Unit 5 - Fourier Analysis of CT Signals + LTI Systems

Fourier Analysis

one of the most widely used techniques in signal processing; about representing signals as sums of pure sinusoids of different frequencies (e^jwt); merely a special case of LT analysis in which sigma = 0

Fourier Analysis allows for?

convenient visualization of the frequency or “spectral” content of a signal

Spectra

basically a plot of a signal vs frequency instead of time; used to analyze bio signals

Sinotrial Node

the heart’s pacemaker; innervated w/ parasympathetic and sympathetic fibers for neural heart rate control

Spectra Analysis of HR(t)

allows selective assessment of the integrity of the parasympathetic and sympathetic nervous systems

Fourier analysis is simplier than what?

LT bc no ROC so can be applied to causal or non-causal signal, symmetry properties can be exploited in FA

Fourier vs LT

fourier: 1D (w), simplier bc no ROC, applied to causal or non-causal signals, sinusoids instead, and studies of signals

LT: 2D (sigma + w), ROC, applied to just causal signals, permits frequencies domain analsysis of exponentially-growing signals, and study of unstable systems

Fourier Analysis is not applicable to what?

exponentially-growing signlas and unstable systems no matter how sinusoids are added, a growing signal is never possible

Fourier analysis is preferred for what?

when teh signals for study are bounded

Basis Set of Vectors

any vector can be represented as a sum of vectors which span the entire vectors space

Orthogonal

vectors perpendicualr to each other

Signals may be similarly represented as what?

a sum of “orthoginal basis of signals”; ex. representing signals as a sum of shifted impulse fcns (convolution and output to any input) and representing signals as sums of sinusoids (multiplication and input limited)

Fourier Series (FS)

an expansion of usuallly a periodic signal on a set of oethogonal basis signals; individual coefficients contain teh amplitude and phase information for thier respective basis signals

Orthogonal Series Expansion

a set of signals orthogonal signal set on [t1, t2] are pairwise orthogonal on [t1, t2] (have to signify time interval); if Kn = 1 for all n, then the set is said to be “orthogonal” (unity length)

Orthogonal Signal Set

can be always be made oethogonal by dividing each signal by its length; ex. sinusoids