Magnetic Resonance Imaging

Fourier Pair

Two functions, the frequency domain form and the corresponding time domain form.

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A DC offset or constant value.

 

 
A delta function at zero.

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Real: cos(2p n t),
Imaginary: -sin(2p n t)

 

 
A delta function at n.

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Comb Function (A series of delta functions separated by T.)

 

 
A comb function with separation 1/T.

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e^-at for t > 0

 

 
Lorentzian
RE: a²/(a² + 4p²n²)
IM: 2a²pn/(a² + 4p²n²)²

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Rectangular pulse of width T starting at 0.

 

 
Sinc
RE: (sin(2p n t))/(2p n t)
IM: -(sin²(p n t))/(p n t)

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Gaussian: exp(-at²)

 

 
Gaussian: exp(-p²n²/a)

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Convolution Theorem

The FT of a convolution of two functions is proportional to the products of the individual Fourier transforms, and vice versa.

If f(w) = FT( f(t) ) and h(w) = FT( h(t) )

then f(w ) g(w) = FT( g(t) f(t) ) and f(w) g(w) = FT( g(t) f(t) )

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With pictures....

FT of a pulsed sine wave

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noise reduction using the convolution theorem

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The Digital FT

Original continuous FID.

 

Sampled FID seen by FT algorithm in computer.

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What does this yield?

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Sampling Rate (f_s)

Number of complex data points recorded per second.

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Over Sampling (f_s > spectral width)

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Under Sampling (f_sampling < spectral width)

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Wrap around problem or artifact

The appearance of one side of the imaged object on the opposite side. In terms of a one dimensional frequency domain spectrum, wrap around is the occurrence of a low frequency peak on the wrong side of the spectrum.

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2-Dimensional FT

An FT performed on a two-dimensional array of data.

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Copyright © 2000 J.P. Hornak.
All Rights Reserved.