EG1110 SIGNALS AND SYSTEMS
Discrete time systems
• u(k) is a discrete time signal (only defined at integer time intervals)
Systems and their properties
• y(k) is a discrete time signal
• Discrete-time system G maps u(k) onto y(k)
Continuous time systems
i.e. Given a discrete-time input signal, G will produce a discrete time output signal
e.g. computer processes, demographic systems.
• u(t) is continuous time signal
• y(t) is continuous time signal
• Continuous time system G maps u(t) onto y(t)
i.e. Given a continous time input signal, G will produce a continuous time output signal
e.g. Mass moving, capacitor discharging....
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Causal systems
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Anti-causal systems
Definition:
• System is anti-causal if it is not causal
A system is causal if (and only if) the current output is only a function of present and past inputs
(and possibly outputs).
⇒ the current output does not depend on future inputs (or outputs)
e.g.
• Current output may depend on future inputs/outputs
e.g.
• In discrete time
y(k) = f (u(k + 1), u(k), u(k − 1), . . .)
• In discrete time
• In continuous time
y(k) = f (u(k), u(k − 1), u(k − 2), . . .)
y(t) = f (u(τ ))
Demographics a good example!
• Most systems are causal!
• In continuous time
y(t) = f (u(τ ))
τ >t
τ ≤t
• All physically realisable systems are causal.
• “Simulations” (computer) can be anti-causal in “virtual time”
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Linear systems
Nonlinear Systems
A system G is said to be linear if the following properties hold:
1.
• A system for which either of the two linearity properties does not hold (normally both properties
fail to hold).
y1 = Gu1
y2 = Gu2





⇒ (y1 + y2) = G(u1 + u2)
• e.g. say G = u2, i.e. we have




This is called the superposition principle
y1 = u21
i.e. response of a system to the sum of two inputs is identical to sum of the responses of the system,
y2 =
had those inputs been applied alone.
2. if y1 = Gu1, then
(1)
u22
For arbitrary u1, u2 then it follows that
αy1 = G(αu1 )
This is called the homogeneity property
y1 + y2 6= (u1 + u2)2
i.e. A scaling of the system’s input results in a corresponding scaling of the system’s output.
e.g. try u1 = 2, u2 = 3 for all time.....
Most of our work will concentrate on linear systems.
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• Example: time-invariant system
Time invariance
• System is time invariant if a time-shift in the input produces a corresponding time-shift in the
y(t) = sin[x(t)]
output.
• Given a system
y(t) = Gu(t)
then this is time invariant if
[1] - Change time variable
[2] - Delay input
From (2):
Let x2(t) = x(t − t0)
y1(t − t0) = sin[x(t − t0)]
y2(t) = sin[x2(t)] = sin[x(t − t0)]
(Time-shifted output)
(Output to delayed input)
Comparing the two parts:
y(t − t0) = Gu(t − t0)
• Essentially time invariance means that an experiment performed at a certain time will give exactly
the same results as that performed at any other time.
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y1(t − t0) = y2(t)
Output produced from delayed input (y2(t)) is identical to time-shifted version of output (y1(t − t0)).
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(2)
• Example: non time-invariant system
y(t) = t sin[x(t)]
(3)
[1] - Change time variable
[2] - Delay input
From (3):
Let
y1(t − t0) = (t − t0) sin[x(t − t0)]
x2(t) = x(t − t0)
(Time shifted output)
y2(t) = t sin[x2(t)] = t sin[x1(t − t0)]
Let
y1(t) = t sin[x1(t)]
(4)
(Response due to time shifted input)
Thus we see that y2(t) 6= y1(t − t0)
Output produced from delayed version of input (y2(t)) NOT identical to
time-shifted version of output (y1(t − t0))
⇒ system time varying not time invariant.
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Static systems
Dynamic systems
• A system whose output at time t only depends on input at time t and not any other time.
• A systems whose current output (at time t) depends on past inputs
• A static system is memoryless
• Dynamics systems have MEMORY
• All static systems are causal
• Most engineering systems are dynamic!
• Examples
• Examples:
y(t) = αu(t)
y(t) = [u(t)]
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y(t) = αu(t) + βu(t1 )
y(t) =
Z t
t1
u(τ )dτ
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t1 < t
t1 < t
Main points of lecture
• A system is represented mathematically as an operator which maps its input to its output.
• Linear time invariant (LTI) systems are an important class of engineering system.
• An interesting, common class of system is that of dynamic systems - systems which have memory
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