Chapter 10

Question 1

What are the reasons we need data analysis and preparation? (4)

Answer 1

• Determination of Signal Time Derivatives
• Noise reduction
• State estimation
• Data consistency checks

Question 2

What is local smoothing in time domain?

Answer 2

• Fitting local polynomial in time to the measured data points

Question 3

What are digital filters?

Answer 3

• Digital filters are a linear combination of input values, i.e. moving average smoother
• Have the advantage that they are symmetric per construction, i.e. do not introduce any
  delay/lag in the smoothed time histories

Question 4

What is the idea behind forward backward filtering?

Answer 4

Filter a given noisy time series twice with a given transfer function, once forward in time, then filtering the forward filtered time series backwards.

Question 5

What is interpolation?

Answer 5

• Local interpolation in time domain is similar to locally smoothing in time domain, just that the center point 𝑎0 is now missing
• Interpolation is done by evaluating a local fitted polynomial at the location of
  the missing data point, i.e. center point
• Interpolation results are good overall, degrade near the endpoints of the signal to be interpolated
• Very sensitive to noise, because no mechanism to distinguish between deterministic signal and noise is given in the interpolation scheme
• Instead, global Fourier smoothing can be used by simply using a target time
  vector with a higher sampling rate in the interpolation of the time history via the inverse Fourier transform

Question 6

What is time delay estimation?

Answer 6

• Problem: Find time lag of two signals that are shifted in time
• Solution:
  – Cross-correlation (or convolution) of both signals for different sample lags reveals the time shift of both signals

Question 7

What are two approaches for numerical differentiation of noisy signals?

Answer 7

1. Choose numerical differentiator incorporating a suitable low-pass digital filter that suppresses the noisy high frequencies.
2. Filter the raw data first (see filter methods) and differentiate the smoothed data
   using one of the differentiator having near ideal response characteristics

Question 8

What are the three types of estimation problems?

Answer 8

Prediction, Filtering, Smoothing

Question 9

What is one important linear filter?

Answer 9

Kalman filter

Question 10

What are the three types of smoothing?

Answer 10

• Fixed interval
• Fixed point
• Fixed lag

Question 11

What is the RTS smoother?

Answer 11

The Rauch-Tung-Striebel Smoother combines the backward filter pass with the (smoothing) pass into a single backward recursion.