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Unformatted text preview: s. Such is n ot t he case w ith 15(t), where we notice t hat v ariations in
f s(t) a re generally a t a higher r ate t han t hose in x (t). T here is still considerable
similarity; b oth signals always remain positive, a nd show no oscillations. B oth
signals have zero or negligible s trength b eyond t = 5. T hus, 15(t) is similar t o
x (t), b ut n ot as similar as 14(t). T his is w hy en = 0.628 for 15(t). T he signal
16(t) is o rthogonal t o x (t), so t hat en = O. T his fact appears t o i ndicate t hat t he
dissimilarity in this case is n ot as s trong as t hat o f h (t) for which en = - 1. T his
conclusion may seem o dd b ecause h (t) a ppears more similar t o x (t) t han does
/6(t). T he dissimilarity between x (t) a nd h (t) is o f t he n ature o f a ntipathy ( the
worst enemy); in a way t hey a re very similar, b ut in opposite ways. O n t he o ther
h and, t he d issimilarity of x (t) w ith 16(t) s tems from t he fact t hat t hey a re almost
of different species o r from different planets; i t is of t he n ature o f being strangers
t o each other. Hence its dissimilarity w ith x (t) r ates lower t han t hat w ith h (t).
D. E xercise E 3.3
Show t hat en o f signals h (t) a nd / 3(t) i n Fig. 3.4 is - 1, t hat o f h (t) a nd ! 4(t) is 0.961, a nd
t hat o f f a(t) a nd !6(t) is zero.
\1 3.2-1 Application t o Signal Detection C orrelation between two signals is a n e xtremely i mportant c oncept which measures t he degree of similarity (agreement or alignment) between t he two signals.
This concept is widely used for signal processing applications in radar, sonar, digital c ommunication, electronic warfare a nd m any o thers.
We explain t his c oncept by a n e xample of r adar where a signal pulse is t ransmitted in order t o d etect a suspected t arget. I f a t arget is p resent, t he pulse will b e 180 3 Signal R epresentation by Orthogonal Sets reflected by it. I f a t arget is n ot p resent, t here will b e no reflected pulse, j ust noise.
T he presence or absence of t he reflected pulse confirms the presence or absence of
a target. T he c rucial p roblem in this procedure is t o detect t he heavily a ttenuated,
reflected pulse ( of known waveform) buried in t he u nwanted noise signal. I n t his situation, correlation of t he received pulse with t he t ransmitted pulse can b e o f g reat
help. A similar s ituation exists in digital communication where we a re required t o
d etect t he presence of one of t he two known waveforms in t he presence of noise.
We now e xplain q ualitatively how signal detection using correlation technique
is accomplished. Consider t he case of binary communication, where two known
waveforms are received in a r andom sequence. Each time we receive a pulse, our
t ask is t o d etermine which of t he two (known) waveforms is received. To make t he
d etection easier, we m ust make t he two pulses as dissimilar as possible. Therefore,
we s hould select t he pulse t hat is t he n egative of t he o ther pulse. This choice give...
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