2020-05-23 20:02:12 +02:00
< h1 > Introduction< / h1 >
2020-05-23 15:57:18 +02:00
< p > This an example of a julia script that can be published using < a href = "http://mpastell.github.io/Weave.jl/latest/usage/" > Weave< / a > . The script can be executed normally using Julia or published to HTML or pdf with Weave. Text is written in markdown in lines starting with " < code > #' < / code > " and code is executed and results are included in the published document.< / p >
< p > Notice that you don' t need to define chunk options, but you can using < code > #+ < / code > . just before code e.g. < code > #+ term= True, caption= ' Fancy plots.' < / code > . If you' re viewing the published version have a look at the < a href = "FIR_design_plots.jl" > source< / a > to see the markup.< / p >
<!-- this setup dependencies, but doesn't appear in the generated document -->
< h1 > FIR Filter Design< / h1 >
< p > We' ll implement lowpass, highpass and ' bandpass FIR filters. If you want to read more about DSP I highly recommend < a href = "http://www.dspguide.com/" > The Scientist and Engineer' s Guide to Digital Signal Processing< / a > which is freely available online.< / p >
< h2 > Calculating frequency response< / h2 >
< p > DSP.jl package doesn' t ( yet) have a method to calculate the the frequency response of a FIR filter so we define it:< / p >
< pre class = 'hljl' >
< span class = 'hljl-k' > using< / span > < span class = 'hljl-t' > < / span > < span class = 'hljl-n' > Plots< / span > < span class = 'hljl-p' > ,< / span > < span class = 'hljl-t' > < / span > < span class = 'hljl-n' > DSP< / span > < span class = 'hljl-t' >
< / span > < span class = 'hljl-nf' > gr< / span > < span class = 'hljl-p' > ()< / span > < span class = 'hljl-t' >
< / span > < span class = 'hljl-k' > function< / span > < span class = 'hljl-t' > < / span > < span class = 'hljl-nf' > FIRfreqz< / span > < span class = 'hljl-p' > (< / span > < span class = 'hljl-n' > b< / span > < span class = 'hljl-oB' > ::< / span > < span class = 'hljl-n' > Array< / span > < span class = 'hljl-p' > ,< / span > < span class = 'hljl-t' > < / span > < span class = 'hljl-n' > w< / span > < span class = 'hljl-t' > < / span > < span class = 'hljl-oB' > =< / span > < span class = 'hljl-t' > < / span > < span class = 'hljl-nf' > range< / span > < span class = 'hljl-p' > (< / span > < span class = 'hljl-ni' > 0< / span > < span class = 'hljl-p' > ,< / span > < span class = 'hljl-t' > < / span > < span class = 'hljl-n' > stop< / span > < span class = 'hljl-oB' > =< / span > < span class = 'hljl-n' > π< / span > < span class = 'hljl-p' > ,< / span > < span class = 'hljl-t' > < / span > < span class = 'hljl-n' > length< / span > < span class = 'hljl-oB' > =< / span > < span class = 'hljl-ni' > 1024< / span > < span class = 'hljl-p' > ))< / span > < span class = 'hljl-t' >
< / span > < span class = 'hljl-n' > n< / span > < span class = 'hljl-t' > < / span > < span class = 'hljl-oB' > =< / span > < span class = 'hljl-t' > < / span > < span class = 'hljl-nf' > length< / span > < span class = 'hljl-p' > (< / span > < span class = 'hljl-n' > w< / span > < span class = 'hljl-p' > )< / span > < span class = 'hljl-t' >
< / span > < span class = 'hljl-n' > h< / span > < span class = 'hljl-t' > < / span > < span class = 'hljl-oB' > =< / span > < span class = 'hljl-t' > < / span > < span class = 'hljl-nf' > Array< / span > < span class = 'hljl-p' > {< / span > < span class = 'hljl-n' > ComplexF32< / span > < span class = 'hljl-p' > }(< / span > < span class = 'hljl-n' > undef< / span > < span class = 'hljl-p' > ,< / span > < span class = 'hljl-t' > < / span > < span class = 'hljl-n' > n< / span > < span class = 'hljl-p' > )< / span > < span class = 'hljl-t' >
< / span > < span class = 'hljl-n' > sw< / span > < span class = 'hljl-t' > < / span > < span class = 'hljl-oB' > =< / span > < span class = 'hljl-t' > < / span > < span class = 'hljl-ni' > 0< / span > < span class = 'hljl-t' >
< / span > < span class = 'hljl-k' > for< / span > < span class = 'hljl-t' > < / span > < span class = 'hljl-n' > i< / span > < span class = 'hljl-t' > < / span > < span class = 'hljl-oB' > =< / span > < span class = 'hljl-t' > < / span > < span class = 'hljl-ni' > 1< / span > < span class = 'hljl-oB' > :< / span > < span class = 'hljl-n' > n< / span > < span class = 'hljl-t' >
< / span > < span class = 'hljl-k' > for< / span > < span class = 'hljl-t' > < / span > < span class = 'hljl-n' > j< / span > < span class = 'hljl-t' > < / span > < span class = 'hljl-oB' > =< / span > < span class = 'hljl-t' > < / span > < span class = 'hljl-ni' > 1< / span > < span class = 'hljl-oB' > :< / span > < span class = 'hljl-nf' > length< / span > < span class = 'hljl-p' > (< / span > < span class = 'hljl-n' > b< / span > < span class = 'hljl-p' > )< / span > < span class = 'hljl-t' >
< / span > < span class = 'hljl-n' > sw< / span > < span class = 'hljl-t' > < / span > < span class = 'hljl-oB' > +=< / span > < span class = 'hljl-t' > < / span > < span class = 'hljl-n' > b< / span > < span class = 'hljl-p' > [< / span > < span class = 'hljl-n' > j< / span > < span class = 'hljl-p' > ]< / span > < span class = 'hljl-oB' > *< / span > < span class = 'hljl-nf' > exp< / span > < span class = 'hljl-p' > (< / span > < span class = 'hljl-oB' > -< / span > < span class = 'hljl-n' > im< / span > < span class = 'hljl-oB' > *< / span > < span class = 'hljl-n' > w< / span > < span class = 'hljl-p' > [< / span > < span class = 'hljl-n' > i< / span > < span class = 'hljl-p' > ])< / span > < span class = 'hljl-oB' > ^-< / span > < span class = 'hljl-n' > j< / span > < span class = 'hljl-t' >
< / span > < span class = 'hljl-k' > end< / span > < span class = 'hljl-t' >
< / span > < span class = 'hljl-n' > h< / span > < span class = 'hljl-p' > [< / span > < span class = 'hljl-n' > i< / span > < span class = 'hljl-p' > ]< / span > < span class = 'hljl-t' > < / span > < span class = 'hljl-oB' > =< / span > < span class = 'hljl-t' > < / span > < span class = 'hljl-n' > sw< / span > < span class = 'hljl-t' >
< / span > < span class = 'hljl-n' > sw< / span > < span class = 'hljl-t' > < / span > < span class = 'hljl-oB' > =< / span > < span class = 'hljl-t' > < / span > < span class = 'hljl-ni' > 0< / span > < span class = 'hljl-t' >
< / span > < span class = 'hljl-k' > end< / span > < span class = 'hljl-t' >
< / span > < span class = 'hljl-k' > return< / span > < span class = 'hljl-t' > < / span > < span class = 'hljl-n' > h< / span > < span class = 'hljl-t' >
< / span > < span class = 'hljl-k' > end< / span >
< / pre >
< pre class = "output" >
FIRfreqz ( generic function with 2 methods)
< / pre >
< h2 > Design Lowpass FIR filter< / h2 >
< p > 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: < span class = "math" > $w(n) = \alpha - \beta\cos\frac{2\pi n}{N-1}$< / span > , where < span class = "math" > $\alpha=0.54$< / span > and < span class = "math" > $\beta=0.46$< / span > < / p >
< pre class = 'hljl' >
< span class = 'hljl-n' > fs< / span > < span class = 'hljl-t' > < / span > < span class = 'hljl-oB' > =< / span > < span class = 'hljl-t' > < / span > < span class = 'hljl-ni' > 20< / span > < span class = 'hljl-t' >
< / span > < span class = 'hljl-n' > f< / span > < span class = 'hljl-t' > < / span > < span class = 'hljl-oB' > =< / span > < span class = 'hljl-t' > < / span > < span class = 'hljl-nf' > digitalfilter< / span > < span class = 'hljl-p' > (< / span > < span class = 'hljl-nf' > Lowpass< / span > < span class = 'hljl-p' > (< / span > < span class = 'hljl-ni' > 5< / span > < span class = 'hljl-p' > ,< / span > < span class = 'hljl-t' > < / span > < span class = 'hljl-n' > fs< / span > < span class = 'hljl-t' > < / span > < span class = 'hljl-oB' > =< / span > < span class = 'hljl-t' > < / span > < span class = 'hljl-n' > fs< / span > < span class = 'hljl-p' > ),< / span > < span class = 'hljl-t' > < / span > < span class = 'hljl-nf' > FIRWindow< / span > < span class = 'hljl-p' > (< / span > < span class = 'hljl-nf' > hamming< / span > < span class = 'hljl-p' > (< / span > < span class = 'hljl-ni' > 61< / span > < span class = 'hljl-p' > )))< / span > < span class = 'hljl-t' >
< / span > < span class = 'hljl-n' > w< / span > < span class = 'hljl-t' > < / span > < span class = 'hljl-oB' > =< / span > < span class = 'hljl-t' > < / span > < span class = 'hljl-nf' > range< / span > < span class = 'hljl-p' > (< / span > < span class = 'hljl-ni' > 0< / span > < span class = 'hljl-p' > ,< / span > < span class = 'hljl-t' > < / span > < span class = 'hljl-n' > stop< / span > < span class = 'hljl-oB' > =< / span > < span class = 'hljl-n' > pi< / span > < span class = 'hljl-p' > ,< / span > < span class = 'hljl-t' > < / span > < span class = 'hljl-n' > length< / span > < span class = 'hljl-oB' > =< / span > < span class = 'hljl-ni' > 1024< / span > < span class = 'hljl-p' > )< / span > < span class = 'hljl-t' >
< / span > < span class = 'hljl-n' > h< / span > < span class = 'hljl-t' > < / span > < span class = 'hljl-oB' > =< / span > < span class = 'hljl-t' > < / span > < span class = 'hljl-nf' > FIRfreqz< / span > < span class = 'hljl-p' > (< / span > < span class = 'hljl-n' > f< / span > < span class = 'hljl-p' > ,< / span > < span class = 'hljl-t' > < / span > < span class = 'hljl-n' > w< / span > < span class = 'hljl-p' > )< / span >
< / pre >
< pre class = "output" >
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
< / pre >
< h2 > Plot the frequency and impulse response< / h2 >
< p > The next code chunk is executed in term mode, see the < a href = "FIR_design.jl" > script< / a > for syntax.< / p >
< pre class = 'hljl' >
< span class = 'hljl-nB' > julia> < / span > < span class = 'hljl-n' > h_db< / span > < span class = 'hljl-t' > < / span > < span class = 'hljl-oB' > =< / span > < span class = 'hljl-t' > < / span > < span class = 'hljl-n' > log10< / span > < span class = 'hljl-oB' > .< / span > < span class = 'hljl-p' > (< / span > < span class = 'hljl-n' > abs< / span > < span class = 'hljl-oB' > .< / span > < span class = 'hljl-p' > (< / span > < span class = 'hljl-n' > h< / span > < span class = 'hljl-p' > ));< / span > < span class = 'hljl-t' >
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
< / span > < span class = 'hljl-nB' > julia> < / span > < span class = 'hljl-n' > ws< / span > < span class = 'hljl-t' > < / span > < span class = 'hljl-oB' > =< / span > < span class = 'hljl-t' > < / span > < span class = 'hljl-n' > w< / span > < span class = 'hljl-oB' > /< / span > < span class = 'hljl-n' > pi< / span > < span class = 'hljl-oB' > *< / span > < span class = 'hljl-p' > (< / span > < span class = 'hljl-n' > fs< / span > < span class = 'hljl-oB' > /< / span > < span class = 'hljl-ni' > 2< / span > < span class = 'hljl-p' > )< / span > < span class = 'hljl-t' >
0.0:0.009775171065493646:10.0< / span >
< / pre >
< pre class = 'hljl' >
< span class = 'hljl-nf' > plot< / span > < span class = 'hljl-p' > (< / span > < span class = 'hljl-n' > ws< / span > < span class = 'hljl-p' > ,< / span > < span class = 'hljl-t' > < / span > < span class = 'hljl-n' > h_db< / span > < span class = 'hljl-p' > ,< / span > < span class = 'hljl-t' >
< / span > < span class = 'hljl-n' > xlabel< / span > < span class = 'hljl-t' > < / span > < span class = 'hljl-oB' > =< / span > < span class = 'hljl-t' > < / span > < span class = 'hljl-s' > " Frequency (Hz)" < / span > < span class = 'hljl-p' > ,< / span > < span class = 'hljl-t' > < / span > < span class = 'hljl-n' > ylabel< / span > < span class = 'hljl-t' > < / span > < span class = 'hljl-oB' > =< / span > < span class = 'hljl-t' > < / span > < span class = 'hljl-s' > " Magnitude (db)" < / span > < span class = 'hljl-p' > )< / span >
< / pre >
< img src = "data:image/png;base64,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
< p > And again with default options< / p >
< pre class = 'hljl' >
< span class = 'hljl-n' > h_phase< / span > < span class = 'hljl-t' > < / span > < span class = 'hljl-oB' > =< / span > < span class = 'hljl-t' > < / span > < span class = 'hljl-nf' > unwrap< / span > < span class = 'hljl-p' > (< / span > < span class = 'hljl-oB' > -< / span > < span class = 'hljl-n' > atan< / span > < span class = 'hljl-oB' > .< / span > < span class = 'hljl-p' > (< / span > < span class = 'hljl-n' > imag< / span > < span class = 'hljl-oB' > .< / span > < span class = 'hljl-p' > (< / span > < span class = 'hljl-n' > h< / span > < span class = 'hljl-p' > ),< / span > < span class = 'hljl-n' > real< / span > < span class = 'hljl-oB' > .< / span > < span class = 'hljl-p' > (< / span > < span class = 'hljl-n' > h< / span > < span class = 'hljl-p' > )))< / span > < span class = 'hljl-t' >
< / span > < span class = 'hljl-nf' > plot< / span > < span class = 'hljl-p' > (< / span > < span class = 'hljl-n' > ws< / span > < span class = 'hljl-p' > ,< / span > < span class = 'hljl-t' > < / span > < span class = 'hljl-n' > h_phase< / span > < span class = 'hljl-p' > ,< / span > < span class = 'hljl-t' >
< / span > < span class = 'hljl-n' > xlabel< / span > < span class = 'hljl-t' > < / span > < span class = 'hljl-oB' > =< / span > < span class = 'hljl-t' > < / span > < span class = 'hljl-s' > " Frequency (Hz)" < / span > < span class = 'hljl-p' > ,< / span > < span class = 'hljl-t' > < / span > < span class = 'hljl-n' > ylabel< / span > < span class = 'hljl-t' > < / span > < span class = 'hljl-oB' > =< / span > < span class = 'hljl-t' > < / span > < span class = 'hljl-s' > " Phase (radians)" < / span > < span class = 'hljl-p' > )< / span >
< / pre >
< img src = "data:image/png;base64,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
