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m20.5 - Classifications of Signal Processing Systems

Module by: C. Sidney Burrus

Summary: Inportant classes of systems are defined: linear, time invariant, stable, causal, and others.

Classifications

The basic classifications of signal processing systems are defined and listed here. We will restrict ourselves to discrete-time systems that have ordered sequences of real or complex numbers as inputs and outputs and will denote the input sequence by x ( n ) x ( n ) and the output sequence by y ( n ) y ( n ) and show the process of the system by x ( n ) → y ( n ) x ( n ) → y ( n ) . Although the independent variable n n could represent any physical variable, our most common usages causes us to generically call it time but the results obtained certainly are not restricted to this interpretation.

  1. Linear. A system is classified as linear if two conditions are true.
    • If x ( n ) → y ( n ) x ( n ) → y ( n ) then a x ( n ) → a y ( n ) a x ( n ) → a y ( n ) for all a a . This property is called homogeneity or scaling.
    • If x 1 ( n ) → y 1 ( n ) x 1 ( n ) → y 1 ( n ) and x 2 ( n ) → y 2 ( n ) x 2 ( n ) → y 2 ( n ) , then ( x 1 ( n ) + x 2 ( n ) ) → ( y 1 ( n ) + y 2 ( n ) ) ( x 1 ( n ) + x 2 ( n ) ) → ( y 1 ( n ) + y 2 ( n ) ) for all x 1 x 1 and x 2 x 2 . This property is called superposition or additivity.
    If a system does not satisfy both of these conditions for all inputs, it is classified as nonlinear. For most practical systems, one of these conditions implies the other. Note that a linear system must give a zero output for a zero input.
  2. Time Invariant, also called index invariant or shift invariant. A system is classified as time invariant if x ( n + k ) → y ( n + k ) x ( n + k ) → y ( n + k ) for any integer k k . This states that the system responds the same way regardless of when the input is applied. In most cases, the system itself is not a function of time.
  3. Stable. A system is called bounded-input bounded-output stable if for all bounded inputs, the corresponding outputs are bounded. This means that the output must remain bounded even for inputs artificially constructed to maximize a particular system's output.
  4. Causal. A system is classified as causal if the output of a system does not precede the input. For linear systems this means that the impulse response of a system is zero for time before the input. This concept implies the interpretation of n n as time even though it may not be. A system is semi-causal if after a finite shift in time, the impulse response is zero for negative time. If the impulse response is nonzero for n → − ∞ n → − ∞ , the system is absolutely non-causal. Delays are simple to realize in discrete-time systems and semi-causal systems can often be made realizable if a time delay can be tolerated.
  5. Real-Time. A discrete-time system can operate in ``real-time" if an output value in the output sequence can be calculated by the system before the next input arrives. If this is not possible, the input and output must be stored in blocks and the system operates in ``batch" mode. In batch mode, each output value can depend on all of the input values and the concept of causality does not apply.

These definitions will allow a powerful class of analysis and design methods to be developed and we start with convolution.

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