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Magnetic Resonance Image and k-space

Fourier transform

出版元 Denis Hoa

The 1D Fourier transform is a mathematical procedure that allows a signal to be decomposed into its frequency components.
On the left side, the sine wave shows a time varying signal.
On the right side, you can observe its equivalent in the frequency domain. The sine wave corresponds to a plot at the same frequency as the sine wave and same amplitude as the maximal amplitude of the sine wave.

 

 

 

Single frequency

The following animation demonstrates the relation between time domain and frequency domain about a sine wave representing a single frequency sound.
Moving scroll bars change frequency and amplitude of the sound so that you can see modifications in time and frequency domain.
To describe a sine wave, we need its amplitude and frequency, but also its phase. The following animation illustrates the consequence of a phase change on the sine wave (shift in time domain)

 

 

 

 

Multiple frequencies

The Fourier transform is a mathematical procedure that decomposes a signal into a sum of sine waves of different frequencies, phases and amplitude. The human ear does the same processing with sounds, which are analyzed as a spectrum of elementary frequencies. Knowing frequency, amplitude and phase of each sine wave, it is possible to reconstruct the signal (inverse Fourier transform).
By moving the scroll bars, you can manipulate up to 3 sine waves of different frequencies, amplitudes and phases.
On the left side, you can see the graph of their sum.

 

 

 

 

Sounds

Even if a signal is very complex, the Fourier transform will always be able to decompose it into its frequency components from which we will reconstruct a signal very similar to the original. (Representing a signal exactly would require an infinite number of frequency components, which is not possible in practice)

Even music can be decomposed by a Fourier Transform. This is employed in spectrum analyser (left side).
Note that low frequency components have the highest amplitude.