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Topic: Bilinear transform

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	THE BILINEAR TRANSFORM
		By inspection of Figures 20 and 21, it is found that the bilinear approximation (42) or (43) also maps the exterior of the unit circle into the lower half-plane.
		Thus, although the bilinear approximation is an approximation, it turns out to exactly preserve the minimum-phase property.
		Clearly, the folding theorem is too generous for applications involving the bilinear transform.
	sepftp.stanford.edu /sep/prof/fgdp/c2/paper_html/node9.html (450 words)

	The Bilinear Transform
		The formula for a general first-order (bilinear) conformal mapping of functions of a
		Bilinear transformations map circles and lines into circles and lines (lines being viewed as circles passing through the point at infinity).
		``The Bark and ERB Bilinear Transforms'', by Julius O. Smith III and Jonathan S. Abel, preprint of version accepted for publication in the IEEE Transactions on Speech and Audio Processing, December, 1999.
	ccrma-www.stanford.edu /~jos/bbt/Bilinear_Transform.html (175 words)

	Finite Differences vs. the Bilinear Transform
		While the bilinear transform ``warps'' the frequency axis, we can say the FDA ``doubly warps'' the frequency axis: It has a progressive, compressive warping in the direction of increasing frequency, like the bilinear transform, but unlike the bilinear transform, it also warps normal to the frequency axis.
		In summary, both the bilinear transform and the FDA preserve order, stability, and positive realness.
		The bilinear transform maps the continuous-time frequency axis in the
	ccrma.stanford.edu /~jos/pasp/Finite_Differences_vs_the_Bilinear.html (530 words)

	Transformations
		Geometric transforms permit the elimination of geometric distortion that occurs when an image is captured.
		A geometric transform is a vector function T that maps the pixel (x,y) to a new position (x',y').
		If the geometric transform does not change rapidly depending on position in the image, low order approximating polynomials, m=2 or m=3, are used, needing at least 6 or 10 pairs of corresponding points.
	www.mcs.csuhayward.edu /~grewe/CS6825/Mat/Math/transforms2.htm (1241 words)

	[No title]
		Because the projective transform is not linear, it cannot be composed as a sequence of translations, shears, scalings and (optionally) rotations.
		The bilinear transform is another nonlinear 4-point transform, which is somewhat better-conditioned than the projective transform, and a little faster to compute because it doesn't require a division.
		Although the bilinear transform does not preserve straight lines, it can be used to approximate a projective transform in situations where the warp is small.
	www.leptonica.com /affine.html (2281 words)

	Digital Filters by Nuhertz Technologies
		Z transform coefficients may be copied directly into the windows clipboard either as shown or formatted for Matlab, or sent directly to the "Custom Filter" text windows.
		None of the techniques are perfect, but in general the Bilinear transformation maps the entire frequency response of the desired filter, the Impulse Invariant transformation preserves the impulse time response, and the Step Invariant preserves the step time response.
		In the case of the Bilinear transformation, the entire continuous frequency spectrum is mapped to the discrete unit circle.
	www.filter-solutions.com /digital.html (1562 words)

	Pulse or digital communications patents new
		The method includes the steps of: a) transforming a temporal domain to a frequency domain by performing a discrete Fourier transform onto at least one of mixed signals inputted from an external device through multi-channel; b) estimating multi-decorrelation...
		In one process, the transformed signal is multiplied by the reciprocal of a variance.
		The image compression apparatus compresses a captured image by performing discrete cosine transformation on image data of the captured image and includes a prediction unit, a quantization unit, and an encoding unit.
	www.freshpatents.com /Pulse-or-digital-communications-dtnewntc375.php (9832 words)

	Filling the gap between the bilinear and the backward-difference transforms: An interactive design approach ...
		The backward difference transform satisfies the first condition, but the second condition is not completely satisfied, since the imaginary axis of the s-plane maps onto the circumference in the z-plane centered at z = 1/2 and having a radius of 1/2.
		is seen that the magnitude corresponding to the bilinear transform approximates the continuous-time low frequency response better than the backwarddifference or any of the transforms resulting from varying a.
		The transforms are obtained by interpolating the bilinear and the backward difference transformations.
	www.findarticles.com /p/articles/mi_qa3792/is_199710/ai_n8775053 (832 words)

	DASE Block Library
		The transfer function is mapped into the z plane using the bilinear transform and is pre-warped to preserve the relative value of the sample rate and the poles/zeros.
		The transfer function is mapped into the z plane using the bilinear transform and is pre-warped to preserve the relative value of the sample rate and the zero.
		The transfer function is mapped into the z plane using the bilinear transform and is pre-warped to preserve the relative value of the sample rate and the zero/pole.
	www.softanalytics.com /Blocks.htm (6959 words)

	DSP
		When an X number of samples before the current sample are multiplied by -1 and an X number of samples after the current sample are multiplied by +1 a 90 degree phase shift can be created.
		The Fast Fourier Transform (FFT) is a method in which point of the same amplitude but different frequencies are used only once in a multiplication instead of for each frequency.
		The bilinear z-transform is used to convert an analog filter into an equivalent digital filter by replacing s as shown:
	www.angelfire.com /sc/erozendaal/dsp.html (494 words)

	Filter Design and Implementation (Signal Processing Toolbox)
		The third step in the analog prototyping technique is the transformation of the filter to the discrete-time domain.
		The bilinear transformation is a nonlinear mapping of the continuous domain to the discrete domain; it maps the s-plane into the z-plane by
		To counteract this nonlinearity, it is necessary to create analog domain filters with "prewarped" band edges, which map to the correct locations upon bilinear transformation.
	www-rohan.sdsu.edu /doc/matlab/toolbox/signal/filter15.html (470 words)

	Bores Signal Processing - Introduction to DSP - IIR filters - design by the bilinear transform
		The problem with which we are faced is to transform the analogue filter design into the sampled data z plane Argand diagram.
		The bilinear transform is a method of squashing the infinite, straight analogue frequency axis so that it becomes finite.
		One way around this is to warp the analogue filter design before transforming it to the sampled data z plane Argand diagram: this warping being designed so that it will be exactly undone by the frequency warping later on.
	www.bores.com /courses/intro/iir/5_warp.htm (419 words)

	The Matched Z-Transform (Site not responding. Last check: 2007-10-26)
		Two transform methods are available: the bilinear transform and the matched z-transform.
		In most circumstances the bilinear transform is superior.
		The ``warping'' inherent in the bilinear transform method upsets this linearity.
	www-users.cs.york.ac.uk /~fisher/mkfilter/mzt.html (372 words)

	bilinear transform
		I have managed to transform the analog bandpass filter to an according digital filter.
		What you need to do, is to transform the seismometer's transfer function into z domain before starting playing with the inverse filter.
		One trivial way would be to transform the 50 Hz limit to a lower relative frequency.
	www.codecomments.com /message595545.html (1405 words)

	The Bilinear Transform
		The formula for a general first-order (bilinear) conformal mapping of functions of a
		Bilinear transformations map circles and lines into circles and lines (lines being viewed as circles passing through the point at infinity).
		``The Bark and ERB Bilinear Transforms'', by Julius O. Smith III and Jonathan S. Abel, preprint of version accepted for publication in the IEEE Transactions on Speech and Audio Processing, December, 1999.
	ccrma.stanford.edu /~jos/bbt/Bilinear_Transform.html (175 words)

	EDN Access -- 09.29.94 Spice does digital filters (Site not responding. Last check: 2007-10-26)
		To simulate the frequency response of a third-order digital filter, you first typically obtain the filter coefficients by applying the bilinear transform to the analog prototype.
		Because the bilinear transform distorts the pole and zero locations of the analog filter, you must prewarp the frequencies before applying the transform to obtain the desired response.
		Spice helps you verify that the prewarped frequencies and the bilinear transform produce the required attenuation in the pass and stop bands.
	www.edn.com /archives/1994/092994/20di1.htm (697 words)

	Relation to functions positive real in the right-half plane
		In general, a bilinear transformation maps circles and lines into circles and lines [Churchill 1960].
		Consequently, we conclude that the largest class of bilinear transforms which convert functions positive real in the outer disk to functions positive real in the right-half plane is characterized by
		The bilinear transform is one which is used to map analog filters intodigital filters.
	www.technick.net /public/code/cp_dpage.php?aiocp_dp=guide_edft_004 (581 words)

	Get More Yet
		In deriving the formula for the bilinear transformation, I had left out a factor of two in the power series for the natural log function.
		In the case of the s-to-z transformation, the key element is the bilinear transform, Equation 2.
		I also showed you how useful the bilinear transform can be in converting a transfer function from the Laplace-transform domain of s to the discrete world of z.
	www.embedded.com /shared/printableArticle.jhtml?articleID=9901083 (1924 words)

	Mathematics of Sampled Data Systems (Site not responding. Last check: 2007-10-26)
		Let us discover what this transformation is, and why it is better to use this transform instead of the Backward Difference Method or Bilinear Transform.
		Recall that the Laplace transform of f(t) is defined as:
		Thus, the z-transform is merely the Laplace Transform of a sampled data sequence and as such, inherits many of the properties of the Laplace Transform.
	lorien.ncl.ac.uk /ming/digicont/digimath/sampled3.htm (556 words)

	Math 132 Applet 3
		A Möbius transformation (also called a fractional linear transformation, projective linear transformation, or a bilinear transformation by some authors) is any map of the form
		The purpose of the rest of the buttons on the applet is to change this equation to a different Möbius transform.
		grid is the "pole" (or "singularity") of the Möbius transformation.
	www.math.ucla.edu /~tao/java/Mobius.html (892 words)

	No Title
		Impulse-invariant transform (Mitra 7.2, p.425-431) and bilinear transform (Mitra 7.3, p.
		In the bilinear transform the whole left subspace of s-plane is mapped into unit circle of z-plane.
		In the second case, we transform a lowpass filter to a highpass filter (see the figure 6).
	www.cis.hut.fi /Opinnot/DISKO_DSP/Laskarit/vast6/vast6.html (853 words)

	Filter Analysis and Design
		One routine commonly used for converting analog filters to digital filters is the bilinear transform.
		Here is the digital filter that results from applying the bilinear transform with a design constant of 3 to the analog filter
		The stable design is achieved by setting the design parameter of the bilinear transformation equal to one of the poles.
	library.wolfram.com /examples/fad (609 words)

	Mathematics of Sampled Data Systems (Site not responding. Last check: 2007-10-26)
		The Bilinear Transform is also known as Tustin's Rule as well as the more familiar Trapezoidal Rule used in numerical integration.
		Using this transformation, the sampling interval can be much larger, although the ensuing algebraic manipulations become more tedious.
		The benefit of using this transformation is only obvious with high order systems that have poles of quite different magnitudes.
	lorien.ncl.ac.uk /ming/digicont/digimath/sampled2.htm (333 words)

	Control System Lessons Start Page
		The two transformations from the s-plane to the z-plane that we have considered so far can be compared here.
		There is nothing about a lead that favors rectangular integration or a lag that favors the bilinear transformation.
		The algorithm using the bilinear transformation method to implement a lag compensator is simple, and is essentially the same as using the rectangular integration method.
	www.facstaff.bucknell.edu /mastascu/eControlHTML/Sampled/Sampled3.html (2632 words)

	Image Processing Transforms - LEADTOOLS' Image Transform Functions for Programmers (Site not responding. Last check: 2007-10-26)
		There are three types of image processing functions in LEADTOOLS, including drawing, filters and transforms.
		Most transforms can be controlled with one or more parameters.
		Place mouse-pointer over the name of the transform to see sample image with transform applied.
	www.leadtools.com /SDK/Raster/Raster-Image-Processing-Transforms.htm (300 words)

	[No title] (Site not responding. Last check: 2007-10-26)
		In this lab, the IIR design techniques that will be looked at is the impulse invariant transformation and the bilinear transform.
		Bilinear transformation The bilinear transformation does a mapping from the s-plane to the z-plane.
		Design two IIR filters using the bilinear transform, which could be used to separate the low tone from the high tone.
	www.ee.unsw.edu.au /signal/ELEC3004lab5.doc (1767 words)

	Dr. Dobb's \| Loose Ends \| August 9, 2006
		The mathematical basis behind Laplace transforms is highly theoretical and too complicated for us to discuss here.
		It's because the Laplace transform gives them access to a huge body of knowledge and techniques such as Nyquist diagrams, Bode plots, root locus plots, the Routh criterion, and more.
		For this reason, the approximation implicit in the bilinear transform gets better and better as we increase the sampling rate.
	www.ddj.com /191901764 (3176 words)

	Xiaochun Li (Site not responding. Last check: 2007-10-26)
		Uniform bounds for the bilinear Hilbert transform II.
		(with L. Grafakos), The disc as a bilinear multiplier.
		(with L. Grafakos), Uniform bounds for the bilinear Hilbert transforms I. Annals of Math., 159 (2004), 889-933.
	www.math.uiuc.edu /~xcli (161 words)

	55:148 Dig. Image Proc. Chapter 4, Part 2
		An example is an attempt to match remotely sensed images of the same area taken after one year, when the more recent image was probably not taken from precisely the same position.
		To inspect changes over the year, it is necessary first to execute a geometric transformation, and then subtract one image from the other.
		finding the point in the digital raster which matches the transformed point and determining its brightness.
	www.icaen.uiowa.edu /~dip/LECTURE/PreProcessing2.html (945 words)

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