Digital Signal Processing/Continuous-Time Fourier Transform

The Continuous-Time Fourier Transform (CTFT) is the version of the fourier transform that is most common, and is the only fourier transform so far discussed in EE wikibooks such as Signals and Systems, or Communication Systems. This transform is mentioned here as a stepping stone for further discussions of the Discrete-Time Fourier Transform (DTFT), and the Discrete Fourier Transform (DFT). The CTFT itself is not useful in digital signal processing applications, so it will not appear much in the rest of this book

CTFT Definition

The CTFT is defined as such:

[CTFT]

${\mathcal {F}}(\omega )=\int f(t)e^{-j\omega t}dt$

Convolution Theorem

Multiplication in the time domain becomes convolution in the frequency domain. Convolution in the time domain becomes multiplication in the frequency domain. This is an example of the property of duality. The convolution theorem is a very important theorem, and every transform has a version of it, so it is a good idea to become familiar with it now (if you aren't previously familiar with it).

Further reading

For more information about the Fourier Transform, see Signals and Systems.