# Data Analysis

June 2, 2021

ObserVIEW is used to analyze time waveform data. Common analysis operations and relevant topics include:

• Sampling
• Fast Fourier transform (FFT)
• Statistical analysis
• Autocorrelation
• Cross-correlation
• Auto spectral density (ASD / PSD)
• Cross spectral density (CSD)

### Taking Data Apart (Analysis)

The English word analysis comes from the Greek word ἀvάλυσις, which means to separate something into parts. There is an unlimited number of analysis methods, including:

• Sampling: taking data apart into samples.
• Fourier analysis: taking data apart into sinusoids.
• Wavelet analysis: taking data apart into wavelets.
• Statistical analysis: taking data apart into statistical properties.

### Putting Data Back Together (Synthesis)

After taking something apart with an analysis, engineers often want to put it back together by approximation or perfectly put it back together by synthesis.

The operation of putting analyzed components back together can be referred to as reconstruction. Is true synthesis (i.e., perfect reconstruction) possible? The answer is yes, but only if you have an ideal low pass filter. An almost ideal filter will result in an almost perfect synthesis.

### Alternate Domains

Data is collected in the time domain and can be analyzed in that domain. However, it is often more convenient and straightforward to use a mathematical transform to project data into a different domain for analysis. Patterns that were not clear in the time domain can become readily apparent in the new one.

The most common alternate domain in engineering applications is the frequency domain. For continuous data, engineers use the Fourier transform to project into this domain.

For discrete-time data, projection is achieved using the discrete-time Fourier transform, the discrete Fourier transform (DFT), or the fast Fourier transform (FFT). Discrete-time data is digital data collected with the assistance of an analog to digital converter or ADC.

### Bases

Every domain is built on a foundation called a basis or, more generally, a frame. For example, in the frequency domain, the basis is made up of sine and cosine waveforms. Data in each domain can then be analyzed into components by projecting the data onto these waveforms.

The set of these components for a given waveform is generally called a transform. Examples of transforms include the Fourier transform, the FFT, the DFT, cosine polynomials, Chebyshev polynomials, and any of the many wavelet transforms. Cosine polynomials. Chebyshev polynomials. Daubechies-p2 wavelets.