Sound and Signals

Draw a sound and change its three numbers

A sound is air being pushed and pulled. A drum skin moves out and squashes the air in front of it, then moves back and thins it out. That pattern of squash and thin travels to your ear at about 340 metres a second. Draw how squashed the air is against time and you have drawn the sound. The line you draw is a signal: a quantity that changes as time passes and carries something worth having.

The simplest signal that repeats is a sine wave, a smooth rise and fall that does the same thing over and over. Three numbers set one completely. The amplitude is how far it swings away from the middle line, and that is loudness. The frequency is how many complete rises and falls it fits into a second. That is how high or low the note sounds. One complete rise and fall is a cycle, and cycles per second has its own name, the hertz, written Hz. The phase says where in its own cycle the wave was at the moment you started watching.

The time one cycle takes is the period. The two are the same fact upside down. 3 Hz means three cycles a second, so one cycle takes a third of a second. That is 333 milliseconds, a millisecond being a thousandth of a second.

Two waves in the same air add up

Air carries more than one sound at once, and it does so by the simplest rule available. Suppose one source is pushing the air out at this instant, and another is pulling it back by a smaller amount. The air moves by the difference. The heights add, instant by instant. That is the whole rule. It has a name, superposition. It is worth having because so much later work depends on nothing else happening.

What comes out can look nothing like either part. Add a tall slow wave to a short fast one and you get a slow wave with ripples riding on it. That single line is what reaches your ear, and your ear takes it apart again.

Chop a wave into a list of numbers

A machine cannot hold a curve. It holds numbers, one after another, and that is all it has ever been able to do. So it measures the wave at one instant, writes the answer down, waits a fixed time, and measures again. Each measurement is a sample. Taking them is sampling. How many you take each second is the sample rate, counted in hertz like a frequency, because it is also a number of things per second.

Between two samples, the recording says nothing. Not a small amount, not an average, nothing at all. Whatever the air did in that gap left no trace. Later processing can estimate or interpolate, but it cannot recover information that the samples never captured.

The sample rate and the gap between the dots

A list of numbers is not a sound. To hear it again, something has to push the air smoothly once more, and it has only the dots to go on. The simplest rule available is to join them with straight lines. That is close enough to what real equipment does to be worth measuring.

The gap between the joined-up line and the wave that was really there is what the sample rate cost you, and it is a number rather than an opinion. The useful way to count is not measurements a second but measurements per cycle. A wave that wiggles twice as fast needs twice as many readings to be caught equally well.

A wheel under a flashing light

Leave sound alone for one step. A wheel with a single white dot painted on its rim is spinning in a dark room. The only light is a lamp that flashes 25 times a second, and between flashes you see nothing at all. Your eye is not watching the wheel. It is being handed 25 snapshots a second, which is sampling, with your eye doing the joining up.

Suppose the wheel turns 24 times a second. Between one flash and the next it gets through 24 twenty-fifths of a turn: nearly the whole way round, but not quite. Your eye has no way of knowing about the nearly-whole turn, because nothing was lit while it happened. All it sees is a dot that has ended up slightly behind where it started.

Predict what a 7 Hz wave comes back as

The wheel and the wave are the same problem in different clothes. Take a wave doing 7 cycles a second and measure it 10 times a second. Between one measurement and the next it gets through seven tenths of a cycle, and the recording has no way of knowing about the whole cycles it missed in between.

So there is a question worth committing to an answer on, before you see it. Join those measurements back up. What is the slowest wave that passes through every single one of them? Most people say 7 Hz, or a mess, or nothing. Press an answer in the lab first, then reveal it, because being wrong here once is what makes the rest of the course stick.

Half the sample rate is the limit

Somewhere between 1 Hz, which came back honestly, and 7 Hz, which came back as 3, the recording started lying. The place where it starts is worth finding rather than being told. So the next lab sweeps every frequency from nothing up to two and a half times the sample rate. It plots what each one comes back as.

Along the bottom is what was really there. Up the side is what the measurements say it was. If sampling were harmless the picture would be a straight diagonal line going up for ever.

Round every reading to the nearest level

Sampling chopped up time. There is a second chop, and it happens to the readings themselves. A machine cannot store a number to unlimited precision. It has a fixed set of levels spread across the range it can measure and every reading is moved to whichever level is nearest. That moving is quantisation.

The distance between neighbouring levels is the step. Nothing can ever be further than half a step from the nearest level, because if it were, some other level would be nearer. So the worst error is half a step, always, and the way to shrink it is to have more levels.

What is kept is no longer a curve but a staircase, and the difference between the two is the rounding error, which can be drawn on its own.

One more bit halves the error

Levels are not chosen freely. A machine stores a number as a row of yes-or-no answers, and one yes-or-no answer is a bit. One bit tells two things apart. Two bits tell four apart, because each of the two answers to the first question can be followed by either answer to the second. Every bit you add doubles the count, so the number of levels is 2 doubled as many times as you have bits. How many bits each sample gets is the bit depth.

That gives a chain: one more bit, twice as many levels, half the step, half the worst rounding error. The signal is unchanged while the error halves, so the signal stands twice as far above it as it did.

The error never goes away. Under every recording there is a permanent low hiss made entirely of rounding, and anything quieter than that hiss is lost in it. That floor has a name: the noise floor.

Measure how much signal is left after noise

Rounding is not the only thing added to a reading. Every real measurement arrives with noise: small random amounts added to each sample by the microphone, the wiring, the warmth of the components, and everything electrical in the building. Noise is different at every reading and unrelated to the signal, which turns out to be the property that matters.

To compare a signal with the noise on top of it you need a fair size for something that spends half its time below the middle line. Adding the readings up gives roughly zero, which is no use. So: square every number, which makes them all positive, average the squares, then take the square root to get back to the original units. Engineers call the result the root mean square, said as it is spelled, backwards.

Do that to the signal, do it to the noise, and divide one by the other. What you have is the signal-to-noise ratio: how many times bigger the thing you want is than the thing you do not.

Average many captures and the noise falls

Here is the property of noise that can be used against it. If the same measurement can be repeated, the signal is in the same place every time and the noise is somewhere different every time. Add up ten captures and the signal adds up ten times over, while the noise partly cancels itself, because the positive amounts in one capture meet negative amounts in another.

Divide by the number of captures and the signal is back to its original height with less noise on it. The improvement is not the number of captures, though: it is the square root of that number. The square root of a number is the value that gives it when multiplied by itself, so the square root of 16 is 4.

Four captures halve the noise. Sixteen quarter it. A hundred divide it by ten. Doing twice as well costs four times as much work, every time, for ever.

Pressure, speed and wavelength

The wave you have been drawing since Step 1 is a drawing of air being squashed and thinned, and it is worth one step to watch that happen. Air is a crowd of molecules. A loudspeaker cone is a wall of that crowd that has started shoving. Push the cone forward and the air in front of it is crowded together: a patch of squashed air, called a compression. Pull it back and it leaves a thinned-out patch, a rarefaction. Each squashed patch shoves the air next to it, which shoves the air next to that, and the pattern runs across the room. The molecules themselves barely travel. It is a shove passing down a queue of people: the shove crosses the whole queue while each person only sways on the spot.

The pattern travels at a speed the air decides, not the sound. In air at 20 °C it is about 343 metres a second, loud or quiet, high note or low. A thunderstorm lets you measure it. The flash arrives almost at once, and the rumble plods along at 343 metres a second. Every three seconds of gap puts the lightning about another kilometre away. Warmer air passes the shove along slightly faster, so the speed creeps up with temperature.

Speed gives a wave a size you can measure with a tape. The period from Step 1 is how long one cycle takes. In that time the pattern moves forward some distance, so one whole cycle occupies a length of air, from one compression to the next. That length is the wavelength. To find it, divide the speed by the cycles a second. A 343 Hz tone in 343 metre-a-second air has a wavelength of one metre. A deep 50 Hz hum is nearly seven metres long, which is longer than most rooms.

Distance, reflections and rooms

Drop a stone in a pond and the ripples spread out in rings, getting lower as they travel. The ripple keeps its energy, but the ring it is spread around keeps growing, so each bit of ring gets less of it. Sound does the same in three dimensions, spreading as a growing sphere. At twice the distance the same push covers four times the area, so the pressure at your ear is half what it was. In the decibels from Step 9, a halving of height is about 6 dB, so every doubling of distance costs about 6 dB.

That clean rule holds in the open, where sound leaves and never comes back. A room is different, because the walls throw it back. One wall far away returns a separate copy, an echo. The walls of a room return thousands of copies, packed so tightly that they smear into a wash of sound. That wash is reverberation, the ringing that follows a shout in an empty stairwell.

Every copy is a wave, and Step 2 said waves add. A reflected copy arrives late, which slides it along against the direct sound. At one frequency the slide lands push on push and the note gets louder. At another it lands push on pull and the note nearly vanishes, which is the walking-past-two-speakers quiz from Step 2 happening off a wall. Which of the two you get depends on where you stand, so the same note is loud in one spot of a room and thin a step away.

The recording chain

Everything between the air and the list of numbers is a short chain of parts, each with one job. A microphone is a drum skin of its own: the air wiggles it, and it turns the wiggle into a wiggling voltage that copies the pressure exactly. That voltage is tiny, so the next stage is an amplifier that makes it bigger. How many times bigger is the gain, and it is a knob somebody has to set. After the amplifier come two parts you have already built the argument for. First the anti-alias filter from Step 7. Then the converter from Steps 3 and 8, which samples the voltage and rounds each sample to a level. Its trade name is the ADC, for analogue-to-digital converter.

Setting the gain is a squeeze between the two failures you have already measured. Set it too low and the signal sits just above the noise floor of Steps 9 and 10, and turning it up afterwards turns the hiss up with it. Set it too high and the loudest peak gets pinned against the top of the converter's range. That is the clipping from Step 8, and nothing later can unbend a flattened peak. The spare space left between the loudest expected peak and the top is called headroom. A careful engineer leaves several decibels of it, because the loudest moment of a take is a surprise by definition.

Anti-aliasing and reconstruction

The rule from Step 7 came with a promise attached: sample at more than twice the fastest wave, and everything survives. The room does not make that promise for you. Squeaky machinery, bats and electrical hash all put waves into the microphone that are faster than half of any rate you chose. So the recorder keeps the promise itself. The anti-alias filter's whole job is to remove everything above half the rate before the sampler sees it. It is the reason the workshop squeak in Step 6 is a story rather than an everyday fault.

A real filter cannot cut like a cliff edge. It fades, over a stretch of frequency called its transition band. Below the stretch, everything passes untouched. Above it, everything is gone. Inside it, waves are partly both. The whole stretch has to fit between the fastest wave you want to keep and half the sample rate. That is the practical reason rates sit comfortably above twice: the gap between 40,000 and 44,100 is not generosity, it is where the filter fades.

Playing back has the same problem in reverse. The output converter, the DAC, holds each number steady until the next one arrives, so what leaves it is the staircase from Step 8. The corners of a staircase are fast wiggles that were never in the music. A smoothing filter after the converter rounds the corners off and leaves the wave the samples described.

Channels, bytes and clocks

A finished recording is just the list of numbers, plus an agreement about how to read it back. You have two ears and they hear different things, so most recordings keep two lists, one for each side. Each list is a channel, and two channels is stereo. Each number in each list is stored in the bits of Step 9. Bits come packed in eights: eight bits is a byte, the unit file sizes are quoted in. Storing sound this way, one plain reading after another, is called PCM, and it is what a .wav file holds.

The cost of a recording is now a multiplication you can do on paper. Numbers a second, times bits per number, times channels, gives bits per second. A compact disc keeps 44,100 numbers a second, 16 bits each, in two channels, which is 1,411,200 bits every second. Divide by eight for bytes, multiply by the length, and you have sized the file before recording a note.

The agreement about reading the list back includes the clock. The numbers say nothing about time except their order, so the player has to tick at the rate the recorder ticked. Play a 48,000-a-second list back at 44,100 ticks a second and every cycle of every wave is stretched. The whole recording comes out longer and deeper, like a singer slowed down to a growl. Play it too fast and you get the squeaky chipmunk version. Time and pitch move together, because both live in the one clock.

Decibels and hearing risk

You have been reading decibels off the pills since Step 9, and here is the whole of the idea. A decibel is not a unit of loudness. It is a way of writing "times bigger" so that multiplying turns into adding. Doubling a wave's height adds about 6 dB, wherever you start from, so ten doublings, which is about a thousand times taller, is about 60 dB. There is one wrinkle to respect. For the energy a sound delivers each second, its power, a doubling adds only about 3 dB, because a wave twice as tall carries four times the energy. Height doublings go in sixes, power doublings in threes, and Lab 29 measures both.

On its own, "20 dB" means nothing. It is "ten times as big" without saying as big as what, so a decibel figure you can trust always names its reference. Sound in the air is measured in dB SPL, against roughly the quietest thing a young ear can catch. A whisper sits near 30, conversation near 60, a rock concert over 100. Recordings are measured in dBFS, against the loudest number the converter can hold. Zero is the top of the range from Step 8, and every honest level is a minus figure below it. Hearing-safety rules are written in dBA, which is dB SPL with the ear's uneven sensitivity folded in. A phone app shows a number too, but unless its microphone has been calibrated the number is a guess.

The safety rules exist because loud sound wears hearing out, quietly and permanently. The widely used workplace guideline puts the limit at 85 dBA for an eight-hour day. Every 3 dB above that doubles the sound energy arriving, so it halves the safe time. At 94 dBA, nine decibels up, three halvings leave one hour.

Impulse response and correlation

Clap once in a big empty hall and listen. What comes back is not your clap. First the slap off the nearest wall, then the farther ones, then a tail that hisses away to nothing. That reply belongs to the hall, not to your hands: clap anywhere in it and the same reply comes back, because the walls have not moved. The hall's reply to one short, sharp sound is called its impulse response, and it is a complete description of what the hall does to sound.

Complete is meant literally. A recording is a row of samples, and each sample is one little push, a tiny scaled clap of its own. The hall answers every push with a copy of its reply, scaled to match, and the copies add, exactly as waves added in Step 2. So adding up shifted, scaled copies of the reply predicts what the hall does to any recording at all. That adding-up has a name, convolution, and it is how film sound puts an actor recorded in a studio into a cathedral.

The matching tool that goes with it is correlation. Take two recordings of the same sound, one delayed, and slide one along the other, scoring at every slide how well the two line up. The slide with the best score is the delay between them. That is how an echo is timed when it is too buried in noise to spot by eye, and timing echoes is range-finding, which Step 12's divide-by-two turned into distance.

Calibration and uncertainty

Every number in this course was measured by machinery on the page, and you could pick the machinery apart. A measurement of a real room has to earn the same trust, and the test is simple to state. Another person, with your notes, should be able to repeat what you did and get the same answer. That means writing down what question you were asking, what you measured with, where the microphone stood and which way it faced, the gain and the rate. It also means keeping the raw numbers themselves.

It also means knowing how far to trust your own equipment. The way to find out is calibration: measure something whose answer is already known, and see what your chain reports. A kitchen scale is calibrated with a known weight. A microphone is calibrated with a small device that plays one exactly known tone into it. Whatever the chain then reports for that tone becomes the correction for everything else.

Being wrong comes in two flavours, and Step 11 met both without naming them. Random scatter is different on every repeat, so repeating and averaging shrinks it, at the square-root price you measured. A lean, called bias, is the same on every repeat, like a scale that always reads a kilogram heavy, and no amount of repeating touches it. Only calibration catches a lean. What remains after both is written down as an honest range of doubt, the uncertainty. Each source of doubt gets its own number, and the largest is the one worth attacking first.

Your own signal, end to end

Everything in this course is in one place here, with nothing marked and nothing to get right. Build a signal out of one or two waves. Then choose how often to measure it, how many bits to keep each measurement in, how much noise to add, and how many captures to average. The chain runs in that order, which is the order a real recorder runs it in.

Two settings fight each other, and finding the fight is the point. Push the measuring rate down and a wave folds into something slower. Push the bits down and a hiss appears under everything. Push the noise up and only more captures will save you.

What you can do now

• Draw any steady sound as a wave, and say which of amplitude, frequency and phase you would change to make it louder, higher or later.
• Explain why two loud sounds can add up to silence, and what has to be exact for it to happen.
• Work out how many numbers a recording will take, from its rate and its length.
• Say what a recording knows about the gap between two samples, which is nothing, and what that costs at a given number of measurements per cycle.
• Predict what a wave above half the sample rate will come back as, using nothing but subtraction, and check yourself against a search that fits the samples.
• Choose a sample rate for a signal whose fastest part you know, and say why exactly twice is not enough.
• Say why a wheel goes backwards on film, and connect it to a sound recording without hand-waving.
• Read a bit depth as a number of doublings, and turn it into a noise floor.
• Measure a signal-to-noise ratio rather than guessing it from a picture.
• Say how many repeated captures it takes to make a measurement ten times cleaner, and why the answer is not ten.
• Turn a frequency into a wavelength, and a travel time into a distance, using nothing but the speed of sound.
• Say why the same note can be loud in one spot of a room and thin a step away.
• Set a recording gain that clears the noise floor and still leaves headroom for the loudest peak, and say where the anti-alias filter must sit.
• Size a recording in bits and bytes before making it, and recognise a clock mismatch by what it does to time and pitch together.
• Read a decibel figure critically: which kind of ratio it is, and measured against what.
• Describe a room by its reply to a clap, and time a buried echo by sliding two recordings past each other.
• Tell random scatter from a steady lean, and say which of the two averaging can fix.

Where this goes

• Filters. The anti-alias filter that keeps coming up is one of these. A filter is arithmetic on the list of numbers you now know how to make: average three neighbouring samples and you have built one.
• The Frequency Domain. Step 1 claimed that any wave is a pile of sine waves added together. That course works out which ones, by hand, on eight samples, and then explains why the fast way of doing it changed what computers are used for.
• Putting Data on a Wave. A wave that never changes carries no information. Change its height, its frequency or its phase on purpose and you can send data on it, which is where radio starts. Everything you have measured here about aliasing, bit depth and noise applies unchanged the moment the signal is a radio signal instead of a sound.