Time-Frequency Analysis and Synthesis of Linear Signal by Franz Hlawatsch

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 SignalSpaces: Time-Frequency Filters, sign Detection and Estimation, andRange-Doppler Estimation is a wonderful reference and will be used as a textual content for complex classes overlaying the subject.

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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].

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