The document covers continuous-time convolution, detailing its properties, graphical methods, and examples for engineering applications. It explains how convolution can be divided into intervals, demonstrating the overlap of functions and area calculations. Additionally, it discusses the implications for communication channels and discrete-time convolution principles in electrical and computer engineering.

1. Continuous-Time Convolution
EE 313 Linear Systems and Signals Fall 2010
Initial conversion of content to PowerPoint
by Dr. Wade C. Schwartzkopf
Prof. Brian L. Evans
Dept. of Electrical and Computer Engineering
The University of Texas at Austin

2. 4 - 2
        

 d
t
f
f
t
f
t
f 2
1
2
1 






Convolution Integral
• Commonly used in engineering, science, math
• Convolution properties
– Commutative: f1(t) * f2(t) = f2(t) * f1(t)
– Distributive: f1(t) * [f2(t) + f3(t)] = f1(t) * f2(t) + f1(t) * f3(t)
– Associative: f1(t) * [f2(t) * f3(t)] = [f1(t) * f2(t)] * f3(t)
– Shift: If f1(t) * f2(t) = c(t), then
f1(t) * f2(t - T) = f1(t - T) * f2(t) = c(t - T).
– Convolution with impulse, f(t) * (t) = f(t)
– Convolution with shifted impulse, f(t) * (t-T) = f(t-T)
important later in modulation

3. 4 - 3
Graphical Convolution Methods
• From the convolution integral, convolution is
equivalent to
– Rotating one of the functions about the y axis
– Shifting it by t
– Multiplying this flipped, shifted function with the other
function
– Calculating the area under this product
– Assigning this value to f1(t) * f2(t) at t
        

 d
t
f
f
t
f
t
f 

 



2
1
2
1

4. 4 - 4
3

2
f()
2
-2 + t 2 + t
g(t-)
*
2
2
t
f(t)
-2 2
3
t
g(t)
Graphical Convolution Example
• Convolve the following two functions:
• Replace t with in f(t) and g(t)
• Choose to flip and slide g() since it is simpler
and symmetric
• Functions overlap like this:

5. 4 - 5
    6
2
3
2
6
2
2
3
2
2
3
)
2
(
3
2
2
2
0
2
2
0
























t
t
t
d
t
t




3

2
f()
2
-2 + t 2 + t
g(t-)
3

2
f()
2
-2 + t 2 + t
g(t-)
Graphical Convolution Example
• Convolution can be divided into 5 parts
I. t < -2
• Two functions do not overlap
• Area under the product of the
functions is zero
II. -2  t < 0
• Part of g(t) overlaps part of f(t)
• Area under the product of the
functions is

6. 4 - 6
Graphical Convolution Example
III. 0  t < 2
• Here, g(t) completely overlaps f(t)
• Area under the product is just
IV. 2  t < 4
• Part of g(t) and f(t) overlap
• Calculated similarly to -2  t < 0
V. t  4
• g(t) and f(t) do not overlap
• Area under their product is zero
  6
2
2
3
2
3
2
0
2
2
0














 


 d
3

2
f()
2
-2 + t 2 + t
g(t-)
3

2
f()
2
-2 + t 2 + t
g(t-)

7. 4 - 7
Graphical Convolution Example
• Result of convolution (5 intervals of interest):



























4
for
0
4
2
for
24
12
2
3
2
0
for
6
0
2
for
6
2
3
2
for
0
)
(
*
)
(
)
(
2
2
t
t
t
t
t
t
t
t
t
g
t
f
t
y
t
y(t)
0 2 4
-2
6

8. 4 - 8
Convolution Demos
• Johns Hopkins University Demonstrations
http://www.jhu.edu/~signals
Convolution applet to animate convolution of simple
signals and hand-sketched signals
Convolve two rectangular pulses of same width gives a
triangle (see handout E)
• Some conclusions from the animations
Convolution of two causal signals gives a causal result
Non-zero duration (called extent) of convolution is the
sum of extents of the two signals being convolved

9. 4 - 9
Transmit One Bit
• Transmission over communication channel (e.g.
telephone line) is analog
h t
)
(t
h
1
p t
)
(
1 t
x
A
‘1’ bit
t
p
)
(
0 t
x
-A
‘0’ bit
Model channel as
LTI system with
impulse response
h(t)
Communication
Channel
input output
x(t) y(t)
t
t
)
(
1 t
y receive
‘1’ bit
)
(
0 t
y
-A Th
receive
‘0’ bit
h+p
t
h+p
h
h
Assume that Th < Tp
A Th

10. 4 - 10
Transmit Two Bits (Interference)
• Transmitting two bits (pulses) back-to-back
will cause overlap (interference) at the receiver
• How do we prevent intersymbol
interference at the receiver?
h t
)
(t
h
1
Assume that Th < Tp
t
p
)
(t
x
A
‘1’ bit ‘0’ bit
p
* =
)
(t
y
-A Th
t
p
‘1’ bit ‘0’ bit
h+p
intersymbol
interference

11. 4 - 11
Transmit Two Bits (No Interference)
• Prevent intersymbol interference by waiting Th
seconds between pulses (called a guard period)
• Disadvantages?
h t
)
(t
h
1
Assume that Th < Tp
* =
t
p
)
(t
x
A
‘1’ bit ‘0’ bit
h+p
t
)
(t
y
-A Th
p
‘1’ bit ‘0’ bit
h+p
h

12. 4 - 12






m
m
n
x
m
h
n
y ]
[
]
[
]
[       

 d
t
x
h
t
y 




h[n] y[n]
x[n]
LTI system
represented
by its impulse
response
h(t) y(t)
x(t)
LTI system
represented
by its impulse
response
Discrete-time Convolution Preview
• Discrete-time
convolution
• For every value of n,
we compute a new
summation
• Continuous-time
convolution
• For every value of t,
we compute a new
integral

13. 4 - 13





1
0
]
[
]
[
]
[
N
m
m
n
x
m
h
n
y
z-1
z-1
z-1
…
…
x[n]
 y[n]
h[0] h[1] h[2] h[N-1]
Discrete-time Convolution Preview
• Assuming that h[n] has finite
duration from n = 0, …, N-1
• Block diagram of an implementation (finite
impulse response digital filter): see slide 2-4

14. 4 - 14
Philosophy
• Pillars of electrical
engineering (related)
Fourier analysis
Probability and
random processes
• Pillars of computer
engineering (related)
System state
Complexity
• Finite-state machines
for digital input/output
Finite number of states
Models all possible input-
output combinations
Can two outputs be true at
the same time?
• Given output observation, work
backwards to inputs to see if output
is possible
• This is called observability
EE 313 is pre-requisite for EE 351K

15. 4 - 15
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