Adding documentation for bilinear function (#527)

* Adding documentation for bilinear function

This commit adds documentation for the bilinear function in the
`Filters` module. docstrings are added in source code for the two
methods of bilinear function with the top level method explained
in detail to providde a clear explanation of what the transform
function is doing.

The function is also added to the export list in `Filters.jl` so that
it is available for Documenter and also for users to use.

The documentation is added to the `docs/src/filters.md` file in the
`Filters` module, Filter design section where the bilinear function
is added as a general digital filter design method.

No code has been changed in this commit, only documentation has been
added and an extra function has been added to the export list.

---------

Co-authored-by: wheeheee <[email protected]>

docs/src/filters.md

@@ -106,6 +106,13 @@ include [`remez`](@ref) which designs equiripple FIR
 filters of all types, and [`iirnotch`](@ref) which designs a
 2nd order "biquad" IIR notch filter.
 
+For a more general application of creating a digital filter from s-domain
+representation of an analog filter, one can use [`bilinear`](@ref):
+
+```@docs
+bilinear
+```
+
 ### [Filter response types](@id response-types)
 
 ```@docs

src/Filters/Filters.jl

@@ -40,6 +40,7 @@ export FilterType,
         Bandstop,
         analogfilter,
         digitalfilter,
+        bilinear,
         iirnotch,
         kaiserord,
         FIRWindow,

src/Filters/design.jl

@@ -412,7 +412,44 @@ analogfilter(ftype::FilterType, proto::FilterCoefficients) =
     transform_prototype(ftype, proto)
 
 # Bilinear transform
+"""
+    bilinear(f::FilterCoefficients{:s}, fs::Real)
+
+Calculate the digital filter (z-domain) ZPK representation of an analog filter defined
+in s-domain using bilinear transform with sampling frequency `fs`. The s-domain
+representation is first converted to a ZPK representation in s-domain and then
+transformed to z-domain using bilinear transform.
+"""
 bilinear(f::FilterCoefficients{:s}, fs::Real) = bilinear(convert(ZeroPoleGain, f), fs)
+
+"""
+    bilinear(f::ZeroPoleGain{:s,Z,P,K}, fs::Real) where {Z,P,K}
+
+Calculate the digital filter (z-domain) ZPK representation of an analog filter defined
+as a ZPK representation in s-domain using bilinear transform with sampling frequency
+`fs`.
+
+Input s-domain representation must be a `ZeroPoleGain{:s, Z, P, K}` object:
+```math
+H(s) = f.k\\frac{(s - \\verb!f.z[1]!) \\ldots (s - \\verb!f.z[m]!)}{(s - \\verb!f.p[1]!) \\ldots (s - \\verb!f.p[n]!)}
+```
+Output z-domain representation is a `ZeroPoleGain{:z, Z, P, K}` object:
+```math
+H(z) = K\\frac{(z - \\verb!Z[1]!) \\ldots (z - \\verb!Z[m]!)}{(z - \\verb!P[1]!) \\ldots (z - \\verb!P[n]!)}
+```
+where `Z, P, K` are calculated as:
+```math
+Z[i] = \\frac{(2 + \\verb!f.z[i]!/\\verb!fs!)}{(2 - \\verb!f.z[i]!/\\verb!fs!)} \\quad \\text{for } i = 1, \\ldots, m
+```
+```math
+P[i] = \\frac{(2 + \\verb!f.p[i]!/\\verb!fs!)}{(2 - \\verb!f.p[i]!/\\verb!fs!)} \\quad \\text{for } i = 1, \\ldots, n
+```
+```math
+K = f.k \\ \\mathcal{Re} \\left[ \\frac{\\prod_{i=1}^m (2*fs - f.z[i])}{\\prod_{i=1}^n (2*fs - f.p[i])} \\right]
+```
+Here, `m` and `n` are respectively the numbers of zeros and poles in the s-domain representation. If `m < n`,
+then additional `n-m` zeros are added at `z = -1`.
+"""
 function bilinear(f::ZeroPoleGain{:s,Z,P,K}, fs::Real) where {Z,P,K}
     ztype = typeof(0 + zero(Z)/fs)
     z = fill(convert(ztype, -1), max(length(f.p), length(f.z)))