Building Signals with Blocks Flashcards
Understand basis expansion and their role in signal processing
what are bases?
elementary signals
a complex signal gets ________ into elementary signals
decomposed
assuming bases are known, what is an equivalent description for the signal?
weights
what must be known for weights to be an equivalent description for the signal?
bases
what are the building blocks of signals?
bases
what do bases represent?
arbitrary signals as weighted products
how are bases scaled?
with different weights
bases must be _____ in order for them to be scaled with different weights
sufficiently different from each other
how are complex/arbitrary signals represented?
by weighted integrals/sums of bases signals
define integral (as defined by this video)
sum of large number of terms; limiting form
define integral (as defined by mathematics)
a function of which a given function is the derivative, i.e. which yields that function when differentiated, and which may express the area under the curve of a graph of the function.
define transform
conversion of signal amplitudes to weighted bases
what are some example transforms?
discrete, fourier, wavelet, and cosine transforms
any signal with N values can be represented by…
N linearly independent bases signals
is there an optimal bases technique?
No
what are some examples of bases techniques?
sinusoids, PCA, Haar, Morlet, Synlet, etc…
what bases are used in Fourier transform analysis?
sinusoids
define sinusoids
Eigenfunctions of Linear Time Invariant systems
define Eigenfunctions
set of independent functions which are the solutions to a given differential equation.
define LTI
Linear Time Invariant system
what is the output if you put a sinusoid into an LTI system?
x[n] = Acos(wn)
also a sinusoid
y[n] = |H(w)|Acos(wn+ angle H(w))
define x[n] = Acos(wn)
sinusoidal input to LTI system
define y[n] = |H(w)|Acos(wn+ angle H(w))
sinusoidal output to LTI system
sinusoidal input to LTI system
x[n] = Acos(wn)
sinusoidal output to LTI system
y[n] = |H(w)|Acos(wn+ angle H(w))
what do LTI systems modify?
the amplitude and the phase of the sinusoid, angle H(w), applied to the input
define wavelet
limited duration oscillatory signals
frequency of a wavelet is inversely related to…
the duration in time of the wavelet
what forms a bases?
a collected of wavelets of characteristic shape, but with different durations and onset locations
how are different wavelet bases made?
by changing the shape of the oscillations
what are some example wavelets?
Haar, Daubechies, Coiflet
define long duration wavelet
- broad in time
- narrow in frequency
- accurately captures frequency
define short duration wavelet
- narrow in time
- broad in frequency
- accurately captures location of transient events
- good for representing signals that differ in time/space
what do good bases do?
concentrate energy into a few significant weights
how do signals get turned to weights?
signals get transformed to weights using bases
what is the goal of signal processing?
to concentrate component signals into relatively small number of bases signals
how do you separate signals?
1) transform signal mixture into weights using bases
2) set the weights of non-relevant signals to zero
3) reconstruct separate signals
how do you denoise a signal?
1) transform noisy signal into weights using bases
2) set small weights to zero
3) reconstruct cleaned signal
how do you compress a signal?
1) transform signal into weights using bases
2) store only the significant weights
3) reconstruct signal from significant weights
define bases expansion
express weighted sum of component signals
