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Unformatted text preview: olution, is e xplained in Sec. 5.21. I n c ircular
convolution, b oth sequences t o b e convolved a re Noperiodic. I f f [k] a nd g[k] a re
b oth Noperiodic, t heir p eriodic (or circular) convolution c[k] is defined as O::;k::;7
otherwise
T he windowed signal is
(O.8)k
fWR[k] = { 0 otherwise N ol c[k] = J[k]@g[k] = z= f [m]g[k  m] (10.76) m =O N ote t hat t he c ircular convolution differs from t he r egular (linear) convolution by
t he fact t hat t he s ummation is over one period ( starting a t a ny point). I n t he linear
convolution, t he s ummation is from  00 t o 0 0. T he r esult of a periodic convolution
is also a n Noperiodic sequence. 652 Suppose we wish t o convolve two finite length sequences i [k] a nd h[k] o f l ength
N j a nd N h, respectively. Let y[kJ b e t he (linear) convolution of t hese sequences;
t hat is, y[k] = i [k] * h[k] (10.77) 2 Yr = Y (rQ o) 0 12 , ~ (10.79) According t o Eq. (10.78) i t follows t hat
(10.80)
w here Fr a nd H r a re t he r th s amples of F (Q) a nd H (Q), respectively. For Eq.
(10.80) t o b e valid, F r a nd H r m ust be compatible for multiplication. I n o ther
w ords, b oth m ust be Nopoint sequences if Y r is a n N opoint sequence. B ut we
know t hat i [k], h[k], a nd y[k] a re N j, N h, a nd No = ( Nj+Nh  I)point sequences.
Hence, we m ust p ad N h  1 zeros t o i [k] a nd p ad N j  1 zeros t o h[k] t o e nsure
t hat F r , H r , a nd Y r a re all No = ( N j + N h  I)point sequences. Once we c ompute
F r a nd H r ( after suitably zeropadding i [k] a nd h[k]), we t ake t he I DFT o f F rHr
t o o btain y[k]. However, we have shown in C hapter 5 [Eq. (5.32a)] t hat F rHr is t he
D FT o f a circular convolution o f i [k] a nd h[k]; t hat is J[k] ® h[k] ~ F rH r . T his
r esult m eans y[k] is equal t o t he c ircular convolution of (suitably p added) i [k] a nd
h[k]. T hus, y[k] is a periodic sequence whose first p eriod is t he l inear convolution
o f ( unpadded) i [k] a nd h[k]. T o summarize, y[k]' which is t he l inear convolution of
I [k] a nd h[kJ, is also equal t o t he c ircular convolution o f s uitably p added i [k] a nd
h[kJ. T his is a n e xtremely i mportant result. T he s ystem r esponse is given by t he
l inear c onvolution of i [k] a nd h[k]. B ut t he p receding result allows us t o c ompute
t his c onvolution as if i t were t he c ircular convolution of (suitably p added) i [k] a nd
h[k]. T his, in t urn, allows us t o use D FT t o p erform t he c omputations.
T he p rocedure t o find t he (linear) convolution o f i [k] a nd h[k], whose lengths
a re N j a nd N h, respectively, c an b e summarized in four s teps as follows:
1. P ad N h  1 zeros t o i [k] a nd N j  1 zeros t o h[k].
2. F ind F r a nd H r , t he D FTs o f t he z eropadded sequences i [k] a nd h[k].
3. M ultiply F r b y H r t o o btain Yr.
4. T he d esired convolution y[k] is t he I DFT o f Yr. o 2 • ~ 01 2 0123456
k...... . • •,
.
. 4 1 2 3 4 56
k...... k...... •
4 . .{j 2 0 12 345678
k ...... F ig. 1 0.14 Line...
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 Spring '13
 Bayliss
 Signal Processing, The Land

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