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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.

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introduction to systems
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
What are signals and what are systems
  • Example 1 Removal of noise from an audio signal
systems working principle
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
slide4
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
  • 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
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
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
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
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
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
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
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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