Sunday, July 17, 2011

Continuous Fourier Transform


Continuous Fourier Transform

  • Definition and notationThe Fourier transform pair in the most general form for a continuous and aperiodic time signal $x(t)$ is (Eqs. 4.8, 4.9):

    \begin{displaymath}X(j\omega)=\int_{-\infty}^{\infty} x(t) e^{-j\omega t} dt,\;\... ...2\pi}\int_{-\infty}^{\infty} X(j\omega) e^{ j\omega t} d\omega \end{displaymath}


    The spectrum $X(j\omega)$ is expressed as a function of $j\omega$ because the spectrum can be treated as the Laplace transform of the signal $X(s)$ evaluated along the imaginary axis ($\sigma=0$):

    \begin{displaymath}X(j\omega)=X(s)\vert _{s=j\omega},\;\;\;\;\; (\mbox{in general,} \;\; s=\sigma+j\omega) \end{displaymath}


    As this notation is closely related to the system analysis concepts such as Laplace transform and transfer function $H(s)$, it is preferred in the field of system design and control. However, in practice, it is more convenient to represent the frequency of a signal by $f=\omega/2\pi$ in cycles/ second or Hertz (Hz, KHz, MHz, GHz, etc.), instead of $\omega$ in radians/second. Replacing $\omega$ by $2\pi f$, we can also express the spectrum as $X(j2\pi f)$ or simply $X(f)$ in this alternative representation:

    \begin{displaymath}X(f)=\int_{-\infty}^{\infty} x(t) e^{-j2\pi f t} dt,\;\;\;\;\;\;\; x(t)=\int_{-\infty}^{\infty} X(f) e^{ j2\pi f t} df \end{displaymath}


    Here the forward and inverse Fourier transform are in perfect symmetry with only a different sign for the exponent, therefore the duality of Fourier transform (Section 4.3.6) between time and frequency domain is better illustrated. As this notation closely relates the signal representations in both time and frequency domains, it is preferred in the field of signal processing. For convenience, the alternative representation of Fourier transform will be used in the following discussion.

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