Properties of Discrete-Time System

1- System with Memory

A system that has memory if its output at time 𝑌0 𝑛 depends on input values other than 𝑋0 𝑛 . Otherwise the system is memoryless. For a discrete signal x[n], time is represented by the discrete increment variable n. An example of a simple memoryless discrete-time system is the equation

𝑦[𝑛] = 5𝑥[𝑛] A memoryless system is also called a static system. A system with memory is also called a dynamic system. An example of a system with memory is the Euler integrator of

𝑦[𝑛] = 𝑦[𝑛 − 1] + 𝐻𝑥[𝑛 − 1]

This equation can be represented as: 

And we see that the output depends on all past values of the input. A second example of a discrete system with memory is one whose output is the average of the last two values of the input. The difference equation describing this system is:

System with Inevitability

A system is said to be invertible if distinct inputs result in distinct outputs. A second definition of inevitability is that the input of an invertible system can be determined from its output. For example, the memoryless system described by

 not invertible The inputs of +2 and -2 produce the same output of +2

Inverse System

Inverse of a System The inverse of a system T is a second system 𝑇𝑖 that, when cascaded with T, yields the identity system.

The identity system is defined by the equation 𝑌[𝑛]=𝑋[𝑛] Consider the two systems of the figure below. System 𝑇𝑖 is the inverse of system T if

Casual System

A system is causal if the output at any time is dependent on the input only at the present time and in the past.

We have defined the unit delay as a system with an input of x[n] and an output of

𝑋[𝑛−1].

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An example of a non-causal system is the unit advance, which has an input of x[n] and an output of 𝑋[𝑛+1]

Another example of a non-causal system is an averaging system, given by

Which requires us to know a future value, 𝑋[𝑛+1] of the input signal in order to calculate the current value, y[n], of the output signal.

We denote the unit advance with the symbol 𝐷−1

We realize it by first delaying a signal and then advancing it. However, we cannot advance a signal more than it has been delayed. Although this system may appear to have no application, the procedure is used in filtering signals “off line,” or in non-real time. If we store a signal in computer memory, we know “future” values of the signal relative to the value that

Stability

BIBO Stability: A system is stable if the output remains bounded for any bounded input. This is the bounded-input bounded-output (BIBO) definition of stability. By definition, a signal x[n] is bounded if there exists a number M such that

Hence, a system is bounded-input bounded-output stable if, for a number R,

For all x[n] such that (1) is satisfied. To determine BIBO stability, R [in general, a function of M in (1)] must be found such that (2) is satisfied. Note that the Euler integrator of

The BIBO is stable; if the signal to be integrated has a constant value of unity, the output increases without limit as n increases.

Time Invariance

Time-invariant System: A system is said to be time invariant if a time shift in the input results only in the same time shift in the output. In this definition, the discrete increment n represents time. For a time-invariant system for which the input x[n] produces the output y[n], the input produces

A test for time invariance is given by:

Linearity

The property of linearity is one of the most important properties that we consider. Once again, we define the system input signal to be x[n] and the output signal to be y[n].

These two criteria can be combined to yield the principle of superposition. A system satisfies the principle of superposition if

Where and are arbitrary constants. A system is linear if it satisfies the principle of superposition. No physical system is linear under all operating conditions. However, a physical system can be tested with the use of (1) to determine ranges of operation for which the system is approximately linear. An example of a linear operation (system) is that of multiplication by a constant K, described by an example of a nonlinear system is the operation of squaring a signal,

A linear time-invariant (LTI) system is a linear system that is also time invariant. LTI systems, for both continuous-time and discrete-time systems, are emphasized in this book. An important class of LTI discrete-time systems are those that are modeled by linear difference equations with constant coefficients.

In this equation, x[n] is the input, y[n] is the output, and the numerical-integration increment H is constant. The general forms of an nth-order linear difference equation with constant coefficients are given by:

This difference equation is said to be of order N. The second version of this general equation is obtained by replacing n with (𝑛+𝑁).