Excursion vs. Frequency

Last week, I was doing a lecture about the basics of audio and I happened to mention one of the rules of thumb that we use in loudspeaker development:

If you have a single loudspeaker driver and you want to keep the same Sound Pressure Level (or output level) as you change the frequency, then if you go down one octave, you need to increase the excursion of the driver 4 times.

One of the people attending the presentation asked “why?” which is a really good question, and as I was answering it, I realised that it could be that many people don’t know this.

Let’s take this step-by-step and keep things simple. We’ll assume for this posting that a loudspeaker driver is a circular piston that moves in and out of a sealed cabinet. It is perfectly flat, and we’ll pretend that it really acts like a piston (so there’s no rubber or foam surround that’s stretching back and forth to make us argue about changes in the diameter of the circle). Also, we’ll assume that the face of the loudspeaker cabinet is infinite to get rid of diffraction. Finally, we’ll say that the space in front of the driver is infinite and has no reflective surfaces in it, so the waveform just radiates from the front of the driver outwards forever. Simple!

Then, we’ll push and pull the loudspeaker driver in and out using electrical current from a power amplifier that is connected to a sine wave generator. So, the driver moves in and out of the “box” with a sinusoidal motion. This can be graphed like this:

Figure 1: The excursion of a loudspeaker driver playing a 1 kHz sine wave at some output level.

As you can see there, we have one cycle per millisecond, therefore 1000 cycles per second (or 1 kHz), and the driver has a peak excursion of 1 mm. It moves to a maximum of 1 mm out of the box, and 1 mm into the box.

Consider the wave at Time = 0. The driver is passing the 0 mm line, going as fast as it can moving outwards until it gets to 1 mm (at Time = 0.25 ms) by which time it has slowed down and stopped, and then starts moving back in towards the box.

So, the velocity of the driver is the slope of the line in Figure 1, as shown in Figure 2.

Figure 2: The excursion and velocity of the same loudspeaker driver playing the same signal.

As the loudspeaker driver moves in and out of the box, it’s pushing and pulling the air molecules in front of it. Since we’ve over-simplified our system, we can think of the air molecules that are getting pushed and pulled as the cylinder of air that is outlined by the face of the moving piston. In other words, its a “can” of air with the same diameter as the loudspeaker driver, and the same height as the peak-to-peak excursion of the driver (in this case, 2 mm, since it moves 1 mm inwards and 1 mm outwards).

However, sound pressure (which is how loud sounds are) is a measurement of how much the air molecules are compressed and decompressed by the movement of the driver. This is proportional to the acceleration of the driver (neither the velocity nor the excursion, directly…). Luckily, however, we can calculate the driver’s acceleration from the velocity curve. If you look at the bottom plot in Figure 2, you can see that, leading up to Time = 0, the velocity has increased to a maximum (so the acceleration was positive). At Time = 0, the velocity is starting to drop (because the excursion is on its was up to where it will stop at maximum excursion at time = 0.25 ms).

In other words, the acceleration is the slope of the velocity curve, the line in the bottom plot in Figure 2. If we plot this, it looks like Figure 3.

Figure 3: The excursion, velocity and acceleration of the same loudspeaker driver playing the same signal.

Now we have something useful. Since the bottom plot in Figure 3 shows us the acceleration of the driver, then it can be used to compare to a different frequency. For example, if we get the same driver to play a signal that has half of the frequency, and the same excursion, what happens?

Figure 4: Comparing the excursion, velocity and acceleration of the same loudspeaker driver playing two different signals with the same excursion. (The red line is the same in Figure 4 as in Figure 3.)

In Figure 4, two sine waves are shown: the black line is 1/2 of the frequency of the red line, but they both have the same excursion. If you take a look at where both lines cross the Time = 0 point, then you can see that the slopes are different: the red line is steeper than the black. This is why the peak of the red line in the velocity curve is higher, since this is the same thing. Since the maximum slope of the red line in the middle plot is higher than the maximum slope of the black line, then its acceleration must be higher, which is what we see in the bottom plot.

Since the sound pressure level is proportional to the acceleration of the driver, then we can see in the top and bottom plots in Figure 4 that, if we halve the frequency (go down one octave) but maintain the same excursion, then the acceleration drops to 25% of the previous amount, and therefore, so does the sound pressure level (20*log10(0.25) = -12 dB, which is another way to express the drop in level…)

This raises the question: “how much do we have to increase the excursion to maintain the acceleration (and therefore the sound pressure level)?” The answer is in the “25%” in the previous paragraph. Since maintaining the same excursion and multiplying the frequency by 0.5 resulted in multiplying the acceleration by 0.25, we’ll have to increase the excursion by 4 to maintain the same acceleration.

Figure 5: Comparing the excursion, velocity and acceleration of the same loudspeaker driver playing two different signals at two different excursions. Notice that some of the vertical scales in the plots have changed. (The red line is the same in Figure 5 as in Figures 4 and 3.)

Looking at Figure 5: The black line is 1/2 the frequency of the red line. Their accelerations (the bottom plots) have the same peak values (which means that they produce the same sound pressure level). This, means that the slopes of their velocities are the same at their maxima, which, in turn, means that the peak velocity of the black line (the lower frequency) is higher. Since the peak velocity of the black line is higher (by a factor of 2) then the slope of the excursion plot is also twice as steep, which means that the peak of the excursion of the black line is 4x that of the red line. All of that is explained again in Figure 6.

Figure 6. A repeat of Figure 5 with some explanations that (hopefully) help.

Therefore, assuming that we’re using the same loudspeaker driver, we have to increase the excursion by a factor of 4 when we drop the frequency by a factor of 2, in order to maintain a constant sound pressure level.

However, we can play a little trick… what we’re really doing here is increasing the volume of our “cylinder” of air by a factor of 4. Since we don’t change the size of the driver, we have to move it 4 times farther.

However, the volume of a cylinder is

π r2 * height

and we’re just playing with the “height” in that equation. A different way would be to use a different driver with a bigger surface area to play the lower frequency. For example, if we multiply the radius of the driver by 2, and we don’t change the excursion (the “height” of the cylinder) then the total volume increases by a factor of 4 (because the radius is squared in the equation, and 2*2 = 4).

Another way to think of this: if our loudspeaker driver was a square instead of a circle, we could either move it in and out 4 times farther OR we would make the width and the length of the square each twice as big to get the a cube with the same volume. That “r2” in the equation above is basically just the “width * length” of a circle…

This is why woofers are bigger than tweeters. In a hypothetical world, a tweeter can play the same low frequencies as a woofer – but it would have to move REALLY far in and out to do it.

Tracking force and pickup compliance

It should not come as a surprise that, when we talk about how a vinyl record works, we can start by looking at the movement of the needle in the groove. If we simplify that connection a little (by reducing the audio signal to one channel, but we’ll come back to that point later), then we can think of this as a needle, sitting on a surface. The needle is at the end of an arm that we call the “cantilever” (because it is fixed on one end and it can move up and down on the other end where the needle is attached) and that cantilever is attached somehow to the tonearm using a springy material of some kind (like rubber, for example).

Figure 1

The simple diagram above shows that arrangement. Of course, I’ve left out a bunch of things, and nothing is to scale, but those details are not important right now.

I’ll make the “spring” in this diagram out of flexible rubber that has some “springiness” or “compliance”. The more compliant the spring, the easier it is to flex. So a stiff spring in not very compliant. (This concept is very important to understand as we go on.)

The audio signal is “encoded” into the surface of the vinyl using bumps and dips that cause the needle to move up and down. I’ve shown this in the simple diagram below.

Figure 2

Notice in that diagram that the needle is in contact with the surface of the vinyl, but the part of the system that connects back to the tonearm has not lifted. This is because the connection between the cantilever and the tonearm assembly is compliant enough to let the cantilever move upwards (or downwards) without moving the rest of the system.

Think of this like driving over a very small bump in the road in your car. The compliance of the tires and the shock absorbers will result in the tire riding over the bump, but the car doesn’t jump as a result.

Remember that the bump in the surface of the vinyl is only passing by, so the needle isn’t raised for long. As a result part of the reason the tonearm doesn’t move upwards (and your car doesn’t jump) is partly because it’s heavy. Its mass results in an inertia that “wants” to stop it from moving up and down. (The other factor that’s involved here is an adjustment in the tonearm called the “tracking force” which is a measurement of how much the tonearm is pushing downwards on the needle.)

Consequently, when that bump comes along, the needle rides on top of it, and the force that is pushing it downwards comes mostly from the “spring” at the other end of the cantilever, as shown below.

Figure 3

If the spring had no compliance (in other words, if it weren’t a spring, and the cantilever were just connected directly to the tonearm) and if the cantilever and needle were strong enough to take the force, then the entire tonearm assembly would jump up and down instead, as shown below. (Imagine riding in a horse-drawn buggy with wooden wheels with steel rims, and no springs on the axles. You’d feel every single rock on the road…)

Figure 4

The tonearm is resting on two points: one is the tip of the needle and the other is at the other end at the pivot point where it also rotates horizontally as you play the album. If we were really dumb turntable designers, then half of the mass of the tonearm would be resting on the needle (and the other half would be resting on the pivot). This would be bad, since your records would wear out very fast. So, a tonearm has some kind of adjustment on it that reduces the amount of weight on the needle. The simplest way to do this is to put a counterweight on the opposite side of the pivot so it’s more like a see-saw at the playground. As you move the counterweight away from the pickup, the downwards force at the needle gets smaller. In fact, you can probably adjust the counterweight so far that the needle-end of the tonearm is lighter, and it is stuck up in the air…

We adjust the amount of downwards force at the needle (called the “tracking force”) to result in a value that is in balance with the compliance of the connection to the cantilever. If the tracking force is too high (or the compliance is too high for the tracking force) then the tonearm will sink like I’ve shown below.

Figure 5

There are lots of things wrong with this. The first is that the needle isn’t at the correct angle to the surface of the vinyl, so it’s not going to move correctly. The second is that the cantilever is at the wrong angle, so it’s not going to move upwards with the same behaviour as it moves downwards, which results in an asymmetrical distortion of the signal. But possibly the most obvious problem is that there’s just too much downwards pressure on the vinyl, so your records will wear out faster.

So, there is a balance between the tracking force and the compliance. That balance ensures that you always have contact between the tip of the needle and the surface of the vinyl as the bumps and dips go by.

Digging into the details

One of the things I do regularly is to measure the magnitude response of a turntable from the surface of the vinyl to the electrical output of the RIAA preamplifier. In order to do this, I play two tracks on a special test record (Brüel & Kjær QR 2010) which has the following audio signals:

  • Track 1
  • 2 seconds of 1 kHz sinusoidal wave, L&R channels (3.16 cm/sec lateral velocity)
  • 20 Hz to 45 kHz sinusoidal tone, log sweep, 5 sec per decade, Left channel
  • Track 2
  • 2 seconds of 1 kHz sinusoidal wave, L&R channels (3.16 cm/sec lateral velocity)
  • 20 Hz to 45 kHz sinusoidal tone, log sweep, 5 sec per decade, Right channel

Sometimes (but very rarely), I notice that the needle will skip (or jump) at the transition between the 1 kHz tone and the start of the sine sweep. If this happens, for track 1, the needle will skip forwards into the sweep.

When this happened the first time I thought “Ah hah! The tracking force isn’t high enough, so the needle is being thrown out of the groove. I just need to adjust it.” But after checking the tracking force with my meter (a very small, very precise and accurate scale), I found out that this was not the problem.

Of course, I could make the problem go away by increasing the tracking force, but then it was too high, and my records (and the needle tip) will wear down faster. This would be covering up the symptom, but not correcting the actual problem.

So, what is the problem? It’s that the compliance of the pickup is too low due to an error in the manufacturing process or the fact that it’s just old and the rubber has stiffened over time. In other words it looks more like the system shown in Figure 4, above.

Let’s take a system where the pickup compliance is too low (so the spring is too stiff), so the tonearm can be tossed up off the vinyl surface. We then combine that with the knowledge of how the needle sits in the groove on the vinyl and which channel is on which side of that groove (which I’ve shown below in Figure 6).

Figure 6

Now we can see that, if there’s a bump in the Left channel, it will push the needle on a 45º angle upwards, and if the tracking force and compliance aren’t working together as they should, then the entire tonearm can be pushed hard enough to cause the needle to lift off the surface of the vinyl, heading in towards the centre of the record (towards the left in Figure 6).

What does the signal actually look like?

Let’s go back and look at a recording of that transition between the 1 kHz tone and the start of the 20 Hz sweep, using a pickup that is behaving properly.

Figure 7

The figure above is a screenshot from Audacity that shows the “raw” signal that I recorded at the input of my sound card which is connected to the output of the RIAA preamplifier. I’ve zoomed in to the moment when the track transitions from the 1 kHz tone to the 20 Hz tone at the start of the sweep.

Let’s now use this to go backwards and try to figure out what the surface of the vinyl looks like. I’ll start by re-creating a “perfect” version of that signal in Matlab by joining a 1 kHz cosine wave to a 20 Hz cosine wave.

Figure 8

You might notice that I’ve changed the value a little. I’m simulating one channel of a tone that has a level of at 5 cm/sec, RMS lateral velocity for two channels, instead of the 3.16 cm/sec from the B&K record. But this doesn’t really matter too much – I’ve just done it to make the numbers look nice and be a little easier to talk about.

I’m simulating a system that has a total gain set so that a modulation velocity of 3.54 cm/sec in one channel will produce 354 mV RMS (500 mV peak) at the output of the RIAA at 1 kHz.

Since the lateral velocity of a two-channel tone is 5 cm/sec, then the velocity of one channel will be 1/sqrt(2) of that value because the groove wall is 45º away from the lateral axis and cos(45º) = 1/sqrt(2).

If we take the signal in Figure 8 and filter it with a RIAA pre-emphasis filter (sometime called an “anti-RIAA” or an “inverse RIAA”) and drop the level by 40 dB (a typical gain for a RIAA preamp), then the signal looks like the plot in Figure 9.

Figure 9

As you can see there, the signal much lower in level overall (because of the -40 dB gain) and the 20 Hz tone is much lower in level than the 1 kHz tone (because of the pre-emphasis filter).

The output of the pickup is a current that is proportional to the velocity of the needle. So, we can move farther backwards in the chain and plot the velocity of the needle over time, shown in Figure 10. As you can see, the shape of this plot looks identical to the one in Figure 9. This is because I’m assuming that the current output of the pickup is in phase with the voltage at the input of the RIAA. (This is a safe assumption for the two frequencies we’re looking at here. If you want to pick a fight with me about this, drop by and do it in person. But you’re buying the beer…)

Figure 10 (I made a mistake in the Y-axis label – it should say cm/sec. I’ll come back and fix that later)

Now comes a jump… the velocity of the needle can be calculated by finding the derivative of the displacement over time, which means that the displacement can be found by integrating the velocity.

If you don’t like calculus, then you can think of it this way: In the old days, if you drove from Struer to Copenhagen, you had to take a ferry to get from the island of Fyn to the island of Zealand. Every once in a while, there would be a policeperson, walking around the parking lot as people waited to board the ferry, handing out speeding tickets to some of the people there. What happened was that the licence plates were recorded with time stamps as they crossed the bridge to Fyn from Jutland – which is about 75 km away from the parking lot. If you arrive at the ferry too early, you must have been speeding, and you get rewarded with an earlier ferry, and an extra charge…

In other words, you can calculate your speed (velocity) by your change (difference) of distance (displacement) over time.

You can also do this backwards: if you know how fast you’re going, you can calculate your displacement over time (you’ll be 100 km away in an hour if you’re driving 100 km/h the whole time, for example). If your velocity changes over time (say you drive a different speed every hour for 10 hours), then you can still calculate your displacement by dividing time into slices (in this case, 1 hour per “slice”) and adding up the individual displacements for the velocity you had during each slice of time. If you divide time into infinitely short slices, then you are integrating instead of adding, but the process is essentially the same.

Back to the story: if we take the signal in Figure 10 and integrate it (and scale it – which isn’t really important for this discussion), we get the curve in Figure 11.

Figure 11

This gives us a good idea of the actual shape of the left wall of the groove in the vinyl for that particular signal.

So, as you can see there, if the connection between the cantilever and the pickup doesn’t have a high enough compliance, it’s no wonder that the needle gets thrown out of the record groove. That’s a heck of a bump to deal with! To be honest, it’s also a little amazing to me that the needle that’s behaving (like the one that produced the output shown in Figure 7) can actually put up with that kind of abuse.

(Special thanks to Jakob Dyreby for helping me to wrap my head around the simulation part of this posting. I did the math, but only after he pointed me in the right direction.)

Post script

Every once in a while, someone will send me a link to a YouTube page that shows an electron microscope “video” of a needle tracking a groove in a vinyl record. If you listen to the explanation of that video, he explains that it’s not really a video. It’s a series of photographs that he took, one by one, and then assembled into a video.

This means that, in that video, the needle isn’t really behaving like it does in real life when the vinyl is moving underneath it.

Imagine setting up a video camera on the side of the road, next to a small speed bump, and making a video of a car driving over it. You’d see that, as the car drives by, the wheels move up into the wheel wells and the car doesn’t get pushed upwards as much, since some of the vertical movement caused by the speed bump is “taken up” by the car’s springs and shock absorbers.

If, instead, you set up a camera, and got the car to move forwards 5 cm and stop – and you take a photo, then the car moves forwards another 5 cm and stops – and you take another photo, and the you repeat this until the car is out of the frame – and then you assemble all of those photos into a video, it would look very different. The car would not remain horizontal when the wheels are on the speed bump because the springs and shock absorbers wouldn’t be compressed at all.

That video is like the second “video” of the car. Of course, it’s still interesting, and it’s well-explained, so no one is playing any tricks on you. But it’s not a video of what actually happens…

Vinyl info and calculators

This is just a collection of information about turntables and vinyl for anyone wanting to dig deeper into It (which might mean that it’s just for me…). I’ll keep adding to this (and completing it) as time goes by.

Glossary

Cantilever

  • The rod or arm that connects the stylus on one end to the “motor” on the other.

Effective Length

  • The straight-line distance between the pivot point of the tonearm and the top of the stylus

Equivalent Mass

  • definition to come

Flutter

  • Higher-frequency modulation of the audio frequency caused by changes in the groove speed. These may be the result of changes in problems such as unstable motor speed, variable compliance on a belt, issues with a spindle bearing, drive wheel eccentricity, and other issues.
  • Flutter describes a modulation in the groove speed ranging from 6 to 100 times a second (6 Hz to 100 Hz).

Frequency Drift

  • Very long-term (or low-frequency) changes in the audio frequency, typically caused by slow changes in the platter rotation speed.
  • Typically, changes with a modulation frequency of less than 0.5 Hz (a period of no less than 2 seconds) are considered to be frequency drift. Faster changes are labelled “Wow”

Groove

  • The v-shaped track pressed into the surface of the vinyl record, in which the stylus sits

Linear Tracking

  • A tonearm that moves linearly, following a path that is parallel to the radius line traced by the stylus. This (in theory) ensures that the tracking error is always 0º, however, in practice this error is merely small.

Modulation Width

  • The distance measured on a line through the spindle from the start of the modulated groove to the end of the modulated groove. This is approximately 3″ or 76 mm.

Mounting Distance

  • The distance between the spindle and the pivot point of the tonearm.

Needle

  • Also known as the stylus. The point that is placed in the groove of the vinyl record.
    Some persons distinguish between the “stylus” (to indicate the chisel on the mastering lathe that creates the groove in the master record), and the “needle” (to indicate the portion of the pickup on a turntable that plays the signal).

Null Radius

  • The radius (distance between the spindle and the stylus) where the tracking error is 0º. A typically-designed and correctly installed radial tracking tonearm has two null radii (see this posting).

Offset Angle

  • The angle between the axis of the stylus and a line drawn between the tonearm pivot and the stylus. See the line diagram below.

Overhang

  • The difference between the Effective Length of the tonearm and the Mounting Distance. This value is used in some equations for calculating the Tracking Error.

Pickup, Electromagnetic

  • Includes three general types: Moving Coil, Moving Magnet, and Variable Reluctance (aka Moving Iron). These produce an output proportional to the velocity of the stylus movement.

Pickup, Piezoelectic

  • Produces an output proportional to the displacement of the stylus.

Pitch

  • The density of the groove count per distance in lines per inch or lines per mm. The pitch can vary from disc to disc, or even within a single track, according to the requirements of the mastering.

Radial Tracking

  • A tonearm that rotates on a pivot point with the stylus tracing a circular path around that pivot.

Radius

  • The distance between the centre of the vinyl disc and the pickup stylus.

RIAA

  • A pre-emphasis / de-emphasis filter designed to fill two functions.
    • The first is a high-frequency attenuation de-emphasis that reduces the playback system’s sensitivity to surface noise. This requires a reciprocal high-frequency pre-emphasis boost.
    • The second is a low-frequency attenuation pre-emphasis that maintains a constant modulation amplitude at lower frequencies to avoid over-excursion of the playback stylus. This requires a reciprocal low-frequency de-emphasis boost.
  • The first of the two plots below, show the theoretical (black lines) and typical (red) response of the pre-emphasis filter. The second of the two plots shows the de-emphasis filter response.

Side Thrust

  • definition to come

Skating Force

  • definition to come

Spindle

  • The centre of the platter around which the record rotates

Stylus

  • Also known as the needle. The point that is placed in the groove of the vinyl record.
    Some persons distinguish between the “stylus” (to indicate the chisel on the mastering lathe that creates the groove in the master record), and the “needle” (to indicate the portion of the pickup on a turntable that plays the signal).

Stylus, Bonded vs. Nude

  • Although the tip of the stylus is typically made of diamond today, in lower-cost units, that diamond tip is mounted or bonded to a metal pin (typically steel, aluminium, or titanium) which is, in turn, connected to the cantilever (the long “arm” that connects back to the cartridge housing). This bonded design is cheaper to manufacture, but it results in a high mass at the stylus tip, which means that it will not move easily at high frequencies.
  • In order to reduce mass, the metal pin is eliminated, and the entire stylus is made of diamond instead. This makes things more costly, but reduces the mass dramatically, so it is preferred if the goal is higher sound performance. This design is known as a nude stylus.

Tracking Error

  • The angle between the tangent to the groove and the alignment of the stylus. In a perfect system, the stylus would align with the tangent to the groove at all radii (distances from the spindle), since this matches the angular rotation of the cutting head when the master was made on a lathe. A linear tracking arm minimises this error. A radial tracking arm can be designed to have two radii with no tracking error (each called a “Null Radius”) but will have some measurable tracking error at all other locations on the disk.
  • One side-effect of tracking error is distortion of the audio signal, typically calculated and expressed as a 2nd-harmonic distortion on a sinusoidal audio signal. However, higher order distortion and intermodulation artefacts also exist.

Warp Wow

  • A modulation of the frequency of the audio signal caused by vertical changes in the vinyl surface (a warped record). This typically happens at a lower frequency, which is why it is “warp wow” and not “warp flutter”.

Wow

  • Low-frequency modulation of the audio frequency caused by changes in the groove speed. These may be the result of changes in problems such as rotation speed of the platter, discs with an incorrectly-placed centre hole, or vertical changes in the surface of the vinyl, and other issues.
  • Wow is a modulation in the groove speed ranging from once every 2 seconds to 6 times a second (0.5 Hz to 6 Hz). Note that, for a turntable, the rotational speed of the disc is within this range. (At 33 1/3 RPM: 1 revolution every 1.8 seconds is equal to approximately 0.556 Hz.)

Disk size limits

Outside starting diameter

  • 7″ discs
    • 6.78″, +0.06″, -0.00″
    • 172.2 mm, +1.524 mm, – 0.0 mm
  • 10″ discs
    • 9.72″, +0.06″, -0.00″
    • 246.9 mm, +1.524 mm, – 0.0 mm
  • 12″ discs
    • 11.72″, +0.06″, -0.00″
    • 297.7 mm, +1.524 mm, – 0.0 mm

Start of modulated pitch diameter

  • 7″ discs
    • 6.63″, +0.00″, -0.03″
    • 168.4 mm, +0.0 mm, – 0.762 mm
  • 10″ discs
    • 9.50″, +0.00″, -0.03″
    • 241.3 mm, +0.0 mm, – 0.762 mm
  • 12″ discs
    • 11.50″, +0.00″, -0.03″
    • 292.1 mm, +0.0 mm, – 0.762 mm

Minimum inside diameter

  • 7″ discs
    • 4.25″
    • 107.95 mm
  • 10″ discs
    • 4.75″
    • 120.65 mm
  • 12″ discs
    • 4.75″
    • 120.65 mm

Lockout Groove diameter

  • 7″ discs
    • 3.88″, +0.00, -0.08
    • 98.552 mm, +0.0 mm, -2.032 mm
  • 10″ discs
    • 4.19″, +0.00, -0.08
    • 106.426 mm, +0.0 mm, -0.762 mm
  • 12″ discs
    • 4.19″, +0.00, -0.08
    • 106.426 mm, +0.0 mm, -0.762 mm

Unmodulated (silent) groove width

  • 2 mil minimum, 4 mil maximum
  • 0.0508 mm minimum, 0.1016 mm maximum

Modulated groove depth

  • 1 mil minimum, 5 mil maximum
  • 0.0254 mm minimum, 0.127 mm maximum
  • The figure below shows the typical, minimum, and maximum groove depths, drawn to scale (with a 13 µm spherical stylus)

Signal levels

  • A typical standard reference level is a velocity of 35.4 mm/sec on one channel.
  • This means that a monophonic signal (identical signal in both channels) with that modulation will have a lateral (side-to-side) velocity of 50 mm/sec.

Calculators and Measurements

Conversion

  • 1 mil = 1 “thou” = 1/1000 inch
  • Lengthmm = Lengthmil * 127/5000

Revolutions per Second

  • RevolutionsPerSecond = RevolutionsPerMinute / 60
  • e.g.
    • 0.556 Rev/Sec @ 33 1/3 RPM
    • 0.75 Rev/Sec @ 45 RPM
    • 1.3 Rev/Sec @ 78 RPM

Seconds per revolution

  • SecondsPerRevolution = 60 / RevolutionsPerMinute
  • e.g.
    • 1.8 Sec/Rev @ 33 1/3 RPM
    • 1.333 Sec/Rev @ 45 RPM
    • 0.769 Sec/Rev @ 78 RPM

Modulation Width

  • MaximumModulationWidth = (StartOfModulatedPitch – MinimumInsideDiameter) / 2
  • e.g. for a 12″ disc
    • (292.1 mm – 120.65 mm) / 2 = 85.725 mm
    • Typically approximately 3″ = 76 mm

Pitch (assuming constant pitch)

  • (RunningTime * RPM) / ModulationWidth
  • e.g.
    • (20 minutes * 33.333 RPM) / 76 mm = 8.77 lines per mm
    • (20 minutes * 33.333 RPM) / 3″ = 222.22 lines per inch

Groove Width

  • GrooveWidthInMil = (1000 / PitchInLinesPerInch + 1) / 2
  • e.g.
    • (1000 / 222 LPI + 1) / 2 = 2.75 mil = 2.75 x 10-3 inches = 0.07 mm

Angular Frequency (of the audio)

  • abbreviated ω (unit: radians per second)
  • ω = 2 * π * FrequencyHz

Angular Speed of Rotation (of of the disk)

  • commonly abbreviated ωr (unit: radians per second)
  • ωr = 2 * π * RevolutionsPerSecond

Displacement Amplitude

  • DisplacementAmplitudePeak = Velocitypeak / ω
  • e.g.
    • 50 mm/sec / (2 * pi * 1000 Hz) = 0.008 mm (peak)

Groove Speed

  • 2 * π * Radius * RevolutionsPerSecond
  • e.g.
    • at the Start of modulated pitch on a 12″ disk turning at 33 1/3 RPM
      • 2 * π * (292.1 mm / 2) * (33.333 / 60) = 509.8 mm/sec
    • at the Minimum inside diameter on a 12″ disk turning at 33 1/3 RPM
      • 2 * π * (120.65 mm / 2) * (33.333 / 60) = 210.6 mm/sec
  • The plot below shows the groove speeds of 12″ 33 1/3 RPM and 7″ 45 RPM for all possible radii.

Wavelength

  • GrooveSpeed / Frequency
  • e.g.
    • 20 Hz at the Start of modulated pitch on a 12″ disk turning at 33 1/3 RPM
      • 509.8 / 20 = 25.5 mm
    • 20 kHz at the Start of modulated pitch on a 12″ disk turning at 33 1/3 RPM
      • 509.8 / 20000 = 0.0255 mm
    • The plot below shows wavelengths of 4 different frequencies for 12″ 33 1/3 RPM records (the longer curves) and 7″ 45 RPM records (the shorter curves)

Tracking Error

  • Tracking Error =
    OffsetAngle – asin ((EffectiveLength2 + Radius2 – MountingDistance2) / (2 * EffectiveLength * Radius))
  • see this posting for an explanation and example

Distortion caused by tracking error

  • Equation is for calculating percentage of second-harmonic distortion of a laterally-modulated monophonic sinusoidal audio signal
  • DistortionPercent = 100 * (PeakVelocity * tan(TrackingError)) / (GrooveSpeed)
  • see this posting for an explanation and examples

Wow and Flutter

  • Typically measured with a 3150 Hz sinusoidal tone, played from the vinyl surface
  • This signal is then de-modulated to determine its change over time. That modulation is then filtered through the response shown below which approximates human sensitivity to frequency modulation of an audio signal. More detailed information is given below
  • The AES6-2008 standard, which is the currently accepted method of measuring and expressing the wow and flutter specification, uses a “2σ” or “2-Sigma” method, which is a way of looking at the peak deviation to give a kind of “worst-case” scenario. In this method, the tone is played from a disc and captured for as long a time as is possible (or feasible). Firstly, the average value of the actual frequency of the output is found (in theory, it’s fixed at 3,150 Hz, but this is never true). Next, the short-term variation of the actual frequency over time is compared to the average, and weighted using the filter shown above. The result shows the instantaneous frequency variations over the length of the captured signal, relative to the average frequency (however, the effect of very slow and very fast changes have been reduced by the filter). Finally, the standard deviation of the variation from the average is calculated, and multiplied by 2 (hence “2-Sigma”, or “two times the standard deviation”), resulting in the value that is shown as the specification. The reason two standard deviations is chosen is that (in the typical case where the deviation has a Gaussian distribution) the actual Wow & Flutter value should exceed this value no more than 5% of the time.

References

All of these are available online. Some of them require you to purchase them (or be a member of an organisation).

  • “Tracking Angle in Phonograph Pickups”
    B. B. Bauer. Electronics magazine, March 1945
  • “Minimising Pickup Tracking Error”
    Dr. John D. Seagrave, Audiocraft Magazine, December 1956, January 1957, and August 1957
  • “Understanding Phono Cartridges”
    S.K. Pramanik, Audio magazine, March 1979
  • “Tonearm Geometry and Setup Demystified”
    Martin D. Kessler and B.V.Pisha, Audio magazine, January 1980
  • “Understanding Tonearms”
    S.K. Pramanik, Audio magazine, June 1980
  • “Analytic Treatment of Tracking Error and Notes on Optimal Pick-up Design”
    H.G.Baerwald, Journal of the Society of Motion Picture Engineers, December 1941
  • “Pickup Arm Design”
    J.K. Stevenson, Wireless World magazine, May 1966, and June 1966
  • “The Optimum Pivot Position on Tonearm”
    S. Takahashi et. al., Audio Engineering Society Preprint no. 1390 (61st Convention, November 1978)
  • “Audible Effects of Mechanical Resonances in Turntables”
    Brüel and Kjær Application Note (1977)
  • “Basic Disc Mastering”; “
    Larry Boden (1981)
  • “Cartridge / Arm / Turntable Followup: Loose Ends and New Developments”
    The Audio Critic, 1:43 (Spring/Fall, 1978)
  • “Have Tone Arm Designers Forgotten Their High-School Geometry?”
    The Audio Critic, 1:31 (Jan./Feb. 1977).
  • “How the Stereo Disc Works”
    Radio-Electronics, (July 1958)
  • “Manual of Analogue Sound Restoration Techniques”
    Peter Copeland (2008)
  • “On the Mechanics of Tonearms”
    Dick Pierce (2005)
  • “Reproduction of Sound in High-Fidelity and Stereo Phonographs”
    Edgar Villchur (1966)
  • Journal of the Audio Engineering Society (www.aes.org)
    • “Centennial Issue: The Phonograph and Sound Recording After One-Hundred Years”
      Vol. 25, No. 10/11 (Oct./Nov. 1977)
    • “Factors Affecting the Stylus / Groove Relationship in Phonograph Playback Systems”
      C.R. Bastiaans; Vol. 15 Issue 4 (Oct. 1967)
    • “Further Thoughts on Geometric Conditions in the Cutting and Playing of Stereo Disk”
      C.R. Bastiaans; Vol. 11 Issue 1 (Jan. 1963)
    • “Record Changers, Turntables, and Tone Arms-A Brief Technical History”
      James H. Kogen; Vol. 25 (Oct./Nov. 1977)
    • “Some Thoughts on Geometric Conditions in the Cutting and Playing of Stereodiscs and Their Influence on the Final Sound Picture”
      Ooms, Johan L., Bastiaans, C. R.; Vol. 7 Issue 3 (Jul. 1959)
    • “The High-Fidelity Phonograph Transducer”
      B.B. Bauer; Vol. 25 Issue 10/11 (Nov. 1977)
  • DIN Standards
    • 45 500: Hi-Fi Technics: Requirements for Disk Recording Reproducing Equipment
    • 45 507: Measuring Apparatus for Frequency Variations in Sound Recording Equipment
    • 45 538: Definitions for Disk Record Reproducing Equipment
    • 45 539: Disk Record Reproducing Equipment: Directives for Measurements, Markings, and Audio Frequency, Connections, Dimensions of Interchangeable Pickups, Requirements of Playback Amplifiers
    • 45 541: Frequency Test Record St 33 and M 33 (33 1/3 rev/min; Stereo and Mono)
    • 45 542: Distortion Test Record St 33 and St 45 (33 1/3 or 45 rev/min; Stereo)
    • 45 543: Frequency Response and Crosstalk Test Record
    • 45 544: Rumble Measurement Test Record St 33 and M 33 (33 1/3 rev/min; Stereo and Mono)
    • 45 545: Wow and Flutter Test Records, 33 1/3 and 45 rev/min
    • 45 546: Stereophonic Disk Record St 45 (45 rpm)
    • 45 547: Stereophonic Disk Record St 33 (33 1/3 rpm)
    • 45 548 Aptitude for Performance of Disk Record Reproducing Equipment
    • 45 549: Tracking Ability Test Record
  • IEC Publications
    • 98: Recommendations for Lateral-Cut Commercial and Transcription Disk Recordings
    • 98: Processed Disk Records and Reproducing Equipment
    • 386: Method of Measurement of Speed Fluctuations in Sound Recording and Reproducing Equipment

Tonearm tracking error and distortion

In the last posting, I reviewed the math for calculating the tracking error for a radial tonearm. The question associated with this is “who cares?”

In the March, 1945 issue of Electronics Magazine, Benjamin Bauer supplied the answer. An error in the tracking angle results in a distortion of the audio signal. (This was also discussed in a 3-part article by Dr. John D. Seagrave in Audiocraft Magazine in December 1956, January 1957, and August 1957)

If the signal is a sine wave, then the distortion is almost entirely 2nd-order (meaning that you get the sine wave fundamental, plus one octave above it). If the signal is not a sine wave, then things are more complicated, so I will not discuss this.

Let’s take a quick look at how the signal is distorted. An example of this is shown below.

In that plot, you can see that the actual output from the stylus with a tracking error (the black curve) precedes the theoretical output that’s actually on the vinyl surface (the red curve) when the signal is positive, and lags when it’s negative. An intuitive way of thinking of this to consider the tracking error as an angular rotation, so the stylus “reads” the signal in the groove at the wrong place. This is shown below, which is merely zooming in on the figure above.

Here, you can see that the rotation (tracking error) of the stylus is getting its output from the wrong place in the groove and therefore has the wrong output at any given moment. However, the amount by which it’s wrong is dependent not only on the tracking error but the amplitude of the signal. When the signal is at 0, then the error is also 0. This is not only the reason why the distortion creates a harmonic of the sine wave, but it also explains why (as we’ll see below) the level of distortion is dependent on the level of the signal.

This intuitive explanation is helpful, but life is unfortunately, more complicated. This is because (as we saw in the previous posting), the tracking error is not constant; it changes according to where the stylus is on the surface of the vinyl.

If you dig into Bauer’s article, you’ll find a bunch of equations to help you calculate how bad things get. There are some minor hurdles to overcome, however. Since he was writing in the USA in 1945, his reference was 78 RPM records and his examples are all in inches. However, if you spend some time, you can convert this to something more useful. Or, you could just trust me and use the information below.

In the case of a sinusoidal signal the level of the 2nd harmonic distortion (in percent) can be calculated with the following equation:

PercentDistortion = 100 * (ω Αpeak α) / (ωr r)

where

  • ω is 2 * pi * the audio frequency in Hz
  • Apeak is the peak amplitude of the modulation (the “height” of the groove) in mm
  • α is the tracking error in radians
  • ωr is the rotational speed of the record in radians per second, calculated using 2 * pi * (RPM / 60)
  • r is the radius of the groove; the distance from the centre spindle to the stylus in mm

Let’s invent a case where you have a constant tracking error of 1º, with a rotational speed of 33 1/3 RPM, and a frequency of 1 kHz. Even though the tracking error remains constant, the signal’s distortion will change as the needle moves across the surface of the record because the wavelength of the signal on the vinyl surface changes (the rotational speed is the same, but the circumference is bigger at the outside edge of the record than the inside edge). The amount of error increases as the wavelength gets smaller, so the distortion is worse as you get closer to the centre of the record. This can be see in the shapes of the curves in the plot below. (Remember that, as you play the record, the needle is moving from right to left in those plots.)

You can also see in those plots that the percentage of distortion changes significantly with the amplitude of the signal. In this case, I’ve calculated using three different modulation velocities. The middle plot is 35.4 mm / sec, which is a typical accepted standard reference level, which we’ll call 0 dB. The other two plots have modulation velocities of -3 dB (25 mm / sec) and + 3 dB (50 mm / sec).

Sidebar: If you want to calculate the Amplitude of the modulation

Apeak = (ModulationVelocity * sqrt(2)) / (2 * pi * FrequencyInHz)

Note that this simplifies the equation for calculating the distortion somewhat.

Also, if you need to convert radians to degrees, then you can multiply by 180/pi. (about 57.3)

Of course, unless you have a very badly-constructed linear tracking turntable, you will never have a constant tracking error. The tracking error of a radial tonearm is a little more complicated. Using the recommended values for the “well known tonearm” that I used in the last posting:

  • Effective Length (l) : 233.20 mm
  • Mounting Distance (d) : 215.50 mm
  • Offset angle (y) : 23.63º

and assuming that this was done perfectly, we get the following result for a 33 1/3 RPM album.

You can see here that the distortion drops to 0% when the tracking error is 0º, which (in this case) happens at two radii (distances between the centre spindle and the stylus).

If we do exactly the same calculation at 45 RPM, you’ll see that the distortion level drops (because the value of ωr increases), as shown below. (But good luck finding a 12″ 45 RPM record… I only have two in my collection, and one of those is a test record.)

Important notes:

Everything I’ve shown above is not to be used as proof of anything. It’s merely to provide some intuitive understanding of the relationship between radial tracking tonearms, tracking error, and the resulting distortion. There is one additional important reason why all this should be taken with a grain of salt. Remember that the math that I’ve given above is for 78 RPM records in 1945. This means that they were for laterally-modulated monophonic grooves; not modern two-channel stereophonic grooves. This means that the math above isn’t accurate for a modern turntable, since the tracking error will be 45º off-axis to the axis of modulation of the groove wall. This rotation can be built into the math as a modification applied to the variable α, however, I’m not going to complicate things further today…

In addition, the RIAA equalisation curve didn’t get standardised until 1954 (although other pre-emphasis curves were being used in the 1940s). Strictly speaking, the inclusion of a pre-emphasis curve doesn’t really affect the math above, however, in real life, this equalisation makes it a little more complicated to find out what the modulation velocity (and therefore the amplitude) of the signal is, since it adds a frequency-dependent scaling factor on things. On the down-side, RIAA pre-emphasis will increase the modulation velocity of the signal on the vinyl, resulting in an increase in the distortion effects caused by tracking error. On the up-side, the RIAA de-emphasis filtering is applied not only to the fundamentals, but the distortion components as well, so the higher the order of the unwanted harmonics, the more they’ll be attenuated by the RIAA filtering. How much these two effects negate each other could be the subject of a future posting; if I can wrap my own head around the problem…

One extra comment for the truly geeky:

You may be looking at the last two plots above and being confused in the same way that I was when I made them the first time. If you look at the equation, you can see that the PercentDistortion is related to α: the tracking error. However, if you look at the plots, you’ll see that I’ve shown it as being related to | α |: the absolute value of the tracking error instead. This took me a while to deal with, since my first versions of the plots were showing a negative value for the distortion. “How can a negative tracking result in distortion being removed?” I asked myself. The answer is that it doesn’t. When the tracking error is negative, then the angle shown in the second figure above rotates counter-clockwise to the left of the vertical line. In this case, then the output of the stylus lags for positive values and precedes for negative values (opposite to the example I gave above), meaning that the 2nd-order harmonic flips in polarity. SINCE you cannot compare the phase of two sine tones that do not have the same frequency, and SINCE (for these small levels of distortion) it’ll sound the same regardless of the polarity of the 2nd-order harmonic, and SINCE (in real-life) we don’t listen to sine tones so we get higher-order THN and IMD artefacts, not just a frequency doubling, THEN I chose to simplify things and use the absolute value.
Post Script to the comment for geeks: This conclusion was confirmed by J.K. Stevenson’s article called “Pickup Arm Design” in the May, 1966 edition of Wireless World where he states “The sign of φ (positive or negative) is ignored as it has no effect on the distortion.” (He uses φ to denote the tracking error angle.)

Penultimate Post Script:

J.K. Stevenson’s article gives an alternative way of calculating the 2nd order harmonic distortion that gives the same results. However, if you are like me, then you think in modulation velocity instead of amplitude, so it’s easier to not convert on the way through. This version of the equation is

PercentDistortion = 100 * (Vpeak tan(α)) / (μ)

where

  • Vpeak is the peak modulation velocity in mm/sec
  • α is the tracking error in radians
  • μ is the groove speed of the record in mm/sec calculated using 2*pi*(rpm/60)*r
  • r is the radius of the groove; the distance from the centre spindle to the stylus in mm

Final Post-Script:

I’ve given this a lot of thought over the past couple of days and I’m pretty convinced that, since the tracking error is a rotation angle on an axis that is 45º away from the axis of modulation of the stylus (unlike the assumption that we’re dealing with a monophonic laterally-modulated groove in all of the above math), then, to find the distortion for a single channel of a stereophonic groove, you should multiply the results above by cos(45º) or 1/sqrt(2) or 0.707 – whichever you prefer. If you are convinced that this was the wrong thing to do, and you can convince me that you’re right, I’ll be happy to change it to something else.

Tonearm alignment and tracking error

The June 1980 issue of Audio Magazine contains an article written by Subir K. Pramanik called “Understanding Tonearms”. This is a must-read tutorial for anyone who is interested in the design and behaviour of radial tonearms.

One of the things Pram talked about in that article concerned the already well-known relationship between tonearm geometry, its mounting position on the turntable, and the tracking error (the angular difference between the tangent to the groove and the cantilever axis – or the rotation of the stylus with respect to the groove). Since the tracking error is partly responsible for distortion of the audio signal, the goal is to minimise it as much as possible. However, without a linear-tracking system (or an infinitely long tonearm), it’s impossible to have a tracking error of 0º across the entire surface of a vinyl record.

One thing that is mentioned in the article is that “Small errors in the mounting distance from the centre of the platter … can make comparatively large differences in angular error” So I thought that I’d do a little math to find out this relationship.

The article contains the diagram shown below, showing the information required to do the calculations we’re interested in. In a high-end turntable, the Mounting Distance (d) can be varied, since the location of the tonearm’s bearing (the location of the pivot point) is adjustable, as can be seen in the photo above of an SME tonearm on a Micro Seiki turntable.

The tonearm’s Effective Length (l) and Offset Angle (y) are decided by the manufacturer (assuming that the pickup cartridge is mounted correctly). The Minimum and Maximum groove radius are set by international standards (I’ve rounded these to 60 mm and 149 mm respectively). The Radius (r) is the distance from the centre of the LP (the spindle) to the stylus at any given moment when playing the record.

In a perfect world, the tracking error would be 0º at all locations on the record (for all values of r from the Maximum to the Minimum groove radii) which would make the cantilever align with the tangent to the groove. However, since the tonearm rotates around the bearing, the tracking error is actually the angle x (in the diagram above) subtracted from the offset angle. “X” can be calculated using the equation:

x = asin ((l2 + r2 – d2) / (2 l r))

So the tracking error is

Tracking Error = y – asin ((l2 + r2 – d2) / (2 l r))

Just as one example, I used the dimensions of a well-known tonearm as follows:

  • Effective Length (l) : 233.20 mm
  • Mounting Distance (d) : 215.50 mm
  • Offset angle (y) : 23.63º

Then the question is, if I make an error in the Mounting Distance, what is the effect on the Tracking Error? The result is below.

If we take the manufacturer’s recommendation of d = 215.4 mm as the reference, and then look at the change in that Tracking Error by mounting the bearing at the incorrect distance in increments of 0.2 mm, then we get the plot below.

So, as you can see there, a 0.2 mm error in the location of the tonearm bearing (which, in my opinion, is a very small error…) results in a tracking error difference of about 0.2º at the minimum groove radius.

If I increase the error to increments of 1 mm (± 5mm) then we get similar plots, but with correspondingly increased tracking error.

If you go back and take a look at the equation above, you can see that the change in the tracking error is constant with the Offset Angle (unlike its relationship with an error in the location of the tonearm bearing, which results in a tracking error that is NOT constant). This means that if you mount your pickup on the tonearm head shell with a slight error in its angle, then this angular error is added to the tracking error as a constant value, regardless of the location of the stylus on the surface of the vinyl, as shown below.

Phase vs Polarity

I know that language evolves. I know that a dictionary is a record of how we use words; not an arbiter of how words should be used. However, I also believe very firmly that if you don’t use words correctly, then you won’t be saying what you mean, and therefore you can be misconstrued.

One of the more common phrases that you’ll hear audio people use is “out of phase” when they mean “180º out of phase” or possibly even “opposite polarity”. I recently heard someone I work with say “out of phase” and I corrected them and said “you mean ‘opposite polarity'” and so a discussion began around the question of whether “180º out of phase” and “opposite polarity” can possibly result in two different things, or whether they’re interchangeable.

Let’s start by talking about what “phase” is. When you look at a sine wave, you’re essentially looking at a two-dimensional view of a three-dimensional shape. I’ve talked about this a lot in two other postings: this one and this one. However, the short form goes something like “Look at a coil spring from the side and it will look like a sine wave.” A coil is a two-dimensional circle that has been stretched in the third dimension so that when you rotate 360º, you wind up back where you started in the first two dimensions, but not the third. When you look at that coil from the side, the circular rotation (say, in degrees) looks like a change in height.

Figure 1
Figure 2

Notice in the two photos above how the rotation of the circle, when viewed from the side, looks only like a change in height related to the rotation in degrees.

Figure 3

The figure above is a classic representation of a sine wave with a peak amplitude of 1, and as you can see there, it’s essentially the same as the photo of the Slinky. In fact, you get used to seeing sine waves as springs-viewed-from-the-side if you force yourself to think of it that way.

Now let’s look at the same sine wave, but we’ll start at a different place in the rotation.

Figure 4

The figure above shows a sine wave whose rotation has been delayed by some number of degrees (22.5º, to be precisely accurate).

If I delay the start of the sine wave by 180 degrees instead, it looks like Figure 5..

Figure 5

However, if I take the sine wave and multiply each value by -1 (inverting the polarity) then it looks like this:

Figure 6

As you can probably see, the plots in Figure 5 and 6 are identical. Therefore, in the case of a sine wave, shifting the phase of the signal by 180 degrees has the same result at inverting the polarity.

What happens when you have a signal that is the sum of multiple sine waves? Let’s look at a simple example below.

Figure 7

The top plot above shows two sine waves, one with a frequency of three times the other, and with 1/3 the amplitude. If I add these two together, the result is the red curve in the lower plot. There are two ways to think of this addition: You can add each amplitude, degree by degree to get the red curve. You can also think of the slopes adding. At the 180º mark, the two downward-going slopes of the two sine waves cause the steeper slope in the red curve.

If we shift the phase of each of the two sine wave components, then the result looks like the plots below.

Figure 8

As you can see in the plots above, shifting the phases of the sine waves is the same as inverting their polarities, and so the resulting total sum (the red curve) is the same as if we had inverted the polarity of the previous total sum.

So, so far, we can conclude that shifting the phase by 180º gives the same result as inverting the polarity.

In the April, 1946 edition of Wireless World magazine, C.E. Cooper wrote an article called “Phase Relationships: ‘180 Degrees Out of Phase’ or ‘Reversed Polarity’?” (I’m not the first one to have this debate…) In this article, it’s states that there is a difference between “phase” and “polarity” with the example shown below.

Figure 9

There is a problem with the illustration in Figure 9, which is the fact that you cannot say that the middle plot has been shifted in phase by 180 degrees because that waveform doesn’t have a “phase”. If you decomposed it to its constituent sines/cosines and shifted each of those by 180º, then the result would look like (c) instead of (b). Instead, this signal has had a delay of 1/2 of a period applied to it – which is a different thing, since it’s delaying in time instead of shifting in phase.

However, there is a hint here of a correct answer… If we think of the black and blue sine waves in the 2-part plots above as sine waves with frequencies 1 Hz and 3 Hz, we can add another “sine wave” with a frequency of 0 Hz, or DC, as shown in Figure 10, below.

Figure 10

In the plot above, the top plot has a DC component (the blue line) that is added to the sine component (the black curve) resulting in a sine wave with a DC offset (the red curve).

If we invert the polarity of this signal, then the result is as shown in Figure 11.

Figure 11

However, if we delay the components by 180º, the result is different, as shown in Figure 12:

Figure 12

The hint from the 1946 article was the addition of a DC offset to the signal. If we think of that as a sine wave with a frequency of 0 Hz, then it can be “phase-shifted” by 180º which results in the same value instead of inverting polarity.

However, to be fair, most of the time, shifting the phase by 180º gives the same result as inverting the polarity. However, I still don’t like it when people say “flip the phase”…

Variations on the Goldberg Variations

As part of a listening session today, I put together a playlist to compare piano recordings. I decided that an interesting way to do this was to use the same piece of music, recorded by different artists on different instruments in different rooms by different engineers using different microphone and techniques. The only constant was the notes on the page in front of the performer.

A link to the playlist is here: LINK TO TIDAL

Playing through this, it’s interesting to pay attention to things like:

  • Overall level of the recording
    • Notice how much (typically) quieter the Dolby Atmos-encoded recording is than the 2.0 PCM encoded ones. However, there’s a large variation amongst the 2.0 recordings.
  • Monophonic vs. stereo recordings
  • Perceived width of the piano
  • Perceived width of the room
  • How enveloping the room is (this might be different from the perceived width, but these two attributes can be co-related, possibly even correlated)
  • Perceived distance to the piano.
    • On some of the recordings, the piano appears to be close. The attack of each note is quite fast, and there is not much reveberation.
    • On some of the recordings, the piano appears to be distant – more reveberant, with a soft, slow attack on each note.
    • On other recordings, it may appear that the piano is both near (because of the fast attack on each hammer-to-string strike) and far (because of the reverberation). (Probably achieved by using a combination of microphones at different distances – or using digital reverb…)
  • The length of the reverberation time
  • Whether the piano is presented as one instrument or a collection of strings (e.g. can you hear different directions to (or locations of) individual notes?)
  • If the piano is presented as a wide source with separation between bass and treble, is the presentation from the pianist’s perspective (bass on the left, treble on the right) or the audience’s perspective (bass on the left, treble on the right… sort of…)

32 is a lot of bits…

Once upon a time, I did a blog posting about why, when we test digital audio systems, we typically use a 997 Hz sine wave instead of a 1000 Hz tone.

The short version of this is the following:

Let’s say that I digitally create a (not-dithered) 1000 Hz sine wave at 0 dB FS in a 16-bit system running at 48 kHz. This means that every second, there are exactly 1000 cycles of the wave, and since there are 48,000 samples per second, this, in turn means that there is one cycle every 48 samples, so sample #49 is identical to sample #1.

So, we are only testing 48 of the possible 2^16 ( = 65,536) quantisation values, right?

Wrong. It’s worse than you think.

If we zoom in a little more, we can see that Sample #1 = 0 (because it’s a sine wave). Sample #25 is also equal to 0 (because 48,000 / 1,000 is a nice number that is divisible by 2).

Unfortunately, 48,000 / 1,000 is a nice number that is also divisible by 4. So what? This means that when the sine wave goes up from 0 to maximum, it hits exactly the same quantisation values as it does on the way from maximum back down to 0. For example, in the figure below, the values of the two samples shown in red are identical. This is true for all symmetrical points in the positive side and the negative side of the wave.

Jumping ahead, this means that, if we make a “perfect” 1 kHz sine wave at 48 kHz (regardless of how many bits in the system) we only test a total of 25 quantisation steps. 0, 12 positive steps, and 12 negative ones.

Not much of a test – we only hit 25 out of a possible 65,546 values in a 16-bit system (or 25 out of 16,777,216 possible values in a 24-bit system).

What if I wanted to make a signal that tested ALL possible quantisation values in an LPCM system? One way to do this is to simply make a linear ramp that goes from the lowest possible value up to the highest possible value, step by step, sample by sample. (of course, there are other ways, but it doesn’t matter… we’re just trying to hit every possible quantisation value…)

How long would it take to play that test signal?

First we convert the number of bits to the number of quantisation steps. This is done using the equation 2^bits. So, you get the following results

Number of BitsNumber of Quantisation Steps
1665,536
2416,777,216
324,294,967,296

If the value of each sample has a different quantisation value, and we play the file at the sampling rate then we can calculate the time it will take by dividing the number of quantisation steps by the sampling rate. This results in the following:

Sampling Rate (kHz)16 Bits24 Bits32 Bits
44.11.5 seconds6.4 minutes27.1 hours
481.4 seconds5.8 minutes24.9 hours
88.20.7 seconds3.2 minutes13.5 hours
960.7 seconds2.9 minutes12.4 hours
176.40.4 seconds1.6 minutes6.8 hours
1920.3 seconds1.5 minutes6.2 hours
352.80.2 seconds47.6 seconds3.4 hours
3840.2 seconds43.7 seconds3.1 hours
705.60.1 seconds23.8 seconds1.7 hours
7680.1 seconds21.8 seconds1.6 hours

So, the moral of the story is, if you’re testing the validity of a quantiser in a 32-bit fixed-point system, and you’re not able to do it off-line (meaning that you’re locked to a clock running at the correct sampling rate) you’d either (1) hope that it’s also a crazy-high sampling rate or (2) that you’re getting paid by the hour.

Why I am thinking about this?

I often get asked for my opinion about audio players; these days, network streamers especially, since they’re in style.

Let’s say, for example, that someone asked me to recommend a network streamer for use with their system. In order to recommend this, I need to measure it to make sure it behaves.

One of the tests I’m going to run is to ensure that every sample value on a file is accurately output from the device. Let’s also make it simple and say that the device has a digital output, and I only need to test 3 LPCM audio file formats (WAV, AIFF and FLAC – since those can be relied to give a bit-for-bit match from file to output). (We’ll also pretend that the digital output can support a 32-bit audio word…)

So, to run this test, I’m going to

  • create test files that I described above (checking every quantisation value at all three bit depths and all 10 sampling rates)
  • play them
  • record them
  • and then compare whether I have a bit-for-bit match from input (the original file) to the output

If you add up all the values in the table above for the 10 sampling rates and the three bit depths, then you get to a total of 4.2 DAYS of play time (playing audio constantly 24 hours a day) per file format.

So, say I wanted to test three file formats for all of the sampling rates and bit depths, then I’m looking at playing & recording 12.6 days of audio – and then I can start the analysis.

REALLY‽

Of course this is silly… I’m not going to test a 32-bit, 44.1 kHz file… In fact, if I don’t bother with the 32-bit values at all, then my time per file format drops from 4.2 days down to 23.7 minutes of play time, which is a lot more feasible, but less interesting if I’m getting paid by the hour.

However, it was fun to calculate – and it just goes to show how big a number 2^32 is…

What is a “virtual” loudspeaker? Part 3

#91.3 in a series of articles about the technology behind Bang & Olufsen

In Part 1 of this series, I talked about how a binaural audio signal can (hypothetically, with HRTFs that match your personal ones) be used to simulate the sound of a source (like a loudspeaker, for example) in space. However, to work, you have to make sure that the left and right ears get completely isolated signals (using earphones, for example).

In Part 2, I showed how, with enough processing power, a large amount of luck (using HRTFs that match your personal ones PLUS the promise that you’re in exactly the correct location), and a room that has no walls, floor or ceiling, you can get a pair of loudspeakers to behave like a pair of headphones using crosstalk cancellation.

There’s not much left to do to create a virtual loudspeaker. All we need to do is to:

  • Take the signal that should be sent to a right surround loudspeaker (for example) and filter it using the HRTFs that correspond to a sound source in the location that this loudspeaker would be in. REMEMBER that this signal has to get to your two ears since you would have used your two ears to hear an actual loudspeaker in that location.
  • Send those two signals through a crosstalk cancellation processing system that causes your two loudspeakers to behave more like a pair of headphones.
Figure 1: A block diagram of the system described above.

One nice thing about this system is that the crosstalk cancellation is only there to ensure that the actual loudspeakers behave more like headphones. So, if you want to create more virtual channels, you don’t need to duplicate the crosstalk cancellation processor. You only need to create the binaurally-processed versions of each input signal and mix those together before sending the total result to the crosstalk cancellation processor, as shown below.

Figure 2: You only need one crosstalk cancellation system for any number of virtual channels.

This is good because it saves on processing power.

So, there are some important things to realise after having read this series:

  • All “virtual” loudspeakers’ signals are actually produced by the left and right loudspeakers in the system. In the case of the Beosound Theatre, these are the Left and Right Front-firing outputs.
  • Any single virtual loudspeaker (for example, the Left Surround) requires BOTH output channels to produce sound.
  • If the delays (aka Speaker Distance) and gains (aka Speaker Levels) of the REAL outputs are incorrect at the listening position, then the crosstalk cancellation will not work and the virtual loudspeaker simulation system won’t work. How badly is doesn’t work depends on how wrong the delays and gains are.
  • The virtual loudspeaker effect will be experienced differently by different persons because it’s depending on how closely your actual personal HRTFs match those predicted in the processor. So, don’t get into fights with your friends on the sofa about where you hear the helicopter…
  • The listening room’s acoustical behaviour will also have an effect on the crosstalk cancellation. For example, strong early reflections will “infect” the signals at the listening position and may/will cause the cancellation to not work as well. So, the results will vary not only with changes in rooms but also speaker locations.

Finally, it’s worth nothing that, in the specific case of the Beosound Theatre, by setting the Speaker Distances and Speaker Levels for the Left and Right Front-firing outputs for your listening position, then you have automatically calibrated the virtual outputs. This is because the Speaker Distances and Speaker Levels are compensations for the ACTUAL outputs of the system, which are the ones producing the signal that simulate the virtual loudspeakers. This is the reason why the four virtual loudspeakers do not have individual Speaker Distances and Speaker Levels. If they did, they would have to be identical to the Left and Right Front-firing outputs’ values.