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
and an imaginary component
-i Sin(2πνTo) 2ATo[Sin(2πνTo)]/(2πνTo).
The real component becomes
using the identity
Cos(x) Sin(x) = 0.5 Sin(2x).
The imaginary component becomes
Adopting the symbolism of Chapter 5 where the pulse has width T instead of 2To we have
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