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Wiener-Khinchin Theorem in .NET Generation PDF417 in .NET Wiener-Khinchin Theorem




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Wiener-Khinchin Theorem use vs .net barcode pdf417 creator todevelop pdf417 on .net GS1 supported barcodes (12.0.12).

The total power in a sig nal is the same whether we compute it in the time domain or in the frequency domain. This result is known as Parseval s theorem: Total Power . h(t). dt = H(f). df (12.0.13).

Frequently one wants to know how much power is contained in the frequency interval between f and f + df. In such circumstances one does not usually distinguish between positive and negative f, but rather regards f as varying from 0 ( zero frequency or D.C.

) to + . In such cases, one de nes the one-sided power spectral density (PSD) of the function h as Ph (f) . H(f). 2 + . H( f). 2 0 f < (12.0.14).

so that the total power .net framework PDF417 is just the integral of Ph (f) from f = 0 to f = . When the 2 function h(t) is real, then the two terms in (12.

0.14) are equal, so Ph (f) = 2 . H(f). .. 12.0 Introduction h(t) 2. (a) Ph ( f ) (one-sided). ( b). Ph( f ) (two-sided) 0. Figure 12.0.1.

Normaliza visual .net pdf417 2d barcode tions of one- and two-sided power spectra. The area under the square of the function, (a), equals the area under its one-sided power spectrum at positive frequencies, (b), and also equals the area under its two-sided power spectrum at positive and negative frequencies, (c).

. Be warned that one occas ionally sees PSDs de ned without this factor two. These, strictly speaking, are called two-sided power spectral densities, but some books are not careful about stating whether one- or two-sided is to be assumed. We will always use the one-sided density given by equation (12.

0.14). Figure 12.

0.1 contrasts the two conventions. If the function h(t) goes endlessly from < t < , then its total power and power spectral density will, in general, be in nite.

Of interest then is the (oneor two-sided) power spectral density per unit time. This is computed by taking a long, but nite, stretch of the function h(t), computing its PSD [that is, the PSD of a function that equals h(t) in the nite stretch but is zero everywhere else], and then dividing the resulting PSD by the length of the stretch used. Parseval s theorem in this case states that the integral of the one-sided PSD-per-unit-time over positive frequency is equal to the mean square amplitude of the signal h(t).

You might well worry about how the PSD-per-unit-time, which is a function of frequency f, converges as one evaluates it using longer and longer stretches of data. This interesting question is the content of the subject of power spectrum estimation, and will be considered below in 13.4 13.

7. A crude answer for. 12. . Fast Fourier Transform now is: The PSD-per-unit visual .net PDF 417 -time converges to nite values at all frequencies except those where h(t) has a discrete sine-wave (or cosine-wave) component of nite amplitude. At those frequencies, it becomes a delta-function, i.

e., a sharp spike, whose width gets narrower and narrower, but whose area converges to be the mean square amplitude of the discrete sine or cosine component at that frequency. We have by now stated all of the analytical formalism that we will need in this chapter with one exception: In computational work, especially with experimental data, we are almost never given a continuous function h(t) to work with, but are given, rather, a list of measurements of h(ti ) for a discrete set of ti s.

The profound implications of this seemingly unimportant fact are the subject of the next section.. CITED REFERENCES AND FUR THER READING: Champeney, D.C. 1973, Fourier Transforms and Their Physical Applications (New York: Academic Press).

Elliott, D.F., and Rao, K.

R. 1982, Fast Transforms: Algorithms, Analyses, Applications (New York: Academic Press)..

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