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 – Ankle Increasing - Head Increasing - Shoulder – Knee frequency content frequency content (e.g. landing, - Trunk – Hip (e.g. head impacts) running) - Hip – Shoulder – Head 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 the signal slightly (more advanced) – 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 – Raw kinematic data (add appropriate column headers) – 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 your formulas.
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