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To illustrate what both @Jazzmaniac and @Phonon are telling you, let's look at the same plot, but for different window lengths. Another change is to look at the plots using the fftshift view --- so t …

It's certainly calculating the right thing. Though instead of sum(x.*(cos(1000*2*pi*t)-i*sin(2*pi*1000*t)))*2/N; you might try sum(x.*exp(-i*2*pi*1000*t))*2/N; If you need to do something simil …

Another take: The Kalman Filter lets you add more information about how the system you're filtering works. In other words, you can use a signal model to improve the output of the filter. Sure, a mov …

EDIT The OP mentions below Deve's answer that this was not the "Generalized Cross Correlation" algorithm referred to. The real one seems to be this one: This paper seems to give equations about h …

The first algorithm that springs to mind is the Goertzel Algorithm. That algorithm usually assumes that the frequency of interest is an integer multiple of the fundamental frequency. However, this pap …

The usual approach to change detection is the CUSUM algorithm. I've done an implementation that just addresses the level (mean) change issue. It's included (in R) below. The black line is the noise- …

When I had a look at rotating machinery, the best reference I could find is Bob Randall's Frequency Analysis. This was generated in conjunction with Brüel & Kjær as they sold lots of nice (and expens …

The MRI system is taking two sets of measurements: the in-phase and quadrature measurements. Both measurements are real. There is no way for a physical system to measure $i5 V$, we measure $5 V$. The …

The key idea is that the random sampling approach enforces more constraints on the resulting signal than the uniform sampling approach does. The POCS (projections onto convex sets) algorithm used for …

Start simple: just use a 1-D median filter of an appropriate length. If I do that with a length of 100 samples, I get the following for your first signal. The top plot shows the original signal (blue …

For most purposes a mixer is a multiplier. The issue regarding sum / difference frequencies is that when you multiply two signals of different frequencies, $\omega_1$ and $\omega_2$ say, then you get …

Jim's answer sent me to think about how to test this statistically. This led me to the Durbin-Watson autocorrelation test. The generalization of it is to form: $$ DW(\tau) = \frac{\displaystyle \su …

Jim Clay's answer is a good one, but I thought I'd expand a little on point 3 about periodicity. A signal, $x\{s\}$, is said to be periodic with period $P$ if $$ x\{s\} = x\{s + P\} $$ For the conti …

For question 1: apply the definition of time invariant: find the output as normal; find the output with the same input but delayed by $T$ $$ y_1(t) = \frac{dx(t)}{dt}\\ y_2(t) = \frac{dx(t-T)}{dt}\\ …

Instantaneous amplitude is usually done in conjunction with analytic signals and the Hilbert transform. See this answer on this site. For signal $s(t)$, given its Hilbert Transform $\hat{s}(t)$ it …