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# Continuous Fourier Transform

• Definitions

The Fourier transform pair in the most general form for a continuous and aperiodic time signal is (Eqs. 4.8, 4.9):

Replacing by in the above, we get the alternative representation:

Note we changed the notation in Eqs. 4.8 and 4.9 to to better illustrate the duality of Fourier transform in both time and frequency domains. In this representation of Fourier transform the forward and inverse transforms are in perfect symmetry with only a different sign for the exponent due to the duality property of the transform (Section 4.3.6). This alternative representation of Fourier transform will be used in the following discussion as it is often more convenient.

• Physical Meaning of Fourier Transform

The complex spectrum of a time signal can be written in polar form

and the inverse transform becomes:

which is a weighted linear combination (integration) of infinite sinusoids with

• Spectrum of Real Signal

Fourier transform is a complex transform. But a physical signal Fourier transform is a complex transform. But any physical signal (temperature, voltage, pressure, etc.) is always real

and its spectrum is conjugate symmetric (Eq. 4.30):

i.e., the real part of is even symmetric , and the imaginary part odd .

Next: Discrete Fourier Transform Up: Fourier_Analysis Previous: Fourier_Analysis
Ruye Wang 2003-11-17