Deploy book to GitHub Pages Travis build: 1328

Commit: 02dd35341f4b48ceaf73dde4350526c4b6b72a31

Author: Travis CI User

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/IFS/IFS.html

@@ -0,0 +1,113 @@
+
+(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)))
+         ;; Calculate coefficients for the odd indices.
+         (twiddle-factors (loop for i from 0 below (length odds)
+                             collect (coefficient 1.0 i fft-length)))
+         ;; Multiply values with coefficients.
+         (odd-terms (mapcar #'* odds twiddle-factors)))
+    ;; Combine the two FFTs.
+    (concatenate 'list 
+                 (mapcar #'+ evens odd-terms)
+                 (mapcar #'- evens odd-terms))))
+
+(defun cooley-tukey-rec (data)
+  "Performs the Fourier Transform using the recursive Cooley-Tukey method."
+  (if (<= (length data) 1)
+      data
+      (loop
+        for i from 0 below (length data)
+        ;; Split even and odd indexed elements into two seperate lists.
+        if (evenp i)
+          collect (elt data i) into evens
+        else
+          collect (elt data i) into odds
+        finally
+          ;; Calculate the Fourier Transform for the two smaller lists and
+          ;; combine them into the Fourier Transform of the full input.
+          (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
+      ;; Split bits into two parts.
+      (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))))
+        ;; Reverse the bits of each part, then swap the results.
+        (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)
+    ;; Take two elements which are stride apart and perform a butterfly on them.
+    (let* ((first-elt-index (+ start i))
+           (second-elt-index (+ start i stride))
+           (first-elt (elt data first-elt-index))
+           (second-elt (elt data second-elt-index))
+           (coeff (coefficient 1.0 i (* 2 stride))))
+    (multiple-value-bind (sum difference) (butterfly first-elt second-elt coeff)
+      ;; Write results back into the list.
+      (setf (elt data first-elt-index) sum)
+      (setf (elt data second-elt-index) difference)))))
+
+(defun cooley-tukey-iter (data)
+  "Performs the Fourier Transform using the iterative Cooley-Tukey method."
+  (loop
+    ;; Bit-shuffle indices.
+    with shuffled-data = (bit-shuffle-indices data)
+    for stride = 1 then (* 2 stride)
+    while (< stride (length shuffled-data))
+    do
+      ;; Compute butterfly groups for the current stride.
+      (loop for i from 0 below (length shuffled-data) by (* 2 stride) do
+        (butterfly-group shuffled-data i stride))
+    finally (return shuffled-data)))
+
+(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))