FT Sinc Detail

The Basics of MRI

FT Pairs: Sinc-Rect

The more advanced student may be wondering about origin of this Fourier pair. The Fourier transform of a rectangular (rect) pulse of amplitude A and width T_o centered at zero time (t) is a sinc function of form

2 A T_o [Sin (2pnT_o)]/(2pnT_o)

When this function is offset by T_o such that the rect now starts at t = 0 and ends at t = 2T_o the Fourier transform becomes

exp(-i2pnT_o) 2AT_o [Sin ( p n T_o)]/(2pnT_o).

Expressing the exponential in terms of sine and cosine we have

[Cos(2pnT_o) -i Sin(2pnT_o)] 2AT_o[Sin(2pnT_o)]/(2pnT_o).

Multiplying through we have a real component

Cos(2pnT_o) 2AT_o[Sin(2pnT_o)]/(2pnT_o),

and an imaginary component

-i Sin(2pnT_o) 2AT_o[Sin(2pnT_o)]/(2pnT_o).

The real component becomes

2AT_o[Sin(2pn2T_o)]/(2pn2T_o)

using the identity

Cos(x) Sin(x) = 0.5 Sin(2x).

The imaginary component becomes

-i 2AT_o[Sin²(2pnT_o)]/(2pnT_o).

Adopting the symbolism of Chapter 5 where the pulse has width T instead of 2T_o we have

RE: (sin(2pnT))/(2pnT)

and

IM: -(sin(pnT))/(pnT).

The Basics of MRI

FT Pairs: Sinc-Rect

Go to the: [chapter | cover ] Copyright © 1996-200 J.P. Hornak. All Rights Reserved.

Go to the: [chapter | cover ]
Copyright © 1996-200 J.P. Hornak.
All Rights Reserved.