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Unformatted text preview: 2, . . . forms an orthogonal basis for L2 (T ). In other words, these signals are pairwise
orthogonal (as shown below), and we can represent any T -periodic CT signal
s ∈ L2 (T ) as a linear combination of these complex exponentials:
∞ ∞ ak exp ak φk (t) = s(t) =
k =−∞ k =−∞ j 2πkt
T . (1) 2
The ﬁrst “=” sign in Eq. (1) needs careful interpretation: unlike the ﬁniteduration DT case, the equality here is not pointwise. Instead, the equality is
understood in the following sense:
M s− ak φk → 0 as N → −∞ and M → ∞.
k =N Nevertheless, the coeﬃcient formula previously derived for orthogonal representations in CN , is still valid:
s, φ k
φk , φk
The inner product of φk and φi is:
τ +T φk , φi =
= j 2πit
j 2π (k − i)t
T if k = i
0 if k = i
exp dt The fact that φk , φi = 0 shows that our vectors are indeed pairwise orthogonal.1
Substituting φk , φk = T back into Eq. (2), we get:
s, φ k
T τ +T s(t) exp −
τ j 2πkt
T dt. (3) Example 1. Let T , t0 , and A be three positive real numbers such that T > t0 > 0.
Consider the following periodic signal:
s(t) = A if |t| ≤
0 if |t| ≤ t0
2 periodically extended with period T , as shown in Fig. 1. Using Eq. (3) with τ =
−T /2, its Fourier series coeﬃcients are:
T t0 /2...
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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 University-West Lafayette.
- Fall '06