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[[Category:LTI systems]]
 
[[Category:LTI systems]]
Examples of:
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 +
== 1. Example of: ==
  
 
'''a.) Linear and non-linear system'''
 
'''a.) Linear and non-linear system'''
 +
 
Linear system: y[n] = x[n]+x[n-1]
 
Linear system: y[n] = x[n]+x[n-1]
 +
 
Non-linear system: y(t) = ln(x(t))
 
Non-linear system: y(t) = ln(x(t))
  
 
'''b.) Casual and non-casual system'''
 
'''b.) Casual and non-casual system'''
 +
 
Causal system: y(t) = 1+ x(t)sin(πt)  
 
Causal system: y(t) = 1+ x(t)sin(πt)  
 +
 
Non-causal system: y(t) = x(-t)
 
Non-causal system: y(t) = x(-t)
  
 
'''c.) System with memory and without memory:'''
 
'''c.) System with memory and without memory:'''
 +
 
System with memory: y(t) = ∫ x(t)dt from 0 to t
 
System with memory: y(t) = ∫ x(t)dt from 0 to t
 +
 
System without memory: y[n] = √(x[n])  
 
System without memory: y[n] = √(x[n])  
  
 
'''d.) Invertible and non-invertible system'''
 
'''d.) Invertible and non-invertible system'''
 +
 
Invertible system: y[n] = x[1-n]  
 
Invertible system: y[n] = x[1-n]  
 +
 
Non-invertible system: y(t) = |x(t)|
 
Non-invertible system: y(t) = |x(t)|
  
 
'''e.) Stable and Unstable system'''
 
'''e.) Stable and Unstable system'''
 +
 
Stable system: y(t) = e^(-t)x(t)u(t)
 
Stable system: y(t) = e^(-t)x(t)u(t)
 +
 
Unstable system: y(t) = x(t) + y(t-1)  
 
Unstable system: y(t) = x(t) + y(t-1)  
  
 
'''f.) Time variant and time invariant system'''
 
'''f.) Time variant and time invariant system'''
 +
 
Time variant system y[n] = x[n]e^[jωn]
 
Time variant system y[n] = x[n]e^[jωn]
 +
 
Time Invariant system y(t) = 2^(x(t))
 
Time Invariant system y(t) = 2^(x(t))
  
 
[[Category:convolution]]
 
[[Category:convolution]]
Graphical Convolution problem:
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 +
 
 +
== 2. Graphical Convolution problem: ==
 +
 
 +
 
 
x(t) = e^(-2t)u(t)
 
x(t) = e^(-2t)u(t)
 +
 
h(t) = u(t)-u(t-1)
 
h(t) = u(t)-u(t-1)
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Find y(t) = x(t) * h(t):
 
Find y(t) = x(t) * h(t):
  
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[[Category:period]]
 
[[Category:period]]
  
3. What is the fundamental period of sin(6/5t)+e^(j3(1-t))?
 
  
sin(6/5t) has period of 5pi/3
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== 3. What is the fundamental period of sin(6/5t)+e^(j3(1-t))? ==
e^(j3(1-t)) = e^(j3)(cos(3t)-jsin(3t)) which has period of 2pi/3
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The fundamental period is the LCM which is 10pi/3
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 +
sin(6/5t) has period of /3
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 +
e^(j3(1-t)) = e^(j3)(cos(3t)-jsin(3t)) which has period of /3
 +
 
 +
The fundamental period is the LCM which is 10π/3
  
 
[[Bonus_point_1_ECE301_Spring2013|Back to first bonus point opportunity, ECE301 Spring 2013]]
 
[[Bonus_point_1_ECE301_Spring2013|Back to first bonus point opportunity, ECE301 Spring 2013]]

Latest revision as of 10:58, 11 February 2013


1. Example of:

a.) Linear and non-linear system

Linear system: y[n] = x[n]+x[n-1]

Non-linear system: y(t) = ln(x(t))

b.) Casual and non-casual system

Causal system: y(t) = 1+ x(t)sin(πt)

Non-causal system: y(t) = x(-t)

c.) System with memory and without memory:

System with memory: y(t) = ∫ x(t)dt from 0 to t

System without memory: y[n] = √(x[n])

d.) Invertible and non-invertible system

Invertible system: y[n] = x[1-n]

Non-invertible system: y(t) = |x(t)|

e.) Stable and Unstable system

Stable system: y(t) = e^(-t)x(t)u(t)

Unstable system: y(t) = x(t) + y(t-1)

f.) Time variant and time invariant system

Time variant system y[n] = x[n]e^[jωn]

Time Invariant system y(t) = 2^(x(t))


2. Graphical Convolution problem:

x(t) = e^(-2t)u(t)

h(t) = u(t)-u(t-1)

Find y(t) = x(t) * h(t):

Solution:Convolusion.jpg


3. What is the fundamental period of sin(6/5t)+e^(j3(1-t))?

sin(6/5t) has period of 5π/3

e^(j3(1-t)) = e^(j3)(cos(3t)-jsin(3t)) which has period of 2π/3

The fundamental period is the LCM which is 10π/3

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