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 To centered at zero time (t) is a sinc function of form

2 A To [Sin (2πνTo)]/(2πνTo)

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

exp(-i2πνTo) 2ATo [Sin ( π ν To)]/(2πνTo).

Expressing the exponential in terms of sine and cosine we have

[Cos(2πνTo) -i Sin(2πνTo)] 2ATo[Sin(2πνTo)]/(2πνTo).

Multiplying through we have a real component

Cos(2πνTo) 2ATo[Sin(2πνTo)]/(2πνTo),

and an imaginary component

-i Sin(2πνTo) 2ATo[Sin(2πνTo)]/(2πνTo).

The real component becomes

2ATo[Sin(2πν2To)]/(2πν2To)

using the identity

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

The imaginary component becomes

-i 2ATo[Sin2(2πνTo)]/(2πνTo).

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

RE: (sin(2πνT))/(2πνT)

and

IM: -(sin(πνT))/(πνT).


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