Lecture 2 The Wigner-Ville Distribution · Lecture 2 –The Wigner-Ville Distribution Properties of...
Transcript of Lecture 2 The Wigner-Ville Distribution · Lecture 2 –The Wigner-Ville Distribution Properties of...
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Lecture 2The Wigner-VilleDistribution
Time-frequency analysis, adaptive filtering and source separation
José Biurrun Manresa15.02.2011
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Lecture 2 –The Wigner-Ville Distribution
Time-Frequency representations
• The Short-Time Fourier Transform (STFT) takes a linearapproach for a time-frequency representation
• It decomposes the signal on elementary components, called atoms
��,� � � � � � � ���
• Each atom is obtained from the window � �� by a translation in time and a translation in frequency (modulation)
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Lecture 2 –The Wigner-Ville Distribution
Time-Frequency representations
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Lecture 2 –The Wigner-Ville Distribution
Time-Frequency representations
• Alternatively, in the frequency domain
��,� � � � � � �����
• The STFT can be considered as the result of passing the signal through a bank of band-pass filters whose frequency response is � � � �� and is deduced from a mother filter � �� by a translation of θ
• Each filter in the bank has a constant bandwidth
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Lecture 2 –The Wigner-Ville Distribution
Time-Frequency representations
• If we consider the square modulus of the STFT, we get the spectrogram, which is th spectral energy density of the locally windowed signal �� � � � � � � � ��
• The spectrogram is a quadratic or bilinear representation
• If the energy of the windows is selected to be one, the energy of the spectrogram is equal to the energy of the signal
• Thus, it can be interpreted as a measure of the energy of the signal contained in the time-frequency domain centered on the point �, ��
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Lecture 2 –The Wigner-Ville Distribution
Time-Frequency representations
• Linear
• STFT
• Wavelet
• Bilinear or Quadratic
• Cohen’s class
• Spectrogram
• Wigner-Ville
• Choi-Williams
• ...
• Affine distributions
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Lecture 2 –The Wigner-Ville Distribution
The energy distributions
• The purpose of the energy distributions is to distribute the energy of the signal over time and frequency
• The energy of a signal � �� can be deduced from the squared modulus of either the signal or its Fourier transform
�� � � � � ��� � � � � ���
• � � � and � � � can be interpreted as energy densities in time and frequency, respectively
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Lecture 2 –The Wigner-Ville Distribution
The energy distributions
• It is natural to look for a joint time and frequency energy density �� �, �� such that
�� ���� �, ������
• As the energy is a quadratic function of the signal, the time-frequency energy distributions will be in general quadratic representations
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Lecture 2 –The Wigner-Ville Distribution
The energy distributions
• Two properties that an energy density should satisfy are the time and frequency marginal conditions
��� �, � �� � � � �
��� �, � �� � � �� �
• If the time-frequency energy density is integrated along one variable, the result is the energy density corresponding to the other variable
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Lecture 2 –The Wigner-Ville Distribution
Cohen’s class
• There are many distributions that statisfy the properties mentioned before
• Therefore, it is possible to impose additional constraints to �� �, � that would result in desirable properties
• Among these properties, the covariance principles are of fundamental importance
• The Cohen’s class is the family of time-frequency energy distributions covariant by translations in time and frequency
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Lecture 2 –The Wigner-Ville Distribution
Cohen’s class
• The spectrogram is an element of the Cohen’s class, since it is a quadratic, time- and frequency-covariant, and preserves energy
� � � � � � �� ⇒ !" �, � � !� ����, �� � � � � ��#� ⇒ !" �, � � !� �, � � ��
! �, � � � � � � � � �
• Taking the square modulus of an atomic decomposition is only a restrictive possibility to define a quadratic representation
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Lecture 2 –The Wigner-Ville Distribution
The Wigner-Ville distribution
• The approach is based on the use of the autocorrelation function for calculating the power spectrum
• To construct the autocorrelation function, the signal is compared to itself for all possible relative shifts, or lags
$�� � � �� � � � % � ��
where � is the shift of the signal with respect to itself
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Lecture 2 –The Wigner-Ville Distribution
The Wigner-Ville Distribution
• In the standard autocorrelation function, time is integrated out of the result, and $�� is only a function of the time lag �
• The Wigner-Ville (and all of Cohen’s class of distribution) uses a variation of the autocorrelation function wher time remains in the result, called instantaneous autocorrelation function
&�� �, � � � � % � 2( �∗ � � � 2(Where � is the time lag and ∗ represents the complex conjugate of the signal �.
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Lecture 2 –The Wigner-Ville Distribution
The Wigner-Ville Distribution
• Instantaneous autocorrelation of four cycle sine plots
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Lecture 2 –The Wigner-Ville Distribution
The Wigner-Ville Distribution
• The Wigner-Ville Distribution (WVD) is defined as
*� �, � � 12,�� � % � 2( �∗ � � � 2( �����
or equivalently
*� �, � � 12,�� � % � 2( �∗ � � � 2( �����
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Lecture 2 –The Wigner-Ville Distribution
The Wigner-Ville Distribution
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Lecture 2 –The Wigner-Ville Distribution
The Wigner-Ville Distribution
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Lecture 2 –The Wigner-Ville Distribution
The Wigner-Ville Distribution
• In an analogy to the STFT, the window is basically a shifted version of the same signal
• It is obtained by comparing the information of the signal with its own information at other times and frequencies
• It possesses several interesting properties, described as follows
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Lecture 2 –The Wigner-Ville Distribution
Properties of the WVD
• Energy conservation: by integrating the WVD of � all over the time-frequency plane, the energy of � is obtained
�� ��*� �, � ����
• Real-valued: the WVD is real-valued across time and frequency
*� �, � ∈ ., ∀�, �
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Lecture 2 –The Wigner-Ville Distribution
Properties of the WVD
• Marginal properties: the energy spectral density and the instantaneous power can be obtained as marginal distributions of *�
�*� �, � �� � � � �
�*� �, � �� � � �� �
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Lecture 2 –The Wigner-Ville Distribution
Properties of the WVD
• Translation covariance: the WVD is time- and frequency-covariant
� � � � � � �� ⇒ *" �, � � *� ����, �� � � � � ��#� ⇒ *" �, � � *� �, � � ��
• Dilation covariance: the WVD also preserves dilation
� � � 0� 0� ; 0 2 0 ⇒ *" �, � � *� 0�, � 0(
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Lecture 2 –The Wigner-Ville Distribution
Properties of the WVD
• Compatibility with filterings: it expresses the fact that if a signal � is the convolution of � and �, the WVD of � is the time-convolution between the WVD of � and the WVD of �
� � � �� �� � � � � �� ⇒
*" �, � � �*� �, � *4 � � �, � ��
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Lecture 2 –The Wigner-Ville Distribution
Properties of the WVD
• Compatibility with modulations : this is the dual property of the previous one : if � is the modulation of � by a function 5, the WVD of � is the frequency-convolution between the WVD of � and the WVD of 5
� � � � ��5 �� ⇒*" �, � � �*� �, � *6 �, � � � ��
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Lecture 2 –The Wigner-Ville Distribution
Properties of the WVD
• Wide-sense support conservation : if a signal has a compact support in time (respectively in frequency), then its WVD also has the same compact support in time (respectively in frequency). This is also called weak finite support
� � � 0, � 2 7 ⇒ *� �, �� � 0, |�| 2 7� � � 0, � 2 9 ⇒ *� �, �� � 0, |�| 2 9
• However, the WVD does not have strong finite support
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Lecture 2 –The Wigner-Ville Distribution
Properties of the WVD
• Unitarity : the unitarity property expresses the conservation of the scalar product from the time-domain to the time-frequency domain (apart from the squared)
�� � �∗ � �����*� �, � *"∗ �, � ����
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Lecture 2 –The Wigner-Ville Distribution
Properties of the WVD
• Instantaneous frequency and group delay : The instantaneous frequency characterizes a local frequency behaviour as a function of time. In a dual way, the local time behaviour as a function of frequency is described by the group delay
• In order to introduce these terms, the concept of analytic signal �: �� must be defined first
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Lecture 2 –The Wigner-Ville Distribution
Properties of the WVD
• For any real valued signal s ��, we associate a complex valued signal �: �� defined as
�: � � � � % ;�7 � ���where �7 � ��� is the Hilbert transform of � ��• �: �� is called the analytic signal associated to � ��
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Lecture 2 –The Wigner-Ville Distribution
Properties of the WVD
• This definition has a simple interpretation in the frequency domain since �: is a single-sided Fourier transform where the negative frequency values have been removed, the strictly positive ones have been doubled, and the DC component is kept unchanged
�: � � 0if� > 0�: � � � 0 if� � 0�: � � 2� � if� 2 0
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Lecture 2 –The Wigner-Ville Distribution
Properties of the WVD
• Instantaneous frequency: the instantaneous frequency of a signal � can be recovered from the WVD as its first order moment (or center of gravity) in frequency
?� � � @�*�A �, � ��@*�A �, � ��
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Lecture 2 –The Wigner-Ville Distribution
Properties of the WVD
• Group delay: the group delay of a signal � can be recovered from the WVD as its first order moment (or center of gravity) in time
�� � � @ �*�A �, � ��@*�A �, � ��
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Lecture 2 –The Wigner-Ville Distribution
Interference in the WVD
• As the WVD is a bilinear function of the signal �, the quadratic superposition principle applies
*�B" �, � � *� �, � %*" �, � % 2C *�," �, �where
*�," �, � � 12,�� � % � 2( �∗ � � � 2( �����
is the cross-WVD of � and �31
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Lecture 2 –The Wigner-Ville Distribution
Interference in the WVD
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Lecture 2 –The Wigner-Ville Distribution
Interference in the WVD
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Lecture 2 –The Wigner-Ville Distribution
Interference in the WVD
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Lecture 2 –The Wigner-Ville Distribution
Interference in the WVD
• These interference terms are troublesome since they may overlap with auto-terms (signal terms) and thus make it difficult to visually interpret the WVD image.
• It appears that these terms must be present or the good properties of the WVD (marginal properties, instantaneous frequency and group delay, localization, unitarity . . . ) cannot be satisfied
• There is a trade-off between the quantity of interferences and the number of good properties
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Lecture 2 –The Wigner-Ville Distribution
Pseudo-WVD
• The definition of the WVD requires the knowledge of
D� �, � � � � % � 2( �∗ � � � 2(from � � �∞ to � � %∞, which can be a problem in practice
• Often a windowed version of D� �, � is used, leading to the Pseudo-WVD (PWVD)
!*� �, � � 12,�� � % � 2( �∗ � � � 2( � �������
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Lecture 2 –The Wigner-Ville Distribution
Pseudo-WVD
• This is equivalent to a frequency smoothing of the WVD, since
!*� �, � � �� � � ��*� �, ����
where � �� is the Fourier transform of � ��• Because of their oscillating nature, the interferences will be
attenuated in the pseudo-WVD compared to the WVD
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Lecture 2 –The Wigner-Ville Distribution
Pseudo-WVD
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Lecture 2 –The Wigner-Ville Distribution
Pseudo-WVD
• However, the consequence of this improved readability is that many properties of the WVD are lost:
• The marginal properties
• The unitarity
• The frequency-support conservation
• The frequency-widths of the auto-terms are increased by this operation
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Lecture 2 –The Wigner-Ville Distribution
References and further reading
• Time Frequency Analysis: Theory and Applications by Leon Cohen. Prentice Hall; 1994. Chapter 8
• Biosignal and Medical Image Processing, Second Edition by John L. Semmlow. CRC press; 2009. chapter 6 pp. 147-151
• The Time Frequency Toolbox tutorial (http://tftb.nongnu.org/tutorial.pdf)
• Material from Signals and System course, FI-UNER
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