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Unformatted text preview: f[k], t hen
c ompute t he D FT according t o E q. (10.68) using t he sample values in t he r ange
k = 0 t o No  1. For instance, t he signal J[k] in Fig. 10.6 does n ot s tart a t k = O.
B ut we c an c onstruct its periodic extension f No [k], as shown in Fig. 1O.2a, a nd use
t he values for k = 0, 1, 2, . .. , 31 i n Eq. (10.69) t o c ompute Fr. T hese values are 5. Circular (or Periodic) Convolution: (1O.74a) o ::; k {~ J[k] = ::; 4 a nd 28::; k ::; 31 5::; k ::; 27 Hence, according t o Eq. 10.69 a nd Nol (10.74b)
where the circular (or periodic) convolution of two Nopoint periodic sequences J[k]
a nd g[k] is defined as
Nol
J[k]@g[k] = L Nol
f[n]g[k  n] = n =O L g[n]J[k  n] (10.75) Fo = L f [k] = 9 k =O a nd
F l = e  jflo + e  j2flo + e  j3flo + e  j4flo + ej2Sflo + ej29flo + ej30flo + e  j3Wo Because n o =
Hence ill, we recognize t hat ej31flo = ejflo , ej30flo = ej2flo , a nd so on. n=O F l = 1 + 2(cos n o
Caution in Interpreting OFT and 1 0FT
E quations (10.68) a nd (10.69) allow us t o c ompute samples o f D TFT a nd
I DTFT for a finite length signal on a digital computer. To avoid certain pitfalls,
we must understand clearly t he n ature of functions synthesized by t he s ums on t he
r ighthand side of these equations. According t o Eq. (10.66), it follows t hat t he s um
on t he r ighthand side of Eq. (10.68) is fNo[k], which is a periodic signal of which
f[k] is t he first cycle. Similarly, t he s um on t he r ighthand side of Eq. (10.69) is
periodic. This is because Fr = No D r. which is periodic. Therefore, b oth t he D FT
equations are periodic. We require only p art of these results (over one cycle) to
compute t he samples of F (n) from f [k] a nd vice versa. T hat is why we placed t he
restriction t hat k or r = 0, 1, 2, '" , No  1 in Eqs. (10.68) a nd (10.69). 7.8865 • E xample 1 0.8
F ind t he D FT o f a 3 point s ignal f[k] i llustrated i n Fig. 10.11a.
F igure 1 0.l1a shows f[k] (solid line) a nd fNo[k] o btained b y p eriodic extension o f f[k]
(shown by d otted lines). I n t his case No = 3 a nd no = 211"
3 F rom E q. (10.69), we o btain
2 Signal f[k] can start at any value o f k .
In deriving t he above results, we assumed t hat t he signal f[k] s tarts a t k = O.
T his restriction, fortunately, is n ot necessary. We now show t hat t his procedure
can be applied t o f [k] s tarting a t a ny instant. Recall t hat t he D FT found by this
procedure is actually t he D FT o f fNo[k], which is a periodic extension o f f[k] w ith
period No. In other words, fNo[k] c an be generated from f[k] by placing f[k] a nd
reproduction thereof end t o e nd a d infinitum. Consider now t he signal J[k] in
Fig. 10.6 in which f[k] begins a t k =  4. T he periodic extension of this signal is + cos 2 no + cos 3 no + cos 4no) = Note t hat Fo = 9 is t he first sample of F (n) , F l = F ( ill) is t he second sample, a nd
so on. T he s amples are spaced 1f / 16 r adians a part, giving a t otal o f 32 s amples in
t he f undamental frequency range. T h...
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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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