Signal Processing and Linear Systems-B.P.Lathi copy

z n m atlab r eturns b 03762 13575 1 9711 13575

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 frequency response. 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 Rate 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[2], " ', 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...
View Full Document

This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.

Ask a homework question - tutors are online