Since a joint TF distribution usually spreads noise and localizes signals, in particular chirps, the receiver may use a TF analysis technique to map the received signal y from the time domain into the joint TF domain. The TF concentration property usually holds after passing through an LTI system (this will be seen later). Instead of pseudo-random signal x, chirp type signals are transmitted as training signals, which have wideband characteristics in the frequency domain but are concentrated in the joint time-frequency (TF) domain. In the following, we introduce a different technique for the system identification problem. With this kind of input signals, noise reduction techniques before system identification do not apply. Since the auto-spectrum of the input signal x is in the denominator in the estimate ( 12.3.2), the input signal is, in general, chosen as a pseudo-random signal with flat spectrum. When this SNR is low, the performance of the estimate in ( 12.3.2) is poor as we will also see later. When the additive noise ε in ( 12.3.1) is a zero-mean Gaussian process and statistically independent of the input signal x, the estimate in ( 12.3.2) is asymptotically unbiased but the performance is limited by the noise variance, or the signal-to-noise ratio (SNR). Where S xy(ω) is the cross-spectrum of x and y, and S xx(ω) is the auto-spectrum of x. Attempts to perform image reconstruction with conventional video cameras have never been totally successful because of the poor quality of video image data. If the entire process is not carried out very carefully, the results will contain false, spurious image components which can easily be misinterpreted. The camera used to acquire both the object image and the system transfer function must be linear over several orders of intensity level, and the image must be spatially oversampled to ensure that all the information is captured. The successful application of transform techniques places severe demands on camera performance. Through the careful application of Fourier-based image-processing techniques, out-of-focus images may be de-fuzzed and resolution approaching the diffraction limit of the microscope can be achieved. Just as in the flat field problem, the most difficult task is obtaining a calibration image which is a faithful representation of the system transfer function. #DTRANSFER MINIMUM SYSTEM SOFTWARE#Fast Fourier transform software and hardware can deconvolve a one million pixel image in a few seconds. By individually transforming the object image and system transfer function into Fourier space, it is possible to deconvolve the system effects through a simple algebraic operation. The corrupting effects introduced by the system may be removed through the application of powerful mathematical transforms. The recorded object image is the convolution of the system transfer function with the original input image. The point source image is captured in precisely the same manner as the object image. The combined effects of the system, including the microscope, CCD and camera electronics are then characterized by imaging a point source which simulates a delta function. An object image is first spatially sampled by the CCD and digitized. These methods have been successfully employed in spectroscopy, astronomy and microscopy ( Agard et al., 1989 ). AIKENS, in Fluorescent and Luminescent Probes for Biological Activity (Second Edition), 1999 37.6.4 Image reconstructionĭigital images produced by a CCD and a light microscope can be enhanced through the application of image-reconstruction techniques. In the most general case, we have a polyphony of ascending and descending melodic lines which run in order to create bubbles whose shape is dictated by the contour lines of the system transfer function. Pictorially speaking, if we draw a parallelism between time-frequency representations and musical scores, we may say that the eigenfunctions of underspread systems give rise to a polyphonic texture which reduces to monophonic lines only in the simple case of systems whose spread function is concentrated on a straight line. This property, for whose validation the analysis of the system eigenfunctions' TFD plays a fundamental role, gives a general framework for interpreting some current data transmission schemes and, most importantly, gives a new perspective on the optimal waveforms for transmissions over time-varying channels. The analytic model for the eigenfunctions of underspread linear operators shown in this article, although approximate, shows that the energy of the system eigenfunctions is mainly concentrated along curves coinciding with level curves of the system transfer function. In Time Frequency Analysis, 2003 13.3.5 Summary and Conclusions
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