You want to look at the innovation sequence; the difference between the filtered value (the estimate or forecast) and the measured value. If you have a perfect filter that removes the noise, the innovation sequence should be white noise (Dirac impulse autocorrelation).
Here is some MATLAB code that illustrate these properties for some white sequences.
N = 1e5;
n1 = -1 + 2*rand(1,N); % uniform noise
n2 = n1 + -1 + 2*rand(1,N); % triangular noise
n3 = randn(1,N); % Gaussian noise
[c1,l1] = xcorr(n1); [c2,l2] = xcorr(n2); [c3,l3] = xcorr(n3);
figure(1), plot(l1,c1,l2,c2,l3,c3) # autocorrelation (Dirac delta)
figure(2), clf, hold on
pwelch(n1,kaiser(N/1000,10),N/2000) # power spectral density (constant)