Discrete-time representation of
            continuous-time signals
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    Physical signals are usually defined in continuous
    time, but signal processing is done more efficiently
    digitally and for discrete-time signals.
    A continuous-time signal x (t) is specified by an
                      c
    (uncountable) infinite number of signal values in
    every interval, whereas a discrete-time signal xd(nTs)
    consists of only one signal value in each sampling
    interval.
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    Key questions:
    • How much information can be preserved when a
     continuous-time signal is represented as a discrete
     signal?
    • How should a continuous-time signal x (t) be
                               c
     discretized most efficiently?
    • After discretization, is it possible to reconstruct
     the continuous-time signal x (t), and how should
                       c
     the reconstruction be done in that case?
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      Sampling and aliasing
      Let a continuous-time signal xc(t) be sampled at
      time instants {nT } to give the discrete-time signal
                       s
      {x (nT )} = {...,x (−2T ),x (−T ),x (0),x (T ),x (2T ),...}
        c   s         c    s  c    s  c    c  s  c  s
      Here:
      T: sampling time (sampling period)
       s
      f = 1: sampling frequency (samples/second, Hz)
       s   Ts
      ω =2πf =2π: sampling frequency (radians/s)
       s       s   Ts
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