Table of contents  
Memory  
Linearity  
Additivity and Homogeneity  
Shift Invariance  
Stability 
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A system is said to be memory less if its output for each value of the independent variable is dependent only on the input signal at that value of an independent variable.
How can we identify if a system has memory?
Note: Consider a system whose output Y(t) depends on input X(t) as: Y(t) = X(t5) + { X(t)  X(t5) }
While at first glance, the system might appear to have memory, it does not. This brings us to the idea that given the description of a system, it need not at all be the most economical one. The same system may have more than one description.
Examples: Assume y[n] and y(t) are respectively outputs corresponding to input signals x[n] and x(t)
Definition:
(It is not necessary for the input and output signals to have the same independent variable for linearity to make sense. The definition for systems with input and/or output signal being discretetime is similar.)
A capacitor, an inductor, a resistor or any combination of these are all linear systems if we consider the voltage applied across them as an input signal and the current through them as an output signal. This is because these simple passive circuit components follow the principle of superposition within their ranges of operation.
Linearity can be thought of as consisting of two properties:
To say a system is linear is equivalent to saying the system obeys both additivity and homogeneity.
(a) We shall first prove homogeneity and additivity imply linearity
(b) To prove linearity implies homogeneity and additivity.
This is easy; put both constants equal to 1 in the definition to get additivity; one of them to 0 to get homogeneity.
Additivity and Homogeneity are Independent Properties:
Examples: Assume y[n] and y(t) are respectively outputs corresponding to input signals x[n] and x(t)
Definition: Say, for a system, the input signal x(t) gives rise to an output signal y(t). If the input signal x(t  t0) gives rise to output y(t  t0), for every t0, and every possible input signal, we say the system is shiftinvariant.
i.e. for every permissible x(t) and every t0
In other words, for a shiftinvariant system, shifting the input signal shifts the output signal by the same offset.
Note this is not to be expected from every system. x(t) and x(t  t_{0}) are different (related by a shift, but different) input signals and a system, which simply maps one set of signals to another, need not at all map x(t) and x(t  t_{0}) to output signal also shift by t_{0}
A system that does not satisfy this property is said to be a shift variant.
Examples: Assume y[n] and y(t) are respectively outputs corresponding to input signals x[n] and x(t)
Example:
Note:This should be true for all bound inputs x(t). It is not necessary for the input and output signal to have the same independent variable for this property to make sense. It is valid for continuous time, discrete time and hybrid systems.
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