By Franz Hlawatsch

Linear sign areas are of basic value in sign and approach concept, conversation concept, and smooth sign processing.

This ebook proposes a time-frequency research of linear sign areas that's in keeping with novel time-frequency representations referred to as the `Wigner distribution of a linear sign house' and the `ambiguity functionality of a linear sign space'.

in addition to being an invaluable show and research software, the Wigner distribution of a linear sign area permits the layout of high-resolution time-frequency filtering tools. This e-book develops such equipment and applies them to the enhancement, decomposition, estimation, and detection of noisy deterministic and stochastic signs. formula of the filtering (estimation, detection) equipment within the time-frequency aircraft yields an instantaneous interpretation of the impact of including or deleting info, altering parameters, and so forth. In a feeling, the previous details and the sign processing projects are delivered to lifestyles within the time-frequency airplane.

the paradox functionality of a linear sign house, nevertheless, is heavily concerning a singular maximum-likelihood multipulse estimator of the diversity and Doppler shift of a slowly fluctuating element objective - an estimation challenge that's vital in radar and sonar. in particular, the paradox functionality of a linear sign house is proper to the matter of optimally designing a suite of radar pulses.

The techniques and strategies offered are amply illustrated by means of examples and images. *Time-Frequency research and Synthesis of Linear Signal**Spaces: Time-Frequency Filters, sign Detection and Estimation, and**Range-Doppler Estimation* is a wonderful reference and will be used as a textual content for complex classes overlaying the subject.

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**Extra info for Time-Frequency Analysis and Synthesis of Linear Signal Spaces: Time-Frequency Filters, Signal Detection and Estimation, and Range-Doppler Estimation**

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I" , I \ / \ \ , ' . I II "' , ---" (e) CWO of two TF shifted Hermite spaces X, Y (both with dimension 16): (a) auto-WD of space X, (b) auto-WD of space y, (c) magnitude of CWD of spaces X, y with effective TF supports of the corresponding auto-WDs indicated by broken lines. 11. basis signals Xl (t), Yl (t). 51) the CWD oftwo orthogonal spaces is identically zero. 2 Discrete-Time Wigner Distribution For a practical implementation of WD analysis on a digital computer, the WD of a signal space must be reformulated in a discrete-time setting.

3) k=l where A is the diagonal eigenvalue matrix containing the eigenvalues Ak in its diagonal. 3) under the orthonormality constraint 46 CHAPTER 2: THE WIGNER DISTRIBUTION OF A LINEAR SIGNAL SPACE af! al = 8kl . It suffices, however, to use a normalization constraint af! ak = 1 since, as will be seen presently, this automatically yields orthonormal vectors ak. Using Lagrange multipliers Vk, the problem then amounts to the unconstrained maximization of N N + L v d 1 - a f! (A-vkI)ak + LVk. k=1 k=1 k=1 k=1 Setting the gradient of aa with respect to ai equal to zero yields the system of equations Aai=lIiai, i=l, ...

29)). 30)). 6 Extensions This section considers two extensions of the WD of a signal space, namely, the cross-WD and a discrete-time WD version [Hlawatsch and Kozek, 1993]. 48), Wh(t,f), is properly Wh (t, f) dt df IIh\\2 1. 10. Spectrogram of a signal space-interpretation as smoothed WD : (a) WD of signal space X, (b) WD of test signal (window) h(t), (c) spectrogram of signal space X. 3) of the auto-WD, 00 WX,y(t,f) ~ :EW1k,X,lk,y(t,f) = k=l Here, W",y(t, f) = 1 i) x(t+ y*(t- E{Wwx,wy(t,f)}. 6: EXTENSIONS 35 is the CWD of two signals x(t), y(t) [Claasen and Mecklenbdiuker, 1980].