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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

Analog signals

Analog signals

Sampling

A/D conversion

Digital Signal

Processing

D/A conversion

& Filtering

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.

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:

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

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