Lec week 7 Filtering

which stage does filtering occur

which stage does filtering occur

after amplification, theres analog filtering, then after averaging, theres digital filtering

describe the 2 locations of filtering depending on file size

filtering before averaging (analog filtering) - larger files

filtering after averaging (digital filtering) - smaller files, soo more preferred.

purpose of filtering

attenuate/reduce unwanted frequencies, which can improve signal to noise ratio by reducing this noise

con of filtering

can distort or lose signal (loss of wanted frequencies, changing data) if used improperly

analog filter

hardware filter

digital filter

filtered using mathematics

frequencies

number of cycles per unit of time/sec.

fouriers theorem

a physical function that varies periodically in time with a frequency f that is expressed as the sum of sinusoidal components of frequencies (f, 2f, 3f, etc)

fourier’s synthesis

complex signal that can be separated into sine waves of different frequencies and amplitudes

how does fourier analysis work

can add sine waves/harmonics of increasing frequency and lower amplitude into the baseline/fundamental frequency (fit into a square wave)

describe the frequency spectra for this waveform

x axis - frequency

y axis - proportion of waveform that has that frequency

only one frequency

describe frequency spectra of this waveform

first bar - baseline frequency

second and third, etc bars - the smaller frequencies that follow from fouriers synthesis

describe whats happening in this image

• greater low frequency contribution to signal than high frequency

• can use a low pass filter to remove the high frequencies based on the ‘noise’ arrow

running average

also called box car filter

essentially a low pass filter than retains low frequencies and removes high frequencies

filtering in time domain where you take the average of some number of points across the data set.

advantage of filtering in time domain

remove high frequencies and can digitally filter at time T using points before and after T (t-1, t+1)

compare this advantage with analog filtering

analog or hardware filters can only filter points before time T because the future hasnt happened yet (has t-1, no t+1)

2 main points of of filtering in time domain

1. removes part of EEG/ERP, so only do so if you consider it noise

2. time-frequency tradeoff → although it makes filtering waveform smoother (remove high freq), you lose temporal precision aka more time smearing

** more smoother waveform means more time smearing

what the ideal low pass and high pass filter

ideal low pass filter: retains low frequencies, removes high frequencies above a frequency cutoff

ideal high pass filter: retains high frequencies, removes low frequencies below a frequency cutoff

ideal filters have what type of function

step function - an absolute cutoff from 0 to 100% or the reverse.

a typical filter has what type of function

transfer function - no absolute frequency cutoff

a typical filter has what kinda of filter cutoff? what is this called

the filter cutoff is not absolute, but rather a transition band

transition band is where frequencies are attenuated (reduced), but not completely removed,

in a typical filter, the frequencies retained and lost are called

passband - some frequencies that should be retained are lost

stopband - some frequencies that should be removed are retained

what does the function of a running average/boxcar filter look like

it has a transfer function - no absolute cut off of frequency

inital rapid attenuation, followed by rings of retained frequencies

solution to removing rings in function in running avg/box car filter?

insteaf of using running avg, use a boxcar filter that makes points closer to time T count more than point further away from time T

dont use an equally weighted box car filter - ex: a 7 point running avg where all 7 points count equally.

band-pass

when both low and high pass filters exist, it is the band that passes

2 ways of determining cut off frequency

1. half-amplitude: a 50% amplitude cutoff of frequency

2. half-power: the frequency at which filtered amplitude is half (-3 dB) the power of the unfiltered power.

describe half power given a 100microvolt eeg signal

(100microvolt)² = 10,000 power

10,000/2 = 5,000 aka half the power

squareroot (5,000) = 70.7 microvolts

filtered amplitude is 71% of the unfiltered amplitude, so 71% signal remains and 29% filtered out

cutoff slope

steepness of the transition band, in units of db of attenuation per octave

typical values of 6, 9, 12 db/octave

greater steepness of transition band = greater time smearing

phase shifting vs zero phase

phase shifting = causal → analog filters shifts peaks in time because only use points before time T

zero phase = noncausal → digital filters dont shift peaks in time because uses points before and after time T

in this diagram, what is the slope describing

changes in gain as frequency increases, in db/octave

so 3 db is half power

what does this diagram say about filtering

these are pass-bands (include both low and high pass filter)

as you filter out more data, theres greater time smearing, which creates a smoother waveform

amplitudes below the cutoff frequency are still attenuated bc of the transition band slope

filtering removes frequencies, thus the last row is the least accurate

is a high pass filter good

very bad bc tend to have more steeper transition band, thus more temporal smearing, and creates/introduce oscillations that did not exist in the unfiltered waveform

removed important information

in this image, a half amplitude cutoff at 6Hz is very high (typical is 0.01), therefore attenuating a lot frequencies

explain what filter is used and why filtering is bad in this image

high pass filter

the red filtered waveform does not represent the unfiltered waveform, but rather has introduce new oscillations and significantly lowered the amplitude of the peaks.

whats the typical high pass filter value

0.01 Hz because low attenuation of frequencies

butterworth filter

a type of analog filter - an ideal electrical filter should reject unwanted frequencies but also not change the wanted frequencies

single pole: one resistor and capacitor - slope is 6dB/octave

double pole: 2 resistor-capacitor ciruits - slope is 12 dB/octave

2 reasons to use filters

1. aliasing error (want signal to be below Nyquist limit), done using hardware aka analog filter

2. improve signal to noise ratio by removing unwanted frequencies

2 problems with filters

1. filtering changes your data (specifically high pass filters)

2. distort the amplitude and timing (time smearing) of ERPs

   → introduce oscillations = high pass filter

   → phase shift = analog filters

name the specific method for avoiding the time smearing when using a running average/box car filter

Use Gaussian filter - have the average be the most weight than the other points, instead of having all the points be equally weighted in a box car filter.

5 filter guidelines

1. for analog filtering, allow a broad range of frequencies to pass to avoid phase shifting, while also keeping below the Nyquist limit (aliasing error)

2. use zero-phase or non-causal digital filters

3. only filter as much is necessary

4. know the characteristics of your filter (the cutoff frequency, the slope)

5. filter settings depend on what the investigator considers is signal and noise

   ** for 1: reminder that broader range of frequencies means not filtering out lots of data

describe the construction of an analog filter

resistors replaced with capacitors to create a capacitive voltage divider (filter)

capacitors block/resists DC and low frequency AC current