<< Chapter < Page Chapter >> Page >
Derives basic properties of the Laplace transform.

Properties of the laplace transform

The properties associated with the Laplace transform are similar to those of the Fourier transform. First, let's set define some notation, we will use the notation L { } to denote the Laplace transform operation. Therefore we can write X ( s ) = L x ( t ) and x ( t ) = L - 1 X ( s ) for the forward and inverse Laplace transforms, respectively. We can also use the transform pair notation used earlier:

x ( t ) X ( s )

With this notation defined, lets now look at some properties.

Linearity

Given that x 1 ( t ) X 1 ( s ) and x 2 ( t ) X 2 ( s ) then for any constants α and β , we have

α x 1 ( t ) + β x 2 ( t ) α X 1 ( s ) + β X 2 ( s )

The linearity property follows easily using the definition of the Laplace transform.

Time delay

The reason we call this the time delay property rather than the time shift property is that the time shift must be positive, i.e. if τ > 0 , then x ( t - τ ) corresponds to a delay. If τ < 0 then we would not be able to use the single-sided Laplace transform because we would have a lower integration limit of τ , which is less than zero. To derive the property, lets evaluate the Laplace transform of the time-delayed signal

L x ( t - τ ) = 0 x ( t - τ ) e - s t d t

Letting γ = t - τ leads to t = γ + τ and d t = d γ . Substituting these quantities into [link] gives

L x ( t - τ ) = - τ x ( γ ) e - s ( γ + τ ) d γ = e - s τ - τ x ( γ ) e - s γ d γ = e - s τ - τ 0 x ( γ ) e - s γ d γ + e - s τ 0 x ( γ ) e - s γ d γ

where we note that the first integral in the last line is zero since x ( t ) = 0 , t < 0 . Therefore the time delay property is given by

L x ( t - τ ) = e - s τ X ( s )

S-shift

This property is the Laplace transform corresponds to the frequency shift property of the Fourier transform. In fact, the derivation of the s -shift property is virtually identical to that of the frequency shift property.

L e - a t x ( t ) = 0 e - a t x ( t ) e - s t d t = 0 x ( t ) e - ( a + s ) t d t = 0 x ( t ) e - ( a + σ + j Ω ) t d t = X ( s + a )

The s -shift property also alters the region of convergence of the Laplace transform. If the region of convergence for X ( s ) is σ > σ m i n , then the region of convergence for L e - a t x ( t ) is σ > σ m i n - Re ( a ) .

Multiplication by t

Let's begin by taking the derivative of the Laplace transform:

d X ( s ) d s = d d s 0 x ( t ) e - s t d t = 0 x ( t ) d d s e - s t d t = - 0 t x ( t ) e - s t d t

So we can write

L t x ( t ) = - d X ( s ) d s

This idea can be extended to multiplication by t n . Letting y ( t ) = t x ( t ) , if follows that

t y ( t ) - d Y ( s ) d s d 2 X ( s ) d s 2

Proceeding in this manner, we find that

t n x ( t ) ( - 1 ) n d n X ( s ) d s n

Time scaling

The time scaling property for the Laplace transform is similar to that of the Fourier transform:

L x ( α t ) = 0 x ( α t ) e - s t d t = 1 α 0 x ( γ ) e - s α γ d γ = 1 α X s α

where in the second equality, we made the substitution t = γ α and d t = d γ α .

Convolution

The derivation of the convolution property for the Laplace transform is virtually identical to that of the Fourier transform. We begin with

L - x ( τ ) h ( t - τ ) d τ = - x ( τ ) L h ( t - τ ) d τ

Applying the time-delay property of the Laplace transform gives

- x ( τ ) L h ( t - τ ) d τ = H ( s ) - x ( τ ) e - s τ d τ = H ( s ) X ( s )

If h ( t ) is the the impulse response of a linear time-invariant system, then we call H ( s ) the system function of the system. The frequency response results by setting s = j Ω in H ( s ) . The system function provides us with a very powerful means of determining the output of a linear time-invariant filter given the input signal. It will also enable us to determine a means of establishing the stability We will discuss stability shortly of a linear-time invariant filter, something which was not possible with the frequency response.

Questions & Answers

what Make things scarce
Soudani Reply
why do we studied economic
Soudani
for the stability and growth of any reign.
Fayaz
what is macro economics.?
Fayaz
Google it
pokemon
which part of economics is interesting..
Fayaz
how does economics define me
clement Reply
what are the factors that determines the demand and supply
sillah
what are the importance of Economics
Betty Reply
tell me something very important about economics..
Fayaz Reply
how may I solve arithmetic mean ,,,all example
Cee Reply
what is agriculture
Itoe Reply
simple method of understanding cost concept
Oludare Reply
what is inflation
Christiana Reply
Inflation is a general increase in price levels
Zuko
is the action of inflating something
Abdifatah
inflation is the persistent increase in general price level of goods and services in an economy over a considerable period of time .
Tetteh
inflation is the general increase of a commodity in a particular period of time.
Turay
inflation is a general increase in price levels of commodities
shehu
what are the types of inflation?
Ebrima
inflation is the period of persistent rise in the general level of the price of goods services over time
Emmanuel
we have creeping inflation, demand pull inflation ,cost push inflation, and galloping inflation .
Emmanuel
I s Nigeria in any kind of in inflation? If yes which kind?
Martins
please can someone help me with the principles of economic?
Jonathan
hi
ahmed
How are you
Jonathan
fine
ahmed
ok
Jonathan
and u
ahmed
so to day what do u have
ahmed
am fine
Jonathan
nothing bro
Jonathan
ok
ahmed
who.?
Fayaz
no discassion just question
Aliraza
how can a location of a firm create difference between producers
joy Reply
what is monetary policy
joy
hello
Abdifatah
is a monetary from policy that's authorized of country encharces
Abdifatah
What would you say about the the mobility of enterprise as a factor of production?
Cathryn Reply
how can I connect myself to this Ambrose platform
kanu Reply
I am good and you I am from sierra Leone and I am new her
kanu Reply
u are welcome bro, here is a good platform for you to be
Alie
That i know,thanks bro.
what the main definition of economic
Uhara Reply
the main definition is given by prof Lionel Robbins as a social science which studies human behavior between ends and scarce which have alternative uses
olajumoke
what covers macro economics.
Fayaz
Essay about Microsoft
Kwena Reply
what is economics
Julie Reply
what do you mean by means in economics
Julie
economic is the wealth of a country.
Moussa
monetary policy is refer to as being expansionary or contractionary.
Abdul
pls who can help me to explain money market and capital market
Au
money market is base on short term loan which is within one year period while capital market is long term loan more than one year...
Muhammad
money market is a market were short term loans are dealt with while capital market is a market were long term loans are traded
Ebrima
What is mean by monetory policy
Lovely
monetary polices are rules that control the rate of monetary exchange in an economic as a whole.
Ebrima
wealth of the nation
Uhara
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Signals, systems, and society. OpenStax CNX. Oct 07, 2012 Download for free at http://cnx.org/content/col10965/1.15
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Signals, systems, and society' conversation and receive update notifications?

Ask