Lect3-signal-processing.pdf
- 1. Examples on System Properties Presented by Dr. Amany AbdElSamea 1
- 2. Outline • Examples on System Properties • Elementary Signals 2
- 3. Example 1 3 Determine System Properties Y(n) = x(n2) Solution: 1- Linear or non-Linear Y1(n) = x1(n2) Y2(n) = x2(n2) Y1(n) + Y2(n) = x1(n2) + x2(n2) 1 x1(n2) + x2(n2) = y’(n) 2 From 1 and 2 y’(n) = Y1(n) + Y2(n) so the system is linear 2- Time-invariant or Time-variant y’(n) = x (n2 -T) 1 Y(n –T) = x ((n-T)2) 2 From 1 and 2 y’(n) ≠ Y(n –T) so the system is time- variant 3-Static or dynamic and causal and non-causal Y(1) = x(1) Y(2) = x(4) Output depend on the future input only so the system is dynamic and non- causal Y(3) =x(9) 4- System is stable since it follow BIBO If we substitute for x(n2) by any value we will obtain a bounded value for Y.
- 4. Example 2 4 Determine System Properties Y(n) = x(n) + n x(n+1) Solution: 1- Linear or non-Linear Y1(n) = x1(n) + n x1 (n+1) Y2(n) = x2(n) + n x2 (n+1) Y1(n) + Y2(n) = x1(n) + n x1 (n+1) + x2(n) + n x2 (n+1) = x1(n) + x2(n) + n (x1 (n+1) + x2 (n+1)) 1 y’(n)= x1(n) + x2(n) + n (x1(n+1) + x2(n+1)) 2 From 1 and 2 y’(n) = Y1(n) + Y2(n) so the system is linear 2- Time-invariant or Time-variant y’(n) = x (n -T) + n x(n-T+1) 1 Y(n –T) = x (n-T) + (n-T) x(n-T+1) 2 From 1 and 2 y’(n) ≠ Y(n –T) so the system is time- variant 3-Static or dynamic and causal and non-causal Output depend on the present and on the future input so the system is dynamic and non- causal 4- System is unstable since In the given system equation, n∞, 𝐲∞ even if x(n) is bounded, Hence the system is unstable
- 5. Example 3 5 Determine System Properties Y(n) = ex(n) Solution: 1- Linear or non-Linear Y1(n) = ex1(n) Y2(n) = ex2(n) Y1(n) + Y2(n) = ex1(n) + ex2(n) 1 y’(n)= ex1(n)+ x2(n) = ex1(n) . ex2(n) 2 From 1 and 2 y’(n) ≠ Y1(n) + Y2(n) so the system is non-linear 2- Time-invariant or Time-variant y’(n) = ex(n-T) 1 Y(n –T) = ex(n-T) 2 From 1 and 2 y’(n) = Y(n –T) so the system is time-invariant 3-Static or dynamic and causal and non-causal Output depend on the present input only so the system is static and causal 4- System is stable since as long as x(n) is bounded y(n) is also bounded
- 11. Complex Sinusoidal Function cont., 11
- 13. General Complex Exponential Function cont., 13
- 14. Relationship between Complex Exponentials and Real Sinusoids 14
- 18. Representing a Rectangular Pulse (Using Unit-Step Functions)
- 25. Properties of Unit-Impulse Function 25
- 26. Examples 26
- 29. Questions