EELE 4310: Digital Signal Processing (DSP)
                         Chapter # 4 : Frequency Analysis of Signals
                                                       (Part Two)
                                                     Spring, 2012/2013
    EELE 4310: Digital Signal Processing (DSP) - Ch.4                         Dr. Musbah Shaat                           1/13
                   Outline
     1    Frequency Analysis of Discrete-Time Signals
    EELE 4310: Digital Signal Processing (DSP) - Ch.4                         Dr. Musbah Shaat                           2/13
                   Frequency Analysis of Discrete-Time Signals
                   The Fourier Series for Discrete-Time Periodic Signals ... 1
             The Fourier series representation of continuous time periodic signal:
                     Can consist of an infinite number of frequency components (possible
                     frequency range −∞ < f < ∞).
                     The frequency spacing between two successive harmonically related
                     frequencies is 1 , where T is the fundamental period.
                                          Tp                p
             The frequency range of discrete-time signals is unique over the
             interval (−π,π) or (0,2π).
             Adiscrete-time signal of fundamental period N can consist of
             frequency components separated by 2π radians or f = 1 cycles.
                                                                         N                             N
      The Fourier series representation of the discrete time periodic signal will
      contain at most N frequency components.
    EELE 4310: Digital Signal Processing (DSP) - Ch.4                         Dr. Musbah Shaat                           3/13
                   Frequency Analysis of Discrete-Time Signals
                   The Fourier Series for Discrete-Time Periodic Signals ... 2
      Discrete-Time Fourier Series (DTFS)
                                                            P
             Synthesis equation : x(t) =                        N−1c ej2πkn/N
                                                          Pk=0 k
             Analysis equation : c = 1                        N−1x(n)e−j2πkn/N
                                               k       N      n=0
             where N is the fundamental period of the signal.
             The Fourier series coefficient {ck} form a periodic sequence when
             extended outside of the range k = 0,1,··· ,N − 1, i.e. c                                            =c
                                                                                                         k+N           k
                               P                                                 P
             c         = 1         N−1x(n)e−j2π(k+N)n/N = 1                          N−1x(n)e−j2πkn/N = c
               k+N         N       n=0                                        N      n=0                                  k
      The spectrum of a signal x(n), which is periodic of period N, is a
      periodic sequence with period N. Consequently, any N consecutive
      samples of the signal or its spectrum provide a complete description of
      the signal in the time and frequency domains.
    EELE 4310: Digital Signal Processing (DSP) - Ch.4                         Dr. Musbah Shaat                           4/13