<< Chapter < Page Chapter >> Page >
Shows a full example of convolution including math and figures.

Basic example

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

Two basic signals that we will convolve together.

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 theorigin, the point of the function that was originally on the origin, is labeled as point t . This step is shown in , h t τ .

Reflected square pulse.
Reflected and shifted square pulse.
h τ and h t τ .
Note that in τ 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, asthe functions become more complicated reflecting and shifting the "right one" will often make the problem much easier.

Regions of integration

We start out with the convolution integral, y t τ x τ h t τ . The value of the function y at time t is given by the amount of overlap(to be precise the integral of theoverlapping 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 .

No overlap.
h t τ on its way "into" x τ
h t τ on its way "out of" x τ
No overlap.
Figures to help understand the regions of intergration
In this case we will have the following four regions. Compare these limits of integration to thefour illustrations of h t τ and x τ in .

    Four limits of integration

  • t 0
  • 0 t 1
  • 1 t 2
  • 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 τ 0 t 1 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 τ t 1 1 1 1 t 1 2 t
The above formulas show the method for calculating convolution; however, do not let the simplicity of thisexample confuse you when you work on other problems. The method will be the same, you will just have to deal withmore math in more complicated integrals.

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 .

Convolution results

Thus, we have the following results for our four regions:

y t 0 t 0 t 0 t 1 2 t 1 t 2 0 t 2
Now that we have found the resulting function for each of the four regions, we can combine them together and graph theconvolution of x t h t .

Shows the system's output in response to the input, x t .

Common sense approach

By looking at we can obtain the system output, y t , by "common" sense.For t 0 there is no overlap, so y t is 0. As t goes from 0 to 1 the overlap will linearly increase with a maximum for t 1 , the maximum corresponds to the peak value in the triangular pulse.As t goes from 1 to 2 the overlap will linearly decrease. For t 2 there will be no overlap and hence no output.

We see readily from the "common" sense approach that the output function y t is the same as obtained above with calculations. When convolving to squarepulses the result will always be a triangular pulse. Its origin, peak value and strech will, of course, vary.

  • Introduction
  • Convolution - Discrete time
  • Convolution - Analog
  • Properties of convolution

Questions & Answers

1.it allows producers to learn and invent 2.it allows agent to take advantage of economic scale 3.it allow agents to focus in areas of
oyebola Reply
can someone explain the motive of money to me?
Alemogho Reply
ask the question you don't know
opportunity cost
the motive of money is to pay for goods or services. it's also for transaction purposes bet it ordinary or unforseen
Money helps to reduce our scarcity problems. It can satisfy your wants to an extent
How dose economics use science methods to explains it problems
Edmond Reply
according to Jhon Robinson the main problem of economics is what to produce? how to produce? whom to produce? economics tries to solve these problem scientifically by using its theories which are scientific in nature,basically micro and macro theories which just explain what the problem is...
and now the solving techinique is also changed. only theories or its matematical +statistical version is not sufficient so the economist had developed Econometrics which includes economic theory+mathematical economics+statistics, for solving economic problem. This method treats economics as a pure
..pure science so this technique is based on pure scientific methods.
Economics is the scientific study of human behavior so there for science is used to solve economics problems using economics theories
economic is not higly subject of taminadu ...
measurement of elasticity of demand
John Reply
Price elasticity of demand= Percentage change in quantity demanded/ Percentage change in price
For example, suppose that a 10 percent increase in the price of an ice-cream cone causes the amount of ice cream you buy to fall by 20 percent. We calculate your elasticity of demand as
Price elasticity of demand =20 percent /10 percent = 2
In this example, the elasticity is 2, reflecting that the change in the quantity demanded is proportionately twice as large as the change in the price.
what is economic
Samuel Reply
economics is a social science that studies the relationship of human behavior between ends and scare...
Why is economics as a subject is so important to human being?
it's significant in study of managing available resources to satisfy human needs and wants
economics is the science of wealth
it is also said that it is the science of scarcity
why study economic
Economic helps to meet our needs and wants nd to make choices.
careers or jobs associated with economics
Economics is more than choices, scarcity, resources etc. It has a lot to do with psychology, social psychology. Understand motivation for consumer decisons.
How does commercial banks create credit
Penda Reply
what is equilibrium price?
Stella Reply
happen when demand curve and supply curve meet
intersection point of demand and supply curve
the equilibrium price is that price at which the quantity demanded is equal to the quantity supplied... i.e the price at which what people want to buy is equal to what the sellers want to sell.
what is difference between demand and quantity demand
the price which is prevailing in the market ,it is the price at which the consumer is getting maximum satisfaction and the producer is getting maximum profit.
at equilibrium price demand for commodity is equal to supply of commodity..
dear sareel, quantity demand means the quantity of commodity which is demanded at a particular price e,g , your demand is for 4 oranges when price of orange is 5,, on the other hand demand means the whole quantity price relationship, quantity demand is shown on a particular point on demand curve,
sarjeel sbb, reply please...
what is elasticity of demand
what is Economics?
Economics can be defined as a social sciences that study human behavior as a relationship between end and scarce mean which have alternative use
What is taxation
economics is the study of scarcity
dear Suleiman how?
economics is a subject matter that studies different economic activities as directed towards maximization of income at level of individual and maximization of social welfare at the level of society as a whole
economics studies us how to allocate scarce resources to full fill the maximum human wants as the resources have alternative uses..and r very scare to fullfill the human wants as the wants of humans are unlimited
demand is what a buyer is able and willing to buy at a specific price where by quantity demanded is amount of demand the buyer is able and willing to buy at a specific price.
thanks che..
how does commercial banks create credit
write models of Aggregate Supply?
abdi Reply
models of aggregate supply?
Then what's the law of elasticity?
wat happen if money circulation failed to be controlled in any county?
Enockz Reply
the amount of money in circulation increases and cause aggregate demand to rise. Thus, hyperinflation.
what's heteroscandacity
price mechanism refers to the manner which price of goods and services affect the demand and supply of goods and services
heteroscandacity: Is the circumstance where there is unequal variances across the observations.
what is price mechanisms?
Che Reply
price mechanism refers to the manner which price of goods and services affect the demand and supply of goods and services
what is demand
kabeer Reply
desire and ability to pay for a commodity is called demand.
i desire to become Prime Minister of India , is it demand ?please answer---
demand refers to the quantity demanded of a commodity that consumers are willing and able to buy at a particular price over a given period of time
or the willingness power of a commoditythat is backed up by the ability to pay
u r 100% right..
is ability to purchase a perticular goods at any price at a given time.
tanz Rather
ur desire to become a prime minister of lndia is nt demand because u have nt made any efforts or willingness, it is called mere wish or lets say mere desire
Thank you so much dear Abigail Mante for giving answer to my question..
U are welcome
it's ok,dear..
The term ceteris paribus implies that
Demand is the quantity demanded of a commodity at which a consumer is willing and able to purchase at various prices over a given period of time.
ceteris parbus explains the law of demand,which states that at a higher price quantity demanded of a commodity is low and at a lower price quantity demanded of a commodity is high all other things being equal.
U know right girl
ceteris paribus is an economics terms used in situations where we considered other factors to remain unchanged
Thanks for the answer
please how does disposable income affect demand?
hell. somebody must tell me what is the meaning if cetrasparibus
plz. guys
the quantity of a commodity that a consumer ready willing and able to purchase at a particular price and in a particular period of time is known as demand
ceterusparibus means constant i,e,(other things do not change or other things remains constant)
right answer yawer saab
things like what@Rather
bilal sheikh why u has been written what
e,g demand is determined by various factors like price of commodity, uses of good, expected changes in near future, nature of commodity , but when we learn price elasticity of demand , we keep other determinants of demand constant(i,e , we do not expect any change in price of good in near future,etc
i think u understand the term ceterus paribus..
bcz I can not understand
ok bro
wht is other factors
income of consumer, taste,
Gold. Medlist in economics and. Lecturer of economics
hello..what's are the factors affecting cross elasticity of demand.? at least five factors
The time it takes to get a product to market. That time limit has a name call "Just in Time."
thanks guys
hey guys what is inelastic demand
when there is no change in quantity demanded due to change in price then elasticity of demand is inelastic
Hi sucker how are you...
thanks for the answer
Any question about economics
yes. what is the main cause of Heteroscedasticity
waht is elasticity of demand
elasticity of demand means change in quantity demanded divided by change in any of its determinent ,e,g , price ed= % change in quantity demanded/ % change in price.
elasticity of demand is the degree of responsiveness of quantity demanded of a commodity to a change in price of the commodity in question ,price of other related commodity n the consumers income
guys what is monopoly
A single seller in the market.
what are the importance of economics
Eric Reply
what causes the demand curve to shift.
other factors except price.it include test of a consumer fashen presence of substitute advatisment
factors other than price
Its helps to determine the price
state three factors that affect demand and supply
1,price of the product 2,expectations in near future 3,uses of product
1.advertisement 2. expectation of future change in price 3.consumers income
1.alternative products on market. 2.changes on price 3.consumer's expectations
what is cheque
adedayo Reply
resources of exchanging of money to any where
it is a writen order to a bank to pay a specific sum of money to a person
what is the meaning of money
Micheal Reply
a device agreed upon by a group of people to be used interchangeably for goods and/or services.
Money is anything that is generally accepted by people in certain country as a medium of exchange and in a settlement of debt
is anything which is legally asepted in any society to be used as a medium of exchange
is it legal to cut money or burn money
Money is a durable commodity, that is acceptable as means payment, settlement of debt and medium of exchange.
No it is not legal,that is an abuse to its path
Money is anything that is generally considered and accepted in exchange of goods and services.
money is anything generally acceptable by people in a particular society fr the buying of goods and services and for the settlement of debts. Money acts as legal tender.
money is anything generally accepted by the public for the settlement. of debt n for the payment of goods and services
in a comparison of the stages of meiosis to the stage of mitosis, which stages are unique to meiosis and which stages have the same event in botg meiosis and mitosis
Leah Reply
Got questions? Join the online conversation and get instant answers!
QuizOver.com Reply

Get the best Algebra and trigonometry course in your pocket!

Source:  OpenStax, Information and signal theory. OpenStax CNX. Aug 03, 2006 Download for free at http://legacy.cnx.org/content/col10211/1.19
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Information and signal theory' conversation and receive update notifications?