Spectral Analysis Option
FlexPro's spectral analysis option offers state of the art procedures for the analysis of both stationary and non-stationary data in a ground breaking easy-to-use wizard interface. The Analysis Wizard can help you visually study the diverse spectral procedures and algorithms and compare them with one another. Once a given spectral procedure is refined and optimized, the results can be immediately saved as objects and documents in FlexPro’s Object List.
Fourier Analyses
FlexPro offers four different options for FFT-based spectra. Three of these procedures are highlighted in the Fourier Spectral Analysis tutorial. This tutorial focuses on high dynamic range Fourier analysis and is particularly recommended if power or amplitude measurements are needed and there are low power signal components present.
The Fourier Spectrum is the main FFT spectral analysis. FlexPro 's best-exact-n FFT makes use of four different fast FFT algorithms. You can thus use any length data sequence without concern for the issues associated with power-of-2 zero padding. FlexPro offers a broad selection of data tapering windows. There are twenty windows with a fixed width windows and nine adjustable tapers, including Chebyshev, VanderMaas and Slepian windows. Zero padding is as simple as specifying the length of the FFT. All Fourier Analyses offer a variety of display formats. In addition to power normalization, you can display dB, normalized dB and amplitude spectra. Peak detection uses a cubic spline-based bin interpolation in order to achieve refinement in peak frequencies. FlexPro also offers critical limits for most Fourier procedures, an asset when you are uncertain if data statistically differ from random noise.
For reduced variance Fourier estimates, the Periodogram spectral procedure averages FFTs from overlapping segments. The Multitaper Spectrum procedure uses the series of orthogonal Slepian data tapers so that information at the edges of the data is utilized, and the variance of the spectral estimate is likewise reduced.
The Peak-Hold Spectrum also segments the data and calculates multiple spectra. However, these are not averaged. Rather, the maximum is formed across all spectra. This procedure is suitable for evaluating non-stationary signals in order to detect resonances during ramp-up, for instance.
Unevenly Spaced Data Fourier Analysis
The Unevenly Spaced Data Fourier Analysis generates a Lomb-Scargle Periodogram. Its primary use in FlexPro is for data with unevenly spaced X values or for data that contains void values. The algorithm was extended to use all continuous data tapering windows. A continuous Chebyshev approximation window was created primarily to service this algorithm.
High Resolution Frequency Estimators
FlexPro offers three options for Spectral estimators. These procedures are often the only alternative for very short data lengths. The Spectral Estimator tutorial is highly recommended for those instances where narrowband harmonics must be estimated to a very high accuracy and when stationary data segments are very short.
The AR Spectral Estimator analysis offers a selection of state-of-the-art autoregressive algorithms. An AR model is fitted to the data and its coefficients are used to generate a continuous spectrum. The best AR spectral methods are excellent frequency estimators, offering this accuracy with quite short data sets. Least-squares methods that offer intrinsic separation of signal and noise through singular value decomposition (SVD) are the most robust of FlexPro’s AR methods. An AR spectral estimator can be generated for any starting and ending frequency and with any desired frequency spacing. Since AR spectral peaks can be exceedingly sharp, FlexPro offers an Adaptive option which uses a Runge-Kutta procedure to integrate the spectrum adaptively. This offers a frequency set containing frequencies concentrated near the peaks. The peak frequencies are computed from the complex roots of the AR model, and are computed to full machine precision.
The ARMA Spectral Estimator is viewed as a good model for signals with noise, since both peaks and nulls can be described. A pole-zero non-linear model is required. True state of the art non-linear fits are offered by FlexPro, and these can include spectral factorization for stability and SVD for signal-noise thresholding.
The EigenAnalysis Spectral Estimator analysis offers the MUSIC (Multiple Signal Classification) and EV (eigenvector) high-performance frequency estimation algorithms. Since these algorithms can produce exceedingly sharp spectral peaks, FlexPro's Adaptive spectrum is particularly valuable. The frequency of each spectral component is automatically refined to full machine precision.
Time-Frequency Spectrum
FlexPro offers three options for non-stationary data whose frequency domain characteristics are changing across time. For a good exposition of FlexPro's capabilities with non-stationary data, the Time-Frequency Spectral Analysis tutorial is highly recommended. The frequency-time tradeoff that is a key part of optimizing these analyses is covered in significant detail.
The STFT (Short Time Fourier Transform) Spectrum analysis produces a 3D time-frequency plot based upon a segmented overlapped FFT. Windowing is normally used to sharpen the resolution in time and minimize spectral leakage. Since the STFT has a uniform time-frequency resolution, amplitudes can be directly read from the spectrum.
The CWT (Continuous Wavelet Transform) Spectrum analysis offers state of the art wavelet spectral analysis. FlexPro offers three adjustable mother wavelets. The number of frequencies is adjustable, as is whether or not the spacing is logarithmic. You can also set whether to use linear or logarithmic frequency division. A high frequency domain resolution Morlet wavelet is available for large data sets
The Peak-Hold STFT spectrum corresponds to the STFT spectrum, but only the global maximum together with its time and frequency data is copied from each single spectrum to the result.
Harmonic Estimation
One of the best ways to characterize signals that consist only of narrowband harmonics and noise is to directly model the oscillations in the time domain. It is also a robust method for harmonic distortion measurements. The Harmonic Estimation tutorial is essential, since the modeling is not a simple one-step procedure.
The Harmonic Estimation analysis uses a powerful composite algorithm that generates a parametric (sinusoids or damped sinusoids) model of the signal. The algorithm has two stages. In the first stage, a procedure is used to estimate the frequencies and component count. The best algorithms use SVD for removing the influence of observation noise. In the second stage a linear fit is made to determine the amplitudes, phases, and damping factors.
Harmonic distortion spectra are available, and are invaluable in optimizing THD measurements.
Two Signal Spectral Procedures
FlexPro offers a variety of two-signal spectral analyses. These are covered in the Cross Spectral Analysis tutorial. The tutorial provides the basics on Fourier Spectral Analysis. It is therefore suggested that you work through the Fourier Spectral Analysis Tutorial first.
The Fourier Cross Spectrum spectral procedure generates spectra that reflect the common power across two distinct signals The Cross Periodogram procedure finds the cross-spectral components using overlapping segments. The Coherence and SNR Spectra option offers coherence computations and Signal to Noise Ratio (SNR) spectra on overlapping Fourier segments.
The Fourier Transfer Function procedure computes the Fourier transform transfer function relating linear system input and output streams.
Non-Linear Models
The Cepstral Analysis is primarily used in echo detection and speech signal applications.
Shock Response Spectrum (SRS)
The Shock Response Spectrum (SRS) is computed from the signal of an accelerometer. The acceleration signal is used for primary excitation of a series of single degree of freedom (SDOF) systems with customizable natural frequencies. The spectrum is formed by the absolute maxima, maxima or minima of these systems’ responses. The shock response spectrum was originally introduced to analyze the damage potential of mechanical impulses, but it can also be used to analyze the damage potential of stationary random vibrations.
Instantaneous Quantities
FlexPro gives you the option to calculate instantaneous quantities (instantaneous amplitude, instantaneous phase and instantaneous frequency) of single-component signals. The instantaneous quantities can also be used to demodulate signals (amplitude demodulation, phase demodulation and frequency modulation).
The algorithm used to determine the instantaneous quantity is based on the Hilbert transform and uses the analytic signal derived from the Hilbert transform. Details and examples of this can be found in the Online Help covering the AnalyticSignal and Hilbert functions.
Analysis Objects
Time-Frequency Spectral Analysis
Uneven Data Fourier Spectral Analysis Object