# From Zero to Infinity: What Makes Numbers Interesting (Spectrum Series) Thus the term may be drawn as an arrow from the th zero to the point on the unit circle, and is an arrow from the th pole. Therefore, each term in Eq. In summary: Figure 8. Figure 8. For example, the dc gain is obtained by multiplying the lengths of the lines drawn from all poles and zeros to the point. The filter gain at half the sampling rate is the product of the lengths of these lines when drawn to the point.

For an arbitrary frequency Hz, we draw arrows from the poles and zeros to the point. Thus, at the frequency where the arrows in Fig.

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Before looking at that, it is a good exercise to try sketching it by inspection of the pole-zero diagram. It is usually easy to sketch a qualitatively accurate amplitude-response directly from the poles and zeros to within a scale factor. Figure: Amplitude response obtained by traversing the entire upper semicircle in Fig. The point of the amplitude obtained in that figure is marked by a heavy dot. For real filters , precisely the same curve is obtained if the lower half of the unit circle is traversed, since.

Thus, plotting the response over positive frequencies only is sufficient for real filters. Graphical Phase Response Calculation The phase response is almost as easy to evaluate graphically as is the amplitude response : If is real, then is either 0 or. Terms of the form can be interpreted as a vector drawn from the point to the point in the complex plane.

The angle of is the angle of the constructed vector where a vector pointing horizontally to the right has an angle of 0. Therefore, the phase response at frequency Hz is again obtained by drawing lines from all the poles and zeros to the point , as shown in Fig.

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The angles of the lines from the zeros are added, and the angles of the lines from the poles are subtracted. Thus, at the frequency the phase response of the two-pole two-zero filter in the figure is. Note that an additional phase of radians appears when the number of poles is not equal to the number of zeros.

### General Introduction

This factor comes from writing the transfer function as. The point of the phase response corresponding to the arrows in that figure is marked by a heavy dot. For real filters , the phase response is odd , so the curve shown here may be reflected through 0 and negated to obtain the plot for negative frequencies. This is because the transfer function is the z transform of the impulse response, and if there is an observable non-canceled pole outside the unit circle, then there is an exponentially increasing component of the impulse response. To see this, consider a causal impulse response of the form.

### Featured Series

The signal has the z transform where the last step holds for , which is true whenever. Thus, the transfer function consists of a single pole at , and it exists for. The pole of moves outside the unit circle, and the impulse response has an exponentially increasing amplitude. Thus, the definition of stability is violated.

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## Constance Reid

Since the z transform exists only for , we see that implies that the z transform no longer exists on the unit circle, so that the frequency response becomes undefined! The above one-pole analysis shows that a one-pole filter is stable if and only if its pole is inside the unit circle. Therefore, all poles must be inside the unit circle for stability. In summary, we can state the following: Isolated poles on the unit circle may be called marginally stable. The impulse response component corresponding to a single pole on the unit circle never decays, but neither does it grow.

Computing Reflection Coefficients to Check Filter Stability Since we know that a recursive filter is stable if and only if all its poles have magnitude less than 1, an obvious method for checking stability is to find the roots of the denominator polynomial in the filter transfer function [Eq. If the moduli of all roots are less than 1, the filter is stable.

## From Zero to Infinity - AbeBooks - Constance Reid:

This test works fine for low-order filters e. It is therefore of interest to use a stability test that is faster and more reliable numerically than polynomial root-finding. Fortunately, such a test exists based on the filter reflection coefficients. It is a mathematical fact [ 48 ] that all poles of a recursive filter are inside the unit circle if and only if all its reflection coefficients which are always real are strictly between -1 and 1.

The full theory associated with reflection coefficients is beyond the scope of this book, but can be found in most modern treatments of linear prediction [ 48 , 47 ] or speech modeling [ 92 , 19 , 69 ]. An online derivation appears in [ 86 ]. Io mica lo sapevo che Constance Reid fosse la sorella di Julia Robinson Constance Reid non era una matematica, anche se naturalmente in caso di dubbi pote Carramba che sorpresa!

What are Numbers Made of? - Infinite Series

Questo libro nacque per caso, dopo che scrisse un articolo sullo Scientific American a proposito della scoperta di due nuovi numeri perfetti da parte di suo cognato Credo insomma che il testo sia molto adatto per avvicinare non tanto alla matematica come materia quanto come modo di pensare i lettori: caldamente consigliato. Jan 01, Sean Maguire rated it really liked it Shelves: math. Even better than Hodges' 'From One to Nine'.

First published in , and updated to keep track of developments in math in the intervening 50 years, Reid does a masterful job of journeying through the fascinating intricacies of the digits. She also offers up my new favourite quote: "It would be difficult for anyone to be more profoundly interested in anything than I am in the theory of primes" G. View 1 comment.

Jul 21, Todd Spicer rated it liked it. Read it five and a half years ago. Got it out again seven weeks ago. Working through it now and really letting the big questions sink in more. Great place to start for concepts dealing with "the shadowy history and philosophy of the modern digits.

Feb 28, Darren Kirby rated it really liked it Shelves: science-math , non-fiction , history. Interesting book. Fascinating at times even. Not overly heavy on math drudgery, easy for non-math nerds to understand. Apr 12, Melissa rated it liked it. So far it's been really good. Cody rated it really liked it Oct 06, Kaushal rated it it was ok Jun 02, Sacha rated it it was amazing Feb 22, Andrea Di Domenico rated it really liked it Sep 19, Matthew rated it it was ok Apr 23, Ross Barnowski rated it really liked it Dec 28, Stephen Miletus rated it it was amazing Dec 14, Brian rated it liked it Nov 08, Gerardo rated it really liked it Jun 03, C P rated it really liked it Jun 27, Keith Pfeiffer rated it it was amazing Jan 08, Therefore, all poles must be inside the unit circle for stability.

In summary, we can state the following: Isolated poles on the unit circle may be called marginally stable. The impulse response component corresponding to a single pole on the unit circle never decays, but neither does it grow. Computing Reflection Coefficients to Check Filter Stability Since we know that a recursive filter is stable if and only if all its poles have magnitude less than 1, an obvious method for checking stability is to find the roots of the denominator polynomial in the filter transfer function [Eq.

If the moduli of all roots are less than 1, the filter is stable. This test works fine for low-order filters e. It is therefore of interest to use a stability test that is faster and more reliable numerically than polynomial root-finding. Fortunately, such a test exists based on the filter reflection coefficients. It is a mathematical fact [ 48 ] that all poles of a recursive filter are inside the unit circle if and only if all its reflection coefficients which are always real are strictly between -1 and 1.

The full theory associated with reflection coefficients is beyond the scope of this book, but can be found in most modern treatments of linear prediction [ 48 , 47 ] or speech modeling [ 92 , 19 , 69 ]. An online derivation appears in [ 86 ]. This recipe is called the step-down procedure , Schur-Cohn stability test , or Durbin recursion [ 48 ], and it is essentially the same thing as the Schur recursion for allpass filters or Levinson algorithm for autocorrelation functions of autoregressive stochastic processes [ 38 ].

Step-Down Procedure Let denote the th-order denominator polynomial of the recursive filter transfer function :. If , the recursion halts prematurely, and the filter is declared unstable. Equivalently, the polynomial is declared non- minimum phase , as defined in Chapter Otherwise, if , the polynomial order is decremented by 1 to yield as follows recall that is monic :. Next is set to , and the recursion continues until is reached, or is found for some.

Whenever , the recursion halts prematurely, and the filter is usually declared unstable at best it is marginally stable , meaning that it has at least one pole on the unit circle. Note that the reflection coefficients can also be used to implement the digital filter in what are called lattice or ladder structures [ 48 ]. As a result, they can be very important for fixed-point implementations such as in custom VLSI or low-cost fixed-point signal processing chips. Testing Filter Stability in Matlab Figure 8. The Durbin recursion is far faster than the method based on root-finding.

Time Constant of One Pole A useful approximate formula giving the decay time-constant 9. The exact relation between and is obtained by sampling an exponential decay :. In particular, a pole outside the unit circle gives rise to an impulse-response component proportional to which grows exponentially over time. However, this was the case for a causal exponential , where is the unit-step function which switches from 0 to 1 at time 0. If the same exponential is instead anticausal , i. Therefore,to unambiguously invert a z transform, we must know its region of convergence. The critical question is whether the region of convergence includes the unit circle: If it does, then each pole outside the unit circle corresponds to an anticausal, finite energy, exponential, while each pole inside corresponds to the usual causal decaying exponential.

Geometric Series The essence of the situation can be illustrated using a simple geometric series. Let be any real or complex number. Then we have. One-Pole Transfer Functions We can apply the same analysis to a one- pole transfer function. Let denote any real or complex number :.

Now consider the rewritten case: where the inverse z transform is the inverse bilateral z transform. In this case, the convergence criterion is , or , and this region includes the unit circle when. In summary, when the region-of-convergence of the z transform is assumed to include the unit circle of the plane, poles inside the unit circle correspond to stable, causal, decaying exponentials , while poles outside the unit circle correspond to anticausal exponentials that decay toward time , and stop before time zero.

Right column: Anticausal exponential decay , pole at. Top: Pole-zero diagram. Bottom: Corresponding impulse response , assuming the region of convergence includes the unit circle in the plane. Poles and Zeros of the Cepstrum The complex cepstrum of a sequence is typically defined as the inverse Fourier transform of its log spectrum [ 60 ].

Applying the Maclaurin series expansion. Since the region of convergence of the z transform must include the unit circle where the spectrum DTFT is defined , we see that the Maclaurin expansion gives us the inverse z transform of all terms of Eq. From Zero to Infinity: What Makes Numbers Interesting (Spectrum Series) From Zero to Infinity: What Makes Numbers Interesting (Spectrum Series) From Zero to Infinity: What Makes Numbers Interesting (Spectrum Series) From Zero to Infinity: What Makes Numbers Interesting (Spectrum Series) From Zero to Infinity: What Makes Numbers Interesting (Spectrum Series) From Zero to Infinity: What Makes Numbers Interesting (Spectrum Series) From Zero to Infinity: What Makes Numbers Interesting (Spectrum Series) From Zero to Infinity: What Makes Numbers Interesting (Spectrum Series)

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