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HW4 - ECE426 SPRING 2007 HOMEWORK SET 4 Special Due Date...

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Unformatted text preview: ECE426 SPRING 2007 HOMEWORK SET 4 Special Due Date: Thursday May 3, 2007 5:00 PM H15 MUSICAL SCALE TONES' An “equal-tempered” musical scale divides an octave (a 2:1 ratio) into notes that are spaced by the same ratio r. Thus if we want a N tone equal tempered scale r=2‘“N) and the frequencies (roughly, the pitches) are given by fk = f0 20““) for k=0:N-1 where f0 is the first tone of the scale. For example, if we are talking about the 0 scale starting at “middle C," then fo=261.62 Hz. However, for this problem we can just consider frequencies normalized to fo=1. For' the usual 12-tone (“Chromatic”) scale, the tones occur at ratios: 1.0000 G DO 1.0595 C# 1.1225 D RE 1.1892 D# - 1.2599 E ME Major third (Nominal 514:1.2500) 1.3348 F FA Fourth (Nominal 4/3=1.3333) 1.4142 F# ' 1.4983 G 80 Fifth (Nominal 3/2=1.5000) 1.5874 G# 16818 A LA Sixth (Nominal 5/3=1.6667 1.7818 A# - 1.8877 B Tl One goal of choosing a particular number of notes per octave is the desire to obtain good approximations to the four low-integer ratios 3/2, 4/3, 5/3, and 5/4, and we see that the 12- tone scale does this fairly well. [Note: we cauld make these exact, but they would only be exact in one key. The equal-tempered scale is actually a classic “engineering compromise’ although the choice was empirical and decided (at least for Western music) hundreds of years ago !] J Now for the problem. Consider equal tempered scales of from N=5 to N=40 tones per octave. Compute the error on the four low—integer ratios for the best available notes. This could be done by exact errors (eg... the fifth with 12-tones has an error 1.5000 4.4983 = 00017, or you could say that it is “flat” (low) by 0.11 %. Consider various choices. - Then consider an overall error measure for all four low-integer ratios, As with filter design, you could consider total squared error, or perhaps maximum absolute error. Make a plot of the total error as a function of N. You should see a relatively small error for N=12 for example. What other values of N might be considered? What engineering considerations might also be involved in the choice? ANALOG PROBLEMS 7 For the following three problems, you should refer to the notes on the three lectures on analog filtering and/or the active filter notes as: http://electronotes.netfirmscom/freehtml H16 AN 10TH ORDER BUTTERWORTH LOW-PASS For this problem, you are asked to design a 10th order analog Butterworth low-pass with a cutoff frequency of 3000 Hz. Use the cascaded Sallen-Key approach. Each of the five second-order sections should be adjusted individually for unit gain. H17 A SALLEN-KEY FILTER WITH A REAL OP~AMP For this problem design a 2”“I order Sailen- -Key low-pass for a cutoff frequency of 300 kHz. Assume first that all elements are ideal. Next assume that the. op-amp is real with a Gain-Bandwidth Product of 1 MHz. This real op-amp will make your network third-order. Solve for the poles of this third—order network and compare them to the ideal case. You will find that the complex-conjugate pair of poles has “degenerated” to a lower frequency and the wrong damping. Plot the frequency responses for the ideal case and the real op—amp case using freqs. On one or more “overdesign” steps, find a better set of starting specifications (pole radius and damping) such that the result with the real op-amp is closer to Butterworth with cutoff of 300 kHz. H18 DESIGNING AN AUDIO MIXER An engineer decides to design a simple audio mixer using a simple op-amp summer, as suggested in the figure below: We are not worried about the inverSion because the signals are audio“ But now we consider that the op-amp is real: Vout = (Gls)( V+ — V.)i. We know that this means that the mixer has a limited bandwidth“ But, do the number of audio inputs connected and the settings of the pots make any difference to the bandwidth achieved? (a) If you have a signal source connected to ajack, but you do not want to include that signal, is it better to disconnect the input plug or do we just set the pot to the bottom? Assume that input signals are zero impedance sources. (b) If the signal plug i_s removed .from the jack, should the pot be set to the top, or to the bottom, or does it not matter? (c) Given the results of (a) and (b), what do you write in the user’s manUal with regard to the best way to handle “unused” inputs? ...
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HW4 - ECE426 SPRING 2007 HOMEWORK SET 4 Special Due Date...

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