Discrete Time Systems & its classifications

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1. 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)

2. 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.

3. 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.

4. Representations of Discrete Time Systems Difference Equations Block diagram

5. Difference Equations Representations of Discrete Time Systems 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.

6. Difference Equations Representations of Discrete Time Systems 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 ]+7 y [ n −1]+2 y [ n −2]= x [ n ]−4 x [ n −1] y[n] = x[n] − x[n − 1]

7. 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.

8. 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

9. 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.

10. 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.

11. Static and Dynamic Systems - Examples

12. 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 n  the output will also gets delayed by n . Mathematically it is specified as follows y(n) = T[x(n)] y(n-n ) = T[x(n-n )]        Where, n  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.

13. 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 x 1 , x 2  are the inputs applied to it and y 1 , y 2  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) .

14. Linear and Non linear Systems - Examples

15. 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.

16. Causal and Non Causal Systems - Examples

17. 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.

18. Stable and Unstable Systems - Examples

19. 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 -1 T = 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.