We can find the input-output relation for a
discrete-time filter much more easily than for analogfilters. The key idea is that a sequence can be written
as a weighted linear combination of unit samples.
Show that
where
is the unit-sample.
If
denotes the
unit-sample
response —the output of a discrete-time
linear, shift-invariant filter to a unit-sampleinput—find an expression for the output.
In particular, assume our filter is FIR, with the
unit-sample response having duration
. If the input has duration
, what is
the duration of the filter's output to thissignal?
Let the filter be a boxcar averager:
for
and zero otherwise.
Let the input be a pulse of unit height and duration
.
Find the filter's output when
,
an odd integer.
A digital filter
A digital
filter has the
depicted unit-sample reponse.
What is the difference equation that defines this
filter's input-output relationship?
What is this filter's transfer function?
What is the filter's output when the input is
?
A special discrete-time filter
Consider a FIR filter governed by the difference equation
Find this filter's unit-sample response.
Find this filter's transfer function.
Characterize this transfer function(
i.e. , what classic filter category
does it fall into).
Suppose we take a sequence and stretch it out by
a factor of three.
Sketch the sequence
for some example
. What is the filter's output to this
input? In particular, what is the output at theindices where the input
is intentionally zero? Now how would you characterize this
system?
Simulating the real world
Much
of physics is governed by differntial equations, and wewant to use signal processing methods to simulate physical
problems. The idea is to replace the derivative with adiscrete-time approximation and solve the resulting
differential equation. For example, suppose we have thedifferential equation
and we approximate the derivative by
where
essentially
amounts to a sampling interval.
What is the difference equation that must be
solved to approximate the differential equation?
When
, the unit step, what will be the simulated output?
Assuming
is a sinusoid, how should the sampling
interval
be chosen so
that the approximation works well?
Derivatives
The derivative of a sequence makes little sense, but still, we can approximate it.
The digital filter described by the difference equation
resembles the derivative formula.
We want to explore how well it works.
Suppose the signal
is a sampled analog signal:
.
Under what conditions will the filter act like a differentiator?In other words, when will
be proportional to
?
The dft
Let's explore the DFT and its properties.
What is the
length-
DFT of
length-
boxcar
sequence, where
?
Consider the special case where
. Find the inverse DFT of the product of
the DFTs of two length-3 boxcars.
If we could use DFTs to perform linear filtering, it
should be true that the product of the input'sDFT and the unit-sample response's DFT equals
the output's DFT. So that you can use what youjust calculated, let the input be a boxcar signal
and the unit-sample response also be a boxcar. Theresult of part (b) would then be the filter's
output
if we could implement
the filter with length-4 DFTs. Does the actualoutput of the boxcar-filter equal the result found
in the
previous part ?
What would you need to change so that the
product of the DFTs of the input and unit-sampleresponse in this case equaled the DFT of the
filtered output?
Questions & Answers
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