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Unformatted text preview: A dt =
−t0 /2 At0
,
T Note, however, that this is not enough to prove that they form an orthogonal basis for
L2 (T ) since L2 (T ) is inﬁnitedimensional. Unlike in ﬁnitedimensional vector spaces, it is not
true that any inﬁnite set of nonzero orthogonal vectors forms an orthogonal basis for L2 (T ).
Proving the fact that signals φk do form an orthogonal basis for L2 (T ) is beyond the scope of
this course. 3
s(t)
A
ttt ttt
0
− t2 −T
2 −T t0
2 T
2 T t Figure 1. Signal s(t) of Example 1. ak =
= 1
T t0 /2 A exp −
−t0 /2 j 2πkt
T dt A
T
j 2πkt
·
exp −
T −j 2πk
T j πkt0
A1
·
· exp
πk 2j
T
A
π kt0
=
.
sin
πk
T
= t0 /2
−t0 /2 − exp − jπkt0
T (Note that this last formula is also valid for k = 0 if we deﬁne sin θ
θ = 1.)
θ =0 Another common way of decomposing CT periodic signals as linear combinations of sinusoidal signals is by using sines and cosines as basis functions, instead
of complex exponentials. The following inﬁnite collection of functions is also an
orthogonal basis for L2 (T ), and is also called a Fourier basis:
c0 (t) = 1,
ck (t) = cos
sk (t) = sin 2πkt
, k = 1, 2 , . . .
T
2πkt
, k = 1, 2 , . . .
T As we did previously, let us ﬁrst prove that these functions are pairwise orthogonal, and ﬁnd their energies over one period. We need to consider all pairwise inner
products–which will now be integrals of products of trigonometric functions. We 4
will therefore need the following formulas:
1
(cos(α − β ) − cos(α + β ))
2
1
sin α cos β =
(sin...
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This note was uploaded on 05/28/2012 for the course ECE 301 taught by Professor V."ragu"balakrishnan during the Fall '06 term at Purdue UniversityWest Lafayette.
 Fall '06
 V."Ragu"Balakrishnan

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