I try to match, but it could not match $I_o-\sinh$ Practical Fourier Transform pair developed by Ben Logan, transform pair also published in The Practical Application of the Fourier Integral Campbell, George A
.Sir J.F. Kaiser describe orally in interview at section What is the spectral window?
the function Interview of J.F.Kaier about this function
the Fourier Transform pair
Where 
This Fourier transform pair widely used in Signal processing, Spectrum analysis and other Engineering and Science fields. please suggest me how Bessel function convert into $\sinh$ function by spherically approximation and integral Fourier transform by Ben Logan. Please describe in between steps from $I_o$ to convert into $\sinh$