What is the right way to use sinc interpolation for a given discrete signal $x[n]$? Following is the sinc interpolation formula:
$$x(t) = \sum_{n=-\infty}^\infty x[n] \mathrm{sinc}\left(\frac{t-nT}{T}\right)$$
A simple explanation with a real world example of how and when to use a sinc interpolator will be very helpful.
I remember being asked to do something like this in a DSP course and it was confusing because I wasn't sure how to model the continuous variable $t$ in MATLAB where everything is digital. You can still make a nice demonstration of sinc interpolation by choosing a low sample rate and use interpolation to create the interpolated signal which is at a higher rate.
Here is a simple MATLAB script that does what I'm talking about. Hopefully it at least helps you see how to use $x[n]$ to create the interpolated signal.
% Create "continuous time" signal, Fc >> f Fc = 1e6; % very high sample rate to simulate "continuous time" Tc = 1/Fc; % sampling period t = (-0.1:Tc:0.1)'; % time axis f = 10; % signal frequency xc = sin(2*pi*f*t); % "continuous time" signal % Create sampled signal Fs = 100; % sampling rate Ts = 1/Fs; % sampling period ratio = round(Ts/Tc); tn = t(1:ratio:end); % sampled time axis xn = xc(1:ratio:end); % sampled signal % Plot the CT signal and sampled signal figure hold on grid on plot(t, xc) stem(tn, xn, 'o') legend('"Continuous time signal"', 'Sampled signal') % Create and plot sinc train sincTrain = zeros(length(t), length(xn)); nind = 1; figure cmap = colormap(jet(length(-floor(length(xn)/2):floor(length(xn)/2)))); ax = axes('colororder', cmap); hold on grid on plot(t, xc, 'k', 'LineWidth', 3) for n = -floor(length(xn)/2):floor(length(xn)/2) sincTrain(:, nind) = xn(nind)*sinc((t - n*Ts)/Ts); p = plot(t, sincTrain(:, nind), 'LineWidth', 2); stem(tn(nind), xn(nind), 'Color', p.Color, 'LineWidth', 2) nind = nind + 1; end xlabel('t') ylabel('Amplitude') set(gca, 'FontSize', 20, 'LineWidth', 3, 'FontWeight', 'bold') xr = sum(sincTrain, 2); % sum(sincTrain, 2) is the interpolated/reconstructed signal, should be equal to xc figure hold on grid on plot(t, xc) plot(t, xr) error = mean(abs(xc - xr).^2); 