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design.jl
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design.jl
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# Filter prototypes, transformations, and transforms
using ..Windows
abstract type FilterType end
#
# Butterworth prototype
#
function Butterworth(::Type{T}, n::Integer) where {T<:Real}
n > 0 || error("n must be positive")
poles = zeros(Complex{T}, n)
for i = 1:div(n, 2)
w = convert(T, 2i-1)/2n
pole = complex(-sinpi(w), cospi(w))
poles[2i-1] = pole
poles[2i] = conj(pole)
end
if isodd(n)
poles[end] = -1
end
ZeroPoleGain{:s}(Complex{T}[], poles, one(T))
end
"""
Butterworth(n)
``n`` pole Butterworth filter.
"""
Butterworth(n::Integer) = Butterworth(Float64, n)
#
# Chebyshev type I and II prototypes
#
function chebyshev_poles(::Type{T}, n::Integer, ε::Real) where {T<:Real}
p = zeros(Complex{T}, n)
μ = asinh(convert(T, 1)/ε)/n
b = -sinh(μ)
c = cosh(μ)
for i = 1:div(n, 2)
w = convert(T, 2i-1)/2n
pole = complex(b*sinpi(w), c*cospi(w))
p[2i-1] = pole
p[2i] = conj(pole)
end
if isodd(n)
w = convert(T, 2*div(n, 2)+1)/2n
pole = b*sinpi(w)
p[end] = pole
end
p
end
function Chebyshev1(::Type{T}, n::Integer, ripple::Real) where {T<:Real}
n > 0 || error("n must be positive")
ripple >= 0 || error("ripple must be non-negative")
ε = sqrt(10^(convert(T, ripple)/10)-1)
p = chebyshev_poles(T, n, ε)
k = one(T)
for i = 1:div(n, 2)
k *= abs2(p[2i])
end
if iseven(n)
k /= sqrt(1+abs2(ε))
else
k *= real(-p[end])
end
ZeroPoleGain{:s}(Complex{T}[], p, k)
end
"""
Chebyshev1(n, ripple)
`n` pole Chebyshev type I filter with `ripple` dB ripple in
the passband.
"""
Chebyshev1(n::Integer, ripple::Real) = Chebyshev1(Float64, n, ripple)
function Chebyshev2(::Type{T}, n::Integer, ripple::Real) where {T<:Real}
n > 0 || error("n must be positive")
ripple >= 0 || error("ripple must be non-negative")
ε = 1/sqrt(10^(convert(T, ripple)/10)-1)
p = chebyshev_poles(T, n, ε)
for i = 1:length(p)
p[i] = inv(p[i])
end
z = zeros(Complex{T}, n-isodd(n))
k = one(T)
for i = 1:div(n, 2)
w = convert(T, 2i-1)/2n
ze = complex(zero(T), -inv(cospi(w)))
z[2i-1] = ze
z[2i] = conj(ze)
k *= abs2(p[2i])/abs2(ze)
end
isodd(n) && (k *= -real(p[end]))
ZeroPoleGain{:s}(z, p, k)
end
"""
Chebyshev2(n, ripple)
`n` pole Chebyshev type II filter with `ripple` dB ripple in
the stopband.
"""
Chebyshev2(n::Integer, ripple::Real) = Chebyshev2(Float64, n, ripple)
#
# Elliptic prototype
#
# See Orfanidis, S. J. (2007). Lecture notes on elliptic filter design.
# Retrieved from http://www.ece.rutgers.edu/~orfanidi/ece521/notes.pdf
# Compute Landen sequence for evaluation of elliptic functions
function landen(k::Real)
niter = 7
kn = Vector{typeof(k)}(undef, niter)
# Eq. (50)
for i = 1:niter
kn[i] = k = abs2(k/(1+sqrt(1-abs2(k))))
end
kn
end
# cde computes cd(u*K(k), k)
# sne computes sn(u*K(k), k)
# Both accept the Landen sequence as generated by landen above
for (fn, init) in ((:cde, :(cospi(u/2))), (:sne, :(sinpi(u/2))))
@eval begin
function $fn(u::Number, landen::Vector{T}) where T<:Real
winv = inv($init)
# Eq. (55)
for i = length(landen):-1:1
oldwinv = winv
winv = 1/(1+landen[i])*(winv+landen[i]/winv)
end
w = inv(winv)
end
end
end
# sne inverse
function asne(w::Number, k::Real)
oldw = NaN
while w != oldw
oldw = w
kold = k
# Eq. (50)
k = abs2(k/(1+sqrt(1-abs2(k))))
# Eq. (56)
w = 2*w/((1+k)*(1+sqrt(1-abs2(kold)*w^2)))
end
2*asin(w)/π
end
function Elliptic(::Type{T}, n::Integer, rp::Real, rs::Real) where {T<:Real}
n > 0 || error("n must be positive")
rp > 0 || error("rp must be positive")
rp < rs || error("rp must be less than rs")
# Eq. (2)
εp = sqrt(10^(convert(T, rp)/10)-1)
εs = sqrt(10^(convert(T, rs)/10)-1)
# Eq. (3)
k1 = εp/εs
k1 >= 1 && error("filter order is too high for parameters")
# Eq. (20)
k1′² = 1 - abs2(k1)
k1′ = sqrt(k1′²)
k1′_landen = landen(k1′)
# Eq. (47)
k′ = one(T)
for i = 1:div(n, 2)
k′ *= sne(convert(T, 2i-1)/n, k1′_landen)
end
k′ = k1′²^(convert(T, n)/2)*k′^4
k = sqrt(1 - abs2(k′))
k_landen = landen(k)
# Eq. (65)
v0 = -im/convert(T, n)*asne(im/εp, k1)
z = Vector{Complex{T}}(undef, 2*div(n, 2))
p = Vector{Complex{T}}(undef, n)
gain = one(T)
for i = 1:div(n, 2)
# Eq. (43)
w = convert(T, 2i-1)/n
# Eq. (62)
ze = complex(zero(T), -inv(k*cde(w, k_landen)))
z[2i-1] = ze
z[2i] = conj(ze)
# Eq. (64)
pole = im*cde(w - im*v0, k_landen)
p[2i] = pole
p[2i-1] = conj(pole)
gain *= abs2(pole)/abs2(ze)
end
if isodd(n)
pole = im*sne(im*v0, k_landen)
p[end] = pole
gain *= abs(pole)
else
gain *= 10^(-convert(T, rp)/20)
end
ZeroPoleGain{:s}(z, p, gain)
end
"""
Elliptic(n, rp, rs)
`n` pole elliptic (Cauer) filter with `rp` dB ripple in the
passband and `rs` dB attentuation in the stopband.
"""
Elliptic(n::Integer, rp::Real, rs::Real) = Elliptic(Float64, n, rp, rs)
#
# Prototype transformation types
#
# returns frequency in half-cycles per sample ∈ (0, 1)
function normalize_freq(w::Real, fs::Real)
w <= 0 && error("frequencies must be positive")
f = 2*w/fs
f >= 1 && error("frequencies must be less than the Nyquist frequency $(fs/2)")
f
end
struct Lowpass{T} <: FilterType
w::T
end
"""
Lowpass(Wn[; fs])
Low pass filter with cutoff frequency `Wn`. If `fs` is not
specified, `Wn` is interpreted as a normalized frequency in
half-cycles/sample.
"""
Lowpass(w::Real; fs::Real=2) = Lowpass{typeof(w/1)}(normalize_freq(w, fs))
struct Highpass{T} <: FilterType
w::T
end
"""
Highpass(Wn[; fs])
High pass filter with cutoff frequency `Wn`. If `fs` is not
specified, `Wn` is interpreted as a normalized frequency in
half-cycles/sample.
"""
Highpass(w::Real; fs::Real=2) = Highpass{typeof(w/1)}(normalize_freq(w, fs))
struct Bandpass{T} <: FilterType
w1::T
w2::T
end
"""
Bandpass(Wn1, Wn2[; fs])
Band pass filter with normalized pass band (`Wn1`, `Wn2`). If
`fs` is not specified, `Wn1` and `Wn2` are interpreted as
normalized frequencies in half-cycles/sample.
"""
function Bandpass(w1::Real, w2::Real; fs::Real=2)
w1 < w2 || error("w1 must be less than w2")
Bandpass{Base.promote_typeof(w1/1, w2/1)}(normalize_freq(w1, fs), normalize_freq(w2, fs))
end
struct Bandstop{T} <: FilterType
w1::T
w2::T
end
"""
Bandstop(Wn1, Wn2[; fs])
Band stop filter with normalized stop band (`Wn1`, `Wn2`). If
`fs` is not specified, `Wn1` and `Wn2` are interpreted as
normalized frequencies in half-cycles/sample.
"""
function Bandstop(w1::Real, w2::Real; fs::Real=2)
w1 < w2 || error("w1 must be less than w2")
Bandstop{Base.promote_typeof(w1/1, w2/1)}(normalize_freq(w1, fs), normalize_freq(w2, fs))
end
#
# Prototype transformation implementations
#
# The formulas implemented here come from the documentation for the
# corresponding functionality in Octave, available at
# https://staff.ti.bfh.ch/sha1/Octave/index/f/sftrans.html
# The Octave implementation was not consulted in creating this code.
# Create a lowpass filter from a lowpass filter prototype
transform_prototype(ftype::Lowpass, proto::ZeroPoleGain{:s, Z, P, K}) where {Z, P, K} =
ZeroPoleGain{:s, Z, P, K}(ftype.w * proto.z, ftype.w * proto.p,
proto.k * ftype.w^(length(proto.p)-length(proto.z)))
# Create a highpass filter from a lowpass filter prototype
function transform_prototype(ftype::Highpass, proto::ZeroPoleGain{:s})
z = proto.z
p = proto.p
k = proto.k
nz = length(z)
np = length(p)
TR = Base.promote_eltype(z, p)
newz = zeros(TR, max(nz, np))
newp = zeros(TR, max(nz, np))
num = one(eltype(z))
for i = 1:nz
num *= -z[i]
newz[i] = ftype.w / z[i]
end
den = one(eltype(p))
for i = 1:np
den *= -p[i]
newp[i] = ftype.w / p[i]
end
abs(real(num) - 1) < np*eps(real(num)) && (num = 1)
abs(real(den) - 1) < np*eps(real(den)) && (den = 1)
ZeroPoleGain{:s}(newz, newp, oftype(k, k * real(num)/real(den)))
end
# Create a bandpass filter from a lowpass filter prototype
function transform_prototype(ftype::Bandpass, proto::ZeroPoleGain{:s})
z = proto.z
p = proto.p
k = proto.k
nz = length(z)
np = length(p)
ncommon = min(nz, np)
TR = Base.promote_eltype(z, p)
newz = zeros(TR, 2*nz+np-ncommon)
newp = zeros(TR, 2*np+nz-ncommon)
for (oldc, newc) in ((p, newp), (z, newz))
for i = 1:length(oldc)
b = oldc[i] * ((ftype.w2 - ftype.w1)/2)
pm = sqrt(b^2 - ftype.w2 * ftype.w1)
newc[2i-1] = b + pm
newc[2i] = b - pm
end
end
ZeroPoleGain{:s}(newz, newp, oftype(k, k * (ftype.w2 - ftype.w1) ^ (np - nz)))
end
# Create a bandstop filter from a lowpass filter prototype
function transform_prototype(ftype::Bandstop, proto::ZeroPoleGain{:s})
z = proto.z
p = proto.p
k = proto.k
nz = length(z)
np = length(p)
npairs = nz+np-min(nz, np)
TR = Base.promote_eltype(z, p)
newz = Vector{TR}(undef, 2*npairs)
newp = Vector{TR}(undef, 2*npairs)
num = one(eltype(z))
for i = 1:nz
num *= -z[i]
b = (ftype.w2 - ftype.w1)/2/z[i]
pm = sqrt(b^2 - ftype.w2 * ftype.w1)
newz[2i-1] = b - pm
newz[2i] = b + pm
end
den = one(eltype(p))
for i = 1:np
den *= -p[i]
b = (ftype.w2 - ftype.w1)/2/p[i]
pm = sqrt(b^2 - ftype.w2 * ftype.w1)
newp[2i-1] = b - pm
newp[2i] = b + pm
end
# Any emaining poles/zeros are real and not cancelled
npm = sqrt(-complex(ftype.w2 * ftype.w1))
for (n, newc) in ((np, newp), (nz, newz))
for i = n+1:npairs
newc[2i-1] = -npm
newc[2i] = npm
end
end
abs(real(num) - 1) < np*eps(real(num)) && (num = 1)
abs(real(den) - 1) < np*eps(real(den)) && (den = 1)
ZeroPoleGain{:s}(newz, newp, oftype(k, k * real(num)/real(den)))
end
transform_prototype(ftype, proto::FilterCoefficients{:s}) =
transform_prototype(ftype, convert(ZeroPoleGain, proto))
"""
analogfilter(responsetype, designmethod)
Construct an analog filter. See below for possible response and
filter types.
"""
analogfilter(ftype::FilterType, proto::FilterCoefficients) =
transform_prototype(ftype, proto)
# Bilinear transform
bilinear(f::FilterCoefficients{:s}, fs::Real) = bilinear(convert(ZeroPoleGain, f), fs)
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)))
ptype = typeof(0 + zero(P)/fs)
p = Vector{typeof(zero(P)/fs)}(undef, length(f.p))
num = one(one(fs) - one(Z))
for i = 1:length(f.z)
z[i] = (2 + f.z[i] / fs)/(2 - f.z[i] / fs)
num *= (2 * fs - f.z[i])
end
den = one(one(fs) - one(P))
for i = 1:length(f.p)
p[i] = (2 + f.p[i] / fs)/(2 - f.p[i]/fs)
den *= (2 * fs - f.p[i])
end
ZeroPoleGain{:z}(z, p, f.k * real(num)/real(den))
end
# Pre-warp filter frequencies for digital filtering
prewarp(ftype::Union{Lowpass, Highpass}) = (typeof(ftype))(prewarp(ftype.w))
prewarp(ftype::Union{Bandpass, Bandstop}) = (typeof(ftype))(prewarp(ftype.w1), prewarp(ftype.w2))
# freq in half-samples per cycle
prewarp(f::Real) = 4*tan(pi*f/2)
# Digital filter design
"""
digitalfilter(responsetype, designmethod)
Construct a digital filter. See below for possible response and
filter types.
"""
digitalfilter(ftype::FilterType, proto::FilterCoefficients) =
bilinear(transform_prototype(prewarp(ftype), proto), 2)
#
# Special filter types
#
"""
iirnotch(Wn, bandwidth[; fs])
Second-order digital IIR notch filter [^Orfandis] at frequency `Wn` with
bandwidth `bandwidth`. If `fs` is not specified, `Wn` is
interpreted as a normalized frequency in half-cycles/sample.
[^Orfandis]: Orfanidis, S. J. (1996). Introduction to signal processing. Englewood Cliffs, N.J: Prentice Hall, p. 370.
"""
function iirnotch(w::Real, bandwidth::Real; fs=2)
w = normalize_freq(w, fs)
bandwidth = normalize_freq(bandwidth, fs)
# Eq. 8.2.23
b = 1/(1+tan(pi*bandwidth/2))
# Eq. 8.2.22
cosw0 = cospi(w)
Biquad(b, -2b*cosw0, b, -2b*cosw0, 2b-1)
end
#
# FIR filter design
#
# Get length and alpha for Kaiser window filter with specified
# transition band width and stopband attenuation in dB
function kaiserord(transitionwidth::Real, attenuation::Real=60)
n = ceil(Int, (attenuation - 7.95)/(π*2.285*transitionwidth))+1
if attenuation > 50
β = 0.1102*(attenuation - 8.7)
elseif attenuation >= 21
β = 0.5842*(attenuation - 21)^0.4 + 0.07886*(attenuation - 21)
else
β = 0.0
end
return n, β/π
end
struct FIRWindow{T}
window::Vector{T}
scale::Bool
end
"""
FIRWindow(window; scale=true)
FIR filter design using window `window`, a vector whose length
matches the number of taps in the resulting filter.
If `scale` is `true` (default), the designed FIR filter is
scaled so that the following holds:
- For [`Lowpass`](@ref) and [`Bandstop`](@ref) filters, the frequency
response is unity at 0 (DC).
- For [`Highpass`](@ref) filters, the frequency response is unity
at the Nyquist frequency.
- For [`Bandpass`](@ref) filters, the frequency response is unity
in the center of the passband.
"""
FIRWindow(window::Vector; scale::Bool=true) = FIRWindow(window, scale)
# FIRWindow(n::Integer, window::Function, args...) = FIRWindow(window(n, args...))
"""
FIRWindow(; transitionwidth, attenuation=60, scale=true)
Kaiser window FIR filter design. The required number of taps is
calculated based on `transitionwidth` (in half-cycles/sample)
and stopband `attenuation` (in dB). `attenuation` defaults to
60 dB.
"""
FIRWindow(; transitionwidth::Real=throw(ArgumentError("must specify transitionwidth")),
attenuation::Real=60, scale::Bool=true) =
FIRWindow(kaiser(kaiserord(transitionwidth, attenuation)...), scale)
# Compute coefficients for FIR prototype with specified order
function firprototype(n::Integer, ftype::Lowpass)
w = ftype.w
[w*sinc(w*(k-(n-1)/2)) for k = 0:(n-1)]
end
function firprototype(n::Integer, ftype::Bandpass)
w1 = ftype.w1
w2 = ftype.w2
[w2*sinc(w2*(k-(n-1)/2)) - w1*sinc(w1*(k-(n-1)/2)) for k = 0:(n-1)]
end
function firprototype(n::Integer, ftype::Highpass)
w = ftype.w
isodd(n) || throw(ArgumentError("FIRWindow highpass filters must have an odd number of coefficients"))
out = [-w*sinc(w*(k-(n-1)/2)) for k = 0:(n-1)]
out[div(n, 2)+1] += 1
out
end
function firprototype(n::Integer, ftype::Bandstop)
w1 = ftype.w1
w2 = ftype.w2
isodd(n) || throw(ArgumentError("FIRWindow bandstop filters must have an odd number of coefficients"))
out = [w1*sinc(w1*(k-(n-1)/2)) - w2*sinc(w2*(k-(n-1)/2)) for k = 0:(n-1)]
out[div(n, 2)+1] += 1
out
end
scalefactor(coefs::Vector, ::Union{Lowpass, Bandstop}) = sum(coefs)
function scalefactor(coefs::Vector, ::Highpass)
c = zero(coefs[1])
for k = 1:length(coefs)
c += ifelse(isodd(k), coefs[k], -coefs[k])
end
c
end
function scalefactor(coefs::Vector, ftype::Bandpass)
n = length(coefs)
freq = middle(ftype.w1, ftype.w2)
c = zero(coefs[1])
for k = 0:n-1
c += coefs[k+1]*cospi(freq*(k-(n-1)/2))
end
c
end
function digitalfilter(ftype::FilterType, proto::FIRWindow)
coefs = firprototype(length(proto.window), ftype)
@assert length(proto.window) == length(coefs)
out = coefs .* proto.window
proto.scale ? rmul!(out, 1/scalefactor(out, ftype)) : out
end
# Compute FIR coefficients necessary for arbitrary rate resampling
function resample_filter(rate::AbstractFloat, Nϕ = 32, rel_bw = 1.0, attenuation = 60)
f_nyq = rate >= 1.0 ? 1.0/Nϕ : rate/Nϕ
cutoff = f_nyq * rel_bw
trans_width = cutoff * 0.2
# Determine resampling filter order
hLen, α = kaiserord(trans_width, attenuation)
# Round the number of taps up to a multiple of Nϕ.
# Otherwise the missing taps will be filled with 0.
hLen = Nϕ * ceil(Int, hLen/Nϕ)
# Ensure that the filter is an odd length
if (iseven(hLen))
hLen += 1
end
# Design filter
h = digitalfilter(Lowpass(cutoff), FIRWindow(kaiser(hLen, α)))
rmul!(h, Nϕ)
end
# Compute FIR coefficients necessary for rational rate resampling
function resample_filter(rate::Union{Integer,Rational}, rel_bw = 1.0, attenuation = 60)
Nϕ = numerator(rate)
decimation = denominator(rate)
f_nyq = min(1/Nϕ, 1/decimation)
cutoff = f_nyq * rel_bw
trans_width = cutoff * 0.2
# Determine resampling filter order
hLen, α = kaiserord(trans_width, attenuation)
# Round the number of taps up to a multiple of Nϕ (same as interpolation factor).
# Otherwise the missing taps will be filled with 0.
hLen = Nϕ * ceil(Int, hLen/Nϕ)
# Ensure that the filter is an odd length
if (iseven(hLen))
hLen += 1
end
# Design filter
h = digitalfilter(Lowpass(cutoff), FIRWindow(kaiser(hLen, α)))
rmul!(h, Nϕ)
end