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ECE114_2011FALL_HW1__[0]

# ECE114_2011FALL_HW1__[0] - The Cooper Union Department of...

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Unformatted text preview: The Cooper Union Department of Electrical Engineering ECE114 Digital Signal Processing Lecture Notes: ROC for Laplace and z—Transforms September 6, 2011 Laplace Transform Laplace transform: 0° H(s)= / h(t)e—“dt Region of convergence (R00) is where Laplace transform integral is absolutely convergent, becomes the condition that: / [h (t)| e'”tdt < 00 —00 where a 1-: Re The general ROC has the form of a vertical strip R1 < Re (3) < R2. Note that in Laplace transform analysis, the point at inﬁnity is not “considered” (i.e, poles, zeros or even being analytic at 00 is not considered; the reason is the kernel of the Laplace transform, e‘st, as a function of s, has an essential singularity at 00, Le, is not analytic at 00 regardless of t). The inverse Laplace transform is: h(t) = ends where F is a vertical line (with upward direction; parametrized as a + jw where a is ﬁxed, R1 < a < R2, and w : —00 —> 00. As a special case, if the jw-axis is part of the ROC, this reduces to: 27F w=—oo h(t) 1 /w H(jw) ejwtdw which is the inverse CTFT (and the Laplace transform becomes H (jw) = f; h (t) e"j“’tdt, the CTFT). A ‘ Note that having jw—axis within the ROC means that the following integral converges: /°°|h(t)|dt<oo 00 Time-domain behavior: a poles in the region Re (3) S R1 (i.e, on the left of the ROC) correspond to modes that manifest in positive time, t > 0; o poles in the region Re (3) 2 R2 (i.e, on the right of the ROC) correspond to modes that manifest in negative time, t < 0. Based on these results, we have the following conclusions: o the ROC of the form R < Re (3), i.e., all the poles are to the LEFT of the ROC, corresponds to a causal signal (system); I the ROC of the form Re (3) < R, i.e., all the poles are to the RIGHT of the ROC, corresponds to an anti—causal signal (system); 0 the ROC includes the jw—axis iff (for a signal) there is a well-deﬁned1 Fourier trans— form, (for a system) there is a well—deﬁned frequency response, and (for a system) we have BIBO stability; 0 if we are in a situation where we consider only causal systems (e.g., characterization of physical analog circuits), then we can say that BIBO stability holds iff all poles are in the LHP. z-Transform z—transform: DO H(z) = Z h(n)z‘” ﬂ=—OO R00 is where Laplace transform integral is absolutely convergent, becomes the condition that: Z lh(n)| M” < oo The general ROC has the form of an annular ring R1 < < R2. Special cases are 0 < or O 5 on the inner condition (i.e., it may or may not include the origin itself), and < 00 or g 00 (i.e., it may extend out to 00, and H (2) may or may not analytic at 00 itself). Note that we MUST consider the presence of poles, zeros (or being analytic) at 00. The inverse z—transform is: h(n) = zn‘ldz where C is a (counterclockwise oriented) circle about the origin, lying within the ROC (i.e., the radius R of 0 satisﬁes R1 < R < R2). As a special case, if the unit circle is within the ROC, then we can parametrize the circle 0 = {z : |z| = l} (counterclockwise) via z = 63"“ where w : —7r —> 7r; observe that z‘ldz = jdw, and the inverse z-transform reduces to: hm) = i / Manama.) =—7r which is the IDTFT; similarly, if the unit circle is within the ROC, we can substitute 2 = :2” into the z—transform and get H (69") = 2‘” h (n) e‘jw”, which is the DTFT. n=—oo 1Here, well—deﬁned means the Fourier transform integral converges absolutely. This forces the resulting Fourier transform H (jw) to have certain “nice” mathematical properties. There may be cases Where we can deﬁne the Fourier transform even when the integral does not converge in the usual sense. For example, for h(t) = 1 Vt, H (jw) = ff; e‘jw‘dt = 27rd (w); recall that, strictly speaking, 6 (w) is not a function. .vl For n 2 1: 2‘” has a zero at 00 of order n, and has a pole at 0 of order n For n 5 —l: 2*” has a zero at 0 of order Inl, and a pole at 00 of order |n|. Time-domain behaviorm‘ @ I \ . . . o poles in the region |z|~g Eggcorrespond to modes that manifest 1n positive time, n 2 1; ' (é. Y~_M/~<~ \ ___ _\vr—~_— '1 o poles in the region R2 5 |z|/\‘correspond to modes that manifest in negative time, TL S _/"’ Pt 1 Based on these results, we have the following conclusions: 0 the ROC of the form R1 < |z| _<_ 00 corresponds to a causal signal (system); if there is a zero of multiplicity m at 00, then the signal support is n 2 m o the ROC of the form R1 < < oo, i.e., there is a (non—removable) singularity at 00, does NOT correspond to a causal system; instead, if there is a pole of multiplicity m, then the signal support is n 2 —m; this is sometimes called a right—sided signal; 0 the ROC of the form 0 3 [2| < R1 corresponds to an anti-causal signal (system); if there is a zero of multiplicity m at 0, then the signal support is n S —m o the ROC of the form 0 < |z| < R1, i.e., there is a (non—removable) singularity at 0, does NOT correspond to an anti-causal system; instead, if there is a pole of multiplicity m, then the signal support is n g m; this is sometimes called a left—sided signal; 0 if the ROC includes the unit circle, i.e., R1 < 1 < R2, then (for a signal) the Fourier transform is well—deﬁned; for a system, it has a well—deﬁned frequency response, and is BIBO stable; 0 if we are in a situation where we consider only causal systems (e..g, digital ﬁlters comprised of delays, constant multipliers and adders only), then we can say that BIBO stability holds iH all poles are inside the unit circle. ...
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