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Introduction to Systems
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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
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 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