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					Digital signal Processing
             By
       Dileep Kumar
   dk_2kes21@yahoo.com




                            1
Lecture 1-2


Basic Concepts




                 2
• Signal:
   A signal is defined as a function of one or more variables
   which conveys information on the nature of a physical
   phenomenon. The value of the function can be a real
   valued scalar quantity, a complex valued quantity, or
   perhaps a vector.

• System:
   A system is defined as an entity that manipulates one or
   more signals to accomplish a function, thereby yielding
   new signals.



                                                          3
• Continuos-Time Signal:
   A signal x(t) is said to be a continuous time signal if it is
   defined for all time t.

• Discrete-Time Signal:
  A discrete time signal x[nT] has values specified only at
  discrete points in time.
• Signal Processing:
  A system characterized by the type of operation that it
  performs on the signal. For example, if the operation is
  linear, the system is called linear. If the operation is non-
  linear, the system is said to be non-linear, and so forth.
  Such operations are usually referred to as “Signal
  Processing”.
                                                             4
     Basic Elements of a Signal Processing
                   System
    Analog input                                    Analog output
    signal                     Analog               signal
                           Signal Processor

                      Analog Signal Processing



Analog                                                       Analog
input                                                        output
signal     A/D                Digital              D/A       signal
         converter        Signal Processor       converter

                     Digital Signal Processing
                                                                5
• Advantages of Digital over Analogue Signal
  Processing:
  A digital programmable system allows flexibility in
  reconfiguring the DSP operations simply by changing the
  program. Reconfiguration of an analogue system usually
  implies a redesign of hardware, testing and verification
  that it operates properly.
  DSP provides better control of accuracy requirements.
  Digital signals are easily stored on magnetic media (tape
  or disk).
  The DSP allows for the implementation of more
  sophisticated signal processing algorithms.
  In some cases a digital implementation of the signal
  processing system is cheaper than its analogue
  counterpart.                                            6
    DSP Applications
            Space photograph enhancement
  Space     Data compression
            Intelligent sensory analysis

  Medical
            Medical image storage and retrieval



           Image and sound compression for
Commercial multimedia presentation.
           Movie special effects
           Video conference calling

            Video and data compression
Telephone   echo reduction
            signal multiplexing
            filtering
                                                  7
           Classification of Signals

•Deterministic Signals
  A deterministic signal behaves in a fixed known way with
  respect to time. Thus, it can be modeled by a known
  function of time t for continuous time signals, or a known
  function of a sampler number n, and sampling spacing T
  for discrete time signals.

• Random or Stochastic Signals:
   In many practical situations, there are signals that either
   cannot be described to any reasonable degree of accuracy
   by explicit mathematical formulas, or such a description is
   too complicated to be of any practical use. The lack of
   such a relationship implies that such signals evolve in time
   in an unpredictable manner. We refer to these signals as
   random.                                                   8
           Even and Odd Signals
 A continuous time signal x(t) is said to an even signal if it
 satisfies the condition
 x(-t) = x(t) for all t
 The signal x(t) is said to be an odd signal if it satisfies the
 condition
 x(-t) = -x(t)
In other words, even signals are symmetric about the
vertical axis or time origin, whereas odd signals are
antisymmetric about the time origin. Similar remarks
apply to discrete-time signals.

Example:



        even                                                 9
                                 odd                 odd
                  Periodic Signals
     A continuous signal x(t) is periodic if and only if there
     exists a T > 0 such that
     x(t + T) = x(t)
     where T is the period of the signal in units of time.
    f = 1/T is the frequency of the signal in Hz. W = 2/T is the
    angular frequency in radians per second.
The discrete time signal x[nT] is periodic if and only if
there exists an N > 0 such that
x[nT + N] = x[nT]
where N is the period of the signal in number of sample
spacings.

    Example:

                                   Frequency = 5 Hz or 10 rad/s
0           0.2        0.4                                   10
Continuous Time Sinusoidal Signals
A simple harmonic oscillation is mathematically
described as
x(t) = Acos(wt + )
This signal is completely characterized by three
parameters:
A = amplitude, w = 2f = frequency in rad/s, and  =
phase in radians.


    A                  T=1/f




                                                11
          Discrete Time Sinusoidal Signals
          A discrete time sinusoidal signal may be expressed as
          x[n] = Acos(wn + )          - < n < 
    Properties:
• A discrete time sinusoid is periodic only if its frequency is a rational
number.
        • Discrete time sinusoids whose frequencies are separated by
        an integer multiple of 2 are identical.
   • The highest rate of oscillation in a discrete time sinusoid is
   attained when w =  ( or w = - ), or equivalently f = 1/2 (or f = -
   1/2).
          1


          0


          -1
               0      2         4          6         8         10 12
         Energy and Power Signals
  •A signal is referred to as an energy signal, if and only if
  the total energy of the signal satisfies the condition
  0<E<
 •On the other hand, it is referred to as a power signal, if
 and only if the average power of the signal satisfies the
 condition
 0 < P< 
•An energy signal has zero average power, whereas a power
signal has infinite energy.

•Periodic signals and random signals are usually viewed as
power signals, whereas signals that are both deterministic and
non-periodic are energy signals.

                                                           13
     Basic Operations on Signals
 (a) Operations performed on dependent
     variables
1. Amplitude Scaling:
let x(t) denote a continuous time signal. The signal y(t)
   resulting from amplitude scaling applied to x(t) is
   defined by
   y(t) = cx(t)
   where c is the scale factor.
In a similar manner to the above equation, for discrete
   time signals we write
   y[nT] = cx[nT]                   2x(t)
            x(t)
                                                        14
 2. Addition:
Let x1 [n] and x2[n] denote a pair of discrete time signals.
  The signal y[n] obtained by the addition of x1[n] + x2[n]
  is defined as
  y[n] = x1[n] + x2[n]
   Example: audio mixer

3. Multiplication:
Let x1[n] and x2[n] denote a pair of discrete-time signals.
  The signal y[n] resulting from the multiplication of the
  x1[n] and x2[n] is defined by
  y[n] = x1[n].x2[n]
Example: AM Radio Signal

                                                               15
(b) Operations performed on independent
      variable
• Time Scaling:
  Let y(t) is a compressed version of x(t). The signal y(t)
  obtained by scaling the independent variable, time t, by
  a factor k is defined by
  y(t) = x(kt)
   – if k > 1, the signal y(t) is a compressed version of
     x(t).
   – If, on the other hand, 0 < k < 1, the signal y(t) is an
     expanded (stretched) version of x(t).



                                                          16
Example of time scaling

      1
                Expansion and compression of the signal
     0.9
                e-t.
     0.8
     0.7           exp(-t)
     0.6
     0.5          exp(-2t)
     0.4
                              exp(-0.5t)
     0.3
     0.2
     0.1
      0
       0                  5                10         15 17
Time scaling of discrete time systems


         10
    x[n]


              5
          0
          -3         -2   -1     0      1     2   3
    x[0.5n]




         10
              5
              0
              -1.5   -1   -0.5   0      0.5   1   1.5
              5
      x[2n]




              0
              -6     -4   -2     0      2     4   6
                                 n                      18
Time Reversal

• This operation reflects the signal about t = 0
  and thus reverses the signal on the time scale.
        5
     x[n]




        0
         0    1      2       3      4       5
        0
                         n
    x[-n]




       -5
        0     1      2       3      4       5
                         n                      19
 Time Shift
A signal may be shifted in time by replacing the
 independent variable n by n-k, where k is an
 integer. If k is a positive integer, the time shift
 results in a delay of the signal by k units of time. If
 k is a negative integer, the time shift results in an
 advance of the signal by |k| units in time.
     x[n]




               1
              0.5
               0 -2
     x[n+3]




               1      0   2    4    6     8     10
              0.5
               0 -2
     x[n-3]




               1      0   2    4    6     8     10
              0.5
               0 -2   0   2   n4    6     8     10   20

				
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