signal and system(c) by fawadjatt-1


									  Signals and Systems

       (Oppenheim & Willsky)
(Section 1.6 Basic System Properties)

          Waseem Khan

Department of Electrical Engineering
         Air University
Systems with and without memory (1.6.1)
 A system whose output at any time instant depends only
 on the input at the same time, is a system without
 memory. e.g. y[n] = (x[n])2
 y(t) = R x(t)      → without memory
 y([n] = abs(x[n]) → without memory
 y([n] = x[n-1]     → with memory (output depends on
                           previous sample)
 y[n] =   ∑ x[k ]   → with memory (output depends on all
          k = −∞                previous samples)

                    → with memory (voltage across capacitor
                           depends on history of current)
Invertibility and Invertible Systems (1.6.2)
  A system is said to be invertible if distinct input to a
  systems leads to distinct outputs.
  e.g. y(t) = 2 x(t). Its inverse system is w(t) = ½ y(t).

y(t) = [x(t)]2 is a non-invertible system.
Causality (1.6.3)
  A system is causal if the output at any time depends only
  on present and past input values.
            Causal                     Non Causal
  y[n] =   ∑ x[k ]
           k = −∞

   y([n] = abs(x[n])                    y[n] = x[n+1]

All memory less systems are causal.
Stability (1.6.4)
   A system is said to be stable if every bounded input to
   the system leads to a bounded output.
   If there exists a finite value ‘A’ such that
        |x(t)| < A      for all ‘t’
   and there exists a finite value ‘B’ such that
        |y(t)| < B      for all ‘t’
   the system is stable.
Examples: y(t) = tan(x(t)) → unstable
y(t) = ex(t)     → stable              y(t) = tx(t) → unstable
y[n] = nx[n]   → unstable                    n
                                  y[n] =   ∑ x[k ]
                                           k = −∞
                                                     → unstable
Time Invariance (1.6.5)
 A system is time-invariant if the behaviour and
 characteristics of the system are fixed and do not
 change with time.

 If input x(t) to a system leads to output y(t), and x(t-t0)
 leads to output y(t-t0), the system is time invariant.

 In DT, if output is y[n] for input x[n] and output is y[n-n0]
 for x[n-n0], the system is TI.

 See examples 1.14, 1.15
Linearity (1.6.6)
 A system that satisfies principle of superposition, is
 called linear system.
 If x1[n] and x2[n] are two inputs of a system and y1[n]
 and y2[n] are the respective outputs, then the system is
 linear if and only if output of the system for input
 is         ay1[n] + by2[n]
Linearity (Example)
             y[n] = (x[n])2
Lets check for linearity.
             y1[n] = (x1[n])2
             y2[n] = (x2[n])2
Then output for input x1[n] + x2[n] is
             y[n]= (x1[n] + x2[n])2
             = (x1[n])2 + (x2[n])2 + 2x1[n] x2[n]
             ≠ (x1[n])2 + (x2[n])2

Hence, this system is not linear.

See examples 1.17~ 1.20 yourself.
Practice Problems


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