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# Fourier Representations

Module by: Mark A. Davenport. E-mail the author

## Fourier Representations

Throughout the course we have been alluding to various Fourier representations. We first recall the appropriate transforms:

• Fourier Series (CTFS): x(t)x(t): continuous-time, finite/periodic on [-π,π][-π,π]
X[k]=12π-ππx(t)e-jktdtX[k]=12π-ππx(t)e-jktdt
(1)
x(t)=12πk=-X[k]ejktx(t)=12πk=-X[k]ejkt
(2)
• Discrete-Time Fourier Transform (DTFT): x[n]x[n]: infinite, discrete-time
X(ejω)=12πn=-x[n]e-jωnX(ejω)=12πn=-x[n]e-jωn
(3)
x[n]=12π-ππX(ejω)ejωndωx[n]=12π-ππX(ejω)ejωndω
(4)
• Discrete Fourier Transform (DFT): x[n]x[n]: finite, discrete-time
X[k]=1Nn=0N-1x[n]e-j2πNknX[k]=1Nn=0N-1x[n]e-j2πNkn
(5)
x[n]=1Nk=0N-1X[k]ej2πNknx[n]=1Nk=0N-1X[k]ej2πNkn
(6)
• Continuous-Time Fourier Transform (CTFT): x(t)x(t): infinite, continuous-time
X(Ω)=12π-x(t)e-jΩtdtX(Ω)=12π-x(t)e-jΩtdt
(7)
x(t)=12π-X(Ω)ejΩtdΩx(t)=12π-X(Ω)ejΩtdΩ
(8)

We will think of Fourier representations in two complimentary senses:

1. “Eigenbasis” representations: Each Fourier transform pair is very naturally related to an appropriate class of LTI systems. In some cases we can think of a Fourier transform as a change of basis.
2. Unitary operators: While we often use Fourier transforms to analyze certain operators, we can also think of a Fourier transform as itself being an operator.

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