Introduction to Systems What are signals and what are systems The system description Classification of systems Deriving the system model – Continuous systems Continuous systems: solution of the differential equation

What are signals and what are systems Example 1 Removal of noise from an audio signal

Systems working principle Taking the voltage from the cartridge playing the ‘78’ rpm record Removing the ‘hiss’ noise by filter Amplifying the information signal Recording the signal to new format

Example 2 Prediction of Share Prices –Problem: Given the price of a share at the close of the market each day, can the future prices be predicted?

The System Description The system description is based on the equations relating the input and output quantities. This way of description is an idealisation, it is a mathematical model which only approximates the true process. This type of approach assumes the real system is hidden in a ‘black’ box and all that is available is a mathematical model relating output and input signals.

Classification of Systems The reason for classifying systems: –If one can derive properties that apply generally to a particular area of the classification then once it is established that a system belongs in this area then these properties can be used with further proof. Continuous /discrete systems Sampling A/D conversion Digital Signal Processing D/A conversion & Filtering Analog signals

Liner/ non-liner Systems The basis of a linear system is that if inputs are superimposed then the responses to these individual inputs are also superimposed. That is: –If an arbitrary input x 1 (t) produce output y 1 (t) and an arbitrary input x 2 (t) produce output y 2 (t), then if the system is linear input x 1 (t)+x 2 (t) will produce output y 1 (t)+y 2 (t). –For a linear system an input (ax 1 (t)+bx 2 (t)) produce an output ay 1 (t)+by 2 (t)), where a, b are constants.

Time invariant /time varying systems The time invariance can be expressed mathematically as follows: –If an input signal x(t) causes a system output y(t) then an input signal x(t-T) causes a system output y(t-T) for all t and arbitrary T. If a system is time invariant and linear it is known as a linear time invariant or LTI system.

Instantaneous/non-instantaneous systems For the system such as y(t)=2x(t), the output at any instant depends upon the input at that instant only, such a system is defined as an instantaneous system. Non-instantaneous systems are said to have a ‘memory’. For the continuous system, the non-instantaneous system must be represent by a differential equation.

Deriving the System Model The steps involved in the construction fo the model: –Identifying the components in the system and determine their individual describing equations relating the signals (variables) associated with them –Write down the connecting equations for the system which relate how the individual components relate to the other. –Eliminate all the variables except those of interest, usually these are input and output variables.

Zero-input and Zero-state responses The zero-state response. This the response to the applied input when all the initial conditions (the system state) is zero The zero-input response. This is the system output due to the initial conditions only. The system input is taken as zero.

Continuous Systems: solution of the differential equation The linear continuous system can in general be described by a differential equation relating the system output y(t) to its input x(t). The nth order equation can be written as: d n y/dt n +a n-1 d n y/dt n +…+a 0 y = b m d m x/dt m +b m-1 d m-1 x/dt m-01 +…+b 0 x It can also be written as: (D n +a n-1 D n-1 +…+a 0 )y=(b m D m + b m-1 D m-1 +…+b 0 )x