File:Spectral leakage caused by "windowing".svg

pkg load signal
% Options
  frame_background_gray = true;

  if frame_background_gray
   graphics_toolkit("qt")         % or graphics_toolkit("fltk")
   frame_background = .94*[1 1 1];
   d = 4;                         % amount to add to text sizes
   ds = 8;                        % amount to small marker size
   dl = 12;                       % amount to large marker size
  else
   graphics_toolkit("gnuplot")    % background will be white regardless of value below
   frame_background = .94*[1 1 1];
   d=0; ds = 0;; dl = 0
  endif

% (https://octave.org/doc/v4.2.1/Graphics-Object-Properties.html#Graphics-Object-Properties)
% Speed things up when using Gnuplot
  set(0, "DefaultAxesFontsize",10+d)
  set(0, "DefaultTextFontsize",12+d)
  set(0, "DefaultAxesYlim",[-2 2])
  set(0, "DefaultAxesYtick",[])
  set(0, "DefaultAxesXtick",[])
  set(0, "DefaultFigureColor",frame_background)
  set(0, "DefaultAxesColor","white")

%=======================================================
samples_per_DFT  = 64;
DFT_display_bins = samples_per_DFT/2;
samples_per_DTFT = 1024;
Hz_per_bin = 1;                                 % Set the DFT bin spacing to 1 Hz
samples_per_sec = Hz_per_bin*samples_per_DFT;   % corresponding sample rate
I = 8;                                          % over-sample the display version of sinusoids

cycles_per_DFT = 13;
cycles_per_sample = cycles_per_DFT/(I*samples_per_DFT);
signal1I = sin(2*pi*cycles_per_sample*(-32*I:96*I));    % pad the [0,64] window with ±32
window1 = signal1I(32*I + (1:samples_per_DFT*I+1));     % extract the window
signal1 = window1(1:I:samples_per_DFT*I);               % decimate the over-sample rate

cycles_per_DFT = 13.50;                                 % repeat steps for higher frequency
cycles_per_sample = cycles_per_DFT/(I*samples_per_DFT);
signal2I = sin(2*pi*cycles_per_sample*(-32*I:96*I));
window2 = signal2I(32*I + (1:samples_per_DFT*I+1)); 
signal2 = window2(1:I:samples_per_DFT*I);

filter = zeros(1,length(signal1I));             % depict a rectangular window
skirt = 10;
filter(32*I + (-skirt+1:samples_per_DFT*I+skirt)) = 1.2;

%=======================================================
hfig = figure("position",[1 -89 1200 800], "color",frame_background);

x1 = .08;               % left margin for annotation
x2 = .02;               % right margin
ws = .05;               % whitespace between plots
y1 = .08                % bottom margin
y2 = .08                % top margin
dy = .08;               % vertical space between rows
height = (1-y1-y2-3*dy)/4;    % space allocated for each of 4 rows

% Compute width of sinusoid plots
xwhite = x1 + ws + ws + x2;     % total whitespace in row containing 3 plots
width1 = (1-xwhite)/3;          % width of all sinusoid plots

y_origin = 1;           % start at top of graph area
%=======================================================
% Plot the unwindowed sinusoid

x_origin = x1;
y_origin = y_origin -y2 -height;        % position of top row
subplot("position",[x_origin y_origin width1 height])

plot((-32*I:96*I), signal1I, "color","black")
xlim([-32 96]*I)   %ylim([-2 2])
set(gca, "xaxislocation","origin")
xlabel("unwindowed")

%=======================================================
% Plot the 13-cycle sinusoid & rectangular window

x_origin = x_origin + width1 + ws;
subplot("position",[x_origin y_origin width1 height])

plot(0:length(window1)-1, window1, "color","blue")
xlim([-32 96]*I)    %ylim([-2 2])
set(gca, "xaxislocation","origin")
hold on
plot((-32*I:96*I), filter, "color","black", "linewidth",2)
xlabel("13 cycles")

%=======================================================
% Plot the 13½-cycle sinusoid & rectangular window

x_origin = x_origin + width1 + ws;
subplot("position",[x_origin y_origin width1 height])

plot(0:length(window2)-1, window2, "color","red")
xlim([-32 96]*I)    %ylim([-2 2])
set(gca, "xaxislocation","origin")
hold on
plot((-32*I:96*I), filter, "color","black", "linewidth",2)
%xlabel("13½ cycles")    % doesn't work

%=======================================================
% Compute and plot Fourier transforms of the 2 sinusoids

x_origin = x1;
y_origin = y_origin -dy -height;
width2   = 1 -x1 -x2;
subplot("position",[x_origin y_origin width2 height])

N = samples_per_DTFT;
Hz_per_bin = samples_per_sec/N;
S1 = abs(fft(signal1,N));
S1 = 20*log10(S1(1:N/2));
S1 = max(0,S1);

plot((0:length(S1)-1)*Hz_per_bin, S1, "color","blue", "linewidth",1);
ylim([0 max(S1)+6])
xlim([0 samples_per_sec/2])

  set(gca, "xtick",0:DFT_display_bins)
% set(gca, "ytick",0:10:ylim(2))    % no, no, no.  It redefines ylim.
  set(gca, "ytick",0:10:(max(S1)+6))
  hold on

S2 = abs(fft(signal2,N));
S2 = 20*log10(S2(1:N/2));
S2 = max(0,S2);

plot((0:length(S2)-1)*Hz_per_bin, S2, "color","red", "linewidth",1);

% Insert delta function for unwindowed transform
stem(13, 35, "^", "MarkerSize",5+ds, "linewidth",2, "color","black", "markeredgecolor","black", "markerfacecolor","black")

ylabel("decibels")
xlabel('\leftarrow  frequency  \rightarrow')

%=======================================================
% Replot 13-cycle sinusoid

y_origin = y_origin -dy -height;
subplot("position",[x_origin y_origin width1 height])

plot(0:length(window1)-1, window1, "color","blue")
xlim([0 length(window1)-1])    %ylim([-2 2])
set(gca, "xaxislocation","origin")
hold on

% Overlay continuous sinusoid with discrete samples
plot(0:I:samples_per_DFT*I-1, signal1, "color","blue", ".", "MarkerSize",5+ds)
xlabel("discrete-time (sampled)")

%=======================================================
% Re-plot the Fourier transform, but truncate it to fit a smaller space

x_origin = x_origin + width1 + ws;
width3 = 1 - x_origin -x2;
subplot("position",[x_origin y_origin width3 height])

Hz_per_bin = samples_per_sec/samples_per_DTFT;
% DFT_display_bins = 32.  Truncate plot to 22  bins:
N = 22.5/32*length(S1);

plot((0:N-1)*Hz_per_bin, S1(1:N), "color","blue", "linewidth",1);
ylim([0 max(S1)+6])
xlim([0 N-1]*Hz_per_bin);
set(gca, "xtick",0:22)
hold on

% Compute and overlay the discrete DFT values
N = samples_per_DFT;
Hz_per_bin = samples_per_sec/N;
S = abs(fft(signal1,N));
S = 20*log10(S(1:N/2));
S = max(0,S);
N = 23;

plot((0:N-1)*Hz_per_bin, S(1:N), "color","blue", ".", "MarkerSize",10+dl);
set(gca, "xtick",(0:N-1)*Hz_per_bin)

%=======================================================
% Replot 13½-cycle sinusoid

x_origin = x1;
y_origin = y_origin -dy -height;
subplot("position",[x_origin y_origin width1 height])

plot(0:length(window2)-1, window2, "color","red")
xlim([0 length(window2)-1])    %ylim([-2 2])
set(gca, "xaxislocation","origin")
hold on

% Overlay continuous sinusoid with discrete samples
plot(0:I:samples_per_DFT*I-1, signal2, "color","red", ".", "MarkerSize",5+ds)
xlabel("discrete-time (sampled)")

%=======================================================
% Re-plot the Fourier transform, but truncate it to fit a smaller space

x_origin = x_origin + width1 + ws;
subplot("position",[x_origin y_origin width3 height])

Hz_per_bin = samples_per_sec/samples_per_DTFT;
% DFT_display_bins = 32.  Truncate plot to 22  bins:
N = 22.5/32*length(S2);

plot((0:N-1)*Hz_per_bin, S2(1:N), "color","red", "linewidth",1);
ylim([0 max(S2)+6])
xlim([0 (N-1)*Hz_per_bin])
set(gca, "xtick",0:22)
hold on

% Compute and overlay the discrete DFT values
N = samples_per_DFT;
Hz_per_bin = samples_per_sec/N;
S = abs(fft(signal2,N));
S = 20*log10(S(1:N/2));
S = max(0,S);
N = 23;

plot((0:N-1)*Hz_per_bin, S(1:N), "color","red", ".", "MarkerSize",10+dl);
set(gca, "xtick",(0:N-1)*Hz_per_bin)
xlabel("DFT bins")