一切權利歸原作者所有。
網站：https://www.cis.rit.edu/htbooks/mri/index.html
原著：Joseph P. Hornak, Ph.D.
翻譯：VictorIsJQS
Chapter 5
FOURIER TRANSFORMSFourier Pairs
傅立葉變換對
To better understand how FT NMR functions, you need to know some common Fourier pairs. A Fourier pair is two functions, the frequency domain form and the corresponding time domain form. Here are a few Fourier pairs which are useful in MRI. The amplitude of the Fourier pairs has been neglected since it is not relevant in MRI.
為了更好地解理傅立葉變換在核磁共振中的作用，需要先了解一些常用的傅立葉變換對。傅立葉變換對是一組函數——頻率域形式和其相對應的時間域形式。在此給出了一些在磁共振成像中非常有用的傅立葉變換對。由於磁共振成像與傅立葉變換對的振幅係數相關不大，因此在示例中我們暫且忽略傅立葉變換對的振幅係數。
*****************（一）*******************
Constant value at all time
時間常數函數
A DC offset or constant value.
直流偏置或常數值函數。
A delta function at zero.
頻率位於 0 的 δ 函數。
*****************（二）*******************
Real: cos(2πνt), Imaginary: -sin(2πνt)
實部：cos(2πνt)，虛部：-sin(2πνt)
A delta function at ν.
頻率位於 ν 的 δ 函數。
*****************（三）*******************
Comb Function (A series of delta functions separated by T.)
梳狀函數（一系列時間域的 δ 函數，每相鄰兩 δ 函數間隔時間 T，即周期為 T）
A comb function with separation 1/T.
傅立葉變換後的頻率域函數也是梳狀函數，其頻率周期為 1/T。
*****************（四）*******************
Exponential Decay: e-at for t > 0.
指數衰減函數：e-at for t > 0.
Lorentzian
RE: a/(a2 + 4π2ν2)
IM: -2πν/(a2 + 4π2ν2)
洛倫茲函數
實部：a/(a2 + 4π2ν2)
虛部： -2πν/(a2 + 4π2ν2)
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【譯者注】
洛倫茲函數
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*****************（五）*******************
A square pulse starting at 0 that is T seconds long.
矩形脈衝，起於 0 時刻，持續 T 秒。
Sinc
RE: (sin(2t))/(2t)
IM: -(sin2(t))/(t)
Sinc 函數
實部：(sin(2t))/(2t)
虛部：-(sin2(t))/(t)
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【譯者注】
Sinc 函數
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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).
*****************（六）*******************
Gaussian: exp(-at2)
高斯函數：exp(-at2)
Gaussian: exp(-22/a)
高斯函數：exp(-22/a)
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【譯者注】
高斯函數
註：此圖來自 Wikipedia.org
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