# 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)

h(t) = a0 + a sin(2πft + θ)

2πft = ω

h(t) = a0 + a sin(ω + θ)

Fourier Transform

• 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

• 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
• Ninth order or less works well for human motion