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What does it mean for a quantity to be ""analog""?

  1. It changes continuously: between any two values there are always more.
  2. It comes from a device that works without a computer or program, such as an amplifier or a record player.
  3. It belongs to an older, mechanical technology, as with analog photography or the vinyl record.
  4. It can only be determined roughly: its value is never exact, only ever an estimate.
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Analog means stepless, with another value between any two; age, mechanics and the absence of a computer have nothing to do with it.

What characterises a digital system?

  1. It works electronically: its states are represented by current or voltage.
  2. It knows only finitely many, clearly separated states.
  3. It is automatically more precise than an analog system.
  4. It stores information as characters and text, so that a person could in principle read it.
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Digital means finitely many states with nothing valid in between, whatever carries them.

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A signal carries the same noise three times; only the number of regions it is divided into grows from two to four to eight. Why do the readings still become less reliable the more regions there are?

  1. Because the noise grows with every additional boundary: each boundary adds its own disturbances to the signal.
  2. Because the distance between regions shrinks until the same disturbance reaches into the neighbouring region.
  3. Because the sensor measures less precisely at fine steps: it can no longer resolve small differences between neighbouring regions cleanly.
  4. Because more regions cost more computing time and readings get lost.

Not the noise grows but the margin shrinks, because the same range is cut into more and narrower regions.

Digitising splits a continuous range of values into finitely many regions. Which statements are true?

  1. All values within the same region count as one and the same state afterwards.
  2. With enough steps, the difference to the analog quantity disappears completely.
  3. Digitising cleans the signal of disturbances, because afterwards only the region counts and no longer the exact value.
  4. The original exact value cannot be recovered from the digital version.
  5. Digitising only works for signals such as light and sound that can be described as waves.

Inside a region every difference is given up for good, and that is exactly what makes the differences between regions reliable.

Digital technology throws information away when digitising and is still more reliable than analog. Why?

  1. Because the few states are far apart: a small disturbance does not change the state and vanishes without trace at the next reading.
  2. Because digital components do not produce noise: they only switch between on and off, and no disturbance arises in doing so.
  3. Because less information is transferred, so fewer errors can happen.
  4. Because computers automatically detect and remove errors as soon as a value does not fit its surroundings.

Digital is reliable because it throws precision away: disturbances smaller than the distance between states vanish at every reading.

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A program reads a brightness value from a sensor and then compares it with a threshold. In terms of digitising, what does the comparison value > 250 do?

value = sensor.read()

if value > 250:
    symbol = ""bright""
else:
    symbol = ""dark""
  1. It converts the analog light into a digital value, since only after the comparison is there a genuine digital state.
  2. It measures the brightness a second time, as a check, so that an outlier from the first measurement does not slip through.
  3. It draws the boundary that reduces many possible numbers to two meaningful states: this is where an alphabet is made.
  4. It increases the precision of the measured value, because a fluctuating number becomes an unambiguous result.
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The sensor digitises the light into numbers; the threshold is the decision that turns numbers into symbols.

A music cassette audibly loses quality with every dubbing; a copied file is identical even after the thousandth copy. Why?

  1. Because files are stored compressed, so less data can break during copying.
  2. Digital copying reads states and sets them afresh; small disturbances below the distance between states vanish completely each time.
  3. Because computers copy far faster than a cassette deck, there is hardly any time for disturbances to creep in.
  4. Because files are not rewritten when copied, only linked; no second version actually comes into being.

Reading a state and setting it afresh erases every disturbance smaller than the distance between states, which is why digital copies do not age.

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What does it mean to ""digitise"" a signal?

  1. To represent it by a finite sequence of numbers.
  2. To put it into a form that can be transmitted over the internet.
  3. To shrink it so it needs less storage.
  4. To record it with an electronic device rather than a mechanical one.
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Digitising means nothing more than representing a signal by a finite sequence of numbers.

A link sends data with light: a program sets the brightness of an LED with numbers from 0 to 255, and a colour sensor at the other end turns the light back into numbers. Where is the boundary between analog and digital?

  1. Already at the sender: because the LED is driven with numbers from 0 to 255, its light is digital too.
  2. Only in the program at the receiving end, when it compares the reading with a threshold.
  3. In the colour sensor: it turns the continuous light into numbers, and from there on everything computes with numbers.
  4. Nowhere: the whole link is digital because there are computers at both ends.

Digital, analog, digital: the boundaries sit at the LED and in the sensor, and the middle belongs to physics.

Which of these things work digitally in the sense of this concept, that is, with finitely many clearly separated states?

  1. A continuously variable light dimmer
  2. A light switch with on and off
  3. A dial thermometer
  4. A die
  5. An hourglass

Whatever snaps into one of finitely many states is digital, with or without a chip.

You design an alphabet of colours for a link that sends data with light and decide how many colours it has. What is the core of this decision?

  1. A trade-off: more colours carry more per reading but push the regions closer together, and how many are safe is decided by measurements.
  2. More colours pay off, because each reading then carries more bits and the transmission gets correspondingly faster.
  3. Two colours are the safe choice, because the largest distance brings the fewest errors and an error costs more than a little speed.
  4. The number is a matter of taste as long as calibration is done properly: calibration puts the boundaries where they are safe.

The size of an alphabet is a trade-off between speed and margin, and the wobble of the readings decides it.

When digitising a photo, the continuous image becomes 64 × 64 points with 256 brightness levels each. Which of the two steps is the sampling?

  1. Rounding each brightness to one of the 256 agreed levels.
  2. Converting the finished level values into binary numbers.
  3. Slicing the continuous image area into the 64 × 64 measurement points.
  4. Storing the image points in a file with header data.

Sampling cuts the axis into points, quantising cuts the value into levels.

A greyscale picture is digitised with 100 × 100 points and 8 bits per point. How large is the resulting file?

  1. 80,000 bytes, because 100 · 100 · 8 makes 80,000.
  2. 10,000 bytes, because 100 · 100 · 8 = 80,000 bits, and 8 bits make one byte.
  3. 30,000 bytes, because every point needs three colour channels of one byte each.
  4. 1,250 bytes, because the 100 · 100 bits are further divided by 8.

Points times bits per point gives bits, and eight bits make a byte.

A digitised photo shows clearly visible blocks, but the colours within the blocks look natural and finely graded. Which step of digitising was too coarse?

  1. The sampling: too few points for the area.
  2. The quantising: too few levels per colour channel.
  3. The transfer: noise accumulated while copying the file.
  4. Both steps equally: blocks always arise from the interplay.

Blocks mean too few points, hard patches in soft shading mean too few levels.

Why not simply digitise a picture at the highest resolution and with 24 bits of colour depth before sending it over a slow link that carries 30 bits per second?

  1. Because a colour sensor at the receiving end measures only three channels, so at most 3 bits per point can arrive.
  2. Because large files pick up more noise in transfer than small ones.
  3. Because more than 8 bits of colour depth cannot be displayed on the receiver's screen.
  4. Because every point and every level costs bits, and at 30 bits per second a large picture would take hours.
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On a slow link every kept bit costs time, so the resolution and the colour depth are a trade-off, not a matter of ""as fine as possible"".

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