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					Digital Signal Processing
            Spring 2012
    Lecture 2: Discrete-Time Signals & Systems


                   Instructor:
                 Engr. Asif Iqbal




               Department of Electrical Engineering
               NUCES-FAST, Peshawar
               asif.iqbal@nu.edu.pk
Part 1: Discrete-Time Signals
• Sequence of numbers
                  , 								 ∞      ∞




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       where n is an integer
• Periodic sampling of an analog




                                        Digital Signal Processing, Lecture 2,
                                                                Spring 2012
  signal
                          , ∞      ∞
       where T is the sampling period




                                                      2
Sequence Operations
• The product and sum of two sequences x[n] and y[n]: sample-
  by-sample product and sum, respectively.




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• Multiplication of a sequence x[n] by a number :




                                                                Digital Signal Processing, Lecture 2,
                                                                                        Spring 2012
  multiplication of each sample value by .

• Delay or shift of a sequence x[n]
                          ,
       where is an integer


                                                                              3
Basic Sequences
• Unit Sample Sequence (discrete-time Impulse or Impulse)
               0, 				   0,            Unit Sample
               1, 				   0.




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                                        ...             ...




                                                                Digital Signal Processing, Lecture 2,
                                                                                        Spring 2012
• Any sequence can be represented as a sum of scaled, delayed
  impulses
                       3             1             3,
• More generally,
              ∑!"
                                                                              4
Unit Step Sequence
• Defined as                       Unit Step
               1, 							 $ 0,
      #
               0, 							   0.




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                                   ...                  ...




                                                              Digital Signal Processing, Lecture 2,
                                                                                      Spring 2012
• Related to the Impluse by
      #        ∑%
                !"         ,
      #                        1         2     ∙∙∙ or
      #        ∑!"
• Conversely
               #      #        1                                            5
Exponential Sequences
• These are very important in representing and analyzing LTI
  discrete-time systems.                         Real




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• Defined as                                     Exponential

               ( %                 ...                    ...




                                                                Digital Signal Processing, Lecture 2,
                                                                                        Spring 2012
• If A and 	are real numbers, the sequence is real.
• If 0        1	and A is positive, the sequence values are
  positive and decrease with increasing n.
• If 0 ) ) 1	, the sequence values alternate in sign, but
  again decrease in magnitude with increasing n.
• If    ) 1	, the sequence values increase with increasing n.
                                                                              6
Combining Basic Sequences
• An exponential sequence that is zero for   0
              ( % , 								 ≥ 0
            =
              0, 														 < 0




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                                                 Digital Signal Processing, Lecture 2,
                                                                         Spring 2012
• A much simpler expression
           = ( % #[ ]




                                                               7
Sinusoidal Sequences
• A general sinusoidal function
               (	cos -       .                      for all




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      With ( and . are real constants
• The ( % with complex 	has real and imaginary parts that are




                                                                   Digital Signal Processing, Lecture 2,
                                                                                           Spring 2012
  exponentially weighted sinusoids.
       If        / 012 and (            ( / 03 , then
                  %
            =(        = ( / 03          / 012 % ,
            = (       /0   12 %43
                                    ,
            = (       	 cos -           + . + j	sin -         +.
                                                                                 8
Complex Exponential Sequence
• When          1,
              = ( /0   12 %43




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              = ( cos -         + . + j	sin -   +.
• By analogy with the continuous-time case, the quantity - is




                                                                    Digital Signal Processing, Lecture 2,
                                                                                            Spring 2012
  called the frequency of the complex sinusoid or complex
  exponential and .	is called the phase.
•     is always an integer (differences between discrete-time and
    continuous-time)


                                                                                  9
An important difference:
Frequency Range
• Consider a frequency -           28
               (/ 0   12 49: %     (/ 012 % / 09:%   (/ 012 % ,




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• More generally, - + 28; , ; being an integer
             = (/ 0   12 49:< %   = (/ 012 % / 09:<% = (/ 012 % ,




                                                                     Digital Signal Processing, Lecture 2,
                                                                                             Spring 2012
• Same for sinusoidal sequences
            = ( cos - + 28;                        (
                                        + . = (	cos	 -        + .)
• So, only consider frequencies in an interval of 28, such as
       −8 < - ≤ 8 or         	0 ≤ - < 28

                                                                            10
Another important difference:
Periodicity
• In the continuous-time case, a sinusoidal signal and a complex
  exponential signal are both periodic.




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• In the discrete-time case, a periodic sequence is defined as
                       >,         for all




                                                                   Digital Signal Processing, Lecture 2,
                                                                                           Spring 2012
       where the period N is necessarily an integer.
• For sinusoid,
       ( cos -                 (
                    + . = (	cos	 -          + - > + .)
                                                    9:!
       which requires that - > = 28 		?;		> =
                                                    12

       where      is an integer                                           11
Another important difference:
Periodicity
• Same for complex exponential sequence
       / 012 %4@     / 012 %




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       which is true for -    28
• So, complex exponential and sinusoidal sequences




                                                                     Digital Signal Processing, Lecture 2,
                                                                                             Spring 2012
  • are not necessarily periodic in with period 28/-
  • and, depending on the value of - , may not be periodic at all.
• Consider
                   :%
  •          cos       ,         with a period of N=8
                    B
                    :%
  •   9   = cos            ,     with a period of N=16
                     C
• Increase in frequency => Increase in period!
                                                                            12
Another important difference:
Frequency
• For a continuous-time sinusoidal signal
         D     (	cos	 Ω D     . ,




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       as Ω increases,      D oscillates more and more rapidly




                                                                     Digital Signal Processing, Lecture 2,
                                                                                             Spring 2012
• For the discrete-time sinusoidal signal
               (	cos	 -        .
  • As - increases from 0 towards 8,     	oscillates more and more
    rapidly
  • As - increases from 8 towards 28, the oscillations become
    slower

                                                                            13
Part 2: Discrete-Time Systems
• A transformation or operator that maps input into output
                            F         G




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                                                               Digital Signal Processing, Lecture 2,
                                                                                       Spring 2012
• Examples:
  • The ideal delay system
                      H    , 																								 ∞    ∞
  • A memory less system
                    9,
                         																													 ∞   ∞
                                                                      14
                                                         Linear Systems




     Digital Signal Processing, Lecture 2,
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15




                             Spring 2012
                                                         • a
                                                               Time-invariant Systems




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16




                             Spring 2012
Causality
• The output sequence value at the index                           depends only
  on the input sequence values for ≤




                                                                                            10-Feb-12
• Example




                                                                                  Digital Signal Processing, Lecture 2,
                                                                                                          Spring 2012
                        H   , 																								 −∞ <   <∞


• Causal for   H   >0
• Non-causal for    H   <0



                                                                                         17
Stability
• A system is stable in the BIBO sense if and only if every
  bounded input sequence produces a bounded output




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  sequence.




                                                                   Digital Signal Processing, Lecture 2,
                                                                                           Spring 2012
• Example
                    9,
                         																													 −∞ <   <∞




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Description: DSP Lecture