Unformatted text preview: e i ts a mplitude spectrum. T he p hase spectrum, however, is changed by  koO. T his added
phase is a linear function of 0 w ith slope  ko. Physical Explanation o f t he Linear Phase
T ime delay in a signal causes a linear phase shift in its spectrum. T he heuristic
explanation of this result is e xactly parallel t o t hat for continuoustime signals given
in Sec. 4.34 (see Fig. 4.20).
FrequencyShifting Property Parseval's Theorem
If f[k] { :=> F(O) t hen E f , t he energy of f [k], is given by If f[k] { :=> (10.51) F(O) t hen
(10.49) In order t o prove this property, we have from Eq. (10.31) T his property is t he d ual of t he timeshifting property. To prove this property, we
have from Eq. (10.31) 00 F*(  0) = L ! *[k]e jOk (1O.52a) k =oo
00 00 k =oo k =oo T his result shows t hat j*[k] { :=> F *(O) (1O.52b) Now k~00If[kll2 = k~OO!*[k]f[k] = k~OO!*[k] [2~ 1" F(O)ejnkdO] T ime and Frequency Convolution Property
If h[k] { :=> F1 (0) a nd h [k] { :=> F2(0) t hen h [k]
a nd * h[k] h [k]h[k]
where { :=> { :=> F l(0)F2(0) 1
271' Fl(O) * F2(0) 2~ 1" F(O) [k~OO!*[k]ejm] (10.50a) = (1O.50b) =~
271' r F(O)F*(O) dO = ~ i2" IF(0)1 2 dO
r
271' i27r 636 10 Fourier Analysis o f DiscreteTime Signals lOA 637 Connection with t he C ontinuousTime Fourier Transform ( FT)
f [kl (a)
F (Q) (b) (a) ~ o (b)
0) _ _ 1 k 211 11 0 11 211 Q ~(t)
4i[kl
( c) ~[kl=f[2kl 1 (c) F iQ) A Decimation (Downsampling) (d) 2
,.: ":..:" f lkl j
rIIII 2 0 10 k 211 (e) · :11 J 11 0 11 211 Q (e) f; [kl
k ( I) Fig. 1 0.8 C onnection b etween t he D TFT a nd t he F ourier t ransform.
16 24 32 211 1 0.4 DTFT Connection with the ContinuousTime Fourier
Transform C onsider a continuoustime signal fe(t) (Fig. 1O.8a) w ith t he Fourier transform
Fe(w). T his signal m ayor may not be bandlimited. For convenience, we shall
assume t he signal t o b e bandlimited t o B Hz (Fig. 1O.8b). This signal is sampled
with a sampling interval T . T he s ampling r ate m ayor m ay n ot b e above t he N yquist
rate. Again, for convenience, we shall assume t hat t he s ampling r ate is a t l east equal
t o t he Nyquist rate; t hat is, T ::::: l /2B. T he s ampled signal l e(t) (Fig. lO.8c) c an
be expressed as
00 L l e(t) = fe(kT)Ii(t  kT) k =oo T he c ontinuoustime Fourier transform o f t he above equation yields
00 Fe(w) = L fe(kT)e  jkTw (10.53) k =oo I n Sec. 5.1 (Fig. 5.le), we have shown t hat Fc(w) is F c(w)IT r epeating periodically
with a period w . = 27r I T, as illustrated in Fig. 1O.8d. L et us construct a discretetime signal f[k] such t hat i ts k th element value is equal t o t he value o f t he k th
s ample of fc(t), as depicted in Fig. 1O.8e; t hat is, f[k] = f c(kT)
Now, F (n), t he D TFT of f[k], is given by F ig. 1 0.9 11 11 2x Q S pectra o f t he d ecimated a nd i nterpolated s ignals. 00 00 k =oo k =oo (10.54)
C omparison o f (10.54) with (10.53) shows t hat F (n) = Fe (~) (10.55) Thus, F (n) c an b e o bt...
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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.
 Spring '13
 Bayliss
 Signal Processing, The Land

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