A Unitary Principle of Optics, Catoptrics, and Dioptrics - download pdf or read online

By Leibniz Gottfried Wilhelm

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Note how the frequency response tends to a Gaussian as the number of convolutions increase. ), and right at the edge of these bounding pupils, convolution filters mix up well-defined interferometric data with invalid background outside the pupils where no fringe data is available or defined. The edge distortion may be so significant that some people shrink the interferogram’s area during spatial filtering to avoid those unreliable pixels near the edge. While this approach may be practical and easy to implement, it implies discarding valid data.

To illustrate this, consider the following five-step band-pass filter with an impulse response function given by h(t) = (1∕8)[????(t) − 2i????(t − 1) − 2????(t − 2) + 2i????(t − 3) + ????(t − 4)]. Taking its DTFT and factorizing the result, we find the following FTF: ][ ]2 [ ] [ H(????) = (1∕8) 1 − ei???? 1 − e−i(????−π∕2) 1 − ei(????−π) . 26. 26, both the pass-band and stop-band regions are clearly represented in the linear plot of |H(????)|, while the stop-band region is out of the chart in the semilog plot. For this reason, we prefer to work exclusively with linear plots of |H(????)| for the filters that we study.

That is H(z)|z∈U = ∞ ∑ h(n) exp(i????) = H(????). 113) n=−∞ Thus, since | exp(i????)| = 1, this means that an LTI system will be BIBO stable if and only if its FTF H(????) is absolutely summable (finite) within the interval ???? ∈ (−π, π), that is } { ∞ ∞ ∑ ∑ | | |h(n)| < ∞. 114) |H(????)| ≤ |h(n) exp(i????)| = n=−∞ n=−∞ To summarize, an LTI system is said to be BIBO stable if any of the following (equivalent) conditions is fulfilled: • Its impulse response {h(n)} is absolutely summable; • Its FTF H(????) =  {h(n)} is absolutely summable; • The unit circle of the Z-domain U(z) is part of the ROC of its transfer function H(z) = {h(n)}.

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A Unitary Principle of Optics, Catoptrics, and Dioptrics by Leibniz Gottfried Wilhelm

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