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Challenge18_Lp%20norm%20challenge

# Challenge18_Lp%20norm%20challenge - T(zI-A-1 b or H...

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Digital Signal Processing Dr. Fred J. Taylor, Professor Lesson Title: L 1 norm bounds Lesson Number [18] Challenge Consider the 2 nd order filter H(z)=z 2 /(z 2 -1/4) having a Direct II architectural form given by: 1 ; 0 4 / 1 ; 1 0 ; 0 4 / 1 1 0 = = = = d c b A T What is the l 1 norm on state x 1 [k]? Response By setting d=0 and c=(1,0) T , the system output is x 1 [k]. The input - x 1 [k] transfer function is therefore H 1 (z)=c
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Unformatted text preview: T (zI-A)-1 b or H 1 (z)=1/(z 2-1/4) = z-2 /(1-1/4z-2 ) The impulse response x 1 [k] can computed using partial fractions or long division. That is, divide (1-1/4z-2 ) into z-2 producing: X 1 (z) = 1z-2 + (1/4)z-4 +(1/4) 2 z-6 +(1/4) 3 z-8 + …. In the time domain the l 1 norm is ||x 1 [k]|| 1 = 1 + ¼ + 1/16 + … = 1/(1-(1/4)) = 4/3 = 1.3333 Lp norms 1...
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