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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

• 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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# 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