-Term paper-
-:CONTENT:-
Ø ABSTRACT
Ø INTRODUCTION
Ø EXAMPLES
Ø CONCLUSION
Ø REFERENCE
ABSTRACT
o Convolution
Integral
Analytic methods are presented for the
systematic evaluation of convolution integrals that contain
piecewise-continuous or piecewise-smooth functions. These methods involve
breaking the convolution integral into a sum of integrals and expressing each
integral in a standard form that can be easily evaluated. Mathematical insight
is gained by looking at the result as a superposition of terms, where each term
is nonzero or active during a specific time interval. Applications to both
causal and noncausal linear systems are considered.
INTRODUCTION
o Continuous
Time Convolution Integral
Just as with discrete signals, the
convolution of continuous signals can be viewed from the input signal, or the output signal. An input signal, x(t), is passed through a system characterized by an impulse
response, h(t), to produce an output signal, y(t). This can be written in the familiar mathematical equation, y(t) = x(t) * h(t). The input
signal is divided into narrow columns, each short enough to act as an impulse to the system. In other
words, the input signal is decomposed into an infinite number of scaled and
shifted delta functions. Each of these impulses produces a scaled and shifted
version of the impulse response in the output signal. The final output signal
is then equal to the combined effect, i.e., the sum of all of the individual
responses.
For this scheme to work, the width of the
columns must be much shorter than the response of the system. Of course,
mathematicians take this to the extreme by making the input segments infinitesimally narrow, turning the
situation into a calculus problem. In this manner, the input viewpoint
describes how a single point (or narrow region) in the input signal affects a
larger portion of output signal.
In comparison, the output viewpoint examines
how a single point in the output signal is determined by the various values
from the input signal. Just as with discrete signals, each instantaneous value
in the output signal is affected by a section of the input signal, weighted by
the impulse response flipped left-for-right. In the discrete case, the signals
are multiplied and summed. In
the continuous case, the signals are multiplied and integrated. In equation form:
o Discrete
Time Convolution Integral
We
know that any discrete-time signal can be represented by a summation of scaled
and shifted discrete-time impulses. Since we are assuming the system to be
linear and time-invariant, it would seem to reason that an input signal
comprised of the sum of scaled and shifted impulses would give rise to an
output comprised of a sum of scaled and shifted impulse responses. This is
exactly what occurs in convolution.
Below we present a more rigorous and mathematical look at the derivation:
Letting
ℋ be a discrete time LTI system, we start
with the following equation and work our way down the theconvoluation sum.
|
y(n)
|
= |
ℋ(x(n) ) | |||
|
= |
| ||||
|
= |
| ||||
|
= |
| ||||
Let
us take a quick look at the steps taken in the above derivation. After our
initial equation we rewrite the function x(n) as a sum of the function times the
unit impulse. Next, we can move around the ℋ operator and the summation because ℋ(˙) is a linear, DT system.
Because of this linearity and the fact that x(k) is a
constant, we pull the constant out and simply multiply it by ℋ(˙) . Finally, we use the fact
that ℋ(˙)
is time invariant in order to reach our final state - the convolution sum!
Above the summation is taken over all
integers. However, in many practical cases either x(n) or h(n) or both are finite, for which case the summations will be
limited. The convolution equations are simple tools which, in principle, can be
used for all input signals.
EXAMPLE
o
Continuous-Time
Convolution
Let
us look at a basic continuous-time convolution example to help express some of
the important ideas. We will convolve together two square pulses, x(t) and h(t), as shown in Figure.
| ||||
|
Figure 1: Two basic signals
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o
Reflect And Shift
Now
we will take one of the functions and reflect it around the y-axis. Then we
must shift the function, such that the origin, the point of the function that
was originally on the origin, is labeled as point t. This step is shown in Figure 2, h(t−τ).
| ||||
|
Figure 2: h(−τ) and h(t−τ). |
Note
that in Figure 2τ is the
1st axis variable while t is a
constant (in this figure). Since convolution is commutative it will never
matter which function is reflected and shifted; however, as the functions
become more complicated reflecting and shifting the "right one" will
often make the problem much easier.
o
Regions Of Integration
We
start out with the convolution integral, y(t) =∫−∞∞x(τ) h(t−τ)
dτ. The value of the
function y at time t is given by the amount of
overlap(to be precise the integral of the overlapping region) between h(t−τ)
and x(τ).
Next,
we want to look at the functions and divide the span of the functions into
different limits of integration. These different regions can be understood by
thinking about how we slide h(t−τ) over x(τ), see Figure 3.
| ||||||||
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Figure 3: Figures to
|
In
this case we will have the following four regions. Compare these limits of
integration to the four illustrations of h(t−τ) and x(τ) in Figure 3.
Four Limits of
Integration
1. t<0
2. 0≤t<1
3. 1≤t<2
4. t≥2
Using the
Convolution Integral
Finally
we are ready for a little math. Using the convolution integral, let us
integrate the product of x(τ) h(t−τ). For our first and fourth
region this will be trivial as it will always be 0. The second region, 0≤t<1, will require the
following math:
|
y(t)
|
= |
∫0tdτ |
|
= |
t |
The
third region, 1≤t<2,
is solved in much the same manner. Take note of the changes in our integration
though. As we move h(t−τ) across our other function, the left-hand edge of the
function, t−1, becomes our lowlimit for the
integral. This is shown through our convolution integral as
|
y(t)
|
= |
∫t−11dτ |
|
= |
1−(t−1) | |
|
= |
2−t |
Note that the value of y(t) at all time
is given by the integral of the overlapping functions. In this example y for a given t equals the gray area in the plots in Figure3.
Convolution
Results
Thus,
we have the following results for our four regions:
y(t)
=
|
0ift<0 |
|
tif0≤t<1 |
|
2−tif1≤t<2 |
|
0ift≥2 |
Now
that we have found the resulting function for each of the four regions, we can
combine them together and graph the convolution of x(t) *h(t).
|
Figure 4: Shows the
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o
DISCRETE
TIME CONVOLUTION
A
quick graphical example may help in demonstrating why convolution works.
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Figure 1: A single
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Figure 2: A scaled
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Figure 3: We now use the
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Figure 4: We now use the
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CONCLUSION
Convolution, one of the most
important concepts in electrical engineering, can be used to determine the
output a system produces for a given input signal. It can be shown that a
linear time invariant system is completely characterized by its impulse
response. The sifting property of the continuous time impulse function tells us
that the input signal to a system can be represented as an integral of scaled
and shifted impulses and, therefore, as the limit of a sum of scaled and
shifted approximate unit impulses. Thus, by linearity, it would seem reasonable
to compute of the output signal as the limit of a sum of scaled and shifted
unit impulse responses and, therefore, as the integral of a scaled and shifted
impulse response. That is exactly what the operation of convolution
accomplishes. Hence, convolution can be used to determine a linear time
invariant system's output from knowledge of the input and the impulse response.
REFERENCES
www.dspguide.com/ch13/2.htm
www.ecn.purdue.edu/VISE/.../convolution.html
web.mac.com/chuck.../iWeb/...convolution01.../forced_convolution-1.pdf
health.ezineseeker.com/convolution-integral-example.htm
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