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Simple adaptive AR filters. We export two functions:

yh = adaptive_filter(y, alg=MSPI; order=4, lr=0.1)

This filters y with an adaptive AR (only poles) filter with specified order and returns yh which is the predicted output from an adaptive line enhancer (ALE). If your noise is wideband and signal narrowband, yh is your desired filtered signal. If the noise is narrowband and the signal is wideband, then y-yh is your desired filtered signal.


  • alg: Stochastic approximation algorithm or weight function. Examples: OMAP, MSPI, OMAS, ADAM, ExponentialWeight, EqualWeight. ExponentialWeight corresponds to the recursive least-squares algorithm (RLS). ADAM corresponds roughly to the normalized least-mean squares (NLMS) algorithm. More options exist if OnlineStats is loaded.
  • y: Input signal
  • order: Filter order
  • lr: Learning rate or weight depending on alg

The function

focused_adaptive_filter(y, band, fs, args...; kwargs...)

allows you to specify a frequency band (tuple) in which to focus the attention of the adaptive filter. fs here denotes the sample rate, e.g., 44100Hz.


using Pkg; Pkg.add("AdaptiveFilters")

Demo app

using AdaptiveFilters, Plots, Interact
inspectdr() # Preferred plotting backend for waveforms

y = [sin.(1:100) .+ 0.1.*randn(100);
         sin.(0.2 .*(1:100)) .+ 0.1.*randn(100)]

function app(req=nothing)
    @manipulate for order = 2:2:10,
                    lr = LinRange(0.01, 0.99, 100),
                    alg = [ExponentialWeight, MSPI, OMAP, OMAS, ADAM]
        yh = adaptive_filter(y, alg, order=order, lr=lr)
        e = y.-yh
        plot([y yh], lab=["Measured signal" "Prediction"], layout=(2,1), show=false, sp=1)
        plot!(e, lab="Error", sp=2, title="RMS: $(mean(abs2, e))")


# Save filtered sound to disk
using WAV
yh = adaptive_filter(y, 4, 0.25, OMAP)
e = y.-yh
wavwrite(e, "filtered.wav", Fs=fs)



A normalized least-mean squares (NLMS) filter can be created like

using AdaptiveFilters, Random
N = 60   # Number of filter taps 
μ = 0.01 # Learning rate
f = NLMS(N, μ)

This filter can then be called like

ŷ, e = f(x, d)

where x is the input signal, d is the desired signal and is the filtered signal. The error e is also returned. This call modifies the internal state of f.

Adaptive line enhancer

The NLMS filter can be used to build an adaptive line enhancer (ALE) by letting the input signal be the desired signal delayed by a number of samples Δ:

y = sin.(0:0.1:100)
yn = y + 0.1*randn(length(y)) # A sinusoid with noise

T = length(y)
YH = zeros(T)
E = zeros(T)

Δ = 1 # Delay in samples

for i = eachindex(y)
    YH[i], E[i] = f(yn[max(i-Δ, 1)], yn[i])

using Plots, Test
@test mean(abs2, y[end-100:end] - YH[end-100:end]) < 1e-3
plot([y yn YH E y-YH], lab=["y" "yn" "yh" "e" "y-yh"])


This is a lightweight wrapper around functionality in OnlineStats.jl which does all the heavy lifting.

Usage from python

  1. First install Julia and install this package in Julia.
  2. Install pyjulia using their instructions.
  3. Now the following should work
$ python3
>>> import julia
>>> from julia import AdaptiveFilters as af
>>> yh = af.adaptive_filter(y)

if that fails, try replacing the first line with

>>> from julia.api import Julia
>>> jl = Julia(compiled_modules=False)

Keyword args etc. work as normal

af.adaptive_filter(y, af.ADAM, order=2)

Example: Adaptive cicada filtering

The following function does a reasonable job at filtering out the sound of cicadas from an audio recording

cicada_filter(y,fs,args...; kwargs...) = y-focused_adaptive_filter(data,(4200,11000),fs,args...; kwargs...)