Sound, signals and sampling

Welcome

0:00Welcome to today's lecture. I'm Professor Kettlewick, your AI professor.

0:05This is Signals and Sound, lecture one: sound, signals and sampling.

0:11Today we'll find out how a sound becomes a list of numbers,

0:17and the one rule you must never break.

What is a signal?

0:22Let's start with something you do every day without thinking: hearing.

0:27When a guitar string vibrates, it pushes the air next to it,

0:33squashing it together and then letting it spread apart.

0:38Those squashes and stretches travel outwards as a pressure wave,

0:43at about 340 metres a second, until they reach your ear.

0:50A microphone does what your eardrum does. A thin diaphragm moves with the pressure,

0:55and the microphone turns that movement into a voltage.

0:59More pressure, higher voltage; less pressure, lower voltage.

1:06That changing voltage is a signal:

1:08a quantity that changes over time and carries information.

1:14In this course, almost every signal starts life as a sound and ends life as a sound.

1:22The simplest signal of all is a sine wave.

1:27It's the sound of a tuning fork, and, as we'll see later in the course,

1:33every other sound can be built from sine waves added together.

1:39So let's learn to read one.

1:43The height of the wave from the middle line to a peak is its amplitude.

1:49A bigger amplitude means a louder sound.

1:55The time it takes to go through one complete cycle is the period,

2:01written T and measured in seconds.

2:04And the number of complete cycles in one second is the frequency,

2:09f, measured in hertz.

2:12Period and frequency are two views of the same thing: f equals one over T.

2:21Let's try one. The note A that orchestras tune to has a frequency of 440 Hz.

2:30What's its period? One over 440 is about 2.27 ms.

2:38440 complete squashes and stretches of the air, every second.

2:47Here it is.

2:49[A pure tone: 440 Hz]

2:51And now half the frequency, 220 Hz.

2:57[A pure tone: 220 Hz]

2:59Lower: in fact exactly one octave lower.

3:04Double the frequency and you go up an octave; halve it and you go down one.

3:11Young, healthy ears hear from about 20 Hz up to about 20,000 Hz: 20 kHz.

3:19The top end falls as we get older,

3:22and many adults hear little above 15 or 16 kHz.

3:29Hold on to that 20 kHz. It's going to matter a great deal in a few minutes.

3:37One more property: phase. Two sine waves can have the same frequency

3:44and amplitude but be at different points in their cycle.

3:49Phase tells us where in its cycle a wave is at time zero, in degrees or radians.

3:58You won't hear the phase of a single tone, but when signals are added together,

4:05phase decides whether they reinforce each other or cancel out.

Why go digital?

4:11So we have a signal: a voltage that varies smoothly and continuously.

4:17We call it analogue, because the voltage is an analogue,

4:22a likeness, of the pressure wave. Why not keep it that way?

4:28Because every time you copy, store or send an analogue signal,

4:34a little noise and distortion creep in, and they never come out again.

4:40Copy a tape, then copy the copy,

4:43and by the fifth generation you can hear the hiss.

4:47A digital signal is a list of numbers,

4:51and a number can be copied a million times without changing.

4:58Numbers can also be processed by a computer: filtered,

5:03mixed, compressed, and sent round the world.

5:07Turning a smooth voltage into numbers takes two steps. First,

5:13we measure it at regular moments in time. That's sampling, and it's today's subject.

5:20Second, we round each measurement to one of a fixed set of levels.

5:25That's quantisation, and it's next lecture.

5:29Sampling chops up time; quantisation chops up the voltage.

Sampling

5:34Imagine photographing the signal's voltage at regular intervals,

5:38and writing down each value. Each measurement is a sample.

5:43The number of samples we take each second is the sampling rate,

5:48or sampling frequency, fs, in hertz.

5:53The time between samples is the sampling period, Ts, which is one over fs.

6:01Once the signal has been sampled, the smooth wave is gone.

6:07All we keep are the dots.

6:10And that is the question for this whole lecture:

6:15from just these dots, can we get the original wave back?

6:21Compact discs use a sampling rate of 44.1 kHz: 44,100 samples every second.

6:28So the time between samples is one over 44,100, about 22.7 microseconds.

6:38How many samples does each cycle of a wave get?

6:41Divide the sampling rate by the signal's frequency.

6:44A 1 kHz tone, sampled at 44.1 kHz, gets 44.1 samples per cycle. Plenty.

6:56A 10 kHz tone gets 4.41. Fewer, but still enough, as we'll see.

7:05Your turn. Pause the video if you like.

7:09At 44.1 kHz, how many samples does each cycle of a 20 kHz tone get?

7:2244.1 divided by 20: 2.205 samples per cycle.

7:30Just over two. That number is no accident.

How fast is fast enough? The sampling theorem

7:34Here is the rule at the heart of digital audio.

7:39To capture a frequency, you need more than two samples per cycle.

7:46Put the other way round: the sampling rate must be more

7:50than twice the highest frequency in the signal. fs must be greater than two fmax.

8:00This is the sampling theorem, often called the Nyquist theorem after Harry Nyquist;

8:06Claude Shannon published the full proof in 1949.

8:11Half the sampling rate, fs over two, is called the Nyquist frequency:

8:17the highest frequency that a given sampling rate can capture.

8:22Why more than two, and not exactly two?

8:28Watch what happens at exactly two samples per cycle.

8:32If the samples happen to land on the peaks, we see the wave.

8:38But if they land on the zero crossings,

8:40every sample is zero, and the wave has vanished completely.

8:45Exactly two isn't safe. Just over two is enough.

8:51Now we can see why a CD samples at 44.1 kHz.

8:58Hearing tops out around 20 kHz, so we need a sampling rate above 40 kHz.

9:0544.1 gives a Nyquist frequency of 22.05 kHz, with a little room to spare,

9:17and we'll see what that room is for in a moment.

9:21The oddly precise number comes from history:

9:25the first digital recorders stored their samples on video tape,

9:30and 44,100 a second fitted the lines of a television picture exactly,

9:37in both the European and the American systems.

9:41Different jobs use different rates.

9:46Telephones sample at 8 kHz, so they can carry nothing above 4 kHz.

9:51That's enough to understand speech, but listen to what it does to music.

9:57[Music, at 44.1 kHz]

10:07[The same music, sampled at 8 kHz]

10:16The words would survive; the sparkle doesn't. Film and video use 48 kHz,

10:23and studios often record at 96 kHz, which gives them more room while they work.

10:30Suppose a bat's call reaches 100 kHz.

10:36What's the lowest sampling rate that could capture it?

10:44More than twice 100 kHz: anything above 200 kHz.

Aliasing and the anti-aliasing filter

10:51So what happens if we break the rule?

10:55Something strange, and you've almost certainly seen it. In films,

11:00the wheels of a speeding wagon or car sometimes seem to turn slowly backwards.

11:07A film camera samples the world 24 times a second.

11:13When a spoke moves almost a full spoke-spacing between one frame

11:18and the next, the camera can't tell that it went nearly all the way forwards;

11:23the nearest match is that it crept a little backwards.

11:31The fast motion is disguised as a slow one.

11:36Sound does exactly the same. Here's a 7 kHz tone, sampled at 8 kHz.

11:43That's far fewer than two samples per cycle. Now look at the dots.

11:50A gentle 1 kHz wave passes through every single one of them.

11:57From the samples alone, there's no way to tell which wave we started with,

12:04and when we play them back, we hear the 1 kHz tone.

12:10This impostor frequency is called an alias, and the effect is aliasing.

12:17Listen to it happen. This tone rises steadily from nothing to 8 kHz,

12:24sampled at 8 kHz with no protection.

12:27[A tone rising steadily, sampled at 8 kHz with no filter]

12:30[Above 4 kHz, the pitch we hear falls back down]

12:33It rises to 4 kHz, the Nyquist frequency,

12:38and then turns round and falls, even though the real tone kept rising.

12:46Everything above the Nyquist frequency folds back down,

12:50like a reflection in a mirror at fs over two.

12:54Here's the crucial point. Once aliasing has happened, it can't be undone.

13:01The alias sits among the real sound,

13:03and nothing done afterwards can tell them apart.

13:08So the only cure is prevention: before a signal is sampled, it goes

13:15through a low-pass filter that removes everything above the Nyquist frequency.

13:21It's called an anti-aliasing filter,

13:23and it's the first block in every analogue to digital system.

13:29If you're studying A Level Electronics,

13:32this is the transmitter in a pulse code modulation system:

13:38the low-pass filter, then the sampling gate driven by the sampling clock,

13:45then the analogue to digital converter, then a parallel-in

13:49serial-out shift register that sends the bits out one at a time.

13:54The low-pass filter comes first for exactly the reason we've just heard.

14:01And that's what the CD's spare room is for.

14:05A real filter can't cut off like a cliff edge;

14:10it needs a range of frequencies to go from letting sound through to blocking it.

14:18On a CD, that range is the gap between 20 kHz,

14:22the top of hearing, and 22.05 kHz, the Nyquist frequency.

A Level recap and three questions

14:32Let's pull that together.

14:34A signal is a quantity that changes over time; in audio,

14:40usually a voltage that follows a pressure wave.

14:44Sampling measures the signal at regular intervals. The sampling rate, fs,

14:51is the number of samples each second, and the sampling period is one over fs.

14:58To capture a frequency, sample at more than twice it: fs greater than two fmax.

15:06Half the sampling rate is the Nyquist frequency.

15:11Frequencies above the Nyquist frequency don't disappear.

15:16They fold back down as aliases, and they can't be removed afterwards,

15:21so an anti-aliasing low-pass filter comes before the sampler.

15:25Three questions to try. The answers are on the website, with full working.

15:33One: a signal contains frequencies up to 12 kHz. What is the minimum sampling rate?

15:41Two: a system samples at 32 kHz.

15:45What is its Nyquist frequency, and could it capture an 18 kHz tone?

15:52Three: in a pulse code modulation transmitter,

15:56why must the low-pass filter come before the sampling gate, and not after it?

16:02If you're studying for A Level, that's the core of today's lecture.

16:09Stay with me if you'd like to go further:

16:13next we'll look at the mathematics of aliasing,

16:16and at how the original wave is rebuilt from the samples.

First year: the mathematics of aliasing

16:23Let's write the sampled sine wave down.

16:28A continuous tone is x of t equals A cos of two pi f t, plus phi.

16:36We keep it only at the sample times, t equals n Ts,

16:41where n counts the samples: zero, one, two, and so on.

16:48So the samples are x of n equals A cos of two pi f n Ts, plus phi.

16:58And since Ts is one over fs, that's A cos of two pi n times f over fs, plus phi.

17:10Notice that the samples depend only on the ratio f over fs:

17:16the frequency measured in cycles per sample.

17:21Now replace f with f plus fs.

17:26The ratio becomes f over fs, plus one, so the angle gains an extra two pi n.

17:34But n is a whole number,

17:36and adding a whole number of complete turns to a cosine changes nothing.

17:42The samples are identical.

17:46A tone at f and a tone at f plus fs can't be told apart once they're sampled;

17:56nor can f plus two fs, or f minus fs, or f plus any whole number of fs.

18:07There's one more twin. Cosine is an even function:

18:11cos of minus theta equals cos of theta. So a tone at minus f gives the same samples

18:20as a tone at f, with its phase flipped.

18:25Put the two facts together, and the frequency fs minus f matches f as well.

18:32That's the mirror at the Nyquist frequency.

18:37So to find the alias of any frequency: shift it by whole multiples of fs

18:43until it lands between minus fs over two and plus fs over two, then take its size.

18:54In symbols, the alias is the size of f minus k fs,

19:00choosing the whole number k that makes it smallest.

19:05Let's practise at 48 kHz. 30 kHz: subtract 48, giving minus 18,

19:11so the alias is 18 kHz. 50 kHz: subtract 48, 2 kHz.

19:1747 kHz: 1 kHz. And 24 kHz, exactly the Nyquist frequency?

19:23It maps to itself, but it's the edge case we met earlier,

19:29where the samples can vanish.

19:32At 44.1 kHz, what does a 25 kHz tone alias to?

19:4525 minus 44.1 is minus 19.1, so 19.1 kHz: right in the range we can hear.

19:56Here's the same idea as a picture. Along the bottom,

20:01the frequency that goes in; up the side, the frequency that comes out.

20:06Up to the Nyquist frequency, out equals in: a straight line.

20:13Then the line folds back down, reaching zero at fs,

20:19rises again to the Nyquist frequency at one and a half fs, and so on: a zigzag.

20:28Every input frequency, however high, lands somewhere between zero and fs over two.

20:36This tells us how hard the anti-aliasing filter has to work.

20:42At 44.1 kHz, anything above 22.05 kHz folds back,

20:49and the fold reaches down into the band we hear: 24.1 kHz lands exactly on 20 kHz.

20:58So the filter must pass everything up to 20 kHz untouched,

21:03and all but silence everything above 24.1 kHz. That's a very steep filter.

21:12Modern converters solve it neatly: they sample far faster,

21:16often 64 times faster, so that a gentle analogue filter is enough,

21:21then filter digitally and throw away the extra samples.

21:27That's oversampling, and we'll meet it properly in lecture four.

First year: getting the wave back

21:33Now the other direction. We have the dots. How do we get a smooth wave back?

21:41You'll often see digital audio drawn as a staircase,

21:45the voltage holding each sample's value until the next one arrives.

21:50A digital to analogue converter can indeed produce a staircase first.

21:57But the sharp edges of those steps are made of frequencies above the Nyquist

22:04frequency: copies, called images, of our signal around fs, two fs, and so on.

22:12A low-pass filter after the converter,

22:15called the reconstruction or anti-imaging filter, removes them.

22:20What's left isn't an approximation of the original wave.

22:24Provided the original contained nothing above the Nyquist frequency,

22:29it is the original wave, apart from the rounding of each sample,

22:34which is next lecture's subject.

22:37There's a beautiful way to see why.

22:40Give every sample its own smooth bump, a shape called a sinc function,

22:48which is one at its own sample time and exactly zero at every other sample time.

22:55Scale each bump by its sample's value, and add them all up.

23:02The bumps fill in the gaps, and the sum passes through every sample

23:07while wiggling no faster than the Nyquist frequency allows.

23:12This is the Whittaker-Shannon interpolation formula,

23:15and it rebuilds a band-limited signal exactly from its samples.

23:21So here's a misconception to drop: digital audio is not steppy.

23:26Below the Nyquist frequency, sampling loses nothing at all.

23:31What it costs is everything above the Nyquist frequency,

23:37which is why we choose the rate to sit above the top of hearing.

23:42It also explains why more samples aren't automatically better.

23:48Once the sampling rate is safely above twice the highest frequency

23:53you care about, extra samples capture frequencies you can't hear.

23:58They make the filters easier to build, rather than the sound itself better.

Summary and what's next

24:05Today, we followed a sound into numbers and back.

24:10A signal is a voltage that follows a pressure wave.

24:15Sampling turns it into a list of numbers, taken fs times a second.

24:21Sample at more than twice the highest frequency,

24:25or frequencies fold back as aliases that can never be removed, so filter first.

24:32And with the right filter after the converter,

24:36the samples give back the original wave exactly.

24:41There's a practice set on the website, with worked answers, for both levels.

24:47Next time: bits and resolution.

24:51We'll round each sample to a fixed set of levels, find out what that costs,

24:57and see why 16 bits give a CD about 96 decibels of range. See you then.