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Introduction to Digital Signal and System Analysis Z Domain Analysis Digital System Input Output Time domain x [n] h [n] y [ n] = x[ n]* h[ n] ↑ ↑ ↑ ↓ ↓ ↓ z z Y z = z-domain X ( ) H ( ) ( ) X ( z) H ( z) Figure 5.2 Input-output relationship in time and z domains Figure 5.2 Input-output relationship in time and z domains Instead of using Eq.(5.2), the inverse z-transform can be made through partial fractions. The following are examples. Example 5.2 A signal has a z-transform 1 X ( z) = , z( z − 1)(2z −1) , find the corresponding original signal x [n] . The z-transform can be represented by partial fractions as X ( z) = 1 z( z −1)(2z −1) = A z + B C + z −1 2z −1 Sharp Minds - Bright Ideas! Please click the advert Employees at FOSS Analytical A/S are living proof of the company value - First - using new inventions to make dedicated solutions for our customers. With sharp minds and cross functional teamwork, we constantly strive to develop new unique products - Would you like to join our team FOSS works diligently with innovation and development as basis for its growth. It is reflected in the fact that more than 200 of the 1200 employees in FOSS work with Research & Development in Scandinavia and USA. Engineers at FOSS work in production, development and marketing, within a wide range of different fields, i.e. Chemistry, Electronics, Mechanics, Software, Optics, Microbiology, Chemometrics. We offer A challenging job in an international and innovative company that is leading in its field. You will get the opportunity to work with the most advanced technology together with highly skilled colleagues. Read more about FOSS at www.foss.dk - or go directly to our student site www.foss.dk/sharpminds where you can learn more about your possibilities of working together with us on projects, your thesis etc. The Family owned FOSS group is the world leader as supplier of dedicated, high-tech analytical solutions which measure and control the quality and production of agricultural, food, pharmaceutical and chemical products. Main activities are initiated from Denmark, Sweden and USA with headquarters domiciled in Hillerød, DK. The products are marketed globally by 23 sales companies and an extensive net of distributors. In line with the corevalue to be ‘First’, the company intends to expand its market position. Dedicated Analytical Solutions FOSS Slangerupgade 69 3400 Hillerød Tel. +45 70103370 www.foss.dk Download free ebooks at bookboon.com 64
Introduction to Digital Signal and System Analysis Z Domain Analysis where A, B, C are constants, which can be determined by the following steps. A B C X ( z) = + + z z −1 2z −1 A( z −1)(2 z −1) + Bz(2z −1) + Cz( z −1) = z( z −1)(2 z −1) A(2z = 2 2 − 3z + 1) + B(2z − z) + C( z z( z −1)(2 z −1) − z) 2 (2A + 2B + C) z + ( −3A − B − C) z + A = z( z −1)(2 z −1) 2 i.e. 2 1 (2A + 2B + C) z + ( −3A − B − C) z + A = z( z −1)(2 z −1) z( z −1)(2 z −1) The numerator of the left hand side is forced to equal to the right hand side, yielding the following simultaneous equations: ⎧2A + 2B + C = 0 ⎪ ⎨ 3A + B + C = 0 ⎪ ⎩ A = 1 Solving the equations, the constants can be found as A=1, B=1, C=-4. Therefore, 1 X ( z) = + z 1 4 − = z z −1 2z −1 −1 ⎧ ⎨1 + ⎩ z + z −1 2z ⎫ ⎬ z − 0.5⎭ . According to basic z-transform pair formulas in Table 5.2, 1 b d [ n] z z −1 b u[ n] 2z z − 0.5 b n 2(0.5) u[ n] Combining the time-shift factor −1 z , the signal in time domain can be obtained as n−1 x[ n] = d [ n −1] + u[ n −1] − 2(0.5) u[ n −1] . 65 Download free ebooks at bookboon.com
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