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

# Digital filters c an process analog signals using a

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Unformatted text preview: ant frequency is 200 Hz. I ·•. 1 1 2.6-6 R epeat P rob. 12.6-5 for a chebyshev filter. 1 2.6-7 Design a digital b andpass B utterworth filter using th~ b ilinear t rans!ormation w ith prewarping t o s atisfy t he following specifications: G p = - 2 d B, G s = - 12 dB, W P1 = 120 r ad/s, w P2 = 300 r ad/s, a nd W S1 = 45 r ad/s, W S1 = 450. T he h ighest significant frequency is 500 Hz. 1 2.6-8 1 2.6-9 R epeat P rob. 12.6-7 for a chebyshev filter. Design a digital b andstop C hebyshev filter using th&lt;;: b ilinear trans~ormation w ith prewarping t o s atisfy t he following specifications: G p = - 1 d B, G s = - 22 d B, W P1 = 40 r ad/s, w P2 = 195 r ad/s, a nd W S1 = 80 r ad/s, W S1 = 120. T he h ighest significant frequency is 200 Hz. 1000 Show t hat t he frequency response of a t hird-order (n = 3) F IR filter w ith t he i mpulse response h[k] t hat is s ymmetric a bout i ts c enter point is given by H[e jwT ] = 2 e- j 5wT 1. (h[1] cos w : + h[O] cos ~T) a nd for t he a ntisymmetric case H[e jwT ] = 2e 1 2.7-2 j (f-1. 5wT ) ( h[l] s in w : + h[O] s in A nother form o f c omb filter is given by t he t ransfer f unction ~T) 800 1000 1 2.8-3 D etermine JH[ejwT]J, u sing t he v on H ann w indow for a fifth-order nonrecursive (FIR) filter in E xample 12.10 ( to a pproximate a n i deallowpass filter) a nd for t he t enth-order nonrecursive filter i n E xample 12.11 ( to a pproximate a n ideal differentiator). 1 2.8-4 A n i deal7r/2 p hase s hifter (also known a s t he H ilbert t ransformer) is given by H a(jw) z -1 s =K-z+1 m aps t he j w-axis in t he s -plane i nto a u nit circle in t he z -plane. T he L HP a nd t he R HP o f t he s -plane m ap i nto t he i nterior a nd t he e xterior, respectively, o f t he u nit circle in t he z-plane. Show t hat if Ha(s) r epresents a s table s ystem, t hen t he c orresponding H[z] also represents a stable system. Hint: s = ( j + j w a nd z = ( K + s )/(K + s). 0 ) _ __ 800 F ig. P 12.8-2 1 2.6-10 T he bilinear t ransformation is a ctually a o ne-to-one m apping r elationship b etween t he s -plane a nd t he z -plane. Show t hat t he t ransformation 1 2.7-1 783 ={ -j j Os. W S. 7I'/T 0 ?w?..-7I'/T ( a) S ketch JHa(jw)J a nd L Ha(jw). ( b) U sing t he i mpulse invariance m ethod, d esign a fourteenth-order ( N = 14) nonrecursive ( FIR) filter t o a pproximate H a(jw). D etermine t he r esulting JH[ejwTlI a nd jwT LH[e, u sing t he r ectangular a nd t he H amming windows. 1 2.8-5 D esign a t enth-order d igital d ifferentiator in E xample 12.11 using t he frequency s ampling m ethod. C ompare t he values o f t his d esign w ith t hose found in Example 12.11 using t he i mpulse invariance m ethod. Q 13.1 785 Introduction 4. S tate e quations can yield a g reat d eal of information a bout a s ystem even when t hey a re n ot solved explicitly. T his c hapter requires some understanding o f m atrix algebra. Section B.6 is a self-contained t reatment o f m atrix a lgebra, which should b e more t han a dequate for t he...
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