Signals and Filtering

Question 1

Errors in measurement = our analysis is prone to errors

Answer 1

• Systematic Errors
  ○ Errors in filming
  § Perspective Error
  § Parallax Error
  § Calibration Error
  ○ Errors in marker position = should place our markers to accurately represent the joint centre that we want to measure
  § Marker in wrong location
  § Marker moving w/ the skin
• Random Errors = when we use filtering
  ○ Often described as “Noise”
  § Errors in digitizing
  § Rounding Errors

Question 2

Correction of errors

Answer 2

• Systematic Errors
  ○ Hopefully will be small if you’re careful
  ○ Not much can be done to correct
• Random Errors
  ○ Data Smoothing
  § Line of best fit is a form of smoothing
• Trying to remove noise but keep info
• Want to maintain our info as much as possible + get rid of the noise = if we start getting rid of too much info we are distorting the signal

Question 3

Signals

Answer 3

• A signal is any property that varies w/ time (ie a time series) + carries info
  Eg
  ○ Sound (oscillations in air, measure air pressure over time)
  ○ Electricity (oscillations in electric current. Measure current over time)
  ○ Radio waves (oscillations in electromagnetic radiation. Measure either changes in frequency (FM) or amplitude (AM) over time
  ○ Movement (changes in position. Measure displacement, velocity, etc, over time)
• Signals can be described by their Amplitude and Frequency
  
  • Amplitude – a measure of how much the signal changes in size
  • Frequency – a measure of how fast the signal changes
• Quicker it goes the higher the frequency, the slower it goes the lower the frequency

Question 4

Amplitude

Answer 4

• How much the signal changes in size
• Indicated by the height of the vertical axis
• Can change these two things independently
• Can change the frequency independent of the amplitude
• Can have the same frequency but have doubled the amplitude

Question 5

Frequency

Answer 5

• How fast the signal changes
• Measured in oscillations per sec

Question 6

Fourier Analysis

Answer 6

• Based on the concept that any function can be represented by a sum of weighted sine + cosine functions
• Allows us to calculate amplitude of each frequency present in a signal
• Can think of this as how “important” each frequency is that makes up the signal

Question 7

Data Smoothing - common methods

Answer 7

• What methods are available for smoothing?
  ○ Digital Filter is a formula applied to data
  § Eg moving average = smooths the curve
  § Polynomial curve fit (eg projectiles)
  § Butterworth Filter
  □ Very common filter in Biomechanics
  □ Available in our digitising software
  □ Enables us to choose a specific cut-off frequency

Question 8

Filter frequency affects data quality if too low

Answer 8

• Best way to check if filter has actually worked is to plot derivatives e.g. if have displacement data (which this is) plot the knee angular velocity + then see how much noise there is + then plot the acceleration data to see how much noise = signals that reflect biomechanical signals

Question 9

Raw Measurements

Answer 9

• Displacement should be good if digitising was careful
• Velocity shows higher errors at high frequency
  ○ When looking at angular velocity = those little changes in the displacements = error is exacerbated/ multiplied as go into angular velocity
• Acceleration likely to show quite substantial errors
• Filtering essential if you want to calculate acceleration
• That’s why need to smooth the knee angle appropriately so that we get better angular velocity + acceleration curves

Question 10

LOW PASS FILTER

Answer 10

• Most of the signal is at frequencies below 5 Hz
• Low Pass Filter removes high frequencies (low frequencies pass through)
• Filter can be applied to remove high frequency “noise”
• Signal is info about what movement is occurring
• “Noise” is an error in the signal
• In biomechanics
  ○ Signal is of relatively low frequencies (5 Hz walking)
  ○ Noise is of relatively high frequencies (30 Hz digitising)

Question 11

Data Smoothing - FILTERING

Answer 11

• Filter removes very small errors in displacement
• Has moderate effect on velocity values
• Always essential when calculating acceleration
• When filtering make sure that its close to your displacement curve, then observe your angular velocity + acceleration curves

Question 12

Nyquist Sampling Theorem

Answer 12

• The sampling rate must be “equal to, or greater than, twice the highest frequency component in the analog signal (time continuous signal).
• If a continuous time signal contains no frequency components higher than P hz, then it can be completely determined by uniform samples taken at a rate fs samples per second where fs ≥ 2P
• The Nyquist theorem defines the minimum sample rate for the highest frequency that you want to measure
• Filming frequency would depend on basically how fast the motion is

Example: fmax = 2 Hz
* Period = 0.5 s
* Frequency = 1/0.5 = 2 Hz
* Sampling rate (Nyquist rate) = 4 Hz = minimum

Question 13

Activity and Signal Frequency

Answer 13

• Movt signals are typically low frequency
  
  • Walking ≈ 5 Hz
  • Running ≈ 12 Hz
  • Sit to stand ≈ 3 Hz
• Low Pass Filter aims to cut out frequencies above this, but preserve frequencies below the cut-off

Question 14

Take-Home Messages

Answer 14

• Digitising enables us to make quantitative measurements from film = are discrete measurements
• Time Series analysis looks at a graph of any discrete variable shown as a function of time.
• Signal is any time series that conveys info
• Properties of a signal are expressed in terms of Amplitude + Frequency
• A signal can be thought of as being composed of many dif sine curves added together
• Fourier Analysis enables us to investigate the size of each frequency making up a signal
• Human Movt typically comprised of low-frequency movts
• A Low Pass Filter enables us to smooth out any errors (noise) in a signal provided those errors are at a higher frequency than the movt being studied
• A low pass filter will be applied to your data after digitising to remove small errors from digitising