Added Cooley-Tukey in Lisp

Author: 0xJonas

CONTRIBUTORS.md

@@ -54,3 +54,4 @@ This file lists everyone, who contributed to this repo and wanted to show up her
 - Amaras
 - Jonathan Dönszelmann
 - Ishaan Verma
+- Delphi1024

contents/cooley_tukey/code/clisp/fft.lisp

@@ -0,0 +1,103 @@
+
+(defun coefficient (time-index freq-index dft-len)
+  "Calculates a single twiddle factor for the Fourier Transform."
+  (exp (- (/ (* #c( 0 1) 2.0 pi time-index freq-index)
+             dft-len))))
+
+(defun dft (data)
+  "Performs the Discrete Fourier Transform"
+  (let ((dft-len (length data)))
+    (loop for freq-index from 0 below dft-len collect
+      (loop for time-index from 0 below dft-len sum
+        (* (coefficient time-index freq-index dft-len) (elt data time-index))))))
+
+(defun merge-sub-ffts (evens odds)
+  "Combines the FFTs of the even and odd indices"
+  (let* ((fft-length (+ (length evens) (length odds)))
+         (indices (loop for i from 0 below (length odds) collect i))
+         (odd-terms (mapcar #'(lambda (v i) (* v (coefficient 1.0 i fft-length)))
+                            odds
+                            indices)))
+    (concatenate 'list 
+                 (mapcar #'+ evens odd-terms)
+                 (mapcar #'- evens odd-terms))))
+
+(defun cooley-tukey-rec (data)
+  "Performs the DFT using the recursive Cooley-Tukey method."
+  (if (<= (length data) 1)
+    data
+    (loop for i from 0 below (length data)
+      if (evenp i)
+        collect (elt data i) into evens
+      else
+        collect (elt data i) into odds
+      finally
+        (return (merge-sub-ffts (cooley-tukey-rec evens)
+                                (cooley-tukey-rec odds))))))
+
+(defun reverse-bits (value num-bits)
+  "Reverses the bits of a value"
+  (if (= num-bits 1)
+    value
+    (let* ((num-low-bits (floor (/ num-bits 2)))
+           (num-high-bits (- num-bits num-low-bits))
+           (bit-mask (- (expt 2 num-low-bits) 1))
+           (lower-half (logand value bit-mask))
+           (upper-half (ash value (- num-low-bits))))
+      (logior (ash (reverse-bits lower-half num-low-bits) num-high-bits)
+              (reverse-bits upper-half num-high-bits)))))
+
+(defun bit-shuffle-indices (data)
+  "Rearanges the elements in a list according to their bit-reversed indices."
+  (loop 
+    with num-bits = (floor (log (length data) 2)) 
+    for i from 0 below (length data)
+    collect (elt data (reverse-bits i num-bits))))
+
+(defun butterfly (a b coeff)
+  "Calculates a single butterfly."
+  (values (+ a (* coeff b)) (- a (* coeff b))))
+
+(defun butterfly-group (data start stride)
+  "Calculates a single group of butterflies."
+  (dotimes (i stride)
+    (let* ((a-index (+ start i))
+           (b-index (+ start i stride))
+           (a (elt data a-index))
+           (b (elt data b-index))
+           (coeff (coefficient 1.0 i (* 2 stride))))
+    (multiple-value-bind (sum difference) (butterfly a b coeff)
+      (setf (elt data a-index) sum)
+      (setf (elt data b-index) difference)))))
+
+(defun cooley-tukey-tailrec (data stride)
+  "Actual iterative implementation of the Cooley-Tukey method."
+  (if (>= stride (length data))
+    data
+    (progn
+      (loop for i from 0 below (length data) by (* 2 stride) do
+        (butterfly-group data i stride))
+      (cooley-tukey-tailrec data (* 2 stride)))))
+
+(defun cooley-tukey-iter (data)
+  "Performs the DFT using the iterative Cooley-Tukey method."
+  (cooley-tukey-tailrec (bit-shuffle-indices data) 1))
+
+(defun approx-eql (list1 list2)
+  (let ((diffs (mapcar #'(lambda (e1 e2) (abs (- e1 e2)))
+                       list1
+                       list2)))
+    (loop for d in diffs always (< d 1e-9))))
+
+(defun test-fft (data)
+  (let ((dft-result (dft data))
+        (rec-result (cooley-tukey-rec data))
+        (iter-result (cooley-tukey-iter data)))
+    (format T "~&DFT and recursive Cooley-Tukey approx. equal: ~a" 
+              (approx-eql dft-result rec-result))
+    (format T "~&DFT and iterative Cooley-Tukey approx. equal: ~a"
+              (approx-eql dft-result iter-result))
+    (format T "~&Recursive Cooley-Tukey and iterative Cooley-Tukey approx. equal: ~a" 
+              (approx-eql rec-result iter-result))))
+
+(test-fft '(0.0 0.25 0.5 0.75 0.0 -0.25 -0.5 -0.75))

contents/cooley_tukey/cooley_tukey.md

@@ -89,6 +89,8 @@ For some reason, though, putting code to this transformation really helped me fi
 [import:3-15, lang:"javascript"](code/javascript/fft.js)
 {% sample lang="rs" %}
 [import:24-37, lang:"rust"](code/rust/fft.rs)
+{% sample lang="lisp" %}
+[import:2-12, lang:"lisp"](code/clisp/fft.lisp)
 {% endmethod %}
 
 In this function, we define `n` to be a set of integers from $$0 \rightarrow N-1$$ and arrange them to be a column.
@@ -142,6 +144,8 @@ In the end, the code looks like:
 [import:17-39, lang="javascript"](code/javascript/fft.js)
 {% sample lang="rs" %}
 [import:39-55, lang:"rust"](code/rust/fft.rs)
+{% sample lang="lisp" %}
+[import:14-36, lang:"lisp"](code/clisp/fft.lisp)
 {% endmethod %}
 
 As a side note, we are enforcing that the array must be a power of 2 for the operation to work.
@@ -255,6 +259,8 @@ Some rather impressive scratch code was submitted by Jie and can be found here:
 [import, lang:"javascript"](code/javascript/fft.js)
 {% sample lang="rs" %}
 [import, lang:"rust"](code/rust/fft.rs)
+{% sample lang="lisp" %}
+[import, lang:"lisp"](code/clisp/fft.lisp)
 {% endmethod %}
 
 <script>