Lecture 4: Linear Time-Invariant Systems
CT Convolution
Evşen Yanmaz
evsen.yanmaz@ozyegin.edu.tr
October 9, 2019
Evşen Yanmaz (OzU)
EE 201 Lecture 4
October 9, 2019
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Overview
1
Representation of CT Signals in terms of Impulses
2
Convolution Integral
3
Properties of Convolution Integral and LTI Systems
Evşen Yanmaz (OzU)
EE 201 Lecture 4
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CT Unit Impulse Response and the Convolution Integral
Staircase approximation to a CT signal
The convolution sum for
discrete systems was based on
the sifting principle: the input
signal can be represented as a
superposition (linear
combination) of scaled and
shifted impulse functions.
This can be generalized to
continuous signals, by thinking
of it as the limiting case of
arbitrarily thin pulses.
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EE 201 Lecture 4
October 9, 2019
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Sifting Property
Any continuous signal can be approximated
by a linear combination of
1
,
0
≤
t
<
∆
∆
thin, delayed pulses: δ∆ (t) =
0, otherwise
Note that this pulse (rectangle) has a unit integral. Then we have:
ˆ = P∞
x(t)
k=−∞ x(k∆)δ∆ (t − k∆)∆
Only one pulse is non-zero
for any value of t. Then as ∆ → 0
P
lim∆→0 ∞
k=−∞ x(k∆)δ∆ (t − k∆)∆
When ∆ → 0, the summation
approaches an integral
R∞
x(t) = −∞ x(τ )δ(t − τ )dτ
This is known as the sifting property of the continuous-time impulse
and there are an infinite number of such impulses δ(t − τ ).
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EE 201 Lecture 4
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Impulse response
Impulse response of a CT system is defined as the output of the
system when its input is the CT impulse signal.
The impulse response is denoted by h(t). Thus, h(t) can be found by
replacing x(t) with δ(t) in the input-output relationship and then
calculating y (t).
Examples:
y (t) = 2x(t)
+ x(t − 1) → h(t) =
Rt
y (t) = τ =−∞ x(τ )dτ → h(t) =
Evşen Yanmaz (OzU)
EE 201 Lecture 4
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Convolution Integral
Impulse response: Input δ(t) → Output: h(t)
Using time-invariance property: Input δ(t − τ ) → Output: h(t − τ )
Using scaling property: Input x(τ )δ(t − τ ) → Output: x(τ )h(t − τ )
Using additivity property:
Input
Z ∞
x(τ )δ(t − τ )dτ →
−∞
Output:
Z
∞
x(τ )h(t − τ )dτ
−∞
Thus, for an LTI system y (t) =
R∞
−∞ x(τ )h(t
− τ )dτ .
The above equation is known as the convolution integral and
applies only to LTI systems.
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EE 201 Lecture 4
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Convolution Integral
This important result implies that whatever the input-output
relationship for an LTI system is, it can be completely described by a
convolution integral.
Note that, although every CT system has an impulse response, the
convolution integral applies only to CT LTI systems.
Algebraically, convolution operation can be written as:
y (t) = x(t) ∗ h(t)
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EE 201 Lecture 4
October 9, 2019
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Graphical Convolution
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Example
Find the convolution of x(t) = u(t) and h(t) = u(t).
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Example
Find the convolution of x(t) = 2(u(t) − u(t − 2)) and
h(t) = u(t) − u(t − 1).
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EE 201 Lecture 4
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Example
Let x(t) = e −at u(t) be the input to an LTI system with h(t) = u(t). Find
y (t) = x(t) ∗ h(t).
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EE 201 Lecture 4
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Exercise
Calculate the convolution of x(t) = e 2t u(−t) and h(t) = u(t − 3). Find
y (t) = x(t) ∗ h(t).
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Properties of Convolution Integral and LTI Systems
Properties of the convolution integral
Commutativity
Associativity
Distributivity
Shifting property
Properties of the CT LTI System
Memory property
Causality
Stability
FIR vs. IIR Systems
Right-sided and left-sided impulse response
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Properties of convolution
Convolution integral has commutativity property,
i.e.,
R∞
x(t) ∗ h(t) = h(t) ∗ x(t) = −∞ h(τ )x(t − τ )dτ
Therefore, we can switch the roles of the system and the signal.
Convolution integral has associativity property, i.e.,
x(t) ∗ (h1 (t) ∗ h2 (t)) = (x(t) ∗ h1 (t)) ∗ h2 (t)
Therefore, we can obtain a single system from two systems that are
connected in series.
Convolution integral has distributivity property, i.e.,
x(t) ∗ (h1 (t) + h2 (t)) = x(t) ∗ h1 (t) + x(t) ∗ h2 (t) = y1 (t) + y2 (t)
Therefore, we can obtain a single system from two systems that are
connected in parallel.
Convolution integral has shifting property, i.e.,
y (t) = x(t) ∗ h(t) → x(t) ∗ h(t − t0 ) = x(t − t0 ) ∗ h(t) = y (t − t0 )
Therefore, if we shift the impulse response or the input signal, the
output is shifted by the same amount.
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Example: Interconnected systems
What is the overall impulse response of the following system?
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Memory and causality property in LTI systems
An LTI system is memoryless if its output depends only on the input
value at the same time, i.e. y (t) = kx(t)
For an impulse response, this can only be true if h(t) = kδ(t)
A causal system only depends on present and past values of the input
signal. We do not use knowledge about future information.
An LTI system is causal if and only if h(t) = 0, t < 0.
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Invertibility property in LTI systems
Is there a system with impulse response h1 (t) such that y (t) = x(t)?
Widely used concept for:
control of physical systems, where the aim is to calculate a control
signal such that the system behaves as specified
filtering out noise from communication systems, where the aim is to
recover the original signal x(t)
The aim is to calculate inverse systems such that
h(t) ∗ h1 (t) = δ(t)
The resulting serial system is therefore memoryless.
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EE 201 Lecture 4
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Stability property in LTI systems
Remember: A system is stable if every bounded input produces a
bounded output.
R∞
For CT, −∞ |h(τ )|dτ < ∞
Example: h(t) = e −t u(t)
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EE 201 Lecture 4
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Further properties
If h(t) = 0 for t < t0 for some t0 ∈ R, then that h(t)
is called right-sided.
Likewise, if h(t) = 0 for t > t0 for some t0 ∈ R, then that h(t)
is called left-sided.
Note that, causal systems are always right-sided, and anti-causal
systems are always left-sided.
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Summary: Convolution Properties
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Wrapping up: Can you answer the following questions?
What is the impulse response? What is step response of a system?
How are they related?
How can you measure the impulse response of a real system?
What is convolution?
What is the sifting principle? How do we represent a CT signal in
terms of impulse function?
What is the outcome of convolving a signal with a step function?
With an impulse function?
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