15 Even and Odd Functions

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• 1-5 Even and Odd Functions
• Even function
• Odd function
• Some properties
• even x even gt even
• odd x odd gt even
• even x odd gt odd
• Areas or integrals

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• Any signal f(t) can be expressed as a sum of
  even and odd components
• Proof
• Example

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• 1.6 Continuous-Time Systems
• A system is a process that transforms input
  signals into output signals

• We will pay our major attention to
  one-dimensional systems (1-input, 1-output)
• However, the classification may involve
  multi-dimensions

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• 1.7 Classification
• 1. Linear and Nonlinear Systems
• Additivity
• Input f1 alone leads to output y1
• Input f2 alone leads to output y2
• If both are inputs for the system, then the
  output is y1 y2
• Homogeneity (scaling)
• Input f(t) along leads to output y(t)
• If input is kf(t), then output is ky(t)

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• A system is linear if the superposition
  principle can be applied to it

• Example

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• Example

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• Example

• Assume
• Verify whether the superposition principle
  applies

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• In the above example, non-linearity is caused by
  the dependence of
• Non-linearity can also be caused by other reasons

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• Example

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• Example 1.9 (What is the difference from the
  first example?

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• Note In the above approach, we assumed an
  inverse system
• Another approach is to introduce a single symbol
  in terms of y.

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• 2. Time-Varying and Time-Invariant Systems
• A system is time-invariant if a time shift in
  the input signal causes an identical time shift
  in the out put signal
• This means

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• Example 1
• Let us apply at the input, then

• Thus the above system is time-invariant
• However, it is nonlinear

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• Example 2
• Let us apply at the input, then

• Thus it is time-variant
• However, it is linear