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Solved Problems-2

Problems-2
Consider a system S with input x[n] and output y[n] related by,
y[n] = x[n]{g[n] + g[n – 1]}
  1. If g[n] = 1, for all n, show that S is time-invariant.
  2. If g[n] = n, show that S is not time-invariant.
  3. If g[n] = 1 + (–1)n, show that S is time-invariant.
Solution
  1. If g[n] = 1, for all n, then Description: Description: Description: 2546.png
For input x[n] = x1[n], output Description: Description: Description: 2552.png …(i)
 
For input x[n] = x1[n – n0], output, Description: Description: Description: 2558.png …(ii)
 
From the condition of time-invariance, the output should be,
Description: Description: Description: 2569.png…(iii)
 
From equations (ii) and (iii), y2[n] = y1[n – n0]
 
Hence, the system is time-invariant.
  1. If g[n] = n, then Description: Description: Description: 2577.png
For input x[n] = x1[n], output Description: Description: Description: 2583.png …(i)
 
For input x[n] = x1[n – n0], output, Description: Description: Description: 2589.png …(ii)
 
From the condition of time-invariance, the output should be,
 
Description: Description: Description: 2595.png…(iii)
 
From equations (ii) and (iii), y2[n] ≠ y1[n – n0]
 
Hence, the system is not time-invariant.
  1. If g[n] = 1+ (–1)n , then Description: Description: Description: 2601.png
This relation is same as that of part (a).
 
Hence the system is time-invariant.
 




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