Non Stationary Signals

NON-STATIONARY SIGNALSComplementos de Processamento de Sinal

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WHAT IS A STATIONARY SIGNAL?

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STATIONARY SIGNALA stationary signal is one that maintains the same statistical measurements for the duration that is being observed.

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STATIONARY SIGNAL

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WHAT IS A NON-STATIONARY SIGNAL?

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NON-STATIONARY SIGNAL

A non-stationary signal is a signal were its statistical properties do vary over time.

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NON-STATIONARY SIGNAL

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TIME-VARIANT SYSTEMS

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A time-variant system is a system where its output depend explicitly upon time.

A non-stationary system has coefficients that do vary with time, and if the coefficients change with time, so does the transfer function, the frequency response and the impulse response of the system.


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Considering an Autoregressive (AR) process,

time-invariant system time-variant system


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Characteristics of a time-variant systemMeanThe short-time mean represents the average of the signal that is being observed. If the mean is constant, this means that the signal is stationary, but if the mean varies for window to window, this could indicate that the signal is non-stationary.


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Characteristics of a time-variant systemVarianceThe variance represents the average power level of the signal.


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Characteristics of a time-variant systemMeasures of activityMeasuring the activity of a signal is analysing its busy-ness, for example by analysing its turning points.


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Characteristics of a time-variant systemAutocorrelation Function (ACF)For a signal to be stationary, the autocorrelation would have to be independent of time. But for a non-stationary the autocorrelation will include a variable for time.


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Characteristics of a time-variant systemAutocorrelation Function (ACF)

wide-sense stationary process

non-stationary processLag


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Characteristics of a time-variant systemPower Spectral Density (PSD)By analysing the spectrogram, if there are a lot of variations in the short-time PSD then that means that the signal is non-stationary. The PSD and the ACF are inter-related by the Fourier transform, meaning that if the PSD is non-stationary the ACF is also non-stationary.


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Characteristics of a time-variant systemHigher-order statistics

The ACF and the PSD can only work on random signals with an order lower or equal to 2.

So we use higher-order statistics to study random signal order greater than 2. Significant variations in could mean that the signal is non-stationary


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Characteristics of a time-variant systemSystem parameters

When a time-variant system model is available, we are able to track changes to its coefficients (ak) over time. So that changes in the model parameters will also affect the output.

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There is only one Problem

How are we going to study a signal that is always changing?


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FIXED SEGMENTATION

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FIXED SEGMENTATION

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The simplest way to break a non-stationary signal in to to quasi-stationary segments would be to consider small windows of fixed duration. Given a signal x(i) for i = 1,2,3,4,, N-1, and considering a duration of M samples, with M

FIXED SEGMENTATION

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Short-time Fourier TransformNow that the signal has been divided into quasi-stationary parts xk(n), we can calculate the Fourier transform for each segment.

FIXED SEGMENTATION

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Short-time Fourier Transform

Segmentation of the signal may be interpreted as the application of a moving window to the signal.

FIXED SEGMENTATION

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Fourier Transform

FIXED SEGMENTATION

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If the time and the frequency variables are continuous

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FIXED SEGMENTATION

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Disadvantages of fixed segmentation The need to calculate a Fourier transform for

every segment.

The duration of the segments (the window has to be short enough to be stationary, but long enough to permit a meaningful analysis).

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ADAPTIVE SEGMENTATION

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Is the process of divided a signal into segments with different durations.

This is done by analysing the prediction error. If the prediction error is high, this means that the signals characteristics have changed.


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There are 3 ways to apply adaptive segmentation to a non-stationary signal.

Spectral Error Measure (SEM)

Autocorrelation Function Distance (ACF)

Generalised Likelihood Ratio (GLR)


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Spectral Error Measure (SEM)This technique uses a fixed window, a moving window and a threshold to divide the signal. By the error in the fixed window and comparing that with the moving window we obtain a spectrum. At the point were the line crosses the threshold, that is the location were the original signal gets segmented.


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Change in the spectral shapeChange in the total power

of the prediction error


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Autocorrelation Function Distance (ACF)

This method is similar to the one before, but instead of using the prediction error it uses the autocorrelation function. If the change in between the ACF of the reference window and the moving window is significant, that means that we should be a segment boundary there.


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Generalised Likelihood Ratio (GLR)This method uses a reference window that continuously grows as long as no new boundary is marked.

The sliding test window is in front of the reference window and is of constant duration.The pooled window formed by concatenating the reference window and the sliding test window.


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LETS RECAPStationary vs Non-stationary signalsTime-variant systemsmeanvariance

autocorrelation functionmeasure activities

power spectral density

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LETS RECAPhigher-order statisticssystem parametersFixed Segmentationshort-time Fourier transformAdaptive Segmentation

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LETS RECAPAdaptive Segmentation

Spectral Error Measure (SEM)Autocorrelation Function Distance (ACF)Generalised Likelihood Ratio (GLR)

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ANY QUESTIONS?


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THANK YOU

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Non Stationary Signals