Convolution Integral
Lecture 5
Convolution Integral:
∞
y (t ) = x(t )* h(t ) = ∫ x(τ )h(t − τ )dτ
−∞
Time-domain analysis:
Convolution
(Lathi 2.4)
System output (i.e. zero-state response) is found by convolving input x(t)
with System’s impulse response h(t).
LTI System
Impulse Response
h(t)
Peter Cheung
Department of Electrical & Electronic Engineering
Imperial College London
y (t ) = x(t )* h(t )
URL: www.ee.imperial.ac.uk/pcheung/teaching/ee2_signals
E-mail: p.cheung@imperial.ac.uk
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E2.5 Signals & Linear Systems
Lecture 5 Slide 1
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Convolution Table (1)
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Lecture 5 Slide 2
Convolution Table (2)
Use table to find convolution results easily:
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Lecture 5 Slide 4
Convolution Table (3)
Example (1)
Find the loop current y(t) of the RLC circuits for input
the initial conditions are zero.
We have seen in slide 4.5 that the system equation is:
The impulse response h(t) was obtained in 4.6:
The input is:
Therefore the response is:
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Lecture 5 Slide 5
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Using distributive property of convolution:
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Lecture 5 Slide 6
When input is complex Example (2)
when all
What happens if input x(t) is not real, but is complex?
If x(t) = xr(t) + jxi(t), where xr(t) and xi(t) are the real and imaginary part of x(t),
then
Use convolution table pair #4:
That is, we can consider the convolution on the real and imaginary components
separately.
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Intuitive explanation of convolution
Convolution using graphical method (1)
Determine graphically y(t) = x(t)*h(t) for x(t) = e-tu(t) and h(t) = e-2tu(t).
Assume the impulse response decays
linearly from t=0 to zero at t=1.
Divide input x(τ) into pulses.
The system response at t is then
determined by x(τ) weighted by h(t- τ)
(i.e. x(τ) h(t- τ)) for the shaded pulse,
PLUS the contribution from all the
previous pulses of x(τ).
The summation of all these weighted
inputs is the convolution integral.
Remember: variable of
integration is τ, not t
y (t ) = x(t )* h(t )
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Convolution using graphical method (2)
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E2.5 Signals & Linear Systems
Interconnected Systems
Parallel connected system
x (t )
Cascade systems &
Commutative property
y(t ) = h1 (t )* x(t ) + h2 (t )* x(t )
y(t ) = [h1 (t )* h2 (t )]* x(t )
x (t )
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Interconnected Systems
Integration:
Also true for differentiation:
Let
Then g(t), the step response is:
Total Response
( x(t) is an impulse, and h(t) is the impulse response of
the system)
Let us put everything together, using our RLC circuit as an example.
Let us assume
In earlier slides, we have shown that
x(t) = 10e−3t u(t), y(0) = 0, y (0) = −5.
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Natural vs Forced Responses
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Additional Example
*
Note that characteristic modes also appears in
zero-state response (because it has an impact
on h(t)).
We can collect the e-t and e-2t terms together, and
call these the NATURAL response.
The remaining e-3t which is NOT a characteristic
mode is the FORCED response.
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