Signal Processing and Linear Systems-B.P.Lathi copy

1 t he sampling theorem 335 repeater station which in

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: which are duals of bandlimited signals. We now prove t hat t he s pectrum F (w) of a signal timelimited t o T seconds can be reconstructed from t he samples of F (w) t aken a t a r ate R > T ( the signal width) samples per H ertz. F igure 5 .13a shows a signal f (t) t hat is t ime limited to T seconds along with its Fourier transform F (w). A lthough F (w) is complex in general, it is adequate for o ur line of reasoning t o show F (w) as a real function. 1 Dn = - F(nwo) To ~his r esult indicates t hat t he coefficients o f t he Fourier series for f ro(t) are ( liTo) times t he sample values of t he s pectrum F (w) t aken a t intervals of w00 T his means t hat t he s pectrum o f t he periodic signal fro (t) is t he s ampled spectrum F (w ), as i llustrated in Fig. 5.13b. Now as long as T < To, t he successive cycles of f (t) a ppearing in fro{t) d o n ot overlap, so t hat f (t) c an be recovered from f ru(t). Such recovery implies indirectly t hat F (w) c an be reconstructed from its samples. These samples are separated by t he f undamental frequency :Fo = l iTo Hz of t he periodic signal fTo(t). T he condition for recovery is t hat To ;::: T; t hat i st 1 :Fo ~ - Hz T Therefore, t o b e able t o r econstruct t he s pectrum F (w) from t he samples of F (w), the samples should b e t aken a t frequency intervals not greater t han :Fo = l iT Hz. I f R is t he s ampling r ate ( samples/Hz), t hen 1 (5.13) R = :Fo ;::: T s amples/Hz Spectral Interpolation T he s pectrum F (w) of a signal f (t) t imelimited t o T seconds can be reconstructed from t he samples of F (w). For this case, using t he d ual of t he a pproach employed t o derive t he signal interpolation formula in Eq. (5.10), we o btain the spectral interpolation formula 1- <a) <a) F (w) = L F(nwo) sinc (W2T - n1T) 21T W o= - • :o To r.=..!..- - ( b) (e) F ig. 5 .13 ~- F (w) = JOO f (t)e-jwtdt = - 00 r io f (t)e-jwtdt 00 ""'" Dnejnwot ~ n =-oo The spectrum F(w) of a unit duration signal f (t) is sampled at the intervals of 1 Hz or 21T rOO/s (the Nyquist rate). The samples are: (5.12) and F(±21Tn) = 0 (n 21T w o=- To = 1, 2, 3, . .. ) We use the interpolation formula (5.14) to construct F(w) from its samples. Since all but one of the Nyquist samples are zero, only one term (corresponding to n = 0) in the summation on the right-hand side of Eq. (5.14) survives. Thus, with F(O) = 1 and T = 1, we obtain r We now c onstruct fro (t), a periodic signal formed by r epeating f (t) every To seconds ( To> T), as depicted in Fig. 5.13b. This periodic signal can be expressed by t he e xponential Fourier series f To (t) = E xample 5 .4 F(O) = 1 Periodic repetition of a signal amounts to sampling its spectrum. (5.14) T n F(w) = sinc (i) (5.15) For a signal of unit duration this is the only spectrum with the sample values F(O) = 1 and F(21Tn) = 0 (n ' " 0). No other spectrum satisfies these conditions. • tThls r esult assumes t hat I (t) does n ot have impulses a t t = 0 o r Fo < l /r. T. I f i mpulses d o o ccur, t hen 338 5 .2 5 Sampling 5.2 Numerica...
View Full Document

This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.

Ask a homework question - tutors are online