Signal and systems

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Signal and systems

• LTI Systems

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Linear Time-Invariant Systems
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Linear
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Memoryless

• A system is memoryless if its output any time
  depends only on the value of input at that same
  time
• For LTI system
• If h(n)0 for n 0

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Invertiblity
x(t)
y(t)
h(t)

• This implies that
• h(n) (n) (n)

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Memoryless

• A system is memoryless if its output any time
  depends only on the value of input at that same
  time
• For LTI system
• If h(t)0 for t 0

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Invertiblity
x(t)
y(t)
h(t)

• X(t)(h(t)h (t))x(t)
• This implies that
• h(t) (t) (t)

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Stability
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Causality
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Causality Stability-Example
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Summary of Convolution
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Summary of Convolution

• The convolution sum operation involves the
  following four steps
• 1. The impulse response hk is time-reversed
  (that is, reflected about the origin) to obtain h
  - k and then shifted by n to form hn - k
  h-(k - n) which is a function of k with
  parameter n.
• 2. Two sequences x k and hn - k are
  multiplied together for all values of k with n
  fixed at some value.
• 3. The product x k h n- k is summed over
  all k to produce a single output sample ynI.
• 4. Steps 1 to 3 are repeated as n varies over - 8
  to 8 to produce the entire output yn.

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Summary of Convolution

• The convolution integral operation involves the
  following four steps
• 1. The impulse response h(t) is time-reversed
  (that is, reflected about the origin) to obtain
  h(t) and then shifted by t to form h(t - t )
  h-(t - t) which is a function of t with
  parameter t.
• 2. The signal x (t) and h (t - t ) are multiplied
  together for all values of r with t fixed at some
  value.
• 3. The product x (t) h (t-t) is integrated over
  all t to produce a single output value y (t).
• 4. Steps 1 to 3 are repeated as t varies over - 8
  to 8 to produce the entire output y (t).