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

P rocedure for determining s tate e quations s

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Unformatted text preview: troduction S ystematic P rocedure for Determining S tate E quations S olution o f S tate E quations L inear Transformation o f S tate Vector C ontrollability and Observability S tate-Space Analysis o f D iscrete-Time S ystems S ummary 785 788 798 811 818 823 830 Answers t o Selected Problems 837 Supplementary Reading 843 Index 844 x B 617 Fourier Analysis o f Discrete-Time Signals 13 S tate-Space Analysis 13.1 13.2 13.3 13.4 13.5 13.6 13.7 583 585 598 603 609 610 Background T he t opics discussed in this c hapter a re n ot e ntirely new t o s tudents t aking t his course. You have a lready s tudied m any o f these topics in earlier courses or are expected t o know t hem from your previous training. Even so, t his b ackground material deserves a review because i t is so pervasive in t he a rea o f signals a nd s ystems. Investing a little t ime i n such a review will p ay big dividends later. Furthermore, t his m aterial is useful n ot only for t his c ourse b ut also for several courses t hat follow. I t will also b e helpful as reference m aterial i n your future professional career. B.1 Complex Numbers C omplex n umbers a re a n e xtension of ordinary n umbers a nd a re a n i ntegral p art o f t he m odern n umber s ystem. Complex numbers, p articularly i maginary n umbers, s ometimes seem mysterious a nd u nreal. T his feeling of unreality derives from t heir u nfamiliarity a nd novelty r ather t han t heir s upposed nonexistence! M athematicians b lundered in calling t hese n umbers "imaginary," for t he t erm immediately prejudices perception. H ad t hese numbers b een c alled by some o ther n ame, t hey would have become demystified long ago, j ust a s i rrational numbers o r n egative numbers were. Many futile a ttempts have b een m ade t o a scribe some physical meaning t o i maginary numbers. However, this effort is needless. In m athematics we assign symbols a nd o perations a ny m eaning we wish as long as i nternal c onsistency is maintained. A healthier approach would have been t o define a symbol i ( with any t erm b ut " imaginary"), w hich has a p roperty i 2 = - 1. T he h istory of m athematics is full of entities which were unfamiliar a nd h eld in abhorrence until familiarity m ade t hem a cceptable. T his fact will become clear from t he following historical note. 8 .1-1 A Historical Note A mong early people t he n umber s ystem c onsisted only of n atural n umbers (positive integers) needed t o c ount t he n umber o f children, c attle, a nd quivers of arrows. These people h ad no need for fractions. Whoever h eard o f two a nd o ne-half children or t hree a nd o ne-fourth cows! 2 B ackground However, w ith t he a dvent of agriculture, people needed to measure continuously varying quantities, such as t he length of a field, t he weight of a quantity o f b utter, a nd so on. T he n umber system, therefore, was extended to include fractions. T he a ncient E gyptians a nd Babylonians knew how to handle fractions, b ut P ythagoras discovered t hat s ome numbers (like the diagonal of a unit square) could not be expressed as a whole n...
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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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