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Continuous_Fourier_series

Continuous_Fourier_series - 1 Continuous-Time Fourier...

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1 Continuous-Time Fourier Series Notes for ECE 301 Signals and Systems Section 1, Fall 2011 Ilya Pollak Purdue University The concepts of orthogonal bases and projections can be extended to spaces of CT signals. Determining whether a series representation converges (and if so, what it converges to) is much more complicated than for Fnite-duration or periodic DT signals. We therefore will only consider one very important example–CT ±ourier series. Consider the set of signals L 2 ( T ) deFned as follows: it is the set of all periodic complex-valued CT signals s ( t ) with period T for which i τ + T τ | s ( t ) | 2 dt < , where τ is an arbitrary real number–i.e., the integral is taken over any period. It turns out that L 2 ( T ) is a vector space (each vector in this case being a continuous- time signal). We deFne the inner product of two signals as follows: a s, g A = i τ + T τ s ( t )( g ( t )) * dt. As in C N , s and g being orthogonal still means a s, g A = 0. As in C N , the norm of a signal s is deFned as the square root of its inner product with itself: b s b ≡ r a s, s A = R i τ + T τ | s ( t ) | 2 dt. Note that the deFnition of L 2 ( T ) guarantees that this quantity is Fnite for any signal in L 2 ( T ). It turns out that the inFnite collection of T -periodic complex exponentials φ k ( t ) = exp p j 2 πkt T P , k = 0 , ± 1 , ± 2 , . . . forms an orthogonal basis for L 2 ( T ). In other words, these signals are pairwise orthogonal (as shown below), and we can represent any T -periodic CT signal s L 2 ( T ) as a linear combination of these complex exponentials: s ( t ) = s k = -∞ a k φ k ( t ) = s k = -∞ a k exp p j 2 T P . (1)
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2 The frst “=” sign in Eq. (1) needs careFul interpretation: unlike the fnite- duration DT case, the equality here is not pointwise. Instead, the equality is understood in the Following sense: v v v v v s M s k = N a k φ k v v v v v 0 as N → −∞ and M → ∞ . Nevertheless, the coe±cient Formula previously derived For orthogonal represen- tations in C N , is still valid: a k = a s, φ k A a φ k , φ k A . (2) The inner product oF φ k and φ i is: a φ k , φ i A = i τ + T τ exp p j 2 πkt T P exp p j
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