how to generate qr code in vb.net Copyright 2008 by The McGraw-Hill Companies, Inc Click here for terms of use in VS .NET

Encoding Code 128 Code Set C in VS .NET Copyright 2008 by The McGraw-Hill Companies, Inc Click here for terms of use

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The multiplicative constant, 10 in this example, gives us a way to compare the relative strength differences between two quantities That is, if the difference between IdB1 and IdB2 is 10 dB, then the intensity of I1 is ten times the strength of I2 Note that since the multiplicative factor in (151) is 20, a difference of 20 dB indicates that one signal has a magnitude 20 times as large as the other The log function is a useful measure of signal strength for two good reasons First, quantities can often vary quite a bit in strength sometimes over many orders of magnitude By using the logarithm we can rescale that variation down to a more manageable number One famous example where this behavior is apparent is the Richter scale used to characterize the strength of earthquakes The details of the Richter scale don t concern us; all that is important for our purposes is that this is a logarithmic quantity This means that each increment on the Richter scale describes an order-of-magnitude increase in strength An earthquake that is a 7 on the Richter scale is 10 times as strong as an earthquake that is a 6 In our case, using logarithms allows us to scale down a wide range of frequencies into a small scale that can be visualized and plotted more easily This works in a way similar to the Richter scale In our case, when the magnitude of the frequency increases by a factor of 10, then log increases by 1 The second reason that using logarithms is useful is that log(AB) = log A + log B (153)
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By turning multiplication into addition, the mathematics of a problem is simpli ed In engineering this can be useful when calculating the overall gain of a composite system, which could be an ampli er or lter By using logarithms, we can simply add together the gain at each stage (in dB) to arrive at the overall gain of the system We call each 10-to-1 change in frequency a decade That is, two frequencies A and B are separated by one decade if A = 10 B If one frequency is twice the other, we say they are an octave apart A = 2 B (155) (154)
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When analyzing the transfer function for a given system, it is important to characterize its low- and high-frequency behavior Given a transfer function
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H (s), we characterize the low-frequency behavior by considering the limit
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(156)
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When doing the analysis in terms of frequency, we let s j and then examine the limit
lim |H ( j )|
(157)
In the high-frequency case, in the s domain we consider the limit
lim H (s)
(158)
Or we let s j and examine
lim |H ( j )|
(159)
EXAMPLE 15-1 Determine the asymptotic behavior of the transfer function H (s) = 2s + 6 s 12
SOLUTION First we do a bit of algebraic manipulation H (s) = 2s + 6 s+3 s+3 2 = H (s) = 2 2 =2 = s 12 s s 12 (s + 3)(s 4) s 4
At low frequencies 1 2 = s 0 s 4 2 lim To examine the function directly in frequency, we set s j and multiply top and bottom of H ( j ) by the complex conjugate H ( j ) = 2 2 = 4 + j 4 + j 4 j 4 j = 8 j2 2 + 16
The low-frequency behavior is lim H ( j ) = lim
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8 j2 8 1 = = 2 + 16 0 16 2
Now let s examine high-frequency behavior We have lim H (s) = lim 2 =0 s s 4
And similarly for What about the behavior of the phase angle First let s calculate it by using H ( j ) = We nd = tan 1 At high frequencies = lim tan 1
8 j2 2 + 16
2 8
= tan 1
= 90 4
On the other hand, at low frequencies we have = lim tan 1
= tan 1 0 = 0 4
Once we understand how to characterize the low- and high-frequency behavior of the transfer function and its phase angle, we are ready to create Bode plots
Creating Bode Plots
A Bode plot is a log linear plot The axes are de ned in the following way:
The horizontal axis is the logarithm of frequency (log10 ) The vertical axis is the frequency response in decibels
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