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7.25 For what values of x are the graphs of (a) f (x) = (x 1)(x + 2) and (b) f (x) = x(x 1)(x + 2) above the x-axis GC Check your answers by means of a graphing calculator. 7.26 Prove Theorem 7.1. [Hint: Solve f (r) = 0 for a0 .] 7.27 Prove Theorem 7.2. [Hint: Make use of the algebra following Problem 7.21.]

(b) Since diagonals AC and BD are equal in length and bisect each other, ABCD is a rectangle. (c) Since the circles are equal, OA > AQ > QB > BO and OAQB is a rhombus.

EXAMPLE 9.4.3 Let us design a first-order digital low-pass filter with a 3-dB cutoff frequency of w, = 0.2% by applying the bilinear transformation to the analog Butterworth filter

I H,,(s) = 1 s j R,.

PROOF:

Because the 3-dB cutoff frequency of the Butterworth filter is R,., for a cutoff frequency w, = 0.2% in the digital filter, we must have

Statements 1. AB > DE, BC > EF 2. AB > DE, BC > EF 3. ABC > DEF 4. AC > DF 5. ^ I > ^ II 6. j B > j E 1. Given

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8.1 INTRODUCTION To a great extent, calculus is the study of the rates at which quantities change. It will be necessary to see how the value f (x) of a function f behaves as the argument x approaches a given number. This leads to the idea of limit. EXAMPLE Consider the function f such that

The preceding section considered the design of digital low-pass filters from analog low-pass filters. There are two approaches that may be used to design other types of frequency selective filters, such as high-pass, bandpass, or bandstop filters. The tirst is to design an analog low-pass filter and then apply a frequency transformation to map the analog filter into the desired frequency selective prototype. This analog prototype is then mapped to a digital filter using a suitable s-plane to z-plane mapping. Table 9-5 provides a list of some analog-to-analog transformations.

2. In a circle, > chords have > arcs. 3. If equals are added to equals, the sums are equal. Definition of > arcs. 4. In a circle, > arcs have > chords. 5. SSS

Table 9-5 The 'kansformation of an Analog Low-pass Filter with a 3-dR Cutoff Frequency L2, to Other F'req"ency selective Filters

6.4 Proving a circle problem stated in words Prove that if a radius bisects a chord, then it is perpendicular to the chord.

This difference equation defines a sequence of operations that are to be performed in order to implement this system. However, note that this system may also be implemented with the following pair of coupled difference equations:

To Prove: OC ' AB Plan: Prove ^ AOD > ^ BOD to show j 1 > j 2. Also, j 1 and j 2 are supplementary.

With this implementation, it is only necessary to provide one memory location to store w(n - I), whereas Eq. (8.1)requires two memory locations, one to store y(n - I) and one to store x(n - 1). This simple example illustrates that there is more than one way to implement a system and that the amount of computation and/or memory required will depend on the implementation. In addition, the implementation may affect the sensitivity of the filter to coefficient quantization. and the amount of round-off noise that appears at the output of the filter. In this chapter, we look at a number of different ways to implement a linear shift-invariant discrete-time system and look at the effect of finite word lengths on these implementations.

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