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

2 A To [Sin (2πνT/2)]/(2πνT/2)

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

exp(-i2πνT/2) 2AT [Sin ( 2π ν T/2)]/(2πνT/2).

Expressing the exponential in terms of sine and cosine we have

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

Multiplying through we have a real component

Cos(2πνT/2) 2AT[Sin(2πνT/2)]/(2πνT/2),

and an imaginary component

-i Sin(2πνT/2) 2AT[Sin(2πνT/2)]/(2πνT/2).

The real component becomes

2AT[Sin(2πν2T/2)]/(2πν2T/2)

and using the identity

Cos(x) Sin(x) = 0.5 Sin(2x) + 0.5 Sin(x-x)

the imaginary component becomes

-i 2AT[Sin2(2πνT/2)]/(2πνT/2).

The final form becomes

g(ν) = (sin(2πνT))/(2πνT) - (sin(πνT))/(πνT).

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