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

From eq 1057 we have lyn1 i fniihn1 1058 a

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Unformatted text preview: ot;: !1 < N o-l Fr = L f[k]e-jrfl.ok r= 0, 1, 2, . .. , No - 1 (10.69) k=O E quations (10.68) a nd (10.69) are precisely t he D FT p air derived in Eqs. (5.18a) a nd (5.18b). These equations relate Fr [the samples of F(!1)] t o f[k] a nd vice versa. Here Fr is t he D FT of J[k] a nd f[k] is t he I DFT (inverse D FT) of Fr. T his relationship is also denoted by t he bidirectional arrow notation of a transform as N o-l fNo[k] = L !1 0 = 211" Drejrfl.ok No r=O (10.62) where Dr = ~ N o-l L f[k]e-jrfl.ok (10.63) No k=O In Eq. (10.63) we used the fact t hat fNo[k] B y definition, t he D TFT o f f[k] is = f[k] for k = 0 ,1,2, . .. , No - 1. Properties o f O FT N o-l F(!1) = L To repeat, Fr , t he D FT o f a n N o-point sequence f[k], is a s et o f uniform samples o f i ts D TFT F(!1) t aken a t frequency intervals o f!1 o = ~. T he sequence f[k] a nd i ts D FT Fr a re related by Eqs. (10.68) a nd (10.69). Observe t hat t here are No e lements in f[k]. Also, there are exactly No elements in Fr (over t he frequency range 211"). T he D FT relationships are finite sums a nd c an be readily computed on a digital c omputer using t he efficient fast Fourier transform ( FFT) a lgorithm. f[k]e-jfl.k (10.64) k=O According t o Eqs. (10.63) a nd (10.64), it follows t hat NoDr is F (r!1 o), t he r th s ample of F(!1). For convenience, we denote this sample by Fr. T hus We list some o f t he i mportant p roperties of D FT proved in C hapter 5. from the preceding discussion, it follows t hat these properties o f D FT also apply t o D TFT samples o f a finite length f[k]. 1. linearity: I f f[k] {=} Fr a nd g[k] {=} G r , t hen (10.70) (10.65) 644 10 Fourier Analysis o f Discrete-Time Signals 2. Conjugate Symmetry: For real f [k] (10.71) There is a conjugate symmetry a bout N o/2, which enables us t o d etermine roughly half the values of Fr from t he o ther half of t he values, when f [k] is real. For instance, in a 7-point D FT, F6 = F l *, F5 = F2 * a nd F4 = F3 *. In a n 8 -point D FT, F 7=Fl*' F 6=F2*' F 5=F3*,andsoon. 3. Time Shifting (Circular Shifting): f [k - n] { =} F re-irflon (10.72) 4. Frequency Shifting: f[k]eikflom { =} Fr - m (10.73) 645 10.6 Signal processing Using D FT a nd F FT depicted in Fig. 10.2a for No = 32. A careful glance a t t his figure shows t hat t his periodic signal c an b e c onstructed b y a ny segment of length 32 a nd r epeating i t periodically b y placing i t e nd t o e nd a d infinitum. We m ay choose a segment over t he r ange k = - 16 t o 15 o r a segment over t he r ange k = 0 t o 31, o r a ny o ther segment o f l ength 32. T he r eader should satisfy himself t hat t he periodic extension of any such segment yields t he s ame periodic signal fNo[k]. Therefore, t he D FT c orresponding t o t he periodic signal fNo[k] is t he D FT o f any o f i ts segment o f No l ength s tarting a t a ny point. So t he signal J[k] m ay s tart a t any point. All we need is t o c onstruct a periodic signal fNo[k] which is a periodic extension of...
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This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.

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