Lesson 9
Introduction Signal Spectral Analysis: Estimation of the power spectral density The problem of spectral estimation is very large and has applications very different from each other Applications: To study the vibrations of a system To study the stability of the frequency of a oscillator To estimate the position and number of signal sources in an acoustic field To estimate the parameters of the vocal tract of a speaker Medical diagnosis Control system design In general To estimate and predict signals in time or in space
Study of radio frequency spectrum in a big city side by side there are the various radio and television channels, the signals cell phone, radar signals, etc. The frequency ranges, if considered with sufficient bandwidth, are occupied by signals totally independent of each other, with different amplitudes and different statistical characteristics To analyze the spectrum, it seems logical to use a selective receiver that measures the energy content in each interval frequency. Non Parametric We will seek the most accurate possible Spectral Analysis estimate of these energies in the time available without making any further assumptions, not looking for models of signal generation
Study of radio frequency spectrum in a big city The non-parametric spectral analysis is a conceptually simple matter if you use the concept of ensemble average. if you have enough realizations of the signal, just calculate the discrete Fourier transform and averaging the powers, component by component. However, rarely you have numerous replicas of the signal; often, you have available a single replica, for an interval of time allotted To determine the power spectrum, you have to use additional assumptions such as stationarity and ergodicity
Analysis of the speech signal consider the spectrum of acoustic signal due to vibration or a voice signal in this case, all of the signal as a whole has unique origins and then there will be the relationship between the contents of the various spectral bands. it must first choose a model for the generation of the signal and then determine Parametric the parameters of the model itself Spectral Analysis For example, it will seek the parameters of a linear filter that, powered by a uniform spectrum signal (white noise) produces a power spectrum similar to the spectrum under analysis
Analysis of the speech signal Obviously, the success of the technique depends on the quality and parametric correctness of the model chosen. Valid models lead to a parsimonious signal description , that is characterized by the minimum number of parameters necessary This will lead to a better estimate of these parameters and then to optimal results the parametric spectral analysis leads to the identification of the model and this opens a subsequent phase of study to understand and then possibly check the status and evolution the system under observation
Formal Problem Definition Let be y= {y(1), y(2), . . . , y(N)} a second order stationary random process, ˆ GOAL: to determine an estimate of its power ( ) spectral density for ω ∈ [−π, +π ] ( ) Observation ˆ We want | ( ) ( ) | 0 The main limitation on the quality of most PSD estimates is due to the quite small number of data samples N usually available Most commonly, N is limited by the fact that the signal under study can be considered wide sense stationary only over short observation intervals
Two possible way(1) There are two main approaches Non Parametric Parametric Spectral Spectral Analysis Analysis assume that the underlying stationary explicitly estimate the covariance stochastic process has a certain structure or the spectrum of the process which can be described using a small without assuming that the number of parameters. In these approaches, process has any particular the task is to estimate the parameters of the structure model that describes the stochastic process
Non Parametric Estimation: Priodogram The periodogram method was introduced by Schuster in 1898. The periodogram method relies on the definition of the PSD in practice the signal y(t) is only available for a finite interval Periodogram Power Specrtal Density Estimation
Correlogram Spectrum estimation can be interpreted as an autocorrelation estimation problem. Correlogram Power Specrtal Density Estimation the estimate of the covariance lag r(k), obtained from the available sample {y(1), y(2), . . . , y(N)}
Estimation of Autocorrelation Sequence(ACS) There are two standard way to obtain an estimate ˆ k ( ) r unbiased estimate biased estimate Both estimators respect the symmetry properties of the ACS The biased estimate is usually preferred, for the following reasons: the ACS sequence decays rather rapidly so that r(k) is quite small for large lags k the ACS sequence is guaranteed to be positive semidefinite . is not the case for the unbised definition if the biased ACS estimator is used in the estimation the correlogram is eqaul to the periodogramm
Statistical Performance Both Periodogram and Correlogram are asymptotically unbiased: Both have large variance, even for large N ˆ can be large for big k | ( ) ( ) | r k r k ˆ | ( ) ( ) | r k r k even if the errors are small, there are ”so many” ˆ that when summed in the PSD error is large | ( ) ( ) | k k
Periodogram Bias Bartlett window. Frequency Domain
Bartlett window Ideally, to have zero bias, we want WB(ω ) = Dirac impulse δ(ω ) The main lobe width decreases as 1/N. For small values of N, WB(ω) may differ quite a bit from δ(ω ) If the unbised estimation the window is rectangular
Summary Bias analysis Note that, unlike WB(ω ), WR(ω ) can assume negative values for some values of ω, thus providing estimate of the PSD that can be negative for some frequencies. The bias manifests itself in different ways Main lobe width causes smearing (or smooting): details in φ(ω) separated in f by less than 1/N are not resolvable. periodogram resolution limit = 1/N Sidelobe level causes leakage For small N, severe bias As N → ∞, WB (ω) → δ(ω), so φ(ω ) is asymptotically unbiased
Periodogram Variance As N → ∞ inconsistent estimate erratic behavior
The Blackman-Tukey method Basic idea : weighted correlogram , with small weight applied to the estimated covariances r(k) with large k lag window Frequency Domain The BT periodogram is a locally smoothed Variance decreases substantially (of the order of M/N) Bias increases slightly (of the order 1/M) The window is chosen so as to ensure that the spectral density of estimated power is always positive
Choice of the BT window Let be It is possible to prove that This means that the more slowly the window decays to zero in one domain, the more concentrated it is in the other domain The equivalent temporal width, N e is determined by the window length (Ne = 2M) for rectangular window, Ne = M for triangular window). Since N e βe = 1 also the bandwidth β e is determined by the window length As M increases, bias decreases and variance increases ⇒ choose M as a tradeoff between variance and bias. As a rule of thumb, we should choose M ≤ N/ 10 in order to reduce the standard deviation of the estimated spectrum at least three times, compared with the periodogram Choose window shape to compromise between smearing (main lobe width) and leakage (sidelobe level). The energy in the main lobe and in the sidelobes cannot be reduced simultaneously, once M is given
The Bartlett method Basic idea : split up the available sample of N observations into L = N/M subsamples of M observations each, then average the periodograms obtained from the subsamples for each value of ω.
Welch method Similar to Bartlett method, but allow overlap of subsequences (gives more subsequences, thus better averaging) and use data window for each periodogram; gives mainlobe-sidelobe tradeoff capability overlap if K = M, no overlap as in Bartlett method if K = M/2, 50% overlap, S = 2M/N data segments The Welch method is approximately equal to Blackman- Tuckey with a non-rectangular lag window
Daniell It can be proved that, for are nearly uncorrelated random variables for The basic idea of the Daniel method is to perform local averaging of 2J + 1 samples in the frequency domain to reduce the variance by about 2J + 1 As J increases: bias increases (more smoothing) variance decreases (more averaging)
Non parametric estimation summary The non-parametric spectral analysis is a conceptually simple matter if you use the concept of ensemble average Goal is to estimate the covariance or the spectrum of the process without assuming that the process has any particular structure Priodogram- Correlogram Asymptotically unbiased, inconsistence None of the methods we have seen solves all the problems of the periodogram Parametic estimation…
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