Sinc Interpolation without using predefined function

I need to write a code that does linear interpolation. (without using interp1) My function has 3 inputs (n,x,n2). Vector n contains the sample points, and x contains the corresponding values, x(n). Vector n2 contains the coordinates of the query points. My code is this:

function y = sinc_interp(n,x,n2)

y = zeros(length(n2),1);

for i=1:length(n2)   
    y(i) = sum(x.*sinc(n2(i)- n));
end

end

I think the formula is ok (It is the Shannon formula). The problem is that for various input functions, it doesn't look good. For example:

a = 0.7;
N = 10;
n = 0:N;
x = n.*a.^n + 1;

n2 = linspace(0,10,1000);

figure(1)
plot(n2,n2.*a.^n2 + 1)

y3 = sinc_interp(n,x,n2);
figure(2)
plot(n2,y3)

Answer 1

[Sam Chak]

Hi @Radu-Andrei

Not an interpolation expert, but I believe your code didn't work correctly because the sampling frequency (at 1 Hz) is relatively low. Additionally, there appears to be a slight error in the implementation of the Whittaker–Shannon interpolation formula, as indicated on this Wikipedia page. Nevertheless, the code has been fixed, and you can adjust the sampling frequency to observe the interpolation effect.

a       = 0.7;              % parameter in continuous function
tFinal  = 10;
Fs      = 20;               % sampling frequency
T       = 1/Fs;

t  = 0:0.01:tFinal;         % for continuous function
ts = 0:T:tFinal;            % for sinc-interpolated

x  = @(t) t.*(a.^t) + 1;    % the continuous function
xn = x(ts);                 % the discrete sequence for sinc-interpolated

% call Whittaker–Shannon interpolation 
y  = sinc_interp(t, T, ts, xn);

plot(t, x(t), 'linewidth', 4, 'Color', '#0e6ea1'),
hold on, grid on 
plot(t, y, '.', 'MarkerSize', 3, 'Color', '#ed9c5a');
xlabel('t')
legend('x(t)', 'sinc')

%% Whittaker–Shannon interpolation Algorithm
function y = sinc_interp(t, T, ts, xn)
    y      = zeros(length(ts), 1);
    for n  = 1:length(ts)   
        y  = y + xn(n)*sinc((t - (n-1)*T)/T);
    end
end

[Radu-Andrei]

Thank you very much! :)