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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 tateSpace Analysis o f D iscreteTime 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 DiscreteTime Signals 13 S tateSpace 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 .11 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 nehalf
children or t hree a nd o nefourth 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.
 Spring '13
 Bayliss
 Signal Processing, The Land

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