←→ ξ(ω)= −∞x(t)e                dt
      (ii) The classical Fourier series:
                                  ∞                                T
                                  iωt                          1           −iω t
                         x(t)=         ξ e j      ←→ ξ =               x(t)e    j dt
                                        j                  j    T
                                j=−∞                                0
                                 D.S.G. POLLOCK: Lectures in Lodz
      (iii) The discrete-time Fourier transform:
                                  π                                  ∞
                              1             iωt                        −iωt
                        x =           ξ(ω)e    dω    ←→ ξ(ω)=              x e
                         t    2π                                            t
                                   −π                               t=−∞
      (iv) The discrete Fourier transform:
                                  T−1                            T−1
                                   iωt                       1         −iω t
                             x =       ξ e j     ←→ ξ =              x e    j
                              t         j                 j   T       t
                                  j=0                            t=0
                                      D.S.G. POLLOCK: Lectures in Lodz
        The relationship between the continuous periodic function and its Fourier series transform
        can be summarised by writing
                                       ∞                               T
                                       iωjt                        1           −iωjt
                             x(t)=         ξ e        ←→ ξ =               x(t)e      dt.               (5.1)
                                            j                  j    T
                                     j=−∞                               0
        The discrete-time Fourier transform reverses the role of the time and frequency domains.
        It maps an absolutely summable sequence x(t)={x ;t =0,±1,±2,...} into a continuous
                                                                   t
        periodic function:
                                       π                                   ∞
                                   1             iωt                        −iωt
                             x =           ξ(ω)e    dω    ←→ ξ(ω)=              x e     .               (5.2)
                              t    2π                                             t
                                        −π                                t=−∞
        The Fourier integral transform maps an absolutely integrable function from one domain to
        a similar function in the other domain:                         
                                    1  ∞         iωt                     ∞        −iωt
                           x(t)=2π −∞ξ(ω)e dω ←→ ξ(ω)= −∞x(t)e                         dt,              (5.3)
        whereas a discrete Fourier transform maps from a finite sequence {x ;t =0,1,2,...,T−1}
                                                                                    t
        in the time domain to a finite sequence {ωj =2πj/T;j =0,1,2,...,T−1} in the frequency
        domain:
                                       T−1                            T−1
                                        iωt                        1        −iω t
                                  x =       ξ e j     ←→ ξ =              x e    j .                    (5.4)
                                   t         j                 j   T        t
                                       j=0                            t=0
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