Filtering (Chapter 4)

Objectives

  • Define how to characterize a signal
  • Examine the Fourier analysis of signals
  • Outline wavelet analysis
  • Explain the sampling theorum
  • Discuss how to ensure cyclic continuity
  • Review various data smoothing techniques

Signal Noise and Data Smoothing

What are the sources of error that may lead to a noisy signal?

Sinusoidal time-varying signal

  • Frequency (f)
  • Amplitude (a)
  • Offset (a0)
  • Phase angle (θ)

Fourier

h(t) = a0 + a sin(2πft + θ)

2πft = ω

h(t) = a0 + a sin(ω + θ)

Fourier

Fourier Transform

Fourier Transform

Fourier Transform

Fourier Transform

Fourier Transform

Fourier Transform

Fourier Transform

Padding data

  • Multiply the power at each frequency by (N+L)/N, where N is the number of nonzero values and L is the number of padded zeros

Padding discontinuities

  • Windowing or subtract a trendline

Sampling Theorum

  • The process signal must be sampled at a frequency greater than twice as high as the highest frequency present in the signal itself
  • Minimum sampling frequency is called: Nyquist sampling frequency
    • Highest voluntary human motion: 10-20 Hz
  • Shannon's reconstruction formula

Smoothing Data

  • Polynomial Smoothing
    • Polynomial Smoothing
    • Ninth order or less works well for human motion
    • Interpolation (Good and bad)
    • Spreadsheet
  • Splines
    • Cubic and quintic splines are most common in biomechanics
  • Fourier Smoothing
    • Inverse transformation
  • Moving Average
  • Digital Filtering
  • End Point Problems

How do we determine the appropriate filter settings (Filtered Data, Power Sprectrum)?


Mini-Project #5