# Biomedical Signal Processing

## Presentation on theme: "Biomedical Signal Processing"— Presentation transcript:

Biomedical Signal Processing
EEG Segmentation & Joint Time-Frequency Analysis Gina Caetano 14/10/2004

Introduction EEG Segmentation 2. Joint Time-Frequency Analysis
Spectral error measure: - Periodogram approach (nonparametric) - Whitening approach (parametric) 2. Joint Time-Frequency Analysis - Linear, nonparametric methods - Nonlinear, nonparametric methods - Parametric methods

EEG Segmentation: Spectral Error Measure
Whitening Approach - Parametric - AR model (reference window) - Linear prediction (test window) - Dissimilarity measure Δ2(n) Segmentation can be based on parametric estimation. The validity of an AR model – estimated from the reference window – is tested by linear prediction in the sliding test window. In AR modeling, the current sample x(n) can be predicted from the p previous samples x(n-1),...,x(n-p) using a Finite Impulse Response filter structure. This technique relies on the fact that a pth order linear prediction error filter Ap(z) decorrelates (“whithens”) the observed signal x(n) as long as it is described by the pth order AR model for which the predictor was designed. Once a change in spectral characteristics occurs in x(n), the output of the linear prediction error filter no longer remains a white process. Dissimilarity measure reflects deviations from the whiteness of the prediction errors e(n).

EEG segmentation AR model of order p describes signal in reference window Power spectrum of e(n) Quadratic spectral error measure Time domain Asymmetric Assuming that a proper model order p has been selected for describing the signal in the REFERENCE window, the prediction error variance σ2e can be estimated by methods presented before. The power spectrum of e(n) is flat, and it serves as a reference to subsequent spectra Se determined from the output of the prediction error filter Ap(z). In the same spirita as the spectral error measure defined from the periodogram, we define a quadratic spectral error measure Δ2(n) in order to quantify deviations between subsequent spectra. The normalization factor differs from that defined for periodogram (with respect to signal power of both test and reference windows) but represents factor suggested in original work. Again, it is advantageous to develop a time-domain expression of Δ2(n) which is better suited for implementation. In order to arrive at such, use result that any power spectrum, even function, can be expressed as a sum of cosines. It is evident that prediction errors deviating from a white process with variance re(0,0) will be penalized:the first term is close to zero as long as the variance re(0,n) of the test window is close to that of the reference window. The second term remains close to zero only as long as e(n) remains white because re(k,n) = σ2eδ(k).

EEG segmentation AR model of order p describes signal in reference window Simpler Asymmetric ad hoc “reverse” test Symmetric Simulations: prediction-based method associated with lower false alarm rate than correlation-method. An even simpler test for detecting a change in spectral characteristics is obtained by neglecting the second term and by modifying the first term such that only the power of the prediction errors are monitored...after insertion of the correlation function estimate for the predictions error. This dissimilarity measure suffers from a serious deficiency in that several AR models with identical prediction error variance. Also, the measure is asymetric as more sensitive to abrupt changes manifested by increase in power, and becomed negative when decrease. The previous two measures based on model-based segmentation incorporate knowledge of one single signal model: the AR model estimated from the reference window and then used for prediction in the test window. None of the two measures are symmetric with respect to detection of signals having increasing or deacresing power. An ad hoc approach is to incorporate a “reverse” test, i.e, to also compute the prediction errors in the reference window using the AR parameters estimated from the test window. Thus a modified error measure involving two AR models is defined. Where et(n) and er(n) denote the prediction errors obtained by AR parameters estimates from the test and reference windows...the same for corresponding error variances. Simulations indicated more symmetric behavior, however no clinical value for EEG segmentation has been established. Simulated signals were by swithching between two AR models with different power spectra. Since the time instants for segment boundaries were a priori known, it was possible to determine detection rate, false alarm rate, and accuracy. Results contrast with previous findings on false alarm rate, that lead pioneers to retract from the prediction-based method (hypersegmentation & misplaced segments), and replaced it by the former correlation-based method. Once EEG segmentation method acceptable performance – group EEG segments in patterns: N’ shortest correlation lags, spectral features (power spectrum), estimated AR parameters.

Joint Time-Frequency Analysis
When in time different frequencies of signal are present Linear, nonparametric methods - Linear filtering operation - Short-time Fourier transform - Wavelet transform Nonlinear, nonparametric methods - Wigner-Ville Distribution (ambiguity function) - General Time-Frequency distributions – Cohen’s class Limitation of Fourier transform is its inability to provide information on when in time different frequencies of a signal are present. Fourier transform only reflects which frequencies exist during the total observation interval-integrates frequency components over all interval. Fourier spectral analysis is adequate for stationary signals whose frequencies are in average equally spread in time. It is inadequate for non-stationary signals with time-dependent spectral content Strong motivation to develop methods which analyze signals in time frequency. Methods divided in three main categories: Linear, nonparametric methods- their time-frequency representations can be obtained from a linear filtering operation. Short-time Fourier Transform is the classical. The wavelet transform is another popular method which belongs to the category of linear methods, and has special hability to characterize transient signals (described in chapter 4) Nonlinear, nonparametric methods - offer an improved time-frequency resolution. The Wigner-Ville distribution, a number of its modifications and it´s limitations are introduced. As ion 1), do not involve modeling assumptions on the signal. Parametric methods – produce time-frequency representations based on assumption that signal derives from a statistical model (AR model as starting point) with time-varying parameters. Methods for AR parameter estimation are modified so that slow changes in parameter values can be tracked and used for computing successive power spectra. Discrete-time context temporarily abandoned to present nonparametric time-frequency methods – easier to understand. Straightforward to translate the continuous-time, short-time Fourier transform to its dicrete-time counterpart. More difficults to WVD. Also, signals in this section are considered deterministic insted of stochastic. Parametric methods - Statistical model with time-varying parameters - AR model parameter estimation (slow changes in time)

Joint Time-Frequency Analysis
When in time different frequencies of signal are present Linear, nonparametric methods - Linear filtering operation - Short-time Fourier transform - Wavelet transform Nonlinear, nonparametric methods - Wigner-Ville Distribution (ambiguity function) - General Time-Frequency distributions – Cohen’s class Limitation of Fourier transform is its inability to provide information on when in time different frequencies of a signal are present. Fourier transform only reflects which frequencies exist during the total observation interval-integrates frequency components over all interval. Fourier spectral analysis is adequate for stationary signals whose frequencies are in average equally spread in time. It is inadequate for non-stationary signals with time-dependent spectral content Strong motivation to develop methods which analyze signals in time frequency. Methods divided in three main categories: Linear, nonparametric methods- their time-frequency representations can be obtained from a linear filtering operation. Short-time Fourier Transform is the classical. The wavelet transform is another popular method which belongs to the category of linear methods, and has special hability to characterize transient signals (described in chapter 4) Nonlinear, nonparametric methods - offer an improved time-frequency resolution. The Wigner-Ville distribution, a number of its modifications and it´s limitations are introduced. As ion 1), do not involve modeling assumptions on the signal. Parametric methods – produce time-frequency representations based on assumption that signal derives from a statistical model (AR model as starting point) with time-varying parameters. Methods for AR parameter estimation are modified so that slow changes in parameter values can be tracked and used for computing successive power spectra. Discrete-time context temporarily abandoned to present nonparametric time-frequency methods – easier to understand. Straightforward to translate the continuous-time, short-time Fourier transform to its dicrete-time counterpart. More difficults to WVD. Also, signals in this section are considered deterministic insted of stochastic. Parametric methods - Statistical model with time-varying parameters - AR model parameter estimation (slow changes in time)

Short-Time Fourier Transform
2D modified Fourier transform ω(t) length resolution in time and frequency Simplest: time-frequency analysis done dividing signal x(t) into short consecutive, possibly overlapping, segments, to which spectral analysis performed. Series of spectra reflect then time-varying nature of signal. Fourier-based spectral analysis applied to each of the short segments: short-time Fourier transform. Fourier transform is modified such that a sliding time window w(t) is included which excerpts each segment to analyze, thereby resulting in a 2D function. The time window is a positive valued function, which can have a rectangular shape, hanning, hamming, Blackman (pag 98). The window length determines the resolution in time and frequency. Short window => good time resolution but poor frequency resolution. Long window => good frequency resolution, but poor time resolution. Analogous to the computation of the periodogram, which was obtained as the squared magnitude of the Fourier transform of the signal, the spectrogram of x(t) is obtained by computing the squared magnitude of the STFT. The spectrogram is a real-valued distribution which thus offers a signal representation in the time-frequency domain. Stochastic (and discrete-time) version was presented for the purpose of segmenting nonstationary signals: running peropdogram (3.186). Spectral analysis based on Fourier transform can never achieve perfect resolution in both time and frequency. Uncertainty principle states that the product of a signal’s time duration and it’s bandwidth must always exceed a lower bound. Δt and ΔΩ are defined as the standard deviation of a random variable, and thus they represent a measure of width in time and frequency, respectively. X(Ω) is the Fourier transform of x(t), t an Ω parameters define the center of gravity of x(t) and X(Ω) respectively, and are obtained in a way analogous to the mean value of a random variable. The equality in the uncertainty principle is only achieved if x(t)=c.exp(-t^2/σ) is a Gaussian signal, σ- width parameter, c- constant. The above particular uncertainty principle only applies to Fourier-based spectral analysis, while other bounds apply to the nonlinear time-frequency representations. Spectrogram Uncertainty Principle Only Fourier-based spectral analysis

Short-Time Fourier Transform
Spectrogram The spectrogram has been used in a wide range of EEG applications thanks to its capability to lucidly disply changes that in the rhythmical activities, difficult to appreciate from the time domain signal. Fig 3.29 Patients with suspected epilepsy undergoing a test on photic stimulation may or may not demonstrate EE response to stimulation. By looking at the spectrogram, it becomes clear if such response is present or not. Figure shows spectrogram obtained during photic stimulation at different rates, which for this particular patient is associated with a well defined peak at each rate. The display format is referred as a compressed spectral array, since successive spectra are presented in compact format, one in front of each other. a) 10 s excerpt of 4-channel EEG recorded during photic stimulation at 12 Hz. b) Spectrogram resulting from all five rates (6, 10, 12, 15, and 20 Hz) exhibits a marked peak. The power spectra are computed from consecutive, 8 seconds intervals.

Short-Time Fourier Transform
Spectrogram EEG Spectrogram Fig a) EEG recorded during heart surgery of an infant, brain activity was recorded. b) Corresponding Spectrogram displays a drastic reduction in high-frequency content after 100 s, which then partially reverts at ~200 s. The change occurs in simultaneous with a decrease in blood pressure. The EEG changes consist in a reduction of frequency content in the interval above 7-8 Hz- likely caused by lack of oxygenated blood that perfuses brain. Diastolic blood pressure

Short-Time Fourier Transform
Spectrogram EEG 1 s Hamming window 2 s Hamming window EEG spectral changes can also occur and evolve much more rapidly on a second-to-second basis, illustrated by the 12-second EEG in : Fig a) EEG recorded during epileptic seizure. Visually, one can easily establish that the temporary increase in amplitude is coupled to a more sinusoidal behaviorr of the signal. But human ey can not appreciate the slow-down in frequency which occurs in the interval 4-8 seconds. The corresponding spectrogram is computed using a Hamming window with a length of b) 1 second, c) 2 second, d) 0.5 second. Evident that the dominant frequency component, initially located around 9 Hz at t = 4s, gradually declines to 6 Hz during the four seconds to t= 8s. Finally, the dominant frequency dissolves into a pattern which resembles that observe prior to the seizure. Meaning unclear. Longest time window exhibits poorest time resolution, and best frequency resolution. Vice versa. One is always faced with a trade-off with respect to resolution in time and frequency. 0.5 s Hamming window

Joint Time-Frequency Analysis
Linear, nonparametric methods - Linear filtering operation - Short-time Fourier transform - Wavelet transform Nonlinear, nonparametric methods - Wigner-Ville Distribution (ambiguity function) - General Time-Frequency distributions – Cohen’s class Limitation of Fourier transform is its inability to provide information on when in time different frequencies of a signal are present. Fourier transform only reflects which frequencies exist during the total observation interval-integrates frequency components over all interval. Fourier spectral analysis is adequate for stationary signals whose frequencies are in average equally spread in time. It is inadequate for non-stationary signals with time-dependent spectral content Strong motivation to develop methods which analyze signals in time frequency. Methods divided in three main categories: Linear, nonparametric methods- their time-frequency representations can be obtained from a linear filtering operation. Short-time Fourier Transform is the classical. The wavelet transform is another popular method which belongs to the category of linear methods, and has special hability to characterize transient signals (described in chapter 4) Nonlinear, nonparametric methods - offer an improved time-frequency resolution. The Wigner-Ville distribution, a number of its modifications and it´s limitations are introduced. As ion 1), do not involve modeling assumptions on the signal. Parametric methods – produce time-frequency representations based on assumption that signal derives from a statistical model (AR model as starting point) with time-varying parameters. Methods for AR parameter estimation are modified so that slow changes in parameter values can be tracked and used for computing successive power spectra. Discrete-time context temporarily abandoned to present nonparametric time-frequency methods – easier to understand. Straightforward to translate the continuous-time, short-time Fourier transform to its dicrete-time counterpart. More difficults to WVD. Also, signals in this section are considered deterministic insted of stochastic. Parametric methods - Statistical model with time-varying parameters - AR model parameter estimation (slow changes in time)

Wigner-Ville Distribution (WVD)
Ambiguity Function Energy Density Spectrum Energy Function Maximum Spectrogram very useful in biomedical applications, but its poor time-frequency resolution prompted development of other techniques as the Wigner-Ville distribution, and which is not constrained to the uncertainty principle related to Fourier transform. Rather than proceeding directly to the definition of the WVD, we will first introduce the ambiguity function which is central to WVD definition, and is an important concept in time-frequency analysis. This function is designed to reflect uncertainty in both time τ and frequency v, that is associated with a signal x(t). Two versions of x(t) which are jointly shifted in time and frequency are introduced. The modulation in frequency is introduced by modulating the signal with the complex exponencial exp(jvt). The ambiguity function nis a 2D-function defined as the correlation between the two time and frequency shifted signal versions, of which one is conjugated. The ambiguity function can be understood as the Fourier transform of the product x*(t-T/2) x(t+T/2) which decribed the deterministic, instantaneous correlation of two values separated by the time lag T. One useful observation is that the Fourier transform of Ax(T,v) yelds the instantaneous, time-dependent correlation. The Fourier transform of Ax(T,0) with respect to T yields the energy density spectrum Sx(Ω). The spectrum Sx(Ω) is the deterministic counterpart to the power spectrum of a random signal. An important property of the ambiguity function is that its maximum is always located in the origin (0,0) in the time T-v domain, and is equal to the energy of the signal x(t). It can also be shown that that the ambiguity function remains concentrated to the origin of the T-v domain although x(t) is subjected to a shift in time and frequency. Decomposing a signal in two components - positive and negative frequencies - it is shown that apart the “auto-term” which is concentrated to the origin, the ambiguity function also includes undesirable terms with identical shape but translated in frequency, reflecting croos-correlation between positive and negative frequencies.

Wigner-Ville Distribution (WVD)
Ambiguity Function Analytic signal Decomposing a signal in two components - positive and negative frequencies - it is shown that apart the “auto-term” which is concentrated to the origin, the ambiguity function also includes undesirable terms with identical shape but translated in frequency, reflecting croos-correlation between positive and negative frequencies. It is possible to remove such cross-correlation without sacrificing signal information. Since real signal have symmetric frequency components, of which one is redundant, we only need to consider positive frequencies of the spectrum. This part can be isolated using the analytic function, and the resulting ambiguity function no longer contains the cross-correlation terms. Analityc signal only accounts with the positive frequencies. And the respective ambiguity function no longer contains the crosscorrelation terms. Considering now the case of a signal composed of two different components x1(t) = s1(t).exp(j.Ω1.t) and x2(t), the ambiguity function will have two “auto-terms” which only depend on the individual signals, and will also include cross-terms which reflect the correlation between the two different components. The separation of these two tupes of terms in the T-v domain turns out to be a very important property to exploit. Fig 3.32 – a) The signal x(t) is the sum of two modulated Gaussians x1(t) and x2(t). Using the analytic representation, the ambiguity domain is shown for b) x1(t), and c) x(t), where the autoterm and the two cross-terms are clearly separated from one another. The properties of the ambiguity domain are illustrated by a signal consisting of two-components which have envelopes s(t) with gaussian shape and shifted in time by t1 and t2, respectively, and which have center frequencies (modulation frequency) Ω1 and Ω2 respectively. Hence the two components are concentrated at points (t1, Ω1) and (t2, Ω2) in the t- Ω domain. In the T-v domain, on the other hand, Ax1(T,v) and Ax2(T,v) are both concentrated at the origin, while the cross-terms are not (clearly separated from origin). Analytic Ambiguity Function

Wigner-Ville Distribution (WVD)
WVD: Continuous-time definition Modulated Gaussian Signal Spectrogram The continuous-time definition of the Wigner-Ville distribution is given by the two-dimensional Fourier transform of the ambiguity function. By inserting the definition of the ambiguity function, the experssion shows explicitly that the Wigner-Ville distribution is related to the signal x(t) , and is obtained by calculating the Fourier transform of the instantaneous time-dependent correlation with respect to the time lag T. The WVD is highly nonlocal transformation, since it weights times which are far away equally to those that are close. The WVD can also be represented as a correlation in the frequency domain, by invoking Parseval’s relation. WVD Properties: real valued Time support (x(t)=0 => W=0) Frequency support (X(Ω)=0 => W=0) Spectrum is obtained by integrating WVD in respect to time : |X(Ω)|^2 Signal energy is obtained by integrating WVD with respect to frequency: |x(t)|^2 The most important property of the WVD is its excellent joint resolutionin time and frequency. The improved resolution is illustrated by an example which compares the WVD to the spectrogram for a modulated gaussian signal. Fig3.33- Corresponding spectral component is much more concentrated in the VWD than in the spectrogram. WVD

Wigner-Ville Distribution (WVD)
WVD: Limitations Two-components Signal Spectrogram Unfortunately, the Wigner-Ville distribution exhibits undesirable properties, important to be aware of. WVD is not always positive valued (only for Gaussian signal). This is not a serious problem. Another more annoying property is the presence of cross-terms due to the fact that the WVD is a quadratic time-frequency distribution. The occurence of cross-terms limits the pratical use of WVD when multi-components signals are considered. The occurence of cross-terms is illustrated for the two-components signal consisting of two modulated Gaussians. The corresponding WVD has the cross-term between the two “auto-terms” (stripped pattern). In this case, the cross-term occurs midway between the frequencies (Ω1+Ω2)/2 and has an amplitude that oscillates with the frequency (Ω1-Ω2). When the signal consists of more than two components, one cross-term will arise for every pair of components, causing difficulties in interpreting the time-frequency distribution. The WVD assigns equal importance to every point in time, so it is highly non-local. In certain cases it may be important to emphasize the signal around time t. Such wighting can be done by multiplying the instantaneous, time-dependent correlation by with a window w(T) centered around the time T. Pseudo Wignerr-Ville distribution. In situations when x(t) is a mono-component signal, it may be of interest to extract the predominant frequency, which defined the main ridge of the time-frequency distribution. This information can be very valuable for the neurophysiologist (more than the fine details of the power spectrum). In order to study the the dominant frequency, we make use of complex signal representation s(t) is the envelope modulated by complex exponencial function with phase fi(t). The first derivative of the phase gives the mean instantaneous frequency. The mean instantaneous frequency can also be determined directly from the Wigner-Ville distribution by calculating the mean ferquency <Ω(t)> for each time instant. This computation is important fro mono-component signal, as variations in the signal’s dominant frequency can be quantified, and give size and duration of the decline in frequency during an epileptic event. But the mean < Ω> looses its interpretation in the presence of multicomponents signals. Wigner-Ville distribution

Joint Time-Frequency Analysis
Linear, nonparametric methods - Linear filtering operation - Short-time Fourier transform - Wavelet transform Nonlinear, nonparametric methods - Wigner-Ville Distribution (ambiguity function) - General Time-Frequency distributions – Cohen’s class Limitation of Fourier transform is its inability to provide information on when in time different frequencies of a signal are present. Fourier transform only reflects which frequencies exist during the total observation interval-integrates frequency components over all interval. Fourier spectral analysis is adequate for stationary signals whose frequencies are in average equally spread in time. It is inadequate for non-stationary signals with time-dependent spectral content Strong motivation to develop methods which analyze signals in time frequency. Methods divided in three main categories: Linear, nonparametric methods- their time-frequency representations can be obtained from a linear filtering operation. Short-time Fourier Transform is the classical. The wavelet transform is another popular method which belongs to the category of linear methods, and has special hability to characterize transient signals (described in chapter 4) Nonlinear, nonparametric methods - offer an improved time-frequency resolution. The Wigner-Ville distribution, a number of its modifications and it´s limitations are introduced. As ion 1), do not involve modeling assumptions on the signal. Parametric methods – produce time-frequency representations based on assumption that signal derives from a statistical model (AR model as starting point) with time-varying parameters. Methods for AR parameter estimation are modified so that slow changes in parameter values can be tracked and used for computing successive power spectra. Discrete-time context temporarily abandoned to present nonparametric time-frequency methods – easier to understand. Straightforward to translate the continuous-time, short-time Fourier transform to its dicrete-time counterpart. More difficults to WVD. Also, signals in this section are considered deterministic insted of stochastic. Parametric methods - Statistical model with time-varying parameters - AR model parameter estimation (slow changes in time)

Cohen’s class General time-frequency distribution
Wigner-Ville distribution pseudoWigner-Ville distribution A general class of time-frequency distributions has been introduced whose degrees of freedom can be exploited for mitigating the cross-term probability. A two-dimensional kernel function weights the ambiguity function in such a way that the undesired terms, being far from the origin are supressed, whereas the auto-terms remaine unchanged. The time-frequency distributions which were described before are all members of cohen’s class. We obtain Wigner –ville Distrib if the kernel function is 1; we obtain the pseudo WVD if the kernel function equals the window function w(T); we obtain the spectrogram using the short-time Fourier transform window function. The supression of cross-terms comes however at one price, smoothing is introduced by the kernel function (because of multiplication with ambiguity function), and it reduces the time-frequency resolution. The kernel function is assumed to have its maximum at the origin g(0,0)=1, which ensures the energy is unaffected. A kernel function should exhibit some other properties: The integration in order with time of the cohen class should give the spectrum (with g(T,0)=1); similarly the signal energy should be obtaine when integrating in order with frequency (with g(0,v) =1). A large number of kernel functions have been presented, the most bpopular beeing the cohen’s class. g(T,v)=exp {-(vT)^2/(4.PI^2.sigma), sigma >0}, sigma is a parameter which determines the degree of cross-term interference reduction and related smoothing effect. For a small value of sigma the kernel is concentrated around the origin in the ambiguity domain, wheres the distribution tends to WVD for large values of sigma. By choosing the parameter sigma, the kernel can be used to reduce the cross-terms while essentially preserving the auto-terms. Spectrogram Choi-Williams distribution

Cohen’s class Choi-Williams distribution Two-components Signal
Wigner-Ville distribution The performance improvement associated with the Choi-Williams distribution is illustrated in the figure, where the Gaussian two-component signal is again analyzed. It is evident that the cross-term is now supressed in the CWD. Choi-William distribution

Cohen’s class Choi-Williams distribution EEG Spectrogram
Loooking agsain at the example of an EEG signal recorded at onset of a seizure, we compare the respective spectropgram, Wigner-Ville distribution and the Choi-Williams distribution.It is noted that the abundance of cross-terms which are present in the WVD are largely supressed in the CWD. Of the three time-frequency representations, the CWD is for this example the one that gives the most clear-cut description of the spectral change. In the CWD we can observe some horizontal and vertical riples , because the exponential kernel doe not offer any cross ter reduction alon the T- and v-axis. The horizontal riple is due to auto-terms that have the same frequency center, and vertical riples to same time center. Here they are hardly influencing the graphic interpretation. Another step to improve performance of time-frequency analysis is to make use of a signal-dependent kernel rather than a fixed one such as exponential. This is motivated because locations of auto-terms and cross-terms in the ambiguity domain are dependent on the signal. The riples mentioned can then be reduced Wigner-Ville distribution Choi-William distribution

Joint Time-Frequency Analysis
Linear, nonparametric methods - Linear filtering operation - Short-time Fourier transform - Wavelet transform Nonlinear, nonparametric methods - Wigner-Ville Distribution (ambiguity function) - General Time-Frequency distributions – Cohen’s class Limitation of Fourier transform is its inability to provide information on when in time different frequencies of a signal are present. Fourier transform only reflects which frequencies exist during the total observation interval-integrates frequency components over all interval. Fourier spectral analysis is adequate for stationary signals whose frequencies are in average equally spread in time. It is inadequate for non-stationary signals with time-dependent spectral content Strong motivation to develop methods which analyze signals in time frequency. Methods divided in three main categories: Linear, nonparametric methods- their time-frequency representations can be obtained from a linear filtering operation. Short-time Fourier Transform is the classical. The wavelet transform is another popular method which belongs to the category of linear methods, and has special hability to characterize transient signals (described in chapter 4) Nonlinear, nonparametric methods - offer an improved time-frequency resolution. The Wigner-Ville distribution, a number of its modifications and it´s limitations are introduced. As ion 1), do not involve modeling assumptions on the signal. Parametric methods – produce time-frequency representations based on assumption that signal derives from a statistical model (AR model as starting point) with time-varying parameters. Methods for AR parameter estimation are modified so that slow changes in parameter values can be tracked and used for computing successive power spectra. Discrete-time context temporarily abandoned to present nonparametric time-frequency methods – easier to understand. Straightforward to translate the continuous-time, short-time Fourier transform to its dicrete-time counterpart. More difficults to WVD. Also, signals in this section are considered deterministic insted of stochastic. Parametric methods - Statistical model with time-varying parameters - AR model parameter estimation (slow changes in time)

Model-based analysis of slowly varying signals
Parametric model of signal Time-varying AR model Slow temporal variations Time-varying noise Two adaptive methods Minimization of prediction error LMS: minimizes forward prediction error variance Gradient Adaptive Lattice: minimizes forward and backward prediction error variances Joint time-frequency analysis can be based on some parametric model of the signal. In this section estimation of parameters of a time-varying AR model is outlined, assuming that the temporal variations are relatively slow.

Model-based analysis of slowly varying signals
LSM Algorithm (AR model, p=8)