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  • Consider the problem of using voltages

  • to represent the information in a black-and-white image.

  • Each (x,y) point in the image has an associated intensity:

  • black is the weakest intensity, white the strongest.

  • An obvious voltage-based representation

  • would be to encode the intensity as a voltage, say 0V for black,

  • 1V for white, and some intermediate voltage

  • for intensities in-between.

  • First question: how much information is there

  • at each point in the image?

  • The answer depends on how well we can distinguish intensities

  • or, in our case, voltages.

  • If we can distinguish arbitrarily small differences,

  • then there's potentially an infinite amount of information

  • in each point of the image.

  • But, as engineers, we suspect there's a lower-bound

  • on the size of differences we can detect.

  • To represent the same amount of information that can be

  • represented with N bits, we need to be able to distinguish

  • a total 2^N voltages in the range of 0V to 1V.

  • For example, for N = 2, we'd need to be able to distinguish

  • between four possible voltages.

  • That doesn't seem too hardan inexpensive volt-meter would

  • let us easily distinguish between 0V, 1/3V, 2/3V and 1V.

  • In theory, N can be arbitrarily large.

  • In practice, we know it would be quite challenging to make

  • measurements with, say, a precision of 1-millionth

  • of a volt and probably next to impossible

  • if we wanted a precision of 1-billionth of a volt.

  • Not only would the equipment start to get very expensive

  • and the measurements very time consuming,

  • but we'd discover that phenomenon like thermal noise

  • would confuse what we mean by the instantaneous voltage

  • at a particular time.

  • So our ability to encode information using voltages

  • will clearly be constrained by our ability

  • to reliably and quickly distinguish

  • the voltage at particular time.

  • To complete our project of representing a complete image,

  • we'll scan the image in some prescribed raster order

  • left-to-right, top-to-bottomconverting intensities

  • to voltages as we go.

  • In this way, we can convert the image

  • into a time-varying sequence of voltages.

  • This is how the original televisions worked:

  • the picture was encoded as a voltage waveform that

  • varied between the representation for black

  • and that for white.

  • Actually the range of voltages was

  • expanded to allow the signal to specify

  • the end of the horizontal scan and the end of an image,

  • the so-called sync signals.

  • We call this a “continuous waveformto indicate that it

  • can take on any value in the specified range at a particular

  • point in time.

  • Now let's see what happens when we try to build a system

  • to process this signal.

  • We'll create a system using two simple processing blocks.

  • The COPY block reproduces on its output whatever

  • voltage appears on its input.

  • The output of a COPY block looks the same as the original image.

  • The INVERTING block produces a voltage of 1-V

  • when the input voltage is V, i.e.,

  • white is converted to black and vice-versa.

  • We get the negative of the input image

  • after passing it through an INVERTING block.

  • Why have processing blocks?

  • Using pre-packaged blocks is a common way

  • of building large circuits.

  • We can assemble a system by connecting the blocks one

  • to another and reason about the behavior

  • of the resulting system without having

  • to understand the internal details of each block.

  • The pre-packaged functionality offered by the blocks

  • makes them easy to use without having to be

  • an expert analog engineer!

  • Moreover, we would expect to be able to wire up

  • the blocks in different configurations

  • when building different systems and be

  • able to predict the behavior of each system

  • based on the behavior of each block.

  • This would allow us to build systems like tinker toys,

  • simply by hooking one block to another.

  • Even a programmer who doesn't understand the electrical

  • details could expect to build systems that perform some

  • particular processing task.

  • The whole idea is that there's a guarantee of predictable

  • behavior:

  • If the components work and we hook them up

  • obeying whatever the rules are for connecting blocks,

  • we would expect the system to work as intended.

  • So, let's build a system with our COPY and INVERTING blocks.

  • Here's an image processing system using a few instances

  • each block.

  • What do we expect the output image to look like?

  • Well, the COPY blocks don't change the image and there are

  • an even number of INVERTING blocks, so, in theory,

  • the output image should be identical to the input image.

  • But in reality, the output image isn't a perfect copy

  • of the input.

  • It's slightly fuzzy, the intensities are slightly off

  • and it looks like sharp changes in intensity have been smoothed

  • out, creating a blurry reproduction of the original.

  • What went wrong?

  • Why doesn't theory match reality?

  • Perhaps the COPY and INVERTING blocks don't work correctly?

  • That's almost certainly true, in the sense that they don't

  • precisely obey the mathematical description of their behavior.

  • Small manufacturing variations and differing environmental

  • conditions will cause each instance of the COPY block

  • to produce not V volts for a V-volt input,

  • but V+epsilon volts, where epsilon represents the amount

  • of error introduced during processing.

  • Ditto for the INVERTING block.

  • The difficulty is that in our continuous-value representation

  • of intensity, V+epsilon is a perfectly correct output value,

  • just not for a V-volt input!

  • In other words, we can't tell the difference between

  • a slightly corrupted signal and a perfectly valid signal

  • for a slightly different image.

  • More importantlyand this is the real killerthe errors

  • accumulate as the encoded image passes through the system

  • of COPY and INVERTING blocks.

  • The larger the system, the larger

  • the amount of accumulated processing error.

  • This doesn't seem so good.

  • It would be awkward, to say the least,

  • if we had to have rules about how many computations could

  • be performed on encoded information

  • before the results became too corrupted to be usable.

  • You would be correct if you thought

  • this meant that the theory we used

  • to describe the operation of our system was imperfect.

  • We'd need a very complicated theory indeed to capture all

  • the possible ways in which the output signal could differ from

  • its expected value.

  • Those of us who are mathematically minded might

  • complain thatreality is imperfect”.

  • This is going a bit far though.

  • Reality is what it is and, as engineers,

  • we need to build our systems to operate reliably

  • in the real world.

  • So perhaps the real problem lies in how we

  • chose to engineer the system.

  • In fact, all of the above are true!

  • Noise and inaccuracy are inevitable.

  • We can't reliably reproduce infinite information.

  • We must design our system to tolerate some amount of error

  • if it is to process information reliably.

  • Basically, we need to find a way to notice that errors have been

  • introduced by a processing step and restore

  • the correct value before the errors have a chance

  • to accumulate.

  • How to do that is our next topic.

Consider the problem of using voltages

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B1 中級

2.2.2.2 アナログシグナリング (2.2.2 Analog Signaling)

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    林宜悉 に公開 2021 年 01 月 14 日
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