final exam studying Flashcards
linspace
generates row vector of n-points (default is 100) between and including a & b
linspace(a, b, n)
transpose (special case)
.’ conjugates complex numbers
to conduct operations on vectors/matrices, the dimensions must be _______
the same
random function
generates random numbers based on input specifications
rand(3)
3 by 3 matrix with random numbers 0-1
rand(10, 1, 5)
10 by 10 of random numbers between 1-5
find function
locates non-zero elements and outputs indices
min/max functions
returns smallest/largest in each vector (for matrices, it’s for each column)
[M, I] = min(x)
what are M and I
M is the values, I is the indices
sortrows function
sorts matrix by specified column from smallest to largest (add ‘descend’ to flip)
symmetric error-bar function
errorbar(x, y, e)
e is the error
asymmetric error-bar function
errorbar(x, y, d, u)
histograms –> d is low error and u is high error
line graph–> d is negative error and u is positive
meshgrid function
creates 2d matrix for input vectors, (repeats the vector over and over)
[ X, Y] = meshgrid(x,y)
how to generate Z with results of meshgrid function?
remember [X, Y] = meshgrid(x,y)
Z = sin(X) .* cos(Y)
linear interpolation
estimating values at certain points based on existing ones
interp1(x,y, new_x)
cubic spline interpolation
smooths curve and minimizes errors (fitting interpolation)
least squares fitting
line of best fit for data based on the slope and intersection (goal is to minimize R)
linear regression
finds the linear function that best fits the data
polyfit function
returns coefficients of degree n that’s the best fit with least squares
polyfit(x, y, n)
polyval function
evaluates polynomial at each point of x
polyval(p, x)
diff function
calculates difference between consecutive elements
resistance and current (I = )
I = V/R
systems (in bioengineering)
groups of things that affect signal of interest
model (in bioengineering)
symbolic representations of process, system, or data collection
signals (at their core)
variations in energy that carry info
types of signals
chemical, mechanical, electrical, and thermal energy
signals for communication in body?
chemical and bioelectrical energy
biotransducer
converts non-electrical energy to electrical signal
sensitivity
change in output for given change in input (dy/dx)
resolution
smallest change in input that changes the measurement
linearity
consistency of input/output relationship
repeatability
expected change in output for repeated changes in input
noise
change in output not related to input, unrelated to signal
measured signal
biosignal of interest + noise
encoding
step of processing signals, transducing body signals so they can be read, can be analog or digital
continous/analog signal
in terms of amplitude, has a defined amplitude at every point
continuous/analog signal
x(t)
continuous/analog domain
uncountable, may or may not be finite
discrete/digital signal
defined over discrete time domain
discrete/digital signal
x[t]
Ts
sample interval
t =
n*Ts = n/fs
fs =
1/Ts
signal measurement system
signal (analog)–> sensor/transducer –> signal processing –> sampling (digital)
sampling
slicing the signal into sequences at time interval (Ts)
systematic error
each measurement deviates from the value by a fixed amount (DC shift)
random error
noise (random variations)
biomedical sources of noise
physiological variability noise, environmental noise/interference, artifact, electronic noise (thermal, shot, white noise)
sinusoidal wave
Asin(wt) = Asin(2pift) = Asin(2pit/T)
w = frq in radians
root mean squared (RMS)
square the signal, calculate the mean, take sqrt
SNR (signal noise ratio)
higher SNR is better system
positive SNR means…
signal > noise
negative SNR means…
noise > signal
aliasing
when signal isn’t sampled enough to capture changes, different signals become indistinguishable and high frq components and mistaken for low frq components
nyquist sampling theorum
to avoid aliasing, the sample rate should e twice the highest frequency component in signal
ex: if highest is 12, signal should be sampled 24 times per cycle
smallest frequency component
defines the period of a signal (T), too low = jagged graph, too high= loopy graph
post-processing
operations after digitization, like shifting, scaling, and noise reduction
2 methods of noise reduction
signal averaging and signal filtering
signal averaging
reduces the noise power, best when the frq spectra of noise and signal overlap
noise = average_odd - average_even
moving average filters
replace each data point with value of average and itself and nearby points
symmetrical moving average filter
3 point moving average:
(Xt-1 + Xt + Xt+1) /3
asymmetrical moving average filter
3 point moving average:
(Xt-2 + Xt-1 + Xt) / 3
weighted moving average
more weight to point being replaced and less to furthest away
((Xt-2 * 1) + (Xt-1 * 2) + (Xt * 3) + (Xt+1 * 2) (Xt+2 *1) / 9
3 classes of signal
periodic, aperiodic, transient (step-like)
periodic signals
every phase is the same, biosignals can be taken as periodic for analysis
aperiodic signals
exists over finite time frame, treated as periodic
transient signal
increases infinitely, hardest to treat
interpolate before averages
[rxy, lags] = xcorr(x, y, ‘coeff’)
correlation
comparison of two signals
correlation coefficient
number that signifies the correlation (-1 to 1)
crosscorrelation
correlating over many shifts/lags
lag
continuous variable of time to shift x(t) with respect to y(t)
peak correlation
tells us where signals are most similar
multiple correlations
shift wave to find best match for signal
autocorrelation
correlating signal with itself
plotting crosscorrelation
[rxy, lags] = xcorr(x,y, ‘coeff’)
figure; plot(lags*Ts, rxy)
max correlation
[max_corr, max_shift] = max(rxy)
crosscovariance
similarity in deviation of two signals about the mean
covariance
similar to correlation but has the mean subtracted from the signal
frequency domain
shows how signal energy is distributed over a range of frequencies
2 components of frequency domain
magnitude (abs(Xf)) and phase (angle(Xf))
fourier series theorum
any periodic signal can be represented as a sum of sinusoids, more sinusoids = better summation
X(f) = fft(x)
bandwidth
range of frequency domain
filters in frequency domain
reduce noise by limiting the bandwidth of range of interest
filter properties
filter type, bandwidth, attenuation slops
multiplication value
dependent on instantaneous frequency, (1 = keep, 0 = pass)
pass band
part of input signal you want to keep, passed to output without change
stop band
part of input signal you want to discard, ideally reduced to 0
cut-off frequency
between pass and stop bands
inverse fourier analysis
going back to time domain from frequency domain
ifft(Xf, n) –> n-point trandformation
low pass filter
passes frequency (below fc) with minimal attenuation and attenuates higher frequency
high pass filter
passes frequency (above fc) with minimal attenuation and attenuates lower frequency
band-pass filter
allows specific interval of frequency to pass and stops the rest
band-stop filter
attenuates specific interval of frequency and passes the rest
notch-filter
attenuates signal within very narrow range of frequencies
filtfilt
digital filtering by processing x data forwards and backgrounds
filfilt(B, A, X) filters in X direction w filter described in A and B
[B, A] = butter(N, Wn)
designs nth order butterworth filter with 0.0 < Wn < 1.0 (cut-off frq) returns numerator B and denominator A
butter(N, Wn, ‘high’)
high pass filter, removes low frequencies
butter(N, Wn, ‘low’)
low pass filter, removes high frequencies
butter(N, Wn, ‘stop’)
band-stop filter of WN [W1, W2]
event time
when signal passes the threshold
relative timing
time passed in between events
single events
time of physiological response to stimulus
latency
time of peak response
event detection
detect by setting and finding the threshold (min or max), jump forward by interval time
false negative
missing an event
identification features of events
amplitude, slopes, peaks
how do you avoid false positives/negatives?
with a good system and measurement system
false positive
detecting an event when none occurred
spatial data
data in space-time domain
digital images
represented as 2D matrix of pixels
resolution (in imaging)
number of pixels per unit length
voxel
3D pixel
dynamic range
number of gray levels in an image (2^bits)
histogram function
represents grayscale distribution (x = gray level, y = frequency)
quantization
digitizing the amplitude
