Digital Signal Processing
                            Joint Ph.D Programme on Fusion Science and Engineering
                                2nd ADVANCED COURSE ON DIAGNOSTICS AND DATA ACQUISITION
                                                        Lisbon, February 2009
                                           Jo˜ao M.Sanches (jmrs@ist.utl.pt)
                                                Instituto Superior T´ecnico
                    Digital Signal Processing              Jo˜ao Sanches/IST/Lisbon               February 2009
                                                   Discrete - Time Signals and Systems
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                                                                                          (a) 1D discrete signal
                                                               (b) Discrete sound wave                                               (c) Image
                                 Digital Signal Processing                                     Jo˜ao Sanches/IST/Lisbon                                       February 2009           1
                                                          Discrete Sequences
                          x = [0;0:57;0:11;−0:16;−0:07;0:04;0:03;−0:01;−0:01;0]
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                                                                                                      −5n
                                                     Figure 1: x(n) = sin(4πn)e
                         Digital Signal Processing                     Jo˜ao Sanches/IST/Lisbon                       February 2009    2
                                       Basic Sequences and Operations
                       • Delay: y(n) = x(n − n0)
                       • Unit step: u(n) = (1 if n ≥ 0,
                                                         0 if n < 0.
                       • Dirac: δ(n) = (0 if n 6= 0, = u(n) − u(n − 1)
                                                   1 if n = 0.
                                                                    n
                       • Real exponential: x(n) = Aa u(n)
                                                                         jω n
                       • Complex exponential: x(n) = Ae 0
                                                                              1 φ jω n            −jω n
                       • Cosine: x(n) = Acos(ω n+φ) = e                              e 0 +e             0
                                                              0               2
                      Digital Signal Processing                 Jo˜ao Sanches/IST/Lisbon                    February 2009  3