small changes to convolutional theorem chapter and code to make thing…

…s more reproduceable (#872)

contents/convolutions/convolutional_theorem/code/julia/convolutional_theorem.jl

@@ -1,4 +1,5 @@
 using FFTW
+using LinearAlgebra
 using DelimitedFiles
 
 # using the convolutional theorem
@@ -8,11 +9,9 @@ end
 
 function main()
 
-    # Random distribution in x
-    x = rand(100)
-
-    # Gaussian signals
-    y = [exp(-((i-50)/100)^2/.01) for i = 1:100]
+    # sawtooth functions for x and y
+    x = [float(i)/200 for i = 1:200]
+    y = [float(i)/200 for i = 1:200]
 
     # Normalization is not strictly necessary, but good practice
     normalize!(x)
@@ -22,6 +21,10 @@ function main()
     fft_output = convolve_fft(x, y)
 
     # outputting convolutions to different files for plotting in external code
-    writedlm("fft.dat", fft_output)
+    # note: we are outputting just the real component because the imaginary
+    #       component is virtually 0
+    writedlm("fft.dat", real(fft_output))
 
 end
+
+main()

contents/convolutions/convolutional_theorem/convolutional_theorem.md

@@ -39,11 +39,20 @@ Also note that the Fourier Transform is a periodic or cyclic operation, so there
 
 ## Example Code
 
+For this example code, we will be using two sawtooth functions as we did in the chapter on [one-dimensional convolutions](../1d/1d.md):
+
 {% method %}
 {% sample lang="jl" %}
 [import, lang:"julia"](code/julia/convolutional_theorem.jl)
 {% endmethod %}
 
+This should produce the following output:
+
+<p>
+    <img class="center" src="../res/cyclic.png" style="width:75%">
+</p>
+
+
 <script>
 MathJax.Hub.Queue(["Typeset",MathJax.Hub]);
 </script>
@@ -54,6 +63,11 @@ MathJax.Hub.Queue(["Typeset",MathJax.Hub]);
 
 The code examples are licensed under the MIT license (found in [LICENSE.md](https://github.com/algorithm-archivists/algorithm-archive/blob/master/LICENSE.md)).
 
+##### Images/Graphics
+
+- The image "[Cyclic](../res/cyclic.png)" was created by [James Schloss](https://github.com/leios) and is licensed under the [Creative Commons Attribution-ShareAlike 4.0 International License](https://creativecommons.org/licenses/by-sa/4.0/legalcode).
+
+
 ##### Text
 
 The text of this chapter was written by [James Schloss](https://github.com/leios) and is licensed under the [Creative Commons Attribution-ShareAlike 4.0 International License](https://creativecommons.org/licenses/by-sa/4.0/legalcode).