EE200_Weber_11-11

EE200_Weber_11-11 - EE 200 Fourier Transforms In previous...

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1 EE 200 Fourier Transforms In previous chapters we saw how a signal that is periodic in the time-domain could be represented as a weighted sum of complex exponentials (the Fourier series expansion.) The Fourier series expansion is a special case of a set of ways to represent signals called the Fourier transforms. x ( t ) = X k e ik " 0 t k = #$ $ % x ( n ) = X k e ik 0 n k = 0 p # 1 $ X k = 1 p x ( m ) e " imk # 0 m = 0 p " 1 $ X m = 1 p x ( t ) e " im 0 t dt 0 p $
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2 EE 200 Fourier Transforms The continuous-time Fourier series representation of a periodic signal x(t) is given by where ω 0 =2 π /p and p is the period of the signal. The expansion coefficients are given by " t # Reals, x ( t ) = X k e ik $ 0 t k = %& & " m # Integers, X m = 1 p x ( t ) e $ im % 0 t dt 0 p &
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3 EE 200 Fourier Transforms If we find the continuous-time Fourier series representation of a signal x , the resulting the sequence of Fourier series coefficients, X m ,can be viewed as a discrete-time signal, X . The continuous-time Fourier series is a system that has a continuous periodic signal for input and a discrete signal for output.
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EE200_Weber_11-11 - EE 200 Fourier Transforms In previous...

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