2.14 The signal f (x) is band limited so that its spectrum, F(v) (i.e., Fourier transform), is zero outside the interval 2pB

a. Expand the spectrum F(v) in a Fourier series on the interval

2pB

b. Show that the coefficients, fk, in this expansion are proportional to samples of f (x) at the interval of 1 2B unit of length. Using these values for fk in the above equation, take the inverse Fourier transform of F(v) to give f (x) and show that the result expresses the signal f (x) in terms of the sample values f ( k 2B ) and an ‘‘interpolation function’’ of the form

This last result is, then, actually a statement of the most popular form of the sampling theorem, in that it expresses the band-limited signal f (x) in terms of discrete samples taken at intervals of 1 2B units of length.

Transcribed Image Text:

F(w)= exp(- k=-00 f 2B