DiscreteTime Signals and Systems

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Discrete-Time Signals and Systems
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Signals

• A function, e.g. sin(t) in continuous-time
  orsin(2 p n / 10) in discrete-time, useful in
  analysis
• A sequence of numbers, e.g. 1,2,3,2,1 which is
  a sampled triangle function, useful in simulation
• A collection of properties, e.g.even, causal,
  and stable, usefulin reasoning about behavior
• A piecewise representation, e.g.
• A functional, e.g. d(t)

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Kronecker Impulse (Function)

• Let dn be a discrete-time impulse function,
  a.k.a. the Kronecker delta function
• Impulse response hn response of a
  discrete-time LTI system to a discrete impulse
  function

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Systems

• Systems operate on signals to produce new signals
  or new signal representations
• Single-input one-dimensional continuous-time
  systems are commonly represented in two ways
• As operators
• As block diagrams

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System Properties

• Let xn, x1n, and x2n be inputs to a linear
  system and let yn, y1n, and y2n be their
  corresponding outputs
• A linear system satisfies
• Additivity x1n x2n ? y1n y2n
• Homogeneity a xn ? a yn for any constant a
• Let xn be the input to time-invariant system
  and yn be its corresponding output. Then,xn
  - m ? yn - m, for any integer m

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Sampling

• Many signals originate as continuous-time
  signals, e.g. conventional music or voice
• By sampling a continuous-time signal at isolated,
  equally-spaced points in time, we obtain a
  sequence of numbers
• n ? , -2, -1, 0, 1, 2,
• Ts is the sampling period.

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Sampling

• Consider audio compact discs (CDs)
• Analog-to-digital (A/D) conversion consists of
  filtering, sampling, and quantization
• Digital-to-analog (D/A) conversion consists of
  interpolation and filtering

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Generating Discrete-Time Signals

• Uniformly sampling a continuous-time signal
• Obtain xn x(n Ts) for -? lt n lt ?.
• How to choose Ts?
• Using a formula
• xn n2 5n 3, for n ? 0would give the
  samples3, -1, -3, -3, -1, 3, ...
• We really do not know what the sequence looks
  like in continuous time because we do not have a
  sampling period associated with it