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# Continuous Fourier Transform

• Definition and notation

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

The spectrum is expressed as a function of because the spectrum can be treated as the Laplace transform of the signal evaluated along the imaginary axis ():

As this notation is closely related to the system analysis concepts such as Laplace transform and transfer function , 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 in cycles/ second or Hertz (Hz, KHz, MHz, GHz, etc.), instead of in radians/second. Replacing by , we can also express the spectrum as or simply in this alternative representation:

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.

• Physical Meaning of Fourier Transform

The time signal squared represents how the energy contained in the signal distributes over time , while its spectrum squared represents how the energy distributes over frequency (therefore the term power density spectrum). Obviously, the same amount of energy is contained in either time or frequency domain, as indicated by Parseval's formula:

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 Signals

Fourier transform is a complex transform. But all physical signals (temperature, voltage, pressure, etc.) are real

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

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

Next: Signal Discretization Up: Image_Processing Previous: Image_Processing
Ruye Wang 2007-11-15