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Unformatted text preview: cedure =1_ e - jsn = e -j3D(ej3D _
= e - j3D ) 2 je- j3D sin 3 0 = 2e j ( ~-3n) sin 3 0
T he term sin 3 0 can be positive as well as negative. Therefore M uch o f t he discussion so far h as b een r ather general. We shall now give some
concrete examples of such filter design. Because we want t he r eader t o b e focussed
on t he procedure, we shall intentionally choose a small value for n ( the filter order)
t o avoid getting distracted by a jungle of data. T he procedure, however, is general
a nd i t c an be applied t o a ny value of n.
T he s teps in t he t ime-domain equivalence design method are: 1. Determine the filter impulse response h[k]
n and L H[d ] is as indicated in Fig. 12.8d. The amplitude response, illustrated in Fig.
12.18c, is shaped like a comb with periodic nulls. The filter can be realized by the structure
in Fig. 12.7b. Since h[k] = b s-k [see Eq. (12.72)], bo = - 1, b l = 1>2 = b3 = b4 = bs = 0,
a nd bs = 1. I n t he first step, we find t he impulse response h[k] o f t he desired filter. According t o t he t ime-domain equivalence criterion in Eq. (12.31),
h[k] = T ha(kT) (12.80) 762 12 F requency Response a nd D igital Filters delayed by N o2-1 t o m ake i t c ausal. T his d elay produces t he d esired linear-phase
S traight t runcation o f d ata a mounts t o using a rectangular window, which h as
a u nit weight over t he window width, a nd zero weight for all o ther k. We saw
t hat a lthough s uch a window gives t he s mallest t ransition b and, i t r esults i n a
slowly decaying oscillatory frequency response in t he s topband. T his b ehavior
c an b e c orrected by using a t apered window o f a s uitable width. T ABLE 1 2.2
S ome W indow F unction a nd T heir C haracteristics Mainlobe
Width Windoww[kJ - M:'Ok:'OM Rolloff
dB/octave P eak Sidelobe
Level in dB 4. . No c. ( Nok_1) 8 .. -6 - 13.3 - 12 - 26.5 - 18 - 31.5 -6 - 42.7
- 58.1 2 Bartlett: 3 lIanning: 0.5 [1 + cos C~o"~1) 4 lIamming: 0.54 + 0.46 cos (~;~1) No 5 Blackman: 0.42 + 0.5 cos (J;~1 ) + 0.08 ( Jo"~ 1 ) 12. .
No - 18 11.211" -6 No 1 8 .. No
8 .. * 3. Filter Frequency Response a nd Realization
K nowing h[O], h[l], h, " ', h[n], we d etermine H[z] using Eq. (12.71) a nd t he
frequency response H [e iwT ] using Eq. (12.74). We now realize t he t runcated
h[k] using t he s tructure i n Fig. 12.7b. M - No-1
2 Rectangular: reet ( Nok_1 ) O ptimality o f t he P rocedure
T he p rocedure outlined here using a r ectangular window function is t he optimum i n t he s ense t hat t he e nergy of t he e rror (difference) between t he d esired
frequency response H a(jw) a nd t he realized frequency response H [e iwT ] is t he m inimum for a given No. T his conclusion follows from t he fact t hat t he r esulting filter
frequency response H [e iwT ] is given by H[eiwT] = I >[k]e- iWkT
k 10 6 763 12.8 Nonrecursive Filter Design 1 :'0 O!:'O 10 K aiser: NO - 59.9 CO! = 8.168) w here ha (t) is t he i mpulse response of t he a nalog filte...
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