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332 Chapter Ten

by a horizontal distance of |B|/T x . This distance must be greater than
the extent of the transformed (unsampled) function in x. Similar re-
sults can be found for all other replicas, but this defines the lower
bound on the sampling rate.

|B|
T x ≤ (10.32)
w
When this bound is used in addition to the Nyquist criterion, the PSD
of the output waveform—after this is also sampled at the generalized
Nyquist rate—will look like Fig. 10.10b. Equation (10.32) fails for B=0,
which is a special case of the LCT where the output is already discrete
and no second sampling process takes place.

10.6 Conclusion
In conclusion we have demonstrated the usefulness of the WDF in
both qualitatively and quantitatively understanding sampling theory.
In particular, the ability of the WDF to describe generalized sampling
is extraordinary. At no point is it necessary to employ a wave integral
in these derivations. In fact, almost all the equations in this chapter
serve as a pretext to the underlying idea; by visualizing a signal’s
energy as a bounded shape in phase space and endowing this shape
with the basic properties of the WDF, we may derive complicated
sampling criteria and interpolation formula.

References
1. H. Nyquist, “Certain topics in telegraph transmission theory,” Trans. AIEE, 47:
617–644 (1928).
2. C. E. Shannon, “Communication in the presence of noise,” Proc. Inst. Radio Eng.
37: 10–21 (1949).
3. E. Whittaker, “On the functions which are represented by the expansions of the
interpolation theory,” Proc. Royal Soc. Edinburgh, Sec. A, 35: 181–194 (1915).
4. V. A. Kotelnikov, “On the carrying capacity of the ether and wire in telecommu-
nications,” Material for the First All-Union Conference on Questions of Com-
munication, Izd. Red. Upr. Svyazi RKKA, Moscow, 1933.
5. G. T. Di Francia, “Resolving power and information,” J. Opt. Soc. Am. 45:
497–499 (1955).
6. S. Haykin, Communication Systems, 4th ed., Wiley, New York, 2001.
7. A. V. Oppenheim, R. Schafer, and J. R. Buck, Discrete Time Signal Processing,2d
ed., Prentice-Hall, Englewood Cliffs, N.J., 1999.
8. R. N. Bracewell, The Fourier Transform and Its Applications, 2d Ed., McGraw-Hill,
New York, 1978.
9. J. W. Goodman, Introduction to Fourier Optics. 2d Ed., McGraw-Hill, New York,
1996.
10. A. Papoulis, The Fourier Integral and Its Applications, McGraw-Hill, New York,
1962.