## 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 (2πνT_{o})]/(2πνT_{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(-i2πνT_{o}) 2AT_{o} [Sin ( π ν T_{o})]/(2πνT_{o}).

Expressing the exponential in terms of sine and cosine we have

[Cos(2πνT_{o}) -i Sin(2πνT_{o})] 2AT_{o}[Sin(2πνT_{o})]/(2πνT_{o}).

Multiplying through we have a real component

Cos(2πνT_{o}) 2AT_{o}[Sin(2πνT_{o})]/(2πνT_{o}),

and an imaginary component

-i Sin(2πνT_{o}) 2AT_{o}[Sin(2πνT_{o})]/(2πνT_{o}).

The real component becomes

2AT_{o}[Sin(2πν2T_{o})]/(2πν2T_{o})

using the identity

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

The imaginary component becomes

-i 2AT_{o}[Sin^{2}(2πνT_{o})]/(2πνT_{o}).

Adopting the symbolism of Chapter 5 where the pulse has width T instead of 2T_{o} we have
RE: (sin(2πνT))/(2πνT)

and

IM: -(sin(πνT))/(πνT).

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