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Signals | Systems | System Response | |||||
Frequency Representation | Fourier Series / Transforms | Background Material |
Most functions are given to us in the time domain. We are most familiar with trigonometric functions like:
y(t) = sin(t) or y(t) = cos(t), which can be generalized to y(t) = sin(2πft) or y(t) = cos(2πft) where
ω = 2πf.
(ω - omega - is the angular frequency with units [rad/s] and f - frequency - is the frequency with units of Hz [1/s])
Sometimes it is useful to be able to represent a function by its frequency content. A graph of the frequency domain plots frequency against the amplitude of a particular frequency (where 0 Hz or radians is DC - direct current).
Power Spectral Density
The power spectral density is an example of a graph of amplitude versus frequency. It is used to see the strength of each frequency for a particular system. Mathematically, it is the Fourier transform of the autocorrelation function.
Figure 2. PSD of systolic blood pressure and RR-interval in a young healthy subject.
(from http://www.cbi.dongnocchi.it/glossary/PowerSpectralDensity.html)
Fourier Series
Every periodic signal can be represented by a series of sines and cosines. As we increase the number of terms in the series, the signal will be approximated more and more accurately. Let's see how this works. Take a sawtooth, for example,
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