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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 “equaltempered” 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:N1 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 12tone (“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 lowinteger 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 equaltempered 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 12tones 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 lowinteger 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 LOWPASS For this problem, you are asked to design a 10th order analog Butterworth lowpass
with a cutoff frequency of 3000 Hz. Use the cascaded SallenKey approach. Each of the
five secondorder sections should be adjusted individually for unit gain. H17 A SALLENKEY FILTER WITH A REAL OP~AMP For this problem design a 2”“I order Sailen Key lowpass for a cutoff frequency of 300
kHz. Assume first that all elements are ideal. Next assume that the. opamp is real with a GainBandwidth Product of 1 MHz. This real
opamp will make your network thirdorder. Solve for the poles of this third—order network
and compare them to the ideal case. You will find that the complexconjugate 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 opamp 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 opamp 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 opamp 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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 Spring '06
 HUTCHINS
 Signal Processing

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