Uniform Apodization Function

An Apodization Function

$$f(x)=1,$$ (1)

having Instrument Function

$$I(x) = \int_{-a}^a e^{-2\pi ikx}\,dx = -{1\over 2\pi ik}(e^{-2\pi ika}-e^{2\pi ikx})$$

$$= {\sin(2\pi ka)\over \pi k} = 2a\mathop{\rm sinc}\nolimits (2\pi ka).$$ (2)

The peak (in units of $a$) is 2. The extrema are given by letting $\beta\equiv 2\pi ka$ and solving

$${d\over d\beta} (\beta\sin\beta)={\sin\beta-\beta\cos\beta\over\beta^2}=0$$ (3)

$$\sin\beta-\beta\cos\beta=0$$ (4)

$$\tan\beta=\beta.$$ (5)

Solving this numerically gives $\beta_0=0$, $\beta_1=4.49341$, $\beta_2=7.72525$, ...for the first few solutions. The second of these is the peak Positive sidelobe, and the third is the peak Negative sidelobe. As a fraction of the peak, they are 0.128375 and $-0.217234$. The Full Width at Half Maximum is found by setting $I(x)=1$

$$\mathop{\rm sinc}\nolimits (x)={\textstyle{1\over 2}},$$ (6)

and solving for $x_{1/2}$, yielding

$$x_{1/2}=2\pi k_{1/2} a=1.89549.$$ (7)

Therefore, with $L\equiv 2a$,

$${\rm FWHM}=2k_{1/2} = {0.603353\over a} ={1.20671\over L}.$$ (8)

See also Apodization Function