Introduction to Systems

1. 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

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

3. 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

4. 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?

5. 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.

6. 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 Analog signals Analog signals Sampling A/D conversion Digital Signal Processing D/A conversion & Filtering

7. 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 x1(t) produce output y1(t) and an arbitrary input x2(t) produce output y2(t), then if the system is linear input x1(t)+x2(t) will produce output y1(t)+y2(t). • For a linear system an input (ax1(t)+bx2(t)) produce an output ay1(t)+by2(t)), where a, b are constants.

8. 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.

9. 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.

10. 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.

11. 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.

12. 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: dny/dtn+an-1dny/dtn+…+a0y = bm dmx/dtm+bm-1dm-1x/dtm-01+…+b0x It can also be written as: (Dn+an-1Dn-1+…+a0)y=(bmDm+ bm-1Dm-1+…+b0)x