Lecture 3 Discrete time systems

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Lecture 3Discrete time systems
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Representation of discrete-time systems
T.
x n
y n
Example Ideal delay system y n x n nd
-8 lt n lt 8
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A memoryless system
The accumulator system
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Moving average
It is a low pass filter!!!
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Systems properties

• Memory
• Causality
• Stability
• Time invariance
• Linearity

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• Memory
• A system is memoryless if yn f ( xn )
• i.e. it sees only present values.
• A system has memory if y n depends on previous
  values
• it can also depend on present and future values!
• Causality
• A system is causal if the output yn depends
  only on present and/or past values.
• On-line systems are causal by definition

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• Time-invariance
• A system is time invariant if a shift in the
  input causes a corresponding shift of the output.
• For all n0 x1 n x n-n0 gives y1n y
  n-n0
• Stability
• A system is stable if every bounded input
  sequence produces a bounded output
• i.e. it never diverges.
• If xn Bx lt 8 then yn By lt 8

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• Linearity
• Linear systems obey the principle of
  superposition.
• 1) Additive property.
• T x1n x2n T x1n T x2n
  y1n y2n
• 2) Scaling property
• T a x1n a T x1n a yn
• Altogether T a x1n b x2n a
  Tx1n b Tx2n
• More generally
• If xn Sk ak xkn then yn Sk ak
  ykn
• where ykn is the system response to the input
  xkn

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Exercise

• Which properties (linearity, causality,
  time-invariance, stability and memory) posses the
  following systems
• a) yn 3 xn 4 xn-1
• b) yn 2 yn-1 xn2
• c) yn n xn
• d) yn cos (xn)
• e) yn log10 (xn)
• f) yn xn4
• g) The accumulator system
• h) The ideal delay system
• i) The moving average system

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• A sequence can be represented as a linear
  combination of delayed impulses

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Linear time-invariant systems (LTI)

• Let hkn be the response to dn-k (an impulse
  at n k)
• If the system is linear
• If the system is time-invariant

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Convolution sum

• Convolution sum

• Linear time-invariant systems can be described
  by the convolution sum!

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• Note that

• A linear time-invariant system can be completely
  characterized by its input response hn

dn
hn
LTI
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Properties of LTI systems (or properties of
convolution)

• Convolution is conmutative
• xn ? hn hn ? xn
• Convolution is distributive
• xn ? (h1n h2n) xn ? h1n xn ?
  h2n

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• Cascade connection
• yn h1n ? h2n ? xn h1n ?
  h2n ? xn

h2
xn
yn
h1

h1?h2
xn
yn
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• Parallel connection
• yn h1n ? xn h2n ? xn h1n
  h2n ? xn

h1
yn
xn

h2

h1h2
xn
yn
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• LTI systems are stable iff
• LTI systems are causal if
• hn 0 n lt 0