title: FIR filter design with Julia author: Matti Pastell date: 21th April 2016 –-

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We'll implement lowpass, highpass and ' bandpass FIR filters. If you want to read more about DSP I highly recommend The Scientist and Engineer's Guide to Digital Signal Processing which is freely available online.

DSP.jl package doesn't (yet) have a method to calculate the the frequency response of a FIR filter so we define it:

using Plots, DSP gr() function FIRfreqz(b::Array, w = range(0, stop=π, length=1024)) n = length(w) h = Array{ComplexF32}(undef, n) sw = 0 for i = 1:n for j = 1:length(b) sw += b[j]*exp(-im*w[i])^-j end h[i] = sw sw = 0 end return h end

FIRfreqz (generic function with 2 methods)

Designing a lowpass FIR filter is very simple to do with DSP.jl, all you need to do is to define the window length, cut off frequency and the window. We will define a lowpass filter with cut off frequency at 5Hz for a signal sampled at 20 Hz. We will use the Hamming window, which is defined as: $w(n) = \alpha - \beta\cos\frac{2\pi n}{N-1}$, where $\alpha=0.54$ and $\beta=0.46$

fs = 20 f = digitalfilter(Lowpass(5, fs = fs), FIRWindow(hamming(61))) w = range(0, stop=pi, length=1024) h = FIRfreqz(f, w)

1024-element Array{Complex{Float32},1}: 1.0f0 + 0.0f0im 0.99546844f0 + 0.095055714f0im 0.98191506f0 + 0.1892486f0im 0.95946306f0 + 0.28172377f0im 0.9283168f0 + 0.37164196f0im 0.8887594f0 + 0.45818728f0im 0.84115064f0 + 0.54057467f0im 0.7859234f0 + 0.618057f0im 0.72357976f0 + 0.6899319f0im 0.65468615f0 + 0.7555481f0im ⋮ 0.00043952762f0 - 0.00041908873f0im 0.0005152718f0 - 0.00040521423f0im 0.0005873293f0 - 0.00037745363f0im 0.0006531789f0 - 0.0003367371f0im 0.0007105166f0 - 0.00028444792f0im 0.0007573364f0 - 0.00022237403f0im 0.0007920005f0 - 0.00015264557f0im 0.0008132961f0 - 7.766036f-5im 0.0008204784f0 - 3.1148685f-18im

The next code chunk is executed in term mode, see the script for syntax.

julia> h_db = log10.(abs.(h)); 1024-element Array{Float32,1}: 0.0 -1.5272748f-6 -6.0314724f-6 -1.3538573f-5 -2.394518f-5 -3.7173842f-5 -5.3121257f-5 -7.165827f-5 -9.257809f-5 -0.00011577748 ⋮ -3.2165928 -3.1834154 -3.1560452 -3.1337893 -3.1161458 -3.102753 -3.0933545 -3.0877802 -3.0859327 julia> ws = w/pi*(fs/2) 0.0:0.009775171065493646:10.0

plot(ws, h_db, xlabel = "Frequency (Hz)", ylabel = "Magnitude (db)")

And again with default options

h_phase = unwrap(-atan.(imag.(h),real.(h))) plot(ws, h_phase, xlabel = "Frequency (Hz)", ylabel = "Phase (radians)")