Lecture 05

For today you should:

1) Read Chapter 4 on NB and make comments/ask questions.

2) Work on exercises.

Today:

1) Noise

2) Spectral density

For next time:

1) Work on exercises.

2) Progress report.

3) Consider case study ideas.

Progress report notes:

1) Include results.

2) Write your own problem set.

3) Practice QMRI.

Question: What are you trying to find out?

Methodology: How are you going about it?

Result: What happened?

Interpretation: How do you interpret the result as an answer to the original question?

Noise

Noise = signal that contains many frequency components; usually there is no clear perceived pitch.

Noise signals are characterized by:

1) Statistical distribution of values (uniform, Gaussian, chi, other)

2) Correlation structure (which is a generalization of serial correlation)

3) Power spectral density: how much power is there at each frequency (on average)?

UU noise is (1) uniform, and (2) uncorrelated, and turns out to have (3) uniform PSD, so it is "white"

UB noise is (1) uniform, and (2) strongly correlated, and turns our to have (3) power proportional to 1/f^2, so it is "red".

If you specify (1) and (2), you can find (3) experimentally or analytically.

For pink noise, I went the other way, choosing (1) and (3).

1) Start with UU noise.

2) Compute FFT.

3) Apply pink filter (1/f in power, 1/sqrt(f) in amp).

4) Inverse FFT.

In the next chapter, we explore the correlation structure of pink noise.

Exercise 4.1: Implement Voss-McCartney.

Additional exercise: Read http://en.wikipedia.org/wiki/Colors_of_noise and implement other colors of noise.

Estimating spectral density

With a periodic signal, every period is the same. If you've seen one, you've seen them all.

With a noise signal, there is no period, and every segment is different. The power-as-a-function-of-frequency relationships hold on average (over a long period of time), but the spectrum of a given segment is...well, noisy.

Bartlett's method is a way to estimate average spectral density as a function of frequency.

Here's the description from Wikipedia:

Bartlett’s method consists of the following steps:

    1. The original N point data segment is split up into K (non-overlapping) data segments, each of length M

    2. For each segment, compute the periodogram by computing the discrete Fourier transform (DFT version which does not divide by M), then computing the squared magnitude of the result and dividing this by M.

    3. Average the result of the periodograms above for the K data segments.

    4. The averaging reduces the variance, compared to the original N point data segment.

The end result is an array of power measurements vs. frequency "bin".

Exercise: Implement this method and plot the "averaged periodogram" for the synthesized and/or noise signals.

Exercise 4.2: Characterize the spectrum of natural noise sources: cymbal crash, applause, rain, etc.

Exercise 4.3: Characterize the spectrum of time series data: stock market, astronomical data, etc.