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

84 t he v alues o f t he coefficient h rk a re s hown

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Unformatted text preview: ; h2(25)=0; (12.93) W e r equire t hat r= 0, 1, 2, , . .. , No - 1 (12.94) N ote t hat t hese a re t he s amples o f t he p eriodic e xtension o f H a(jw) ( or H[e jwT ]). B ecause we force t he f requency r esponse o f t he f ilter t o b e e qual t o t he d esired f requency r esponse a t No e quidistant f requencies i n t he s pectrum, t his m ethod is k nown a s t he f requency s ampling o r t he s pectral s ampling m ethod. I n o rder t o f ind t he f ilter t ransfer f unction, we first d etermine t he f ilter i mpulse r esponse h[kJ. T hus, o ur p roblem is t o d etermine t he filter i mpulse r esponse f rom t he k nowledge o f t he No u niform s amples o f t he p eriodic e xtension o f t he f ilter frequency r esponse H[e iwT ]. B ut H[e jwT ] is t he D TFT o f h[k] [see Eq. (10.92)]. Hence, a s s hown i n Sec. 10.6-2 [Eqs. (10.68) a nd (10.69)], h[kJ a nd H[eirwoT] ( the No u niform s amples o f H[e iwT ]) a re t he D FT p air w ith = woT. H ence, t he d esired h[kJ is t he I DFT o f H[e iwT ], g iven b y no 772 12 F requency Response a nd D igital Filters 773 12.8 Nonrecursive Filter Design ....... k = 0, 1, 2, , . .. , N o - 1 (12.95) N ote t hat H[eirwoTI = H a(jrwO) a re known [Eq. (12.94)1. We c an use I FFT t o c ompute t he N o values of h [kl. F rom these values of h[kJ, we c an d etermine t he f ilter transfer function H [zl a s -=1! - AlI.::lL -.fu!. T 7T 7T N o-l H [zl = L h [klz-k 2T l1.!!.. • 7T 7T F ig. 1 2.21 (12.96) M -k 7T f Lowpass filter design using the frequency sampling method. k=O N o-l Linear Phase (Constant Delay) Filters H [zl = We desire t hat H a(jw) = H [eiwTI. T he filter featured in Eqs. (12.95) a nd (12.96) satisfies t his c ondition only a t t he N o values o f w. B etween samples, t he f requency response, especially t he p hase response, could deviate considerably. I f w e w ant a linear phase characteristic, t he p rocedure is slightly modified. First, we s tart w ith a zero phase (or a c onstant phase ± ~ ) response. For such a frequency response h [kl is a n even (or o dd) function of k (see P rob. 4.1-1). I n e ither case, h [kl is centered a t k = 0. To realize t his n oncausal filter, we need t o delay h[kl by ( No - 1)/2 u nits. Such a d elay a mounts t o m ultiplying H [eiwTI w ith e -i¥wT. T hus, t he delay o f h [kl does n ot a lter t he filter a mplitude response, b ut t he p hase response changes by - (No - 1)wT / 2, which is a linear function o f w. Hence, we are a ssured t hat t he filter is realizable (causal) a nd h as a linear phase. T hus, if we wish t o realize a frequency response H [eiwTI, we b egin w ith H [eiWTle-i¥wT a nd find t he I DFT o f its N o samples. T he r esulting I DFT a t k = 0, 1, 2, 3, . .. , No - 1 is t he desired impulse response, which is causal, a nd t he c orresponding phase response i s linear. Note t hat w oT = 'flo, a nd t he N o uniform samples o f H [ eiwTle-i¥wT a re L (12.99) h [klz-k k=O T his p rocedure will now b e e xplained by a n example. • E xample 1 2.12 Using the frequency sampling method, design a sixth-order nonrecursive ideallowpass filter of cutoff frequency f r r ad/s. The...
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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.

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