Upsampling Interpolation Illustration

pkg load signal

% big values take too long for unoptimized calculation, so
% for illustration, we will use 440 instead of 44100.

Fs = 44; % the sampling frequency

Fdisp = 64*Fs; % resolution of the continuous plot
F = 8*Fs; % the upsampling frequency
T = 1/Fs; % the sampling time

max = 5;  % generate and calculate from 0 to here
tdisp = 0:1/Fdisp:max; % the time-samples for "continuous" plotting
t = 0:1/F:max; % the time-samples for upsampled plotting
ts = 0:T:max;  % the time at the sample points

hz = 20; % the sine wave will be at this frequency
         % this must be < Fs/2 to avoid aliasing
x = @(t) sin(2*pi*hz*t);
xn = x(ts);  % the discrete sequence

% perform interpolation from xn to x using sinc:
interpolatedSinc = 0;
for n=1:length(ts)
    % n-1 due to the 1-based indexing of matlab
    interpolatedSinc = interpolatedSinc + xn(n) * sinc((t-(n-1)*T)/T);
end

# using zero order hold (that is, repeating the samples)
#  (truncated end to match t length)
interpolatedZOH =  repmat(xn(:)', F/Fs)(1:(length(xn)-1)*F/Fs+1);

# Connect-the-dots: Draw a line between 2 adjacent samples
interpolatedLine =  interp1(ts,xn,t);


graphT = 0.2; % plot this much time from the center

% the sine wave with the samples
subplot(4,1,1);
hold off;
plot(tdisp, x(tdisp), 'k'); hold on;  % plot the sine wave
stem(ts, xn);  hold on; % plot sampled
xlim([max/2-graphT/2,max/2+graphT/2]); % pick graphT from the middle
                                       %   (bec the edges are not bw-limited)
ylim([-1,1])

% using sinc function
subplot(4,1,2);
hold off;
stem(ts, xn);  hold on; % plot sampled
stem(t, interpolatedSinc, 'k:.'); % plot interpolated in black
xlim([max/2-graphT/2,max/2+graphT/2]); % pick graphT from the middle
                                       %   (bec the edges are not bw-limited)
ylim([-1,1])

% using ZOH
subplot(4,1,3);
hold off;
stem(ts, xn);  hold on; % plot sampled
stem(t, interpolatedZOH, 'k:.');
xlim([max/2-graphT/2,max/2+graphT/2]); % pick graphT from the middle
                                       %   (bec the edges are not bw-limited)
ylim([-1,1])


% using connect-the-dots
subplot(4,1,4);
hold off;
stem(ts, xn);  hold on; % plot sampled
stem(t, interpolatedLine, 'k:.');
xlim([max/2-graphT/2,max/2+graphT/2]); % pick graphT from the middle
                                       %   (bec the edges are not bw-limited)
ylim([-1,1])