Whittaker-Shannon Sampling Theorem: The Infinite Interval Version

Remark: Note that even though the expansion coefficients can be determined from these integrals, it is not necessary to do so. Instead, one can obtain the expansion coefficients $\alpha_{j\ell}$ from $f(t)$ directly. One need not evaluate the integral at all. The key to success lies in the sifting property, Eq.(2.62).

Suppose one knows that $f(t)$ has a Fourier transform which is non-zero only in the interval $j\varepsilon <\omega <(j+1)\varepsilon$ . This is no severe restriction because $f$ is square integrable, and one can set $j=0$ , provided we make $\varepsilon$ large enough. This implies that

$$\alpha_{j\ell} = 0~~\qquad~~\textrm{if}~~j\not= 0\,.$$

(Why?) Consequently, we have

$$f(t)= \sum^\infty_{\ell = -\infty} \alpha_{0\ell} P_{0\ell}(t)~~\qquad~~-\infty <t <\infty\,,$$

where the wave packets $P_{0\ell}$ are given by Eq.(2.64). It is easy to determine the expansion coefficients. Using the sifting property, Eq.(2.62), one obtains

$$f\left(\frac{2\pi}{\varepsilon} k\right) =\alpha_{0k} \sqrt{\frac{\varepsilon}{2\pi}}~~\qquad~~ k=0,\pm 1,\dots\,.$$

This means that the expansion coefficients $\alpha_{0k}$ are determined from the values of $f$ sampled at the equally spaced points $t= \frac{2\pi}{\varepsilon}k$ . These sampled values of $f$ determined its representation

$$f(t) = \sum^\infty_{\ell =-\infty} f\left(\frac{2\pi}{\varepsilon}\ell \right)\sqrt{\frac{2\pi}{\varepsilon}} P_{0\ell}(t)$$

in terms of the set of orthonormal wave packets. This representation of $f$ in terms of its sampled values is $100\%$ accurate. It is called the Whittaker-Shannon sampling theorem. It is a generalization of the special case, Eq.(2.13) mentioned on page .

Exercise 24.2 (WAVE PACKET TRAINS)
Consider the wave packet

$$Q_{j\ell}(t) = {1 \over \sqrt {2\pi \varepsilon}}\int^{(j+{1 \ove... ...ver 2})\varepsilon} e^{i\omega t} e^{-2\pi i \ell \omega /\varepsilon}d\omega.$$

Express the summed wave packets

(a)

$\sum\limits^\infty_{j= - \infty} Q_{j\ell}(t)$

(b)

$\sum\limits^\infty_{\ell = - \infty} Q_{j\ell}(t)$

(c)

$\sum\limits^\infty_{\ell = - \infty} \sum\limits^\infty_{j = -\infty} Q_{j\ell}(t)$

in terms of appropriate Dirac delta functions, if necessary.

Lecture 16