kaiser performance: use Bessels.jl?

[anowacki]

When resampling signals with a very low or high arbitrary rate, the call to kaiser is what dominates. Profiling this, it seems like most of the time here is taken up by the call to SpecialFunctions.besseli. This allocates but also the computation is just fundamentally slow for these arguments.

Using DSP v0.7.8:

julia> @btime resample(x, rate) setup=(x=rand(1000); rate=1/50) seconds=3;
  37.353 ms (58055 allocations: 4.02 MiB)

However, I see that Bessels.jl aims to provide Bessel functions in pure Julia which are faster than SpecialFunctions.jl's, and as accurate.

Using the following version of DSP.Windows.kaiser which uses Bessels.besseli we get an 8-fold improvement in runtime for this particular signal length and resampling rate:

julia> function DSP.kaiser(n::Integer, α::Real; padding::Integer=0, zerophase::Bool=false)
           pf = 1.0/Bessels.besseli(0,pi*α)
           DSP.Windows.makewindow(n, padding, zerophase) do x
               pf*Bessels.besseli(0, pi*α*(sqrt(1 - (2x)^2)))
           end
       end

julia> @btime resample(x, rate) setup=(x=rand(1000); rate=1/50) seconds=3;
  4.444 ms (37 allocations: 3.14 MiB)

If you are happy to add an extra dependency on Bessels.jl, and you feel it is stable enough to rely on in the future, then I would be happy to open a PR so long as this passes tests.

[ViralBShah]

@oscardssmith Is Bessels.jl stable enough and in good shape for DSP.jl to rely upon?

[oscardssmith]

IMO Bessels.jl at this point is pretty stable. Nice to see a 10x improvement. You could probably get a little more by using Bessels.besseli0 instead.

[ViralBShah]

@martinholters Would it be ok to add a dependency on Bessels.jl?

[martinholters]

Fine with me.