# Data Filtering by malj

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```									     Signal Processing

ESS 5314-001
Lecture 5

Reading: Robertson et al. 2004, Ch 11
Noise
• Component of final signal not due to process itself
(e.g. walking)
• Sources of Noise
– Electronic noise in device (e.g. 60 Hz from lights or
computer)
– Spatial precision of TV scan
– Film digitizing system
– Human error in film digitizing

• Noise leads to error and the need for data filtering
and smoothing
Harmonic Analysis
• AC – Alternating Component
– Continuously changing with time
– Periodic (rhythmic), Random, or both

• DC – Direct Component
– Bias value (offset if you will) about which the
AC component fluctuates
– Figure 2.13 in text
Frequency Content
• Periodic Signals
– Rhythmic – repeating - pulsing
– Discrete Frequency (evident from spectral analysis)
– e.g. sine wave

• Non-periodic Signals
– More chaotic in nature – no obvious pattern
– Power across continuous spectrum
Periodic Signals
• Can be described by their lowest (f1), highest (f2)
and fundamental (f0) frequencies

• Harmonics
– Higher frequency components
– Multiples of f0

• Thus – a perfectly periodic signals can be broken
down into its harmonic components
Fourier Series
• Fourier Series
– Summing the harmonic components of a signal
together with proper amplitudes
V(t) = Vdc + V1sin(ω0t + θ1) + V2sin(2ω0t + θ2) + … + Vnsin(nω0t + θn)

where:        ω0 = 2πf0
θn = phase angle of nth harmonic
thus,

V(t) = (4V/ π) (sin ω0t + 1/3 sin 3ω0t + 1/5 sin 5ω0t + …)

…however many harmonics it takes
Spectral Analysis
• Spectral Density Analysis
• Harmonic Plots                            Looking at signal in
frequency spectrum
• Spectral Plots

• Gives amplitude or power of each frequency
component in the signal plotted against frequency
• Power Spectral Density Analysis
– Utilizes Fourier transformation or harmonic analysis
– Use it to determine
A) Necessary sampling frequency
B) Appropriate cutoff frequency for filter
Spectral Analysis
Sampling Theorem
• Capture
– Film or TV or other non-image based
• Sampling Processes

• Play back
– Slo-Mo
• Jumpiness
• See distinct steps
– Normal Speed (24 Hz film, 60 Hz TV)
• No jumpiness – smooth motion

– WHY?           The human eye retains an image for ~1/15 second
Thus, fills in any blanks and smoothes the jumps
Our eyes deceive us…

• The human eye retains an image for ~1/15
second

• Short term memory allows the observer to
smooth the jumps
Sampling Theorem
Shannon’s Sampling Theorem (Nyquist)
• Sample at least two times (2X) as high as the
highest frequency in the signal itself.

So how fast should I sample?
1. Sample as fast as you can initially (computer will limit)
2. Analyse your signal in the power spectrum (spectral
analysis)
3. Determine frequency containing almost 100% of the
signal
•   Use at least 2 times this to sample your data (10 times if
possible, although sometimes this is overkill)
4. Determine the 95% frequency (frequency at which 95%
of the signal lies below)
•   Set this as your filter cutoff frequency (low pass)
Signal Alias

Winter, 1990
Signal versus Noise
• Goal of sampling and filtering
– Maximize signal to noise ratio

• Video/Kinematic Data
– Certain body segments may have different frequency content
depending on the movement

–   Heel
Increasing       - Shoulder
–   Knee         frequency content
frequency content
(e.g. landing,                       - Trunk
running)                           - Hip
–   Shoulder
Non-stationary Signals
• Other movements may be periodic (cyclic),
but the frequency content shifts over time.

– Impulsive movements
• e.g. landing (heel strike, jumping-landing)

– Phases of movement
• e.g. process of damping the impact forces and
vibrations (heel strike transient, toe contact, heel
contact, bottom-most position)
Walking
• 2 Hz walking frequency = Stride frequency of 1 Hz
– Thus 1 stride per second
– Expect harmonics at 2 Hz, 3 Hz, 4 Hz, etc.
• Substantial research and analysis
– Highest harmonics occur in heel and toe
– 99.7% of signal power is contained in the lower 7
harmonics (i.e., below 6 Hz)
• Thus…a greater noise component lies above 6 Hz
– And thus lower signal to noise ratio
• So…
Filtering and Smoothing

• Goal is to maximize signal to noise ratio
– e.g. want as much signal below 6 Hz as possible

• Filter Cutoff
– Frequency limit at which you filter your data
– Select filter cutoff at the peak of signal power
– Ex. 6 Hz for walking
Digital Filter
• Attenuates selected frequency in signal
• Assumption
– Signal contains low frequency
– Noise contains high frequency

• Low Pass Filter
– Allows signal to pass through unattenuated
– High frequency noise is removed
Effect on Data
Low Pass Filter
Linear Envelop

Butterworth 2nd order, cutoff = 2 Hz
Low Pass Filter
Linear Envelop

Butterworth 2nd order, cutoff = 3 Hz
Low Pass Filter
Linear Envelop

Butterworth 2nd order, cutoff = 10 Hz
Effects on Derivatives
• First…why do we care about derivatives?
–S
• Displacement (directly from coordinate data)
– dS / dt
• Velocity (derivative of displacement)
– dV / dt = d2S / dt2
• Acceleration (2nd derivative of displacement)
– F = m∙a
• Where we get resultant joint moments, joint loads, etc.
Effects on Derivatives
• See Figure 2.25 in Winter (2005)
• Derivatives
– Slopes along the curve at each data point
• 2 inflections in derivative for every inflection in
original curve
– Thus curve gets bumpy quick
– “Derivation is inherently noisy”
• So if original curve is noisy…
– Derivative will be even more noisy
– Eventually meaningless
Effect on
Data

Winter, 1990
Effects on Derivatives
Butterworth Filter
X1i = (a0Xi) + (a1Xi-1) + (a2Xi-2) + (b1X1i-1) + (b2X1i-2)

X1         = Filtered data
X          = Raw data
i          = sample #
a#, b#     = filter coefficients (constants)

The filtered data point is just a function of the
immediate and previous raw and filtered data points.
Filter Coefficients

• Filter coefficients are determined by the:
– Filter type
– Cutoff frequency (fc)
– Sampling frequency (fs)

• See Tables 2.1 and 2.2 for coefficient values
(Winter, 2005)
Cutoff Coefficients
Butterworth Low-Pass Filter

Winter, 1990
Cutoff Coefficients
Critically Damped Low-Pass Filter

Winter, 1990
Filters
• Low Pass
– Retains frequency below cutoff
• High Pass
– Retains frequency above cutoff
• Band Pass
– Retains frequency between cutoffs (limits)
• Notch
– Filters out frequency at designated limit
– e.g. 60 Hz notch to filter out electrical noise
• Others
The Filtering Process
• Filtering (single pass) creates a phase shift that
does not match original data in time
– Single Pass = 2nd Order
• Run filter again in reverse direction
– Dual Pass = 4th Order
– Resets filtered data in the original time series

• Passing twice affects the frequency spectrum of
– Need to adjust cutoff filter accordingly
– For example: 6Hz / 0.802 = 7.48Hz (new cutoff)
Assignment
• In Excel…
• Create a “4th order, 0-phase lag” low pass digital filter
– Use earlier slide from Winter, 1990, p.38 for filter coefficients
• Input you will need:
– Sampling frequency (fs) = 69.9Hz
– Use a Cutoff frequency (fc) = 6Hz (for starters)
• At the end, let this be a variable you can change
– Use the Data File provided (it is kinematics data)
– x-y data for toe, heel, ankle, knee, hip, shoulder, WBCOM “markers”
• Output: Filtered data
– A new column for each column of original data
• Presentation:
– Graph raw and smoothed data (with 6Hz cutoff) – Vertical
Position will do
– Email or bring soft copy of spreadsheet (this will contain your
functions)
– Organization will count. I will need to be able to figure out
where your cutoff and sampling frequency values are, and check