<< Chapter < Page Chapter >> Page >

The Fourier series expansion results in transforming a periodic, continuous time function, x ˜ ( t ) , to two discrete indexed frequency functions, a ( k ) and b ( k ) that are not periodic.

The fourier transform

Many practical problems in signal analysis involve either infinitely long or very long signals where the Fourier series is not appropriate.For these cases, the Fourier transform (FT) and its inverse (IFT) have been developed. This transform has been used with great success invirtually all quantitative areas of science and technology where theconcept of frequency is important. While the Fourier series was used before Fourier worked on it, the Fourier transform seems to be his original idea.It can be derived as an extension of the Fourier series by letting the length or period T increase to infinity or the Fourier transform can be independently defined and then the Fourier series shown to be a special case of it. Thelatter approach is the more general of the two, but the former is more intuitive [link] , [link] .

Definition of the fourier transform

The Fourier transform (FT) of a real-valued (or complex) function of the real-variable t is defined by

X ( ω ) = - x ( t ) e - j ω t d t

giving a complex valued function of the real variable ω representing frequency. The inverse Fourier transform (IFT) is given by

x ( t ) = 1 2 π - X ( ω ) e j ω t d ω .

Because of the infinite limits on both integrals, the question of convergence is important. There are useful practical signals that donot have Fourier transforms if only classical functions are allowed because of problems with convergence. The use of delta functions(distributions) in both the time and frequency domains allows a much larger class of signals to be represented [link] .

Properties of the fourier transform

The properties of the Fourier transform are somewhat parallel to those of the Fourier series and are important in applying it tosignal analysis and interpreting it. The main properties are given here using the notation that the FT of a real valued function x ( t ) over all time t is given by F { x } = X ( ω ) .

  1. Linear: F { x + y } = F { x } + F { y }
  2. Even and Oddness: if x ( t ) = u ( t ) + j v ( t ) and X ( ω ) = A ( ω ) + j B ( ω ) then
    u v A B | X | θ
    even 0 even 0 even 0
    odd 0 0 odd even 0
    0 even 0 even even π / 2
    0 odd odd 0 even π / 2
  3. Convolution: If continuous convolution is defined by:
    y ( t ) = h ( t ) * x ( t ) = - h ( t - τ ) x ( τ ) d τ = - h ( λ ) x ( t - λ ) d λ
    then F { h ( t ) * x ( t ) } = F { h ( t ) } F { x ( t ) }
  4. Multiplication: F { h ( t ) x ( t ) } = 1 2 π F { h ( t ) } * F { x ( t ) }
  5. Parseval: - | x ( t ) | 2 d t = 1 2 π - | X ( ω ) | 2 d ω
  6. Shift: F { x ( t - T ) } = X ( ω ) e - j ω T
  7. Modulate: F { x ( t ) e j 2 π K t } = X ( ω - 2 π K )
  8. Derivative: F { d x d t } = j ω X ( ω )
  9. Stretch: F { x ( a t ) } = 1 | a | X ( ω / a )
  10. Orthogonality: - e - j ω 1 t e j ω 2 t = 2 π δ ( ω 1 - ω 2 )

Examples of the fourier transform

Deriving a few basic transforms and using the properties allows a large class of signals to be easily studied. Examples of modulation, sampling,and others will be given.

  • If x ( t ) = δ ( t ) then X ( ω ) = 1
  • If x ( t ) = 1 then X ( ω ) = 2 π δ ( ω )
  • If x ( t ) is an infinite sequence of delta functions spaced T apart, x ( t ) = n = - δ ( t - n T ) , its transform is also an infinite sequence of delta functions of weight 2 π / T spaced 2 π / T apart, X ( ω ) = 2 π k = - δ ( ω - 2 π k / T ) .
  • Other interesting and illustrative examples can be found in [link] , [link] .

Note the Fourier transform takes a function of continuous time into a function of continuous frequency, neither function being periodic. If “distribution" or“delta functions" are allowed, the Fourier transform of a periodic function will be a infinitely long string of delta functions with weights that are the Fourierseries coefficients.

The laplace transform

The Laplace transform can be thought of as a generalization of the Fourier transform in order to include a larger class of functions, to allow theuse of complex variable theory, to solve initial value differential equations, and to give a tool for input-output description of linearsystems. Its use in system and signal analysis became popular in the 1950's and remains as the central tool for much of continuous time systemtheory. The question of convergence becomes still more complicated and depends on complex values of s used in the inverse transform which must be in a “region of convergence" (ROC).

Definition of the laplace transform

The definition of the Laplace transform (LT) of a real valued function defined over all positive time t is

F ( s ) = - f ( t ) e - s t d t

and the inverse transform (ILT) is given by the complex contour integral

f ( t ) = 1 2 π j c - j c + j F ( s ) e s t d s

where s = σ + j ω is a complex variable and the path of integration for the ILT must be in the region of the s plane where the Laplace transform integral converges. This definition is often called thebilateral Laplace transform to distinguish it from the unilateral transform (ULT) which is defined with zero as the lower limit of the forwardtransform integral [link] . Unless stated otherwise, we will be using the bilateral transform.

Notice that the Laplace transform becomes the Fourier transform on the imaginary axis, for s = j ω . If the ROC includes the j ω axis, the Fourier transform exists but if it does not, only the Laplace transform of the function exists.

There is a considerable literature on the Laplace transform and its use in continuous-time system theory. We will develop most of these ideas forthe discrete-time system in terms of the z-transform later in this chapter and will only briefly consider only the more important propertieshere.

The unilateral Laplace transform cannot be used if useful parts of the signal exists for negative time. It does not reduce to the Fouriertransform for signals that exist for negative time, but if the negative time part of a signal can be neglected, the unilateral transform willconverge for a much larger class of function that the bilateral transform will. It also makes the solution of linear, constant coefficient differentialequations with initial conditions much easier.

Properties of the laplace transform

Many of the properties of the Laplace transform are similar to those for Fourier transform [link] , [link] , however, the basis functions for the Laplace transform are not orthogonal. Some of the more important ones are:

  1. Linear: L { x + y } = L { x } + L { y }
  2. Convolution: If y ( t ) = h ( t ) * x ( t ) = h ( t - τ ) x ( τ ) d τ
    then L { h ( t ) * x ( t ) } = L { h ( t ) } L { x ( t ) }
  3. Derivative: L { d x d t } = s L { x ( t ) }
  4. Derivative (ULT): L { d x d t } = s L { x ( t ) } - x ( 0 )
  5. Integral: L { x ( t ) d t } = 1 s L { x ( t ) }
  6. Shift: L { x ( t - T ) } = C ( k ) e - T s
  7. Modulate: L { x ( t ) e j ω 0 t } = X ( s - j ω 0 )

Examples can be found in [link] , [link] and are similar to those of the z-transform presented later in these notes. Indeed, note the parallals anddifferences in the Fourier series, Fourier transform, and Z-transform.

Questions & Answers

can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
I'm not sure why it wrote it the other way
I got X =-6
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
Commplementary angles
Idrissa Reply
im all ears I need to learn
right! what he said ⤴⤴⤴
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
is it 3×y ?
Joan Reply
J, combine like terms 7x-4y
Bridget Reply
im not good at math so would this help me
Rachael Reply
I'm not good at math so would you help me
what is the problem that i will help you to self with?
how do you translate this in Algebraic Expressions
linda Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
what is the k.e before it land
what is the function of carbon nanotubes?
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
what is system testing
what is the application of nanotechnology?
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
anybody can imagine what will be happen after 100 years from now in nano tech world
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
silver nanoparticles could handle the job?
not now but maybe in future only AgNP maybe any other nanomaterials
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
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, Digital signal processing and digital filter design (draft). OpenStax CNX. Nov 17, 2012 Download for free at http://cnx.org/content/col10598/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Digital signal processing and digital filter design (draft)' conversation and receive update notifications?