Parseval’s theorem

Using Parseval s theorem, from equation (9.126) the //2-optimal eontrol problem ean be expressed in the frequeney domain as... 

Since the function A k) is the Fourier transform of (x, t), the two functions obey Parseval s theorem as given by equation (B.28) in Appendix B... 

Show that the square pulse A k) in equation (1.21) and the corresponding function W(x, t) obey Parseval s theorem. 

Parseval s theorem is also known as the completeness relation and may be used to verify that the set of functions e " for —oo oo are complete, as (fiscussed in... 

Applying Parseval s theorem for Fourier series to the scries 22.3) and (22.4) we find that the series... 

Of a special interest here is a charge in aperiodic motion, as in a collisional encounter. In that case, the theory of Fourier transforms is used to describe the continuous spectra that result. Specifically, starting from Eq. 2.60 and making use of Parseval s theorem, Eq. 2.52, the total energy radiated in the aperiodic event per unit solid angle and per unit frequency interval is obtained as... 

The total power P in the excitation is obtained from Parseval s theorem ... 

The rest of the optimization is identical to the procedure described for the white power spectrum. The total excitation power in a (-1,1) binary sequence is, according to Parseval s theorem [Eq. (66)] always constant. 

Parseval s theorem demonstrates that the total power is the same whether it is computed in the time domain or the frequency domain. It is often of interest to determine the power in some small interval (f+ df), as a function of the frequency. This is called the spectrum of the signal h(t). 

To obtain Parseval s theorem for the function /(x) in equation (B.17), we first take the complex conjugate of /(x)... 

Finally, integration over the variable k yields Parseval s theorem for the Fourier integral,... 

From Parseval s theorem, a least-squares criterion in the time domain is equivalent to the least-squares criterion in the Fourier domain (22) ... 

As time increases from —oo to 0, the half width of the wave packet y(x, t) continuously decreases and the maximum amplitude continuously increases. At t = 0 the half width attains its lowest value of y/lja and the maximum amplitude attains its highest value of 1 / and both values are in agreement with the wave packet in equation (1.20). As time increases from 0 to oo, the half width continuously increases and the maximum amplitude continuously decreases. Thus, as increases, the wave packet y(x, z) remains gaussian in shape, but broadens and flattens out in such a way that the area under the square y(x, 01 of the wave packet remains constant over time at a value of (2A/7ra), in agreement with Parseval s theorem (1.18). 

The basic question for all practical considerations is that of the relation between Ns and Nf. If we know this relation, we can predict Wy from the experimental quantity Ng, In order to derive such a relation, we use Parseval s theorem in the theory of Fourier transforms this reads in our context... 

Inserting Eq. (C.26) into Eq. (C.25) then leads after a short calculation (or immediately from Parseval s theorem [69]) to... 

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