13.2 Wigner distribution and ambiguity function

Time-frequency analysis is all about understanding how signals change over time. The Wigner distribution and ambiguity function are powerful tools that let us see a signal's energy in both time and frequency at once.

These methods reveal hidden patterns in complex signals like speech or radar. They're super useful for analyzing stuff that changes quickly, but they can be tricky to interpret due to cross-terms and interference.

Wigner Distribution and Ambiguity Function

Definition and Properties of Wigner Distribution

Top images from around the web for Definition and Properties of Wigner Distribution

• 

  Wigner distribution function - Wikipedia, the free encyclopedia View original

  Is this image relevant?

• 

  Wigner distribution function - Wikipedia, the free encyclopedia View original

  Is this image relevant?

• 

  Wigner distribution function - Wikipedia, the free encyclopedia View original

  Is this image relevant?

• 

  Wigner distribution function - Wikipedia, the free encyclopedia View original

  Is this image relevant?

1 of 2

Top images from around the web for Definition and Properties of Wigner Distribution

• 

  Wigner distribution function - Wikipedia, the free encyclopedia View original

  Is this image relevant?

• 

  Wigner distribution function - Wikipedia, the free encyclopedia View original

  Is this image relevant?

• 

  Wigner distribution function - Wikipedia, the free encyclopedia View original

  Is this image relevant?

• 

  Wigner distribution function - Wikipedia, the free encyclopedia View original

  Is this image relevant?

1 of 2

• Wigner distribution represents a signal in both time and frequency domains simultaneously
• Defined as the Fourier transform of the signal's autocorrelation function with respect to the time lag variable
• Real-valued function of time and frequency
• Satisfies marginal properties integrating over time yields the signal's power spectrum, integrating over frequency yields the signal's instantaneous power
• Can be interpreted as a joint time-frequency energy density function

Ambiguity Function and its Relationship to Wigner Distribution

• Ambiguity function is the 2D Fourier transform of the Wigner distribution
• Represents the signal in the time-frequency shift domain (Doppler-delay domain)
• Measures the similarity between the signal and its time-frequency shifted versions
• Ambiguity function is complex-valued its magnitude is invariant to time and frequency shifts
• Wigner distribution can be obtained from the ambiguity function via an inverse 2D Fourier transform

Applications of Wigner Distribution and Ambiguity Function

• Time-frequency analysis analyzing the time-varying spectral content of non-stationary signals (speech, music, radar)
• Quadratic time-frequency distributions Wigner distribution is a member of the Cohen's class of quadratic time-frequency distributions
• Signal detection and parameter estimation using the ambiguity function (radar, sonar)
• Quantum mechanics Wigner distribution is used to represent the phase-space distribution of quantum states

Interference and Cross-Terms

Cross-Terms in Wigner Distribution

• Cross-terms appear in the Wigner distribution when the signal consists of multiple components
• Result from the bilinear nature of the Wigner distribution
• Appear as oscillatory structures in the time-frequency plane between the auto-terms (true signal components)
• Can lead to difficulties in interpreting the Wigner distribution and identifying the true signal components
• Example cross-terms between two sinusoidal components appear as a third component at their average frequency

Interference and its Impact on Interpretation

• Interference refers to the interaction between cross-terms and auto-terms in the Wigner distribution
• Can obscure the true time-frequency structure of the signal
• Makes it challenging to distinguish between true signal components and artifacts introduced by cross-terms
• Interference patterns depend on the relative phase and amplitude of the signal components
• Example interference between a linear chirp and a sinusoid results in a complex pattern of cross-terms

Cohen's Class of Time-Frequency Distributions

• Cohen's class is a general framework for constructing quadratic time-frequency distributions
• Includes the Wigner distribution as a special case
• Allows for the suppression of cross-terms by applying a 2D kernel function in the ambiguity domain
• Different kernel functions lead to different time-frequency distributions with varying cross-term suppression and resolution trade-offs
• Examples of Cohen's class distributions Choi-Williams distribution, Born-Jordan distribution, Zhao-Atlas-Marks distribution