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<pre><a name="line-1"></a><span class='hs-comment'>-----------------------------------------------------------------------------</span>
<a name="line-2"></a><span class='hs-comment'>-- |</span>
<a name="line-3"></a><span class='hs-comment'>-- Module : ForSyDe.Shallow.FilterLib</span>
<a name="line-4"></a><span class='hs-comment'>-- Copyright : (c) SAM Group, KTH/ICT/ECS 2007-2008</span>
<a name="line-5"></a><span class='hs-comment'>-- License : BSD-style (see the file LICENSE)</span>
<a name="line-6"></a><span class='hs-comment'>-- </span>
<a name="line-7"></a><span class='hs-comment'>-- Maintainer : forsyde-dev@ict.kth.se</span>
<a name="line-8"></a><span class='hs-comment'>-- Stability : experimental</span>
<a name="line-9"></a><span class='hs-comment'>-- Portability : portable</span>
<a name="line-10"></a><span class='hs-comment'>--</span>
<a name="line-11"></a><span class='hs-comment'>-- This is the filter library for ForSyDe heterogeneous MoCs - CT-MoC, SR-MoC,</span>
<a name="line-12"></a><span class='hs-comment'>-- and Untimed-MoC.</span>
<a name="line-13"></a><span class='hs-comment'>--</span>
<a name="line-14"></a><span class='hs-comment'>-- The filters at CT-MoC are based on the filters implemented at SR-MoC and Untimed-MoC, </span>
<a name="line-15"></a><span class='hs-comment'>-- which means a signal in CT-MoC is always digitalized by a A\/D converter, processed by </span>
<a name="line-16"></a><span class='hs-comment'>-- digital filters at SR or Untimed domain, and converted back into a CT domain signal by </span>
<a name="line-17"></a><span class='hs-comment'>-- a D\/A converter. A CT-filter is composed of one A\/D converter, one digital filter, and </span>
<a name="line-18"></a><span class='hs-comment'>-- one D\/A converter.</span>
<a name="line-19"></a><span class='hs-comment'>--</span>
<a name="line-20"></a><span class='hs-comment'>-- The implementation of the filters at untimed and synchronous domains follows the</span>
<a name="line-21"></a><span class='hs-comment'>-- way in a paper /An introduction to Haskell with applications to digital signal processing, </span>
<a name="line-22"></a><span class='hs-comment'>-- David M. Goblirsch, in Proceedings of the 1994 ACM symposium on Applied computing./,</span>
<a name="line-23"></a><span class='hs-comment'>-- where the details of the FIR\/IIR, AR, and ARMA filters are introduced. The ARMA filter</span>
<a name="line-24"></a><span class='hs-comment'>-- is the kernel part of our general linear filter 'zLinearFilter' in Z-domain at SR\/Untimed</span>
<a name="line-25"></a><span class='hs-comment'>-- MoC, and is also the kernel digital filter for the linear filter 'sLinearFilter' in </span>
<a name="line-26"></a><span class='hs-comment'>-- S-domain at CT-MoC.</span>
<a name="line-27"></a><span class='hs-comment'>-----------------------------------------------------------------------------</span>
<a name="line-28"></a><span class='hs-keyword'>module</span> <span class='hs-conid'>ForSyDe</span><span class='hs-varop'>.</span><span class='hs-conid'>Shallow</span><span class='hs-varop'>.</span><span class='hs-conid'>FilterLib</span> <span class='hs-layout'>(</span>
<a name="line-29"></a> <span class='hs-comment'>-- *FIR filter</span>
<a name="line-30"></a> <span class='hs-varid'>firFilter</span><span class='hs-layout'>,</span>
<a name="line-31"></a> <span class='hs-comment'>-- *AR and ARMA filter trim</span>
<a name="line-32"></a> <span class='hs-varid'>arFilterTrim</span><span class='hs-layout'>,</span> <span class='hs-varid'>armaFilterTrim</span><span class='hs-layout'>,</span>
<a name="line-33"></a> <span class='hs-comment'>-- *The solver mode</span>
<a name="line-34"></a> <span class='hs-conid'>SolverMode</span><span class='hs-layout'>(</span><span class='hs-keyglyph'>..</span><span class='hs-layout'>)</span><span class='hs-layout'>,</span>
<a name="line-35"></a> <span class='hs-comment'>-- *The general linear filter in S-domain</span>
<a name="line-36"></a> <span class='hs-varid'>sLinearFilter</span><span class='hs-layout'>,</span>
<a name="line-37"></a> <span class='hs-comment'>-- *The general linear filter in Z-domain</span>
<a name="line-38"></a> <span class='hs-varid'>zLinearFilter</span><span class='hs-layout'>,</span>
<a name="line-39"></a> <span class='hs-comment'>-- *s2z domain coefficient tranformation</span>
<a name="line-40"></a> <span class='hs-varid'>s2zCoef</span><span class='hs-layout'>,</span>
<a name="line-41"></a> <span class='hs-comment'>-- *The Z-domain to ARMA coefficient tranformation</span>
<a name="line-42"></a> <span class='hs-varid'>h2ARMACoef</span>
<a name="line-43"></a> <span class='hs-layout'>)</span>
<a name="line-44"></a> <span class='hs-keyword'>where</span>
<a name="line-45"></a>
<a name="line-46"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>ForSyDe</span><span class='hs-varop'>.</span><span class='hs-conid'>Shallow</span><span class='hs-varop'>.</span><span class='hs-conid'>MoCLib</span>
<a name="line-47"></a><span class='hs-comment'>--import ForSyDe.Shallow.CTLib</span>
<a name="line-48"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>ForSyDe</span><span class='hs-varop'>.</span><span class='hs-conid'>Shallow</span><span class='hs-varop'>.</span><span class='hs-conid'>CoreLib</span>
<a name="line-49"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>ForSyDe</span><span class='hs-varop'>.</span><span class='hs-conid'>Shallow</span><span class='hs-varop'>.</span><span class='hs-conid'>PolyArith</span>
<a name="line-50"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Data</span><span class='hs-varop'>.</span><span class='hs-conid'>List</span> <span class='hs-layout'>(</span><span class='hs-varid'>zipWith5</span><span class='hs-layout'>)</span>
<a name="line-51"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Control</span><span class='hs-varop'>.</span><span class='hs-conid'>Monad</span><span class='hs-varop'>.</span><span class='hs-conid'>Instances</span> <span class='hs-conid'>()</span> <span class='hs-comment'>-- Monad instance for (-> r)</span>
<a name="line-52"></a>
<a name="line-53"></a><a name="firFilter"></a><span class='hs-comment'>-- |The FIR filter. Let '[x_n]' denote the input signal, '[y_n]' denote the ouput</span>
<a name="line-54"></a><span class='hs-comment'>-- signal, and '[h_n]' the impulse response of the filter. Suppose the length of</span>
<a name="line-55"></a><span class='hs-comment'>-- the impulse responses is M samples. The formula for '[y_n]' is </span>
<a name="line-56"></a><span class='hs-comment'>-- $sum_{k=0}^{M-1} h_k*x_{n-k}$.</span>
<a name="line-57"></a><span class='hs-definition'>firFilter</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Num</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>a</span><span class='hs-keyglyph'>]</span> <span class='hs-comment'>-- ^Coefficients of the FIR filter</span>
<a name="line-58"></a> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Signal</span> <span class='hs-varid'>a</span> <span class='hs-comment'>-- ^Input signal</span>
<a name="line-59"></a> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Signal</span> <span class='hs-varid'>a</span> <span class='hs-comment'>-- ^Output signal</span>
<a name="line-60"></a><span class='hs-definition'>firFilter</span> <span class='hs-varid'>hs</span> <span class='hs-varid'>xs</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>mealySY</span> <span class='hs-varid'>stateF</span> <span class='hs-layout'>(</span><span class='hs-varid'>outF</span> <span class='hs-varid'>hs</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>repeatN</span> <span class='hs-layout'>(</span><span class='hs-varid'>length</span> <span class='hs-varid'>hs</span><span class='hs-layout'>)</span> <span class='hs-num'>0</span><span class='hs-layout'>)</span> <span class='hs-varid'>xs</span>
<a name="line-61"></a> <span class='hs-keyword'>where</span>
<a name="line-62"></a> <span class='hs-varid'>stateF</span> <span class='hs-varid'>xs0</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>fixedList</span> <span class='hs-varid'>xs0</span> <span class='hs-varid'>x</span>
<a name="line-63"></a> <span class='hs-varid'>outF</span> <span class='hs-varid'>hs</span> <span class='hs-varid'>xs0</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>iprod</span> <span class='hs-varid'>hs</span> <span class='hs-varop'>$</span> <span class='hs-varid'>fixedList</span> <span class='hs-varid'>xs0</span> <span class='hs-varid'>x</span>
<a name="line-64"></a>
<a name="line-65"></a><a name="arFilterTrim"></a><span class='hs-comment'>-- |The autoregressive filter is the simplest IIR filter. It is characterized </span>
<a name="line-66"></a><span class='hs-comment'>-- by a list of numbers '[a_1,a_2,...,a_M]' called the autoregression </span>
<a name="line-67"></a><span class='hs-comment'>-- coefficients and a single number 'b' called the gain. 'M' is the order of </span>
<a name="line-68"></a><span class='hs-comment'>-- the filter. Let '[x_n]' denote the input signal, '[y_n]' denote the ouput</span>
<a name="line-69"></a><span class='hs-comment'>-- signal. The formula for '[y_n]' is $\sum_{k=1}^M {a_k*y_{n-k}+b*x_n}$. </span>
<a name="line-70"></a><span class='hs-comment'>-- Although it is an IIR filter, here we only get the same length of ouput </span>
<a name="line-71"></a><span class='hs-comment'>-- signal as the input signal.</span>
<a name="line-72"></a><span class='hs-definition'>arFilterTrim</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Num</span> <span class='hs-varid'>a</span><span class='hs-layout'>,</span> <span class='hs-conid'>Fractional</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span>
<a name="line-73"></a> <span class='hs-keyglyph'>[</span><span class='hs-varid'>a</span><span class='hs-keyglyph'>]</span> <span class='hs-comment'>-- ^Coefficients of the AR filter.</span>
<a name="line-74"></a> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>a</span> <span class='hs-comment'>-- ^The gain</span>
<a name="line-75"></a> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Signal</span> <span class='hs-varid'>a</span> <span class='hs-comment'>-- ^Input signal</span>
<a name="line-76"></a> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Signal</span> <span class='hs-varid'>a</span> <span class='hs-comment'>-- ^Output signal</span>
<a name="line-77"></a><span class='hs-definition'>arFilterTrim</span> <span class='hs-keyword'>as</span> <span class='hs-varid'>b</span> <span class='hs-varid'>xs</span> <span class='hs-keyglyph'>=</span>
<a name="line-78"></a> <span class='hs-varid'>mealySY</span> <span class='hs-layout'>(</span><span class='hs-varid'>stateF</span> <span class='hs-keyword'>as</span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>outF</span> <span class='hs-keyword'>as</span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>repeatN</span> <span class='hs-layout'>(</span><span class='hs-varid'>length</span> <span class='hs-keyword'>as</span><span class='hs-layout'>)</span> <span class='hs-num'>0</span><span class='hs-layout'>)</span> <span class='hs-varid'>xs</span>
<a name="line-79"></a> <span class='hs-keyword'>where</span>
<a name="line-80"></a> <span class='hs-varid'>stateF</span> <span class='hs-keyword'>as</span> <span class='hs-varid'>b</span> <span class='hs-varid'>xs0</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>fixedList</span> <span class='hs-varid'>xs0</span> <span class='hs-varop'>$</span> <span class='hs-varid'>outF</span> <span class='hs-keyword'>as</span> <span class='hs-varid'>b</span> <span class='hs-varid'>xs0</span> <span class='hs-varid'>x</span>
<a name="line-81"></a> <span class='hs-varid'>outF</span> <span class='hs-keyword'>as</span> <span class='hs-varid'>b</span> <span class='hs-varid'>xs0</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>b</span><span class='hs-varop'>*</span><span class='hs-varid'>x</span> <span class='hs-varop'>+</span> <span class='hs-layout'>(</span><span class='hs-varid'>iprod</span> <span class='hs-keyword'>as</span> <span class='hs-varid'>xs0</span><span class='hs-layout'>)</span>
<a name="line-82"></a>
<a name="line-83"></a><a name="armaFilterTrim"></a><span class='hs-comment'>-- |The ARMA filter combines the FIR and AR filters. Let '[x_n]' denote the </span>
<a name="line-84"></a><span class='hs-comment'>-- input signal, '[y_n]' denote the ouput signal. The formula for '[y_n]' is</span>
<a name="line-85"></a><span class='hs-comment'>-- $\sum_{k=1}^M {a_k*y_{n-k}+b*x_n} + sum_{i=0}^{N-1} b_i*x_{n-i}$. The ARMA</span>
<a name="line-86"></a><span class='hs-comment'>-- filter can be defined as the composition of an FIR filter having the impulse</span>
<a name="line-87"></a><span class='hs-comment'>-- reponse '[b_0,b_1,...,b_N-1]' and an AR filter having the regression </span>
<a name="line-88"></a><span class='hs-comment'>-- coefficients '[a_1,a_2,...,a_M]' and a gain of '1'. Although it is an IIR </span>
<a name="line-89"></a><span class='hs-comment'>-- filter, here we only get the same length of ouput signal as the input signal.</span>
<a name="line-90"></a><span class='hs-definition'>armaFilterTrim</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Num</span> <span class='hs-varid'>a</span><span class='hs-layout'>,</span> <span class='hs-conid'>Fractional</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span>
<a name="line-91"></a> <span class='hs-keyglyph'>[</span><span class='hs-varid'>a</span><span class='hs-keyglyph'>]</span> <span class='hs-comment'>-- ^Coefficients of the FIR filter</span>
<a name="line-92"></a> <span class='hs-keyglyph'>-></span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>a</span><span class='hs-keyglyph'>]</span> <span class='hs-comment'>-- ^Coefficients of the AR filter.</span>
<a name="line-93"></a> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Signal</span> <span class='hs-varid'>a</span> <span class='hs-comment'>-- ^Input signal</span>
<a name="line-94"></a> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Signal</span> <span class='hs-varid'>a</span> <span class='hs-comment'>-- ^Output signal</span>
<a name="line-95"></a><span class='hs-definition'>armaFilterTrim</span> <span class='hs-varid'>bs</span> <span class='hs-keyword'>as</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>arFilterTrim</span> <span class='hs-keyword'>as</span> <span class='hs-num'>1</span> <span class='hs-varop'>.</span> <span class='hs-varid'>firFilter</span> <span class='hs-varid'>bs</span>
<a name="line-96"></a>
<a name="line-97"></a>
<a name="line-98"></a><a name="SolverMode"></a><span class='hs-comment'>-- |The solver mode.</span>
<a name="line-99"></a><a name="SolverMode"></a><span class='hs-keyword'>data</span> <span class='hs-conid'>SolverMode</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>S2Z</span> <span class='hs-comment'>-- ^Tustin tranfer from s-domain to z-domain</span>
<a name="line-100"></a> <span class='hs-keyglyph'>|</span> <span class='hs-conid'>RK4</span> <span class='hs-comment'>-- ^Runge Kutta 4 with fixed simulation steps</span>
<a name="line-101"></a> <span class='hs-keyword'>deriving</span> <span class='hs-layout'>(</span><span class='hs-conid'>Show</span><span class='hs-layout'>,</span> <span class='hs-conid'>Eq</span><span class='hs-layout'>)</span>
<a name="line-102"></a>
<a name="line-103"></a><a name="sLinearFilter"></a><span class='hs-comment'>-- |The general linear filter in S-domain at CT-MoC. As the kernel</span>
<a name="line-104"></a><span class='hs-comment'>-- implementation is in Z-domain, the smapling rate should be specified. </span>
<a name="line-105"></a><span class='hs-comment'>-- It is used on the S-transformation with the following forms, with </span>
<a name="line-106"></a><span class='hs-comment'>-- coefficients for the descending powers of 's' and m < n.</span>
<a name="line-107"></a><span class='hs-comment'>--</span>
<a name="line-108"></a><span class='hs-comment'>-- > b_0*s^m + b_1*s^m-1 + ... + b_m-1*s^1 + b_m*s^0</span>
<a name="line-109"></a><span class='hs-comment'>-- >H(s) = ------------------------------------------------ (Eq 1)</span>
<a name="line-110"></a><span class='hs-comment'>-- > a_0*s^n + a_1*s^n-1 + ... + a_n-1*s^1 + a_n*s^0</span>
<a name="line-111"></a><span class='hs-comment'>--</span>
<a name="line-112"></a><span class='hs-comment'>-- If we multiply both the numerator and the denominator with s^-n, we get </span>
<a name="line-113"></a><span class='hs-comment'>-- another equivelent canonical form</span>
<a name="line-114"></a><span class='hs-comment'>--</span>
<a name="line-115"></a><span class='hs-comment'>-- > b_0*s^m-n + b_1*s^m-n-1 + ... + b_m-1*s^1-n + b_m*s^-n</span>
<a name="line-116"></a><span class='hs-comment'>-- >H(s) = ----------------------------------------------------- (Eq 2)</span>
<a name="line-117"></a><span class='hs-comment'>-- > a_0*s^0 + a_1*s^-1 + ... + a_n-1*s^1-n + a_n*s^-n</span>
<a name="line-118"></a><span class='hs-comment'>--</span>
<a name="line-119"></a><span class='hs-comment'>-- To instantiate a filter, with sampling interval 'T ', we use</span>
<a name="line-120"></a><span class='hs-comment'>--</span>
<a name="line-121"></a><span class='hs-comment'>-- > filter1 = sLinearFilter T [1] [2,1]</span>
<a name="line-122"></a><span class='hs-comment'>-- </span>
<a name="line-123"></a><span class='hs-comment'>-- to get a filter with the transfer function</span>
<a name="line-124"></a><span class='hs-comment'>-- </span>
<a name="line-125"></a><span class='hs-comment'>-- > 1</span>
<a name="line-126"></a><span class='hs-comment'>-- >H(s) = --------</span>
<a name="line-127"></a><span class='hs-comment'>-- > 2*s + 1</span>
<a name="line-128"></a><span class='hs-comment'>-- </span>
<a name="line-129"></a><span class='hs-comment'>-- and</span>
<a name="line-130"></a><span class='hs-comment'>--</span>
<a name="line-131"></a><span class='hs-comment'>-- > filter2 = sLinearFilter T [2,1] [1,2,2]</span>
<a name="line-132"></a><span class='hs-comment'>--</span>
<a name="line-133"></a><span class='hs-comment'>-- to get another filter with the transfer function</span>
<a name="line-134"></a><span class='hs-comment'>-- </span>
<a name="line-135"></a><span class='hs-comment'>-- > 2*s +1</span>
<a name="line-136"></a><span class='hs-comment'>-- >H(s) = ----------------</span>
<a name="line-137"></a><span class='hs-comment'>-- > s^2 + 2*s + 2</span>
<a name="line-138"></a><span class='hs-comment'>--</span>
<a name="line-139"></a><span class='hs-comment'>-- There are two solver modes, 'S2Z' and 'RK4'.</span>
<a name="line-140"></a><span class='hs-comment'>-- Caused by the precision problem, the time interval in CT uses Rational data</span>
<a name="line-141"></a><span class='hs-comment'>-- type and the digital data types in all the domains are Double.</span>
<a name="line-142"></a><span class='hs-definition'>sLinearFilter</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Num</span> <span class='hs-varid'>a</span><span class='hs-layout'>,</span> <span class='hs-conid'>Fractional</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span>
<a name="line-143"></a> <span class='hs-conid'>SolverMode</span> <span class='hs-comment'>-- ^Specify the solver mode</span>
<a name="line-144"></a> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Rational</span> <span class='hs-comment'>-- ^Fixed simulation interval</span>
<a name="line-145"></a> <span class='hs-keyglyph'>-></span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>a</span><span class='hs-keyglyph'>]</span> <span class='hs-comment'>-- ^Coefficients of the polynomial numerator in s-domain</span>
<a name="line-146"></a> <span class='hs-keyglyph'>-></span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>a</span><span class='hs-keyglyph'>]</span> <span class='hs-comment'>-- ^Coefficients of the polynomial denominator in s-domain</span>
<a name="line-147"></a> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Signal</span> <span class='hs-layout'>(</span><span class='hs-conid'>SubsigCT</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span><span class='hs-comment'>-- ^Input CT-signal</span>
<a name="line-148"></a> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Signal</span> <span class='hs-layout'>(</span><span class='hs-conid'>SubsigCT</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span><span class='hs-comment'>-- ^Output CT-signal</span>
<a name="line-149"></a><span class='hs-definition'>sLinearFilter</span> <span class='hs-varid'>filterMode</span> <span class='hs-varid'>step</span> <span class='hs-varid'>bs</span> <span class='hs-keyword'>as</span> <span class='hs-varid'>inS</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>outS</span>
<a name="line-150"></a> <span class='hs-keyword'>where</span>
<a name="line-151"></a> <span class='hs-comment'>-- A2D conversion</span>
<a name="line-152"></a> <span class='hs-varid'>inSDigital</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>a2dConverter</span> <span class='hs-varid'>step</span> <span class='hs-varid'>inS</span>
<a name="line-153"></a> <span class='hs-comment'>-- D2A conversion</span>
<a name="line-154"></a> <span class='hs-varid'>outS</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>d2aConverter</span> <span class='hs-conid'>DAhold</span> <span class='hs-varid'>step</span> <span class='hs-varid'>outSDigital</span>
<a name="line-155"></a> <span class='hs-comment'>-- Digital filter</span>
<a name="line-156"></a> <span class='hs-varid'>outSDigital</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>filterMode</span> <span class='hs-varop'>==</span> <span class='hs-conid'>S2Z</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>armaFilterTrim</span> <span class='hs-varid'>bs'</span> <span class='hs-varid'>as'</span> <span class='hs-varid'>inSDigital</span>
<a name="line-157"></a> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>otherwise</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>rk4FilterDigital</span> <span class='hs-varid'>step</span> <span class='hs-keyword'>as</span> <span class='hs-varid'>bs</span> <span class='hs-varid'>inSDigital</span>
<a name="line-158"></a> <span class='hs-keyword'>where</span> <span class='hs-layout'>(</span><span class='hs-varid'>bs'</span><span class='hs-layout'>,</span><span class='hs-varid'>as'</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>h2ARMACoef</span> <span class='hs-varop'>$</span> <span class='hs-varid'>s2zCoef</span> <span class='hs-varid'>step</span> <span class='hs-varid'>bs</span> <span class='hs-keyword'>as</span>
<a name="line-159"></a>
<a name="line-160"></a><a name="rk4FilterDigital"></a><span class='hs-comment'>-- |Digital filter using Runge Kutta 4 solver.</span>
<a name="line-161"></a><span class='hs-definition'>rk4FilterDigital</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Fractional</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>=></span>
<a name="line-162"></a> <span class='hs-conid'>Rational</span> <span class='hs-keyglyph'>-></span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>a</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>-></span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>a</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Signal</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Signal</span> <span class='hs-varid'>a</span>
<a name="line-163"></a><span class='hs-definition'>rk4FilterDigital</span> <span class='hs-varid'>step</span> <span class='hs-keyword'>as</span> <span class='hs-varid'>bs</span> <span class='hs-varid'>inSDigital</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>outSDigital</span>
<a name="line-164"></a> <span class='hs-keyword'>where</span>
<a name="line-165"></a> <span class='hs-comment'>-- Below are the skeletons of the RK-4 solver, with</span>
<a name="line-166"></a> <span class='hs-comment'>-- input signal 'inSDigital' and output signal 'outSDigital'</span>
<a name="line-167"></a> <span class='hs-comment'>-- Coefficients handling</span>
<a name="line-168"></a> <span class='hs-varid'>as''</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>dropWhile</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>\</span><span class='hs-varid'>x</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>x</span><span class='hs-varop'>==</span><span class='hs-num'>0.0</span><span class='hs-layout'>)</span> <span class='hs-keyword'>as</span>
<a name="line-169"></a> <span class='hs-varid'>a0</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>head</span> <span class='hs-varid'>as''</span>
<a name="line-170"></a> <span class='hs-comment'>-- Normalized the coefficients</span>
<a name="line-171"></a> <span class='hs-varid'>as'</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>reverse</span> <span class='hs-varop'>$</span> <span class='hs-varid'>tail</span> <span class='hs-varop'>$</span> <span class='hs-varid'>map</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>\</span><span class='hs-varid'>x</span> <span class='hs-keyglyph'>-></span> <span class='hs-comment'>-</span><span class='hs-varid'>x</span><span class='hs-varop'>/</span><span class='hs-varid'>a0</span><span class='hs-layout'>)</span> <span class='hs-varid'>as''</span>
<a name="line-172"></a> <span class='hs-varid'>bs'</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>reverse</span> <span class='hs-varop'>$</span> <span class='hs-varid'>map</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>\</span><span class='hs-varid'>x</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>x</span><span class='hs-varop'>/</span><span class='hs-varid'>a0</span><span class='hs-layout'>)</span> <span class='hs-varid'>bs</span>
<a name="line-173"></a> <span class='hs-comment'>-- Order of the filter</span>
<a name="line-174"></a> <span class='hs-varid'>orderFilter</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>length</span> <span class='hs-varid'>as'</span>
<a name="line-175"></a> <span class='hs-comment'>-- The last state function, '0' is for the time </span>
<a name="line-176"></a> <span class='hs-varid'>fXn</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>iprod</span> <span class='hs-layout'>(</span><span class='hs-num'>0</span><span class='hs-conop'>:</span><span class='hs-varid'>as'</span><span class='hs-layout'>)</span>
<a name="line-177"></a> <span class='hs-comment'>-- The functions for the observalbe state matrix 'A'</span>
<a name="line-178"></a> <span class='hs-varid'>stateFunctions</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>ffn'</span> <span class='hs-varid'>orderFilter</span> <span class='hs-varop'>++</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>fXn</span><span class='hs-keyglyph'>]</span>
<a name="line-179"></a> <span class='hs-comment'>-- Initial states</span>
<a name="line-180"></a> <span class='hs-varid'>initialStates</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>repeatN</span> <span class='hs-varid'>orderFilter</span> <span class='hs-num'>0.0</span>
<a name="line-181"></a> <span class='hs-varid'>inputSteps</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>signal</span> <span class='hs-varop'>$</span> <span class='hs-varid'>repeat</span> <span class='hs-varid'>step'</span>
<a name="line-182"></a> <span class='hs-comment'>-- The states signal</span>
<a name="line-183"></a> <span class='hs-varid'>statesSignal</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>rks4InSY</span> <span class='hs-num'>0.0</span> <span class='hs-varid'>initialStates</span> <span class='hs-varid'>stateFunctions</span>
<a name="line-184"></a> <span class='hs-varid'>inputSteps</span> <span class='hs-varid'>inSDigital</span> <span class='hs-varop'>--xs</span>
<a name="line-185"></a> <span class='hs-comment'>-- The ouput digital signal </span>
<a name="line-186"></a> <span class='hs-varid'>outSDigital</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>mapSY</span> <span class='hs-layout'>(</span><span class='hs-varid'>iprod</span> <span class='hs-varid'>bs'</span><span class='hs-layout'>)</span> <span class='hs-varid'>statesSignal</span>
<a name="line-187"></a> <span class='hs-comment'>-- The fixed simulation step</span>
<a name="line-188"></a> <span class='hs-varid'>step'</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>fromRational</span> <span class='hs-varid'>step</span>
<a name="line-189"></a>
<a name="line-190"></a><a name="ffn'"></a><span class='hs-comment'>-- The length of the function list is 'n-1' for nth order filter</span>
<a name="line-191"></a><span class='hs-definition'>ffn'</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Num</span> <span class='hs-varid'>t</span><span class='hs-layout'>,</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>t1</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-varid'>t</span> <span class='hs-keyglyph'>-></span> <span class='hs-keyglyph'>[</span><span class='hs-keyglyph'>[</span><span class='hs-varid'>t1</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>t1</span><span class='hs-keyglyph'>]</span>
<a name="line-192"></a><span class='hs-definition'>ffn'</span> <span class='hs-varid'>n</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>ffn</span> <span class='hs-num'>0</span> <span class='hs-varid'>n</span>
<a name="line-193"></a>
<a name="line-194"></a><a name="ffn"></a><span class='hs-comment'>-- Construct the functions for the diagonal '1'</span>
<a name="line-195"></a><span class='hs-definition'>ffn</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Num</span> <span class='hs-varid'>t1</span><span class='hs-layout'>,</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>t</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Int</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>t</span> <span class='hs-keyglyph'>-></span> <span class='hs-keyglyph'>[</span><span class='hs-keyglyph'>[</span><span class='hs-varid'>t1</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>t1</span><span class='hs-keyglyph'>]</span>
<a name="line-196"></a><span class='hs-definition'>ffn</span> <span class='hs-keyword'>_</span> <span class='hs-num'>1</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>[]</span>
<a name="line-197"></a><span class='hs-definition'>ffn</span> <span class='hs-varid'>m</span> <span class='hs-varid'>n</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>ff1</span> <span class='hs-varid'>m</span> <span class='hs-conop'>:</span> <span class='hs-varid'>ffn</span> <span class='hs-layout'>(</span><span class='hs-varid'>m</span><span class='hs-varop'>+</span><span class='hs-num'>1</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>n</span><span class='hs-comment'>-</span><span class='hs-num'>1</span><span class='hs-layout'>)</span>
<a name="line-198"></a>
<a name="line-199"></a><a name="ff1"></a><span class='hs-definition'>ff1</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>t</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Int</span> <span class='hs-keyglyph'>-></span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>t</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>t</span>
<a name="line-200"></a><span class='hs-definition'>ff1</span> <span class='hs-varid'>m</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>iprod</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>[</span><span class='hs-num'>0</span><span class='hs-layout'>,</span><span class='hs-num'>0</span><span class='hs-keyglyph'>]</span> <span class='hs-varop'>++</span> <span class='hs-layout'>(</span><span class='hs-varid'>repeatN</span> <span class='hs-varid'>m</span> <span class='hs-num'>0</span><span class='hs-layout'>)</span> <span class='hs-varop'>++</span> <span class='hs-keyglyph'>[</span><span class='hs-num'>1</span><span class='hs-keyglyph'>]</span> <span class='hs-varop'>++</span> <span class='hs-layout'>(</span><span class='hs-varid'>repeat</span> <span class='hs-num'>0</span><span class='hs-layout'>)</span> <span class='hs-layout'>)</span>
<a name="line-201"></a>
<a name="line-202"></a><a name="rks4InSY"></a><span class='hs-comment'>-- |RK-4 to solve the 1st-order ODEs, with input signal.</span>
<a name="line-203"></a><span class='hs-definition'>rks4InSY</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Num</span> <span class='hs-varid'>a</span><span class='hs-layout'>,</span> <span class='hs-conid'>Fractional</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span>
<a name="line-204"></a> <span class='hs-varid'>a</span> <span class='hs-comment'>-- ^The initial time</span>
<a name="line-205"></a> <span class='hs-keyglyph'>-></span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>a</span><span class='hs-keyglyph'>]</span> <span class='hs-comment'>-- ^The initial state values</span>
<a name="line-206"></a> <span class='hs-keyglyph'>-></span> <span class='hs-keyglyph'>[</span><span class='hs-layout'>(</span><span class='hs-keyglyph'>[</span><span class='hs-varid'>a</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span><span class='hs-keyglyph'>]</span> <span class='hs-comment'>-- ^List of the functions of the ODEs.</span>
<a name="line-207"></a> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Signal</span> <span class='hs-varid'>a</span> <span class='hs-comment'>-- ^Input signal of steps</span>
<a name="line-208"></a> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Signal</span> <span class='hs-varid'>a</span> <span class='hs-comment'>-- ^Input signal</span>
<a name="line-209"></a> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Signal</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>a</span><span class='hs-keyglyph'>]</span> <span class='hs-comment'>-- ^Next state signal</span>
<a name="line-210"></a><span class='hs-definition'>rks4InSY</span> <span class='hs-varid'>x0</span> <span class='hs-varid'>ys0</span> <span class='hs-varid'>fFs</span> <span class='hs-varid'>hs</span> <span class='hs-varid'>us</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>scanl3SY</span> <span class='hs-varid'>stateF</span> <span class='hs-varid'>ys0</span> <span class='hs-varid'>xs</span> <span class='hs-varid'>hs</span> <span class='hs-varid'>us</span>
<a name="line-211"></a> <span class='hs-keyword'>where</span>
<a name="line-212"></a> <span class='hs-varid'>stateF</span> <span class='hs-varid'>ysn</span> <span class='hs-varid'>xn</span> <span class='hs-varid'>h</span> <span class='hs-varid'>ut</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>zipWith</span> <span class='hs-layout'>(</span><span class='hs-varop'>+</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>repeatN</span> <span class='hs-varid'>orderODE'</span> <span class='hs-num'>0.0</span> <span class='hs-varop'>++</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>ut</span><span class='hs-varop'>*</span><span class='hs-varid'>h</span><span class='hs-keyglyph'>]</span><span class='hs-layout'>)</span> <span class='hs-varop'>$</span> <span class='hs-comment'>--Input value</span>
<a name="line-213"></a> <span class='hs-varid'>rks4</span> <span class='hs-varid'>h</span> <span class='hs-varid'>xn</span> <span class='hs-varid'>fFs</span> <span class='hs-varid'>ysn</span>
<a name="line-214"></a> <span class='hs-varid'>xs</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>scanldSY</span> <span class='hs-layout'>(</span><span class='hs-varop'>+</span><span class='hs-layout'>)</span> <span class='hs-varid'>x0</span> <span class='hs-varid'>hs</span>
<a name="line-215"></a> <span class='hs-comment'>-- Order -1 of the ODEs</span>
<a name="line-216"></a> <span class='hs-varid'>orderODE'</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>length</span> <span class='hs-varid'>ys0</span> <span class='hs-comment'>-</span> <span class='hs-num'>1</span>
<a name="line-217"></a>
<a name="line-218"></a><a name="rks4"></a><span class='hs-comment'>-- |One step RK-4 for the 1st-order ordinary differential equations (ODEs).</span>
<a name="line-219"></a><span class='hs-definition'>rks4</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Num</span> <span class='hs-varid'>a</span><span class='hs-layout'>,</span> <span class='hs-conid'>Fractional</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span>
<a name="line-220"></a> <span class='hs-varid'>a</span> <span class='hs-comment'>-- ^The step</span>
<a name="line-221"></a> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>a</span> <span class='hs-comment'>-- ^Initial value of time</span>
<a name="line-222"></a> <span class='hs-keyglyph'>-></span> <span class='hs-keyglyph'>[</span><span class='hs-keyglyph'>[</span><span class='hs-varid'>a</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>a</span><span class='hs-keyglyph'>]</span> <span class='hs-comment'>-- ^List of the funcitons of the ODEs.</span>
<a name="line-223"></a> <span class='hs-keyglyph'>-></span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>a</span><span class='hs-keyglyph'>]</span> <span class='hs-comment'>-- ^List of the value at the current state</span>
<a name="line-224"></a> <span class='hs-keyglyph'>-></span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>a</span><span class='hs-keyglyph'>]</span> <span class='hs-comment'>-- ^List of the value at the next state</span>
<a name="line-225"></a><span class='hs-definition'>rks4</span> <span class='hs-varid'>h</span> <span class='hs-varid'>x0</span> <span class='hs-varid'>fFs</span> <span class='hs-varid'>ys0</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>ys1</span>
<a name="line-226"></a> <span class='hs-keyword'>where</span>
<a name="line-227"></a> <span class='hs-varid'>h_2</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>h</span><span class='hs-varop'>/</span><span class='hs-num'>2.0</span>
<a name="line-228"></a> <span class='hs-varid'>ks1</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>map</span> <span class='hs-layout'>(</span><span class='hs-varid'>h</span><span class='hs-varop'>*</span><span class='hs-layout'>)</span> <span class='hs-varop'>$</span> <span class='hs-varid'>map'</span> <span class='hs-layout'>(</span><span class='hs-varid'>x0</span><span class='hs-conop'>:</span><span class='hs-varid'>ys0</span><span class='hs-layout'>)</span> <span class='hs-varid'>fFs</span> <span class='hs-comment'>-- -- map (h *) $ applyFt x0 fFs ys0</span>
<a name="line-229"></a> <span class='hs-varid'>ks2</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>map</span> <span class='hs-layout'>(</span><span class='hs-varid'>h</span><span class='hs-varop'>*</span><span class='hs-layout'>)</span> <span class='hs-varop'>$</span> <span class='hs-varid'>map'</span> <span class='hs-layout'>(</span><span class='hs-varid'>x0</span><span class='hs-varop'>+</span><span class='hs-varid'>h_2</span><span class='hs-conop'>:</span><span class='hs-varid'>zipWith</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>\</span><span class='hs-varid'>y</span> <span class='hs-varid'>k</span><span class='hs-keyglyph'>-></span> <span class='hs-varid'>y</span><span class='hs-varop'>+</span><span class='hs-varid'>k</span><span class='hs-varop'>/</span><span class='hs-num'>2.0</span><span class='hs-layout'>)</span> <span class='hs-varid'>ys0</span> <span class='hs-varid'>ks1</span><span class='hs-layout'>)</span> <span class='hs-varid'>fFs</span>
<a name="line-230"></a> <span class='hs-varid'>ks3</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>map</span> <span class='hs-layout'>(</span><span class='hs-varid'>h</span><span class='hs-varop'>*</span><span class='hs-layout'>)</span> <span class='hs-varop'>$</span> <span class='hs-varid'>map'</span> <span class='hs-layout'>(</span><span class='hs-varid'>x0</span><span class='hs-varop'>+</span><span class='hs-varid'>h_2</span><span class='hs-conop'>:</span><span class='hs-varid'>zipWith</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>\</span><span class='hs-varid'>y</span> <span class='hs-varid'>k</span><span class='hs-keyglyph'>-></span> <span class='hs-varid'>y</span><span class='hs-varop'>+</span><span class='hs-varid'>k</span><span class='hs-varop'>/</span><span class='hs-num'>2.0</span><span class='hs-layout'>)</span> <span class='hs-varid'>ys0</span> <span class='hs-varid'>ks2</span><span class='hs-layout'>)</span> <span class='hs-varid'>fFs</span>
<a name="line-231"></a> <span class='hs-varid'>ks4</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>map</span> <span class='hs-layout'>(</span><span class='hs-varid'>h</span><span class='hs-varop'>*</span><span class='hs-layout'>)</span> <span class='hs-varop'>$</span> <span class='hs-varid'>map'</span> <span class='hs-layout'>(</span><span class='hs-varid'>x0</span><span class='hs-varop'>+</span><span class='hs-varid'>h</span><span class='hs-conop'>:</span><span class='hs-varid'>zipWith</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>\</span><span class='hs-varid'>y</span> <span class='hs-varid'>k</span><span class='hs-keyglyph'>-></span> <span class='hs-varid'>y</span><span class='hs-varop'>+</span><span class='hs-varid'>k</span><span class='hs-layout'>)</span> <span class='hs-varid'>ys0</span> <span class='hs-varid'>ks3</span><span class='hs-layout'>)</span> <span class='hs-varid'>fFs</span>
<a name="line-232"></a> <span class='hs-varid'>ys1</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>zipWith5</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>\</span><span class='hs-varid'>y0</span> <span class='hs-varid'>k1</span> <span class='hs-varid'>k2</span> <span class='hs-varid'>k3</span> <span class='hs-varid'>k4</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>y0</span> <span class='hs-varop'>+</span> <span class='hs-varid'>k1</span><span class='hs-varop'>/</span><span class='hs-num'>6</span> <span class='hs-varop'>+</span> <span class='hs-varid'>k2</span><span class='hs-varop'>/</span><span class='hs-num'>3</span> <span class='hs-varop'>+</span> <span class='hs-varid'>k3</span><span class='hs-varop'>/</span><span class='hs-num'>3</span> <span class='hs-varop'>+</span> <span class='hs-varid'>k4</span><span class='hs-varop'>/</span><span class='hs-num'>6</span><span class='hs-layout'>)</span>
<a name="line-233"></a> <span class='hs-varid'>ys0</span> <span class='hs-varid'>ks1</span> <span class='hs-varid'>ks2</span> <span class='hs-varid'>ks3</span> <span class='hs-varid'>ks4</span>
<a name="line-234"></a>
<a name="line-235"></a><a name="zLinearFilter"></a><span class='hs-comment'>-- |The general linear filter in Z-domain.</span>
<a name="line-236"></a><span class='hs-definition'>zLinearFilter</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Fractional</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>=></span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>a</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>-></span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>a</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Signal</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Signal</span> <span class='hs-varid'>a</span>
<a name="line-237"></a><span class='hs-definition'>zLinearFilter</span> <span class='hs-varid'>bs</span> <span class='hs-keyword'>as</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>armaFilterTrim</span> <span class='hs-varid'>bs'</span> <span class='hs-varid'>as'</span>
<a name="line-238"></a> <span class='hs-keyword'>where</span>
<a name="line-239"></a> <span class='hs-varid'>bs'</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>map</span> <span class='hs-layout'>(</span><span class='hs-layout'>(</span><span class='hs-keyglyph'>\</span><span class='hs-varid'>x</span> <span class='hs-varid'>y</span><span class='hs-keyglyph'>-></span> <span class='hs-varid'>y</span><span class='hs-varop'>/</span><span class='hs-varid'>x</span> <span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>head</span> <span class='hs-keyword'>as</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-varid'>bs</span>
<a name="line-240"></a> <span class='hs-varid'>as'</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>map</span> <span class='hs-layout'>(</span><span class='hs-layout'>(</span><span class='hs-keyglyph'>\</span><span class='hs-varid'>x</span> <span class='hs-varid'>y</span><span class='hs-keyglyph'>-></span> <span class='hs-comment'>-</span><span class='hs-varid'>y</span><span class='hs-varop'>/</span><span class='hs-varid'>x</span> <span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>head</span> <span class='hs-keyword'>as</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-varop'>$</span> <span class='hs-varid'>tail</span> <span class='hs-keyword'>as</span>
<a name="line-241"></a>
<a name="line-242"></a><a name="s2zCoef"></a><span class='hs-comment'>-- |s2z domain coefficient tranformation with a specified sampling rate.</span>
<a name="line-243"></a><span class='hs-comment'>-- The Tustin transformation is used for the transfer, with</span>
<a name="line-244"></a><span class='hs-comment'>--</span>
<a name="line-245"></a><span class='hs-comment'>-- > 2(z - 1) </span>
<a name="line-246"></a><span class='hs-comment'>-- > s = ---------- (Eq 3)</span>
<a name="line-247"></a><span class='hs-comment'>-- > T(z + 1)</span>
<a name="line-248"></a><span class='hs-comment'>--</span>
<a name="line-249"></a><span class='hs-comment'>-- in which, 'T' is the sampling interval.</span>
<a name="line-250"></a><span class='hs-definition'>s2zCoef</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Num</span> <span class='hs-varid'>a</span><span class='hs-layout'>,</span> <span class='hs-conid'>Fractional</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span>
<a name="line-251"></a> <span class='hs-conid'>Rational</span> <span class='hs-comment'>-- ^Sampling rate in Z-domain</span>
<a name="line-252"></a> <span class='hs-keyglyph'>-></span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>a</span><span class='hs-keyglyph'>]</span> <span class='hs-comment'>-- ^Coefficients of the polynomial numerator in s-domain</span>
<a name="line-253"></a> <span class='hs-keyglyph'>-></span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>a</span><span class='hs-keyglyph'>]</span> <span class='hs-comment'>-- ^Coefficients of the polynomial denominator in s-domain</span>
<a name="line-254"></a> <span class='hs-keyglyph'>-></span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>[</span><span class='hs-varid'>a</span><span class='hs-keyglyph'>]</span><span class='hs-layout'>,</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>a</span><span class='hs-keyglyph'>]</span><span class='hs-layout'>)</span> <span class='hs-comment'>-- ^Tuple of the numerator and denominator </span>
<a name="line-255"></a> <span class='hs-comment'>-- coefficients in Z-domain</span>
<a name="line-256"></a><span class='hs-definition'>s2zCoef</span> <span class='hs-varid'>sampleT</span> <span class='hs-varid'>bs</span> <span class='hs-keyword'>as</span> <span class='hs-keyglyph'>=</span> <span class='hs-layout'>(</span><span class='hs-varid'>reverse</span> <span class='hs-varid'>bs'</span><span class='hs-layout'>,</span> <span class='hs-varid'>reverse</span> <span class='hs-varid'>as'</span><span class='hs-layout'>)</span>
<a name="line-257"></a> <span class='hs-keyword'>where</span>
<a name="line-258"></a> <span class='hs-layout'>(</span><span class='hs-varid'>bs'</span><span class='hs-layout'>,</span><span class='hs-varid'>as'</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>getCoef</span> <span class='hs-varid'>hZ</span>
<a name="line-259"></a> <span class='hs-varid'>bsInv</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>reverse</span> <span class='hs-varid'>bs</span>
<a name="line-260"></a> <span class='hs-varid'>asInv</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>reverse</span> <span class='hs-keyword'>as</span>
<a name="line-261"></a> <span class='hs-varid'>numerator'</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>foldl</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>\</span><span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>addPoly</span> <span class='hs-varid'>x</span> <span class='hs-varop'>$</span> <span class='hs-varid'>scalePoly</span> <span class='hs-layout'>(</span><span class='hs-varid'>fst</span> <span class='hs-varid'>y</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>snd</span> <span class='hs-varid'>y</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-262"></a> <span class='hs-layout'>(</span><span class='hs-conid'>Poly</span> <span class='hs-keyglyph'>[</span><span class='hs-num'>0</span><span class='hs-keyglyph'>]</span><span class='hs-layout'>)</span> <span class='hs-varop'>$</span> <span class='hs-varid'>zip</span> <span class='hs-varid'>bsInv</span> <span class='hs-varid'>sList</span>
<a name="line-263"></a> <span class='hs-varid'>denominator'</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>foldl</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>\</span><span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>addPoly</span> <span class='hs-varid'>x</span> <span class='hs-varop'>$</span> <span class='hs-varid'>scalePoly</span> <span class='hs-layout'>(</span><span class='hs-varid'>fst</span> <span class='hs-varid'>y</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>snd</span> <span class='hs-varid'>y</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-264"></a> <span class='hs-layout'>(</span><span class='hs-conid'>Poly</span> <span class='hs-keyglyph'>[</span><span class='hs-num'>0</span><span class='hs-keyglyph'>]</span><span class='hs-layout'>)</span> <span class='hs-varop'>$</span> <span class='hs-varid'>zip</span> <span class='hs-varid'>asInv</span> <span class='hs-varid'>sList</span>
<a name="line-265"></a> <span class='hs-varid'>hZ</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>divPoly</span> <span class='hs-varid'>numerator'</span> <span class='hs-varid'>denominator'</span>
<a name="line-266"></a> <span class='hs-comment'>-- Tustin transform</span>
<a name="line-267"></a> <span class='hs-varid'>s</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>PolyPair</span> <span class='hs-layout'>(</span><span class='hs-conid'>Poly</span> <span class='hs-keyglyph'>[</span><span class='hs-comment'>-</span><span class='hs-num'>2</span><span class='hs-layout'>,</span><span class='hs-num'>2</span><span class='hs-keyglyph'>]</span><span class='hs-layout'>,</span><span class='hs-conid'>Poly</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>fromRational</span> <span class='hs-varid'>sampleT</span><span class='hs-layout'>,</span><span class='hs-varid'>fromRational</span> <span class='hs-varid'>sampleT</span><span class='hs-keyglyph'>]</span><span class='hs-layout'>)</span>
<a name="line-268"></a> <span class='hs-varid'>sList</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>map</span> <span class='hs-layout'>(</span><span class='hs-varid'>powerPoly</span> <span class='hs-varid'>s</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>[</span><span class='hs-num'>0</span><span class='hs-keyglyph'>..</span><span class='hs-keyglyph'>]</span>
<a name="line-269"></a>
<a name="line-270"></a><a name="h2ARMACoef"></a><span class='hs-comment'>-- |The Z-domain to ARMA coefficient tranformation. It is used on the </span>
<a name="line-271"></a><span class='hs-comment'>-- Z-transfer function</span>
<a name="line-272"></a><span class='hs-comment'>--</span>
<a name="line-273"></a><span class='hs-comment'>-- > b_0*z^m-n + b_1*z^m-n-1 + ... + b_m-1*z^1-n + b_m*z^-n</span>
<a name="line-274"></a><span class='hs-comment'>-- >H(z) = ----------------------------------------------------- (Eq 4)</span>
<a name="line-275"></a><span class='hs-comment'>-- > a_0*z^0 + a_1*z^-1 + ... + a_n-1*z^1-n + a_n*z^-n</span>
<a name="line-276"></a><span class='hs-comment'>--</span>
<a name="line-277"></a><span class='hs-comment'>-- which is normalized as</span>
<a name="line-278"></a><span class='hs-comment'>--</span>
<a name="line-279"></a><span class='hs-comment'>-- > b_0/a_0*z^m-n + b_1/a_0*z^m-n-1 + ... + b_m/a_0*z^-n</span>
<a name="line-280"></a><span class='hs-comment'>-- >H(z) = ------------------------------------------------------- (Eq 5)</span>
<a name="line-281"></a><span class='hs-comment'>-- > 1 + a_1/a_0*z^-1 + ... + a_n-1/a_0*z^1-n + a_n/a_0*z^-n</span>
<a name="line-282"></a><span class='hs-comment'>--</span>
<a name="line-283"></a><span class='hs-comment'>-- The implementation coudl be</span>
<a name="line-284"></a><span class='hs-comment'>--</span>
<a name="line-285"></a><span class='hs-comment'>-- >y(k) = b_0/a_0*x_k+m-n + b_1/a_0*x_k+m-n-1 + ... + b_m/a_0*x_k-n</span>
<a name="line-286"></a><span class='hs-comment'>-- > (Eq 6)</span>
<a name="line-287"></a><span class='hs-comment'>-- > - a_1/a_0*y_k-1 - ... - a_n/a_0*y_k-n</span>
<a name="line-288"></a><span class='hs-comment'>--</span>
<a name="line-289"></a><span class='hs-comment'>-- Then, we could get the coefficients of the ARMA filter.</span>
<a name="line-290"></a><span class='hs-definition'>h2ARMACoef</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Num</span> <span class='hs-varid'>a</span><span class='hs-layout'>,</span> <span class='hs-conid'>Fractional</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span>
<a name="line-291"></a> <span class='hs-layout'>(</span><span class='hs-keyglyph'>[</span><span class='hs-varid'>a</span><span class='hs-keyglyph'>]</span><span class='hs-layout'>,</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>a</span><span class='hs-keyglyph'>]</span><span class='hs-layout'>)</span> <span class='hs-comment'>-- ^Coefficients in Z-domain</span>
<a name="line-292"></a> <span class='hs-keyglyph'>-></span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>[</span><span class='hs-varid'>a</span><span class='hs-keyglyph'>]</span><span class='hs-layout'>,</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>a</span><span class='hs-keyglyph'>]</span><span class='hs-layout'>)</span> <span class='hs-comment'>-- ^Coefficients of the ARMA filter</span>
<a name="line-293"></a><span class='hs-definition'>h2ARMACoef</span> <span class='hs-layout'>(</span><span class='hs-varid'>bs</span><span class='hs-layout'>,</span><span class='hs-keyword'>as</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-layout'>(</span><span class='hs-varid'>scalePolyCoef</span> <span class='hs-varid'>a0_1</span> <span class='hs-varid'>bs</span><span class='hs-layout'>,</span>
<a name="line-294"></a> <span class='hs-varid'>scalePolyCoef</span> <span class='hs-layout'>(</span><span class='hs-num'>0</span><span class='hs-comment'>-</span><span class='hs-varid'>a0_1</span><span class='hs-layout'>)</span> <span class='hs-varop'>$</span> <span class='hs-varid'>tail</span> <span class='hs-keyword'>as</span><span class='hs-layout'>)</span>
<a name="line-295"></a> <span class='hs-keyword'>where</span>
<a name="line-296"></a> <span class='hs-varid'>a0_1</span> <span class='hs-keyglyph'>=</span> <span class='hs-num'>1.0</span><span class='hs-varop'>/</span> <span class='hs-varid'>head</span> <span class='hs-keyword'>as</span>
<a name="line-297"></a>
<a name="line-298"></a><span class='hs-comment'>-- Helper functions</span>
<a name="line-299"></a>
<a name="line-300"></a><a name="map'"></a><span class='hs-definition'>map'</span> <span class='hs-keyglyph'>::</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-></span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>a</span><span class='hs-keyglyph'>-></span><span class='hs-varid'>b</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>-></span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>b</span><span class='hs-keyglyph'>]</span>
<a name="line-301"></a><span class='hs-definition'>map'</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>flip</span> <span class='hs-varop'>$</span> <span class='hs-varid'>sequence</span>
<a name="line-302"></a>
<a name="line-303"></a>
<a name="line-304"></a><a name="iprod"></a><span class='hs-comment'>-- |Computes the inner product.</span>
<a name="line-305"></a><span class='hs-definition'>iprod</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>b</span> <span class='hs-keyglyph'>=></span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>b</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>-></span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>b</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>b</span>
<a name="line-306"></a><span class='hs-definition'>iprod</span> <span class='hs-varid'>xs</span> <span class='hs-varid'>ys</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>sum</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>x</span><span class='hs-varop'>*</span><span class='hs-varid'>y</span> <span class='hs-keyglyph'>|</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-layout'>,</span> <span class='hs-varid'>y</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'><-</span> <span class='hs-varid'>zip</span> <span class='hs-varid'>xs</span> <span class='hs-varid'>ys</span><span class='hs-keyglyph'>]</span>
<a name="line-307"></a>
<a name="line-308"></a><a name="repeatN"></a><span class='hs-comment'>-- |Repeat an element for a given times.</span>
<a name="line-309"></a><span class='hs-definition'>repeatN</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Int</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-></span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>a</span><span class='hs-keyglyph'>]</span>
<a name="line-310"></a><span class='hs-definition'>repeatN</span> <span class='hs-varid'>n</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>take</span> <span class='hs-varid'>n</span> <span class='hs-varop'>.</span> <span class='hs-varid'>repeat</span>
<a name="line-311"></a>
<a name="line-312"></a><a name="fixedList"></a><span class='hs-comment'>-- |Maintain a fixed length of list like Fifo, except the outputs are ignored.</span>
<a name="line-313"></a><span class='hs-definition'>fixedList</span> <span class='hs-keyglyph'>::</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>a</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-></span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>a</span><span class='hs-keyglyph'>]</span>
<a name="line-314"></a><span class='hs-definition'>fixedList</span> <span class='hs-varid'>xs</span> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>take</span> <span class='hs-layout'>(</span><span class='hs-varid'>length</span> <span class='hs-varid'>xs</span><span class='hs-layout'>)</span> <span class='hs-varop'>$</span> <span class='hs-varid'>y</span><span class='hs-conop'>:</span><span class='hs-varid'>xs</span>
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distrib
>
Fedora
>
14
>
x86_64
>
media
>
updates
>
by-pkgid
>
a47f0719970f9f829128f311a437816d
>
files
>
342
