Customer center

We are a boutique essay service, not a mass production custom writing factory. Let us create a perfect paper for you today!

Example research essay topic: F X S 2 - 866 words

NOTE: Free essay sample provided on this page should be used for references or sample purposes only. The sample essay is available to anyone, so any direct quoting without mentioning the source will be considered plagiarism by schools, colleges and universities that use plagiarism detection software. To get a completely brand-new, plagiarism-free essay, please use our essay writing service.
One click instant price quote

To avoid confusion, it is customary to write the Fourier transform and its inverse so that they exhibit reversibility: F (s) = f (x) exp (-i 2 xs) dx f (x) = F (s) exp (i 2 xs) ds so that f (x) = f (x) exp (-i 2 xs) dx exp (i 2 xs) ds as long as the integral exists and any discontinuities, usually represented by multiple integrals of the form [f (x+) + f (x-) ], are finite. There are functions for which the Fourier transform does not exist; however, most physical functions have a Fourier transform, especially if the transform represents a physical quantity. Other functions can be treated with Fourier theory as limiting cases. Many of the common theoretical functions are actually limiting cases in Fourier theory. Usually functions or waveforms can be split into even and odd parts as follows: f (x) = E (x) + O (x) where E (x) = [f (x) + f (-x) ] O (x) = [f (x) - f (-x) ] and E (x), O (x) are, in general, complex. In this representation, the Fourier transform of f (x) reduces to 2 E (x) cos (2 xs) dx - 2 iO (x) sin (2 xs) dx It follows then that an even function has an even transform and that an odd function has an odd transform. (Bracewell, 14).

An important case is that of an Hermitian function, one in which the real part is even and the imaginary part is odd, i. e. , f (x) = f (-x). The Fourier transform of an Hermitian function is even. In addition, the Fourier transform of the complex conjugate of a function f (x) is F (-s), the reflection of the conjugate of the transform.

The cosine transform of a function f (x) is defined as Fc (s) = 2 f (x) cos 2 sx dx. This is equivalent to the Fourier transform if f (x) is an even function. In general, the even part of the Fourier transform of f (x) is the cosine transform of the even part of f (x). The cosine transform has a reverse transform given by f (x) = 2 Fc (s) cos 2 sx ds. Likewise, the sine transform of f (x) is defined by FS (s) = 2 f (x) sin 2 sx dx. As a result, i times the odd part of the Fourier transform of f (x) is the sine transform of the odd part of f (x).

Combining the sine and cosine transforms of the even and odd parts of f (x) leads to the Fourier transform of the whole of f (x): f (x) = CE (x) - i SO (x) where, C, and S stand for -i times the Fourier transform, the cosine transform, and the sine transform respectively, or F (s) = FC (s) - iFS (s) (Bracewell, 17 - 18). Since the Fourier transform F (s) is a frequency domain representation of a function f (x), the s characterizes the frequency of the decomposed co sinusoids and sinusoids and is equal to the number of cycles per unit of x (Bracewell, 18 - 21). If a function or waveform is not periodic, then the Fourier transform of the function will be a continuous function of frequency (Brigham, 4). It is known that most geophysical signals can be expressed as a decomposition of the signal into sine and cosine functions of different frequencies (also referred to as harmonics). This is called Fourier analysis. This concept is usually exposed in a calculus or physics course where sine and cosine functions expressed as a Fourier series are used to represent a periodic function of time. (In 1822, Joseph Fourier was the first person who attempted to prove the convergence of such a series. ) There are the usual conditions placed on the signal, i.

e. : 1) it cannot be multi valued at any given time, 2) it cannot have an infinite number of discontinuities, or maxima or minima, and 3) it must be bounded within its period. The frequencies of the trigonometric functions are the spectral components of the Fourier series. These frequencies are predetermined by the periodicity, T of the function and are equal to n/Therefore, the frequency spectrum is composed of discrete line spectra. When a signal is not periodic, the spectrum is not discrete and the Fourier series must be generalized into the Fourier integral or Fourier transform. As long as the integral of the absolute value of the signal converges, the continuous signal s (t) can be expressed as the Fourier integral, where Second equation defines the Fourier transform of s (t); equation 3. 1 is the inverse Fourier transform that recovers s (t) back from S (f). These equations are at the heart of spectral analysis and they are so tightly connected that they are usually called the Fourier transform pair.

It is customary to use a lower case symbol for the time (or space) domain function and an upper case symbol for the corresponding function of frequency. S (f) and s (t) are referred to as the frequency domain and time domain representations of the signal, respectively.


Free research essays on topics related to: transform, functions, s 2, 2 f, f x

Research essay sample on F X S 2

Writing service prices per page

  • $18.85 - in 14 days
  • $19.95 - in 3 days
  • $23.95 - within 48 hours
  • $26.95 - within 24 hours
  • $29.95 - within 12 hours
  • $34.95 - within 6 hours
  • $39.95 - within 3 hours
  • Calculate total price

Our guarantee

  • 100% money back guarantee
  • plagiarism-free authentic works
  • completely confidential service
  • timely revisions until completely satisfied
  • 24/7 customer support
  • payments protected by PayPal

Secure payment

With EssayChief you get

  • Strict plagiarism detection regulations
  • 300+ words per page
  • Times New Roman font 12 pts, double-spaced
  • FREE abstract, outline, bibliography
  • Money back guarantee for missed deadline
  • Round-the-clock customer support
  • Complete anonymity of all our clients
  • Custom essays
  • Writing service

EssayChief can handle your

  • essays, term papers
  • book and movie reports
  • Power Point presentations
  • annotated bibliographies
  • theses, dissertations
  • exam preparations
  • editing and proofreading of your texts
  • academic ghostwriting of any kind

Free essay samples

Browse essays by topic:

Stay with EssayChief! We offer 10% discount to all our return customers. Once you place your order you will receive an email with the password. You can use this password for unlimited period and you can share it with your friends!

Academic ghostwriting

About us

© 2002-2024 EssayChief.com