Adding a signal to a time series

how 1 go adding signal (such sin wave) time series?



instance, period occur every time step

 

if there's 1 period of sine wave @ every time step, contribution sine wave constant. value of constant depends on phase of sine wave relative time sample.


if above isn't clear, draw on graph paper. assume each vertical line on graph paper sample point. draw sine wave period identical vertical lines of graph paper (one cycle per space between vertical lines). value of sine wave @ each vertical line?

if vertical lines represent samples of time series, sine wave's value @ each sample point?

if change phase of sine wave respect vertical lines (sample points), keep sine's period @ 1 cycle per sample interval, changes? example, if phase 1/4-waveform (90 degrees), changes @ each sample point?

reference, "nyquist frequency" , "shannon sampling theorem", , see "aliasing" means dsp term.


if want add sine wave period less sampling frequency, use sin() function calculate based on ratio of sine frequency sampling frequency. example: sine wave of 100 hz, , sampling frequency of 1 khz, have 10 sample points per sine period. since sin() circular, means 2pi = 10 sine-wave samples.

equations aren't hard work out if apply basic algebra. think it's worthwhile know how develop equations, because give better sense of how signals related sampled data.

 

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