Discrete Time Systems & its classifications

15EE55C – DIGITAL SIGNAL PROCESSING AND
ITS APPLICATIONS
DISCRETE TIME SYSTEMS
Dr. M. Bakrutheen AP(SG)/EEE
Mr. K. Karthik Kumar AP/EEE
DEPARTMENT OF ELECTRICAL AND ELECTRONICS ENGINEERING
NATIONAL ENGINEERING COLLEGE, K.R. NAGAR, KOVILPATTI – 628 503
(An Autonomous Institution, Affiliated to Anna University – Chennai)

SYSTEMS
 System is a device or combination of devices, which can operate on signals and
produces corresponding response.
 Input to a system is called as excitation and output from it is called as response.
 For one or more inputs, the system can have one or more outputs.
 Example: Communication System
 A system is defined mathematically as a unique operator or transformation that maps
an input signal in to an output signal.
 This is defined as y(n) = T[x(n)] where x(n) is input signal, y(n) is output signal, T[]
is transformation that characterizes the system behavior.

CONTINUOUS AND DISCRETE TIME SYSTEMS
 One of the most important distinctions to understand is the difference
between discrete time and continuous time systems.
 A system in which the input signal and output signal both have
continuous domains is said to be a continuous system.
 One in which the input signal and output signal both have discrete
domains is said to be a continuous system.
 Of course, it is possible to conceive of signals that belong to neither
category, such as systems in which sampling of a continuous time signal
or reconstruction from a discrete time signal take place.

REPRESENTATIONS OF DISCRETE TIME SYSTEMS
 Difference Equations
 Block diagram

DIFFERENCE EQUATIONS REPRESENTATIONS OF
 One of the most important concepts of DSP is to be able to properly
represent the input/output relationship to a given DT system.
 A linear constant-coefficient difference equation (LCCDE) serves as a
way to express just this relationship in a discrete-time system.
 Writing the sequence of inputs and outputs, which represent the
characteristics of the DT system, as a difference equation help in
understanding and manipulating a system.
 An equation that shows the relationship between consecutive values of a
sequence and the differences among them.
 They are often rearranged as a recursive formula so that a systems
output can be computed from the input signal and past outputs.

DIFFERENCE EQUATIONS REPRESENTATIONS OF
 In discrete-time systems, essential features of input and output signals appear only at
specific instants of time, and they may not be defined between discrete time steps or
they may be constant.
 These systems are also called the sequential systems.
 X and x(n) are used to represent the input.
 They are described by difference equations.
 A general Nth-order linear constant-coefficient differential equation can be written as
 Example
 y[n]+7y[n−1]+2y[n−2]=x[n]−4x[n−1]
 y[n] = x[n] − x[n − 1]

BLOCK DIAGRAM REPRESENTATIONS OF DISCRETE
TIME SYSTEMS
 In order to introduce a block diagram representation of discrete time
systems, we need to define some basic blocks that can be interconnected
to form complex systems.

CLASSIFICATIONS OF DISCRETE TIME SYSTEMS
 In the analysis as well as in the design of systems, it is desirable to classify
the systems according to the general properties that they satisfy.
 For a system to possess a given property, the property must hold for every
possible input signal to the system.
 If a property holds for some input signals but for others, the system does not
possess the property.
 General Categories are:
 Static systems
 Time - invariant systems
 Linear systems
 Causal systems
 Stable systems

STATIC AND DYNAMIC SYSTEMS - STATIC
 In static system the outputs at present instant depends only on present
inputs.
 These systems are also called as memory less systems as the system
output at give time is dependent only on the inputs at that same time.

STATIC AND DYNAMIC SYSTEMS - DYNAMIC
 Dynamic systems are those in which the output at present instant
depends on past inputs and past outputs.
 These are also called as systems with memory as the system output
needs to store information regarding the past inputs or outputs.

STATIC AND DYNAMIC SYSTEMS - EXAMPLES

TIME VARIANT AND TIME INVARIANT SYSTEMS
 A system is said to be time variant system if its response varies with time.
 If the system response to an input signal does not change with time such
system is termed as time invariant system. The behavior and characteristics
of time variant system are fixed over time.
 In time invariant systems if input is delayed by time n0 the output will also
gets delayed by n0. Mathematically it is specified as follows
y(n) = T[x(n)]
y(n-n0) = T[x(n-n0)]
Where, n0 is the time delay.
 Time invariance minimizes the complexity involved in the analysis of
systems. Most of the systems in practice are time invariant systems.

LINEAR AND NON LINEAR SYSTEMS
 A linear system is one which satisfies the principle of superposition and homogeneity or
scaling
 Consider a linear system characterized by the transformation operator T[]. Let x1, x2 are
the inputs applied to it and y1, y2 are the outputs. Then the following equations hold for a
linear system
y1(n) = T[x1(n)], y2 = T[x2(n)]
 Principle of homogeneity:
 T [a*x1(n)] = a*y1(n), T [b*x2(n)] = =b*y2(n)
 Principle of superposition:
 T [x1(n)] + T [x2(n)] = y1(n)+y2(n)
 Linearity:
 T [a*x1(n)] + T [b*x2(n)] = a*y1(n)+b*y2(n) .

LINEAR AND NON LINEAR SYSTEMS - EXAMPLES

CAUSAL AND NON CAUSAL SYSTEMS
 The principle of causality states that the output of a system always succeeds input.
 A system for which the principle of causality holds is defined as causal system.
 If an input is applied to a system at time n=0 then the output of a causal system is
zero for n<0.
 If the output depends on present and past inputs then system is casual otherwise non
casual.
 A system in which output (response) precedes input is known as Non causal system.
 If an input is applied to a system at time n=0 s then the output of a non causal system
is non zero for n<0.
 Such systems are referred as non-anticipative as the system output does not
anticipate future values of input.
 Non causal systems do not exist in practice.

CAUSAL AND NON CAUSAL SYSTEMS - EXAMPLES

STABLE AND UNSTABLE SYSTEMS
 Most of the control system theory involves estimation of stability of
systems.
 Stability is an important parameter which determines its applicability.
 Stability of a system is formulated in bounded input bounded output
sense i.e. a system is stable if its response is bounded for a bounded
input (bounded means finite).
 An unstable system is one in which the output of the system is
unbounded for a bounded input.
 The response of an unstable system diverges to infinity.

STABLE AND UNSTABLE SYSTEMS - EXAMPLES

STABLE AND UNSTABLE SYSTEMS
 A system is said to be invertible if distinct inputs lead to distinct outputs.
 For such a system there exists an inverse transformation (inverse
system) denoted by T-1[] which maps the outputs of original systems to
the inputs applied.
 Accordingly we can write
TT-1 = T-1T = I
Where I = 1 one for single input and single output systems.
 A non-invertible system is one in which distinct inputs leads to same
outputs. For such a system an inverse system will not exist.

Discrete Time Systems & its classifications

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