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<pre><a name="line-1"></a><span class='hs-comment'>{-# LANGUAGE ScopedTypeVariables, FlexibleContexts #-}</span>
<a name="line-2"></a><span class='hs-comment'>-----------------------------------------------------------------------------</span>
<a name="line-3"></a><span class='hs-comment'>-- |</span>
<a name="line-4"></a><span class='hs-comment'>-- Module : ForSyDe.DFT</span>
<a name="line-5"></a><span class='hs-comment'>-- Copyright : (c) SAM Group, KTH/ICT/ECS 2007-2008</span>
<a name="line-6"></a><span class='hs-comment'>-- License : BSD-style (see the file LICENSE)</span>
<a name="line-7"></a><span class='hs-comment'>-- </span>
<a name="line-8"></a><span class='hs-comment'>-- Maintainer : forsyde-dev@ict.kth.se</span>
<a name="line-9"></a><span class='hs-comment'>-- Stability : experimental</span>
<a name="line-10"></a><span class='hs-comment'>-- Portability : portable</span>
<a name="line-11"></a><span class='hs-comment'>--</span>
<a name="line-12"></a><span class='hs-comment'>-- This module includes the standard Discrete Fourier Transform (DFT)</span>
<a name="line-13"></a><span class='hs-comment'>-- function, and a fast Fourier transform (FFT) algorithm, for</span>
<a name="line-14"></a><span class='hs-comment'>-- computing the DFT, when the input vectors' length is a power of 2.</span>
<a name="line-15"></a><span class='hs-comment'>-----------------------------------------------------------------------------</span>
<a name="line-16"></a><span class='hs-keyword'>module</span> <span class='hs-conid'>ForSyDe</span><span class='hs-varop'>.</span><span class='hs-conid'>DFT</span><span class='hs-layout'>(</span><span class='hs-varid'>dft</span><span class='hs-layout'>,</span> <span class='hs-varid'>fft</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-17"></a>
<a name="line-18"></a>
<a name="line-19"></a><span class='hs-keyword'>import</span> <span class='hs-keyword'>qualified</span> <span class='hs-conid'>Data</span><span class='hs-varop'>.</span><span class='hs-conid'>Param</span><span class='hs-varop'>.</span><span class='hs-conid'>FSVec</span> <span class='hs-keyword'>as</span> <span class='hs-conid'>V</span>
<a name="line-20"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Data</span><span class='hs-varop'>.</span><span class='hs-conid'>Param</span><span class='hs-varop'>.</span><span class='hs-conid'>FSVec</span>
<a name="line-21"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Data</span><span class='hs-varop'>.</span><span class='hs-conid'>TypeLevel</span> <span class='hs-layout'>(</span><span class='hs-conid'>Nat</span><span class='hs-layout'>,</span> <span class='hs-conid'>IsPowOf</span><span class='hs-layout'>,</span> <span class='hs-conid'>D2</span><span class='hs-layout'>)</span>
<a name="line-22"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Data</span><span class='hs-varop'>.</span><span class='hs-conid'>Complex</span>
<a name="line-23"></a>
<a name="line-24"></a>
<a name="line-25"></a><a name="dft"></a><span class='hs-comment'>-- | The function 'dft' performs a standard Discrete Fourier Transformation</span>
<a name="line-26"></a><span class='hs-definition'>dft</span> <span class='hs-keyglyph'>::</span> <span class='hs-keyword'>forall</span> <span class='hs-varid'>s</span> <span class='hs-varop'>.</span> <span class='hs-conid'>Nat</span> <span class='hs-varid'>s</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>FSVec</span> <span class='hs-varid'>s</span> <span class='hs-layout'>(</span><span class='hs-conid'>Complex</span> <span class='hs-conid'>Double</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>FSVec</span> <span class='hs-varid'>s</span> <span class='hs-layout'>(</span><span class='hs-conid'>Complex</span> <span class='hs-conid'>Double</span><span class='hs-layout'>)</span>
<a name="line-27"></a><span class='hs-definition'>dft</span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>V</span><span class='hs-varop'>.</span><span class='hs-varid'>map</span> <span class='hs-layout'>(</span><span class='hs-varid'>bigX_k</span> <span class='hs-varid'>v</span><span class='hs-layout'>)</span> <span class='hs-varid'>nVector</span>
<a name="line-28"></a> <span class='hs-keyword'>where</span>
<a name="line-29"></a> <span class='hs-varid'>lT</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>V</span><span class='hs-varop'>.</span><span class='hs-varid'>lengthT</span> <span class='hs-varid'>v</span>
<a name="line-30"></a> <span class='hs-varid'>lV</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>V</span><span class='hs-varop'>.</span><span class='hs-varid'>genericLength</span> <span class='hs-varid'>v</span>
<a name="line-31"></a> <span class='hs-comment'>-- FIXME: dft does not type-check without this type signature:</span>
<a name="line-32"></a> <span class='hs-comment'>-- learn why!</span>
<a name="line-33"></a> <span class='hs-varid'>nVector</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>FSVec</span> <span class='hs-varid'>s</span> <span class='hs-varid'>a</span>
<a name="line-34"></a> <span class='hs-varid'>nVector</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>kVector</span> <span class='hs-varid'>lT</span>
<a name="line-35"></a> <span class='hs-varid'>fullCircle</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>V</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'>n</span> <span class='hs-keyglyph'>-></span> <span class='hs-comment'>-</span><span class='hs-num'>2</span><span class='hs-varop'>*</span><span class='hs-varid'>pi</span><span class='hs-varop'>*</span><span class='hs-varid'>n</span><span class='hs-varop'>/</span><span class='hs-varid'>lV</span><span class='hs-layout'>)</span> <span class='hs-varid'>nVector</span>
<a name="line-36"></a> <span class='hs-varid'>bigX_k</span> <span class='hs-varid'>v</span> <span class='hs-varid'>k</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>sumV</span> <span class='hs-layout'>(</span><span class='hs-conid'>V</span><span class='hs-varop'>.</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-varid'>v</span> <span class='hs-layout'>(</span><span class='hs-varid'>bigW</span> <span class='hs-varid'>k</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-37"></a> <span class='hs-varid'>bigW</span> <span class='hs-varid'>k</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>V</span><span class='hs-varop'>.</span><span class='hs-varid'>map</span> <span class='hs-layout'>(</span><span class='hs-varop'>**</span> <span class='hs-varid'>k</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-conid'>V</span><span class='hs-varop'>.</span><span class='hs-varid'>map</span> <span class='hs-varid'>cis</span> <span class='hs-varid'>fullCircle</span><span class='hs-layout'>)</span>
<a name="line-38"></a> <span class='hs-varid'>sumV</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>V</span><span class='hs-varop'>.</span><span class='hs-varid'>foldl</span> <span class='hs-layout'>(</span><span class='hs-varop'>+</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-num'>0</span><span class='hs-conop'>:+</span> <span class='hs-num'>0</span><span class='hs-layout'>)</span>
<a name="line-39"></a>
<a name="line-40"></a>
<a name="line-41"></a><a name="fft"></a><span class='hs-comment'>-- | The function 'fft' implements a fast Fourier transform (FFT) algorithm, </span>
<a name="line-42"></a><span class='hs-comment'>-- for computing the DFT, when the size N is a power of 2.</span>
<a name="line-43"></a><span class='hs-definition'>fft</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Nat</span> <span class='hs-varid'>s</span><span class='hs-layout'>,</span> <span class='hs-conid'>IsPowOf</span> <span class='hs-conid'>D2</span> <span class='hs-varid'>s</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span>
<a name="line-44"></a> <span class='hs-conid'>FSVec</span> <span class='hs-varid'>s</span> <span class='hs-layout'>(</span><span class='hs-conid'>Complex</span> <span class='hs-conid'>Double</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>FSVec</span> <span class='hs-varid'>s</span> <span class='hs-layout'>(</span><span class='hs-conid'>Complex</span> <span class='hs-conid'>Double</span><span class='hs-layout'>)</span>
<a name="line-45"></a><span class='hs-definition'>fft</span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>V</span><span class='hs-varop'>.</span><span class='hs-varid'>map</span> <span class='hs-layout'>(</span><span class='hs-varid'>bigX</span> <span class='hs-varid'>v</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>kVector</span> <span class='hs-varid'>lT</span><span class='hs-layout'>)</span>
<a name="line-46"></a> <span class='hs-keyword'>where</span> <span class='hs-varid'>lT</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>V</span><span class='hs-varop'>.</span><span class='hs-varid'>lengthT</span> <span class='hs-varid'>v</span>
<a name="line-47"></a>
<a name="line-48"></a><a name="kVector"></a><span class='hs-definition'>kVector</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'>Nat</span> <span class='hs-varid'>s</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-varid'>s</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>FSVec</span> <span class='hs-varid'>s</span> <span class='hs-varid'>a</span>
<a name="line-49"></a><span class='hs-definition'>kVector</span> <span class='hs-varid'>s</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>V</span><span class='hs-varop'>.</span><span class='hs-varid'>iterate</span> <span class='hs-varid'>s</span> <span class='hs-layout'>(</span><span class='hs-varop'>+</span><span class='hs-num'>1</span><span class='hs-layout'>)</span> <span class='hs-num'>0</span>
<a name="line-50"></a>
<a name="line-51"></a>
<a name="line-52"></a><a name="bigX"></a><span class='hs-definition'>bigX</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Nat</span> <span class='hs-varid'>s</span><span class='hs-layout'>,</span> <span class='hs-conid'>IsPowOf</span> <span class='hs-conid'>D2</span> <span class='hs-varid'>s</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span>
<a name="line-53"></a> <span class='hs-conid'>FSVec</span> <span class='hs-varid'>s</span> <span class='hs-layout'>(</span><span class='hs-conid'>Complex</span> <span class='hs-conid'>Double</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Integer</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Complex</span> <span class='hs-conid'>Double</span>
<a name="line-54"></a><span class='hs-comment'>-- since there are no output length constraints (no vector is being returned)</span>
<a name="line-55"></a><span class='hs-comment'>-- we can simply obtain the list inside the vector and work with it directly</span>
<a name="line-56"></a><span class='hs-definition'>bigX</span> <span class='hs-varid'>v</span> <span class='hs-varid'>k</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>bigX'</span> <span class='hs-layout'>(</span><span class='hs-conid'>V</span><span class='hs-varop'>.</span><span class='hs-varid'>genericLength</span> <span class='hs-varid'>v</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-conid'>V</span><span class='hs-varop'>.</span><span class='hs-varid'>fromVector</span> <span class='hs-varid'>v</span><span class='hs-layout'>)</span> <span class='hs-varid'>k</span>
<a name="line-57"></a> <span class='hs-keyword'>where</span> <span class='hs-varid'>bigX'</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Integer</span> <span class='hs-keyglyph'>-></span> <span class='hs-keyglyph'>[</span><span class='hs-conid'>Complex</span> <span class='hs-conid'>Double</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Integer</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Complex</span> <span class='hs-conid'>Double</span>
<a name="line-58"></a> <span class='hs-comment'>-- The first argument is the length of the list (bigN)</span>
<a name="line-59"></a> <span class='hs-varid'>bigX'</span> <span class='hs-keyword'>_</span> <span class='hs-layout'>(</span><span class='hs-varid'>x0</span><span class='hs-conop'>:</span><span class='hs-keyglyph'>[</span><span class='hs-varid'>x1</span><span class='hs-keyglyph'>]</span><span class='hs-layout'>)</span> <span class='hs-varid'>k</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>even</span> <span class='hs-varid'>k</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>x0</span> <span class='hs-varop'>+</span> <span class='hs-varid'>x1</span> <span class='hs-varop'>*</span> <span class='hs-varid'>bigW</span> <span class='hs-num'>2</span> <span class='hs-num'>0</span>
<a name="line-60"></a> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>odd</span> <span class='hs-varid'>k</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>x0</span> <span class='hs-comment'>-</span> <span class='hs-varid'>x1</span> <span class='hs-varop'>*</span> <span class='hs-varid'>bigW</span> <span class='hs-num'>2</span> <span class='hs-num'>0</span>
<a name="line-61"></a> <span class='hs-varid'>bigX'</span> <span class='hs-varid'>l</span> <span class='hs-varid'>xs</span> <span class='hs-varid'>k</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>bigF_even</span> <span class='hs-varop'>+</span> <span class='hs-varid'>bigF_odd</span> <span class='hs-varop'>*</span> <span class='hs-varid'>bigW</span> <span class='hs-varid'>l</span> <span class='hs-varid'>k</span>
<a name="line-62"></a> <span class='hs-keyword'>where</span> <span class='hs-varid'>bigF_even</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>bigX'</span> <span class='hs-varid'>halfl</span> <span class='hs-layout'>(</span><span class='hs-varid'>evens</span> <span class='hs-varid'>xs</span><span class='hs-layout'>)</span> <span class='hs-varid'>k</span>
<a name="line-63"></a> <span class='hs-varid'>bigF_odd</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>bigX'</span> <span class='hs-varid'>halfl</span> <span class='hs-layout'>(</span><span class='hs-varid'>odds</span> <span class='hs-varid'>xs</span><span class='hs-layout'>)</span> <span class='hs-varid'>k</span>
<a name="line-64"></a> <span class='hs-varid'>halfl</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>l</span> <span class='hs-varop'>`div`</span> <span class='hs-num'>2</span>
<a name="line-65"></a>
<a name="line-66"></a><a name="bigW"></a><span class='hs-definition'>bigW</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Integer</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Integer</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Complex</span> <span class='hs-conid'>Double</span>
<a name="line-67"></a><span class='hs-definition'>bigW</span> <span class='hs-varid'>bigN</span> <span class='hs-varid'>k</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>cis</span> <span class='hs-layout'>(</span><span class='hs-comment'>-</span><span class='hs-num'>2</span> <span class='hs-varop'>*</span> <span class='hs-varid'>pi</span> <span class='hs-varop'>*</span> <span class='hs-layout'>(</span><span class='hs-varid'>fromInteger</span> <span class='hs-varid'>k</span><span class='hs-layout'>)</span> <span class='hs-varop'>/</span> <span class='hs-layout'>(</span><span class='hs-varid'>fromInteger</span> <span class='hs-varid'>bigN</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-68"></a>
<a name="line-69"></a><a name="evens"></a><span class='hs-definition'>evens</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>
<a name="line-70"></a><span class='hs-definition'>evens</span> <span class='hs-conid'>[]</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>[]</span>
<a name="line-71"></a><span class='hs-definition'>evens</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>v1</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>v1</span><span class='hs-keyglyph'>]</span>
<a name="line-72"></a><span class='hs-definition'>evens</span> <span class='hs-layout'>(</span><span class='hs-varid'>v1</span><span class='hs-conop'>:</span><span class='hs-keyword'>_</span><span class='hs-conop'>:</span><span class='hs-varid'>v</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>v1</span> <span class='hs-conop'>:</span> <span class='hs-varid'>evens</span> <span class='hs-varid'>v</span>
<a name="line-73"></a>
<a name="line-74"></a><a name="odds"></a><span class='hs-definition'>odds</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>
<a name="line-75"></a><span class='hs-definition'>odds</span> <span class='hs-conid'>[]</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>[]</span>
<a name="line-76"></a><span class='hs-definition'>odds</span> <span class='hs-keyglyph'>[</span><span class='hs-keyword'>_</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>[]</span>
<a name="line-77"></a><span class='hs-definition'>odds</span> <span class='hs-layout'>(</span><span class='hs-keyword'>_</span><span class='hs-conop'>:</span><span class='hs-varid'>v2</span><span class='hs-conop'>:</span><span class='hs-varid'>v</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>v2</span> <span class='hs-conop'>:</span> <span class='hs-varid'>odds</span> <span class='hs-varid'>v</span>
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<a name="line-79"></a>
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