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
differentiate between demand and supply
giving examples
In economics, a perfect market refers to a theoretical construct where all participants have perfect information, goods are homogenous, there are no barriers to entry or exit, and prices are determined solely by supply and demand. It's an idealized model used for analysis,
When MP₁ becomes negative, TP start to decline.
Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 •
Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of lab
Kelo
Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 •
Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of labour (APL) and marginal product of labour (MPL)
Quantity demanded refers to the specific amount of a good or service that consumers are willing and able to purchase at a give price and within a specific time period. Demand, on the other hand, is a broader concept that encompasses the entire relationship between price and quantity demanded
Ezea
ok
Shukri
how do you save a country economic situation when it's falling apart
Economic growth as an increase in the production and consumption of goods and services within an economy.but
Economic development as a broader concept that encompasses not only economic growth but also social & human well being.
Shukri
production function means
Jabir
What do you think is more important to focus on when considering inequality ?
sir...I just want to ask one question... Define the term contract curve? if you are free please help me to find this answer 🙏
Asui
it is a curve that we get after connecting the pareto optimal combinations of two consumers after their mutually beneficial trade offs
Awais
thank you so much 👍 sir
Asui
In economics, the contract curve refers to the set of points in an Edgeworth box diagram where both parties involved in a trade cannot be made better off without making one of them worse off. It represents the Pareto efficient allocations of goods between two individuals or entities, where neither p
Cornelius
In economics, the contract curve refers to the set of points in an Edgeworth box diagram where both parties involved in a trade cannot be made better off without making one of them worse off. It represents the Pareto efficient allocations of goods between two individuals or entities,
Cornelius
Suppose a consumer consuming two commodities X and Y has
The following utility function u=X0.4 Y0.6. If the price of the X and Y are 2 and 3 respectively and income Constraint is birr 50.
A,Calculate quantities of x and y which maximize utility.
B,Calculate value of Lagrange multiplier.
C,Calculate quantities of X and Y consumed with a given price.
D,alculate optimum level of output .
the market for lemon has 10 potential consumers, each having an individual demand curve p=101-10Qi, where p is price in dollar's per cup and Qi is the number of cups demanded per week by the i th consumer.Find the market demand curve using algebra. Draw an individual demand curve and the market dema
suppose the production function is given by ( L, K)=L¼K¾.assuming capital is fixed find APL and MPL. consider the following short run production function:Q=6L²-0.4L³ a) find the value of L that maximizes output b)find the value of L that maximizes marginal product