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Unformatted text preview: if the system is asymptotically stable and if the input signal is DTFtransformable. We shall not belabor this
method further because it is clumsier and more restricted than the ztransform method
discussed in the next chapter. In the ztransform, we generalize the frequency variable j n
t o (7 + j n so t hat the resulting exponentials can grow or decay with k. This procedure is
tHere Y(fl) is a function of variable ein . Hence, x = ei n for the purpose of comparison with the
expression in Sec. B.55. 1 0.62 Computation o f Direct and Inverse DTFT S pectral analysis o f d igital signals requires d etermination o f D TFT a nd I DTFT
[ determining F (n) from J[k) a nd vice versa). T his d etermination could b e accomplished by using t he D TFT e quations [Eqs. (lO.30) a nd (10.31)) directly on a digital
computer. However, t here a re two difficulties i n i mplementation o f t hese equations
on a digital computer.
1. E quation (lO.31) involves summing a n infinite n umber o f t erms, which is n ot
possible because i t requires infinite computer time.
2. E quation (lO.30) requires integration which can only b e p erformed approximately on a c omputer b ecause a computer approximates a n i ntegral by a s um. T he first problem can be s urmounted e ither by restricting t he analysis only t o
a finite length I[k) o r b y t runcating I[k) b y a s uitable window. T he e rror because 642 10 Fourier Analysis o f DiscreteTime Signals II
.. .. ... InIIJ. ....
r f [k) 643 10.6 Signal processing Using D FT a nd F FT Therefore, NoDr a re t he samples o f F(!1) t aken uniformly a t t he frequency intervals
o f !10 . B ut b ecause!1 o = 211"/No, t here a re exactly No n umber of these samples of
F(!1) over t he f undamental frequency interval o f 211". According t o Eqs. (10.62) and
(10.65), i t follows t hat (a) .. k  (10.66)
where [from Eqs. (10.63) a nd (10.65)]
N ol
Fr = . .. 11'J1111 ..
1
I
'oJ f[k]ejrfl.ok (10.67) k=O Because fNo [k] = f[k] for k = 0, 1, 2, . .. , (No  1), we can express t he Eq. (10.66)
as
f[k] = ~
No F ig. 10.10 D FT computation of a finite length signal.
of windowing may be reduced by using a wider a nd a t apered window. We shall
see t~at if f[k] has a finite duration, t he samples o f F(!1) c an be computed using
a fillIte sum (rather t han a n integral). This solves t he second problem. Moreover,
f[k] is uniquely determined from these samples of F(!1).
In order t o derive appropriate relationships, consider t he signal f[k] s tarting
a t k = 0, a nd w ith a finite length No, a s shown in Fig. 1O.lOa. Let us construct a
periodic signal f No [k] by repeating f [k] periodically a t intervals of No, a s illustrated
in F~g. 1O.lOb. We c an represent t he periodic signal fNo [k] by t he discretetime
FourIer series (DTFS) as [see Eqs. (10.8) a nd (10.9)] L N ol L Frejrfl.ok k = 0, 1, 2, . .. , No  1 (10.68) r =O Moreover, we need t o d etermine F r , t he samples of t he D TFT, only over t he interval
211". Therefore, Eq. (10.67) c an be expressed as o :&qu...
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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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