For example, the cepstrum converts a signal to the frequency domain through Fourier transform, takes the logarithm, then applies another Fourier transform.

( For example, in computations, it is often convenient to only implement a fast Fourier transform corresponding to one transform direction and then to get the other transform direction from the first.

For example, JPEG compression uses a variant of the Fourier transformation ( discrete cosine transform ) of small square pieces of a digital image.

For periodic functions, both the Fourier transform and the DTFT comprise only a discrete set of frequency components ( Fourier series ), and the transforms diverge at those frequencies.

For instance, if we impose some requirements on a distribution f, we can attempt to translate these requirements in terms of the Fourier transform of f. The Paley – Wiener theorem is an example of this.

For example, the fact that the Fourier transform is invariant to rotations.

For more details on the sequences of operations used for various quantum algorithms, see universal quantum computer, Shor's algorithm, Grover's algorithm, Deutsch-Jozsa algorithm, amplitude amplification, quantum Fourier transform, quantum gate, quantum adiabatic algorithm and quantum error correction.

For example, the Fourier transform of such simple functions as polynomials does not exist in the classical sense.

For a voltage pulse, starting at and moving in the positive-direction, then the transmitted pulse at position can be obtained by computing the Fourier Transform,, of, attenuating each frequency component by, advancing its phase by, and taking the inverse Fourier Transform.

For example, in Fourier analysis, functions are expressed in terms of a certain infinite sum of trigonometric functions.

For an overview of those and other related operations, refer to Fourier analysis or List of Fourier-related transforms.

For such a function, we can calculate its Fourier series on any interval that includes the points where ƒ is not identically zero.

For most of the last century, it was recognized that Fourier equation is in contradiction with the theory of relativity because it admits an infinite speed of propagation of heat signals.

For example, according to Fourier equation, a pulse of heat at the origin would be felt at infinity instantaneously.

For homogeneous turbulence ( i. e., statistically invariant under translations of the reference frame ) this is usually done by means of the energy spectrum function, where k is the modulus of the wavevector corresponding to some harmonics in a Fourier representation of the flow velocity field u ( x ):

For stochastic signals, the squared magnitude of the Fourier transform typically does not approach a limit, but its expectation does ; see periodogram.

For any positive Λ ′ less than Λ, define S < sub > Λ ′</ sub > ( a functional over field configurations φ whose Fourier transform has momentum support within ) as

For a Circulant matrix, the singular value decomposition is given by the Fourier transform,

For sound generation it has three hybrid synth engines that combine additive, subtractive, Fourier and other synthesis methods.

For a piecewise C < sup > 1 </ sup > function the Fourier series converges to the function at every point except at the jump discontinuities.

For each positive integer N ≥ 1, let S < sub > N </ sub > f be the Nth partial Fourier series

He was awarded the IEEE Medal of Honor in 1982 " For his contributions to the spectral analysis of random processes and the fast Fourier transform ( FFT ) algorithm.

For although I had crossed a corner of the hall on my way to the toilet I still could not tell for sure how far to the rear the darkness extended.

For a blood-chilling ring of terror to the very sound of his name was the tool he needed for the job he'd promised to do.

For better or for worse, we all now live in welfare states, the organizing principle of which is collective responsibility for individual well-being.

For ten minutes they ran beneath the squall, raising their arms and, for the first time, shouting and capering.

For this love of the boy for his mother is a hopeless and forbidden love, doomed by its nature.

Later Helion wrote of this phase: `` For years I built for myself a subtle instrument of relationships -- colors and forms without a name.

For innocence, of all the graces of the spirit, is I believe the one most to be prayed for.

For a freshman Congressman to read political Lessons to graybeard Democrats was poor policy for one who needed to make friends.

For some time, despondency in some Northern quarters had been displayed in two ways -- an eagerness for peace and a dissatisfaction with Lincoln.

For them only a little more needed to be learned, and then all physical knowledge could be neatly sorted, packaged and put in the inventory to be drawn on for the solution of any human problem.

For a time, urging Breasted to give up his public relations work and take up writing instead, he hoped to persuade him to become his assistant in research for the labor novel ; ;

For a time it appeared that a common European army might be created, but the project for a European Defense Community was rejected by the French National Assembly in 1954.

In his recent book, Hurray For Anything ( 1957 ), one of the most important short poems -- and it is the title poem for one of the long jazz arrangements -- is written for recital with jazz.

For some time the Communists honored the distinction between the Soviet zone of Germany and the Soviet sector of Berlin by promulgating separately the laws for the two areas.

For it was the millions of buffalo and prairie chicken and the endless seas of grass that symbolized for a whole generation of Americans the abundant supply that was to take many of them westward when the Ohio and Mississippi valleys began to fill.

For the most part, however, the new version is contemporary and, as such, should be the means for many to attain a clearer comprehension of the meaning of those words recorded so many hundreds of years ago by the first followers of Christ.

For those who `` like poetry but never get around to reading it '', the Library of Congress makes it possible for poets to be heard reading their own work.

For it is the family that, in China, has always provided social security for the indigent, the sick, the down-and-out members of the clan.

For years he wore hand-me-down suits and homemade paper collars, was even driven to scrounging for cigarette butts in Vienna's gutters.

For an instant John longed for the sound of the bells of Noyon-la-Sainte, the touch of his mother's hand, the lilt of Charles's voice in the square raftered rooms, his father's bass tones rumbling to the canons, and the sight of the beloved bishop.

For further information, contact the Bank for Cooperatives serving the region, or the Farm Credit Administration, Research and Information Division, Washington 25, D.C..

For the dignity, the influence, and the power of the legislative branch of our Government -- it is a privilege for us to do honor to this great man who represents not alone his own district but all the people of our country.

For what Sam Rayburn's life in this House teaches us is that loyalty and character are not divisive and there is no such thing as being for your country and neglecting your district.