Signals and Filtering in Movement Analysis

Nyquist Sampling Theorem and Frequency Analysis

• Definition of Nyquist Sampling Theorem

  • The Nyquist sampling theorem states that to accurately reconstruct a signal, it must be sampled at a frequency at least twice ($2 \times$) the highest frequency content present within that signal.

  • Sampling a signal at a frequency that is too low results in aliasing effects, which create a "false frequency" or error in the data.

  • This theorem is the primary determinant for setting both the sampling frequency during data collection and the cut-off frequency during post-processing and filtering.

• Nyquist Sampling Theorem in Force Data

  • Ground Reaction Force (GRF) data is typically sampled at a rate of $1000\,Hz$. This is significantly higher than the requirements dictated by the Nyquist theorem for standard movement.

  • The frequency content of a GRF signal is generally below $50\,Hz$, with $95\%$ of the signal content falling below $30\,Hz$.

  • Analysis shows that the power spectrum at $100\,Hz$ is effectively the same as at $1000\,Hz$.

  • Consequences of Sampling at the Limit: If force data is sampled exactly at the Nyquist limit ($100\,Hz$ for a $50\,Hz$ signal), errors occur:

    • The peak value may be completely missing or severely attenuated (e.g., a drop from $1345\,N$ to $1315\,N$).

    • A phase shift may occur, potentially delaying the signal (e.g., a shift of $6\,ms$ later).

• Importance of Knowing Signal Frequency

  • Understanding the frequency spectrum assists in decisions regarding the collection and processing of signals.

  • Signal Types:

    1. Alternating Signal: These change continuously over time and can be periodic, random, or a combination. They may also include a direct current (DC) component.

    2. Frequency Content: A pure sine or cosine waveform represents a single frequency. Complex waveforms are the mathematical sum of various sine and cosine waves.

  • Spectrum-Based Decisions: The signal spectrum (frequency content) determines:

    • The required sampling rate before Analog-to-Digital (A-D) conversion.

    • The necessary length of the record to be converted.

    • The specific frequency of filtering required.

Fourier Transformation and Spectrum Analysis

• Principles of Fourier Transformation

  • Periodic Signals: These have discrete frequencies. Analysis requires expressing the signal content in multiples of the fundamental frequency ($f_0$).

  • Harmonics: These are the higher frequencies that are multiples of the fundamental frequency.

  • Fourier Series: This is defined as the sum of the proper amplitudes of these harmonics.

  • Non-periodic Signals: These have a continuous spectrum defined by a lowest frequency ($f_1$) and a highest frequency ($f_2$).

• Mathematical Processes

  • Discrete Fourier Transform (DFT): A mathematical process used to calculate the amplitude or power of each frequency component, which is then plotted against frequency. It assumes each harmonic has a constant amplitude and phase, though in real-life signals, these are often average values over a period of time.

  • Fast Fourier Transform (FFT): A computational method used to convert a time-domain signal into the frequency domain. It is necessary because of the large number of calculations required for a discrete Fourier transformation; the FFT avoids repetition and time-consuming calculations.

Applications of Spectrum Analysis

• Analog-to-Digital (A/D) Converters

  • An electrical signal is fed into the input terminal of an A/D converter.

  • The computer controls the rate at which the signal is sampled, with the optimum rate determined by the Nyquist sampling theorem.

  • The analog input signal is transformed into a series of short-duration pulses.

  • The amplitude and polarity of these pulses are translated into a digital format using binary code ($0, 1$). A $12-bit$ code translates to $2^{12} = 4096$ discrete levels.

• Deciding Recording Length

  • Recording length is determined by the lowest frequency present in a signal.

  • Example: For measuring the Center of Pressure (COP) and Center of Mass (COM) during quiet standing, a record length of greater than $1\,min$ is required.

• Signal vs. Noise in Kinematic Data

  • Every marker axis in motion capture is associated with its own frequency.

  • Walking Example: Walking signals typically occur at less than $6\,Hz$ (representing the 7th harmonic).

• Calculations of Velocity and Acceleration

  • Time differentiation of a signal that contains higher-frequency noise leads to errors.

  • 1st Derivative (Velocity): The amplitude of harmonics increases with frequency. The 1st derivative of the 3rd harmonic is $3\times$ that of the 1st harmonic.

  • 2nd Derivative (Acceleration): The increase repeats; the 3rd harmonic becomes $9\times$ the 1st harmonic in the second derivative.

Techniques for Filtering Signals

• Overview of Filtering

  • The primary aim of filtering is to remove noise (unwanted signal components).

  • Methods include:

    1. Curve-fitting techniques.

    2. Digital filtering.

    3. Residual analysis (used for choosing the cutoff frequency).

• Filtering Method 1: Curve-Fitting Techniques

  • Polynomials: These assume a trajectory has a predetermined shape. A smooth signal results from fitting the assumed mathematical expression to the raw noisy data using a "best fit" approach. Formula: $x(t) = a_0 + a_1t + a_2t^2 + a_3t^3 + \dots + a_nt^n$.

  • Splines: Piece-wise polynomials joined together at "knots." The curve is broken into sections starting and ending with an inflection point. Improper selection of inflection points results in errors.

    • Cubic Spline: The most common type; it balances closeness of fit with calculation speed.

    • Quintic Spline.

    • Least Squares Splines: Use input parameters to control the balance of smoothness and fit.

  • Software Examples:

    • Visual 3D: Uses a cubic interpolation spline that goes through data points; it is used specifically for gap filling and does not filter the data.

    • Vicon (Woltring/Generalized Cross-Validatory Spline): Estimates smoothing parameters based on all data points and predicts a best-fit curve using statistical optimization. It does not go through every data point, allowing it to both fill gaps and filter the data.

• Filtering Method 2: Digital Filtering

  • Attenuation is based on the different frequency content of the signal versus the noise.

  • Phase Shift: Digital filtering can cause a phase shift of the output signal relative to the input. A $2nd$ order filter causes a $90^\circ$ phase lag (phase distortion).

  • Zero Phase Shift: Refiltering the signal in the opposite direction results in a $0$ net phase shift, effectively creating a $4th$ order filter.

  • Low Pass Filters: These allow low-frequency data (where most human movement signals reside) to pass through unattenuated while high-frequency noise is attenuated.

  • Critically Damped Filter: Has no overshoot in response to step- or impulse-type inputs but features a slower rise time. This is rarely seen in human movement analysis.

  • Butterworth Low Pass Filter: The preferred filter for human movement. It allows lower-frequency data through, removing small random digitizing errors and soft-tissue artifacts. It has a shorter rise time but may have a slight overshoot in response to impulse-type inputs.

• Filtering Method 3: Residual Analysis

  • This is a better alternative to simple harmonic analysis for determining the cutoff frequency ($f_c$).

  • It analyzes the difference between the filtered and unfiltered signals over a wide range of cutoff frequencies.

  • It seeks a balance between signal distortion and the amount of noise removed.

  • Different cutoff frequencies should be determined for each separate marker displacement.

Case Studies and Practical Applications

• Critical Importance of Nyquist and Filtering: The GPS Example

  • Published Research Claim: A study on match running demands claimed to record data using a portable GPS at a sampling rate of $15\,Hz$. They applied a $4th$ order Butterworth filter with a $1\,Hz$ cutoff frequency.

  • The Reality: The actual physical sampling rate of the GPS unit (SPI HPU, GPSports) was only $5\,Hz$. The researchers "up-sampled" the data linearly from $5\,Hz$ to $15\,Hz$.

  • The Error: Using residual analysis, it was found that the researchers over-filtered the data. A $1\,Hz$ filter removed most of the actual signal content, potentially invalidating the results.

• Inverse Dynamics and Filtering

  • Inverse dynamics involves combining GRF, kinematic, and anthropometric data.

  • Segmental acceleration is calculated via numerical differentiation of position data, which amplifies errors.

  • Filtering Requirements: Kinematic data and force data must be filtered at the same cutoff frequency to maintain consistency in joint moment calculations.

  • Bisseling & Hof (2006) Study Examples:

    • SM100: Low pass filter with $f_c = 100\,Hz$.

    • SM20: Low pass filter with $f_c = 20\,Hz$.

    • AM: Accelerometer-based method where both kinematic and accelerometer data are low pass filtered at $100\,Hz$.