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
A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?