Why Square Waves Matter

A while ago I wrote a post about what Square waves are, how they are formed, and how they are really just made up of a bunch of increasingly higher-pitched sign waves.

I'm sure that was more than enough graphs and lines to scare anyone into submission. That's not such a bad thing.

But one thing that I didn't get into is why square waves are important to the modern sound engineer.

Distortion - nothing's perfect
Whether you want it or not, Distortion is at the heart of every piece of reproduced music. It doesn't matter if that reproduction is "live" (as in a microphone running through a mixing console at a rock concert) or if it was recorded 15 years ago and played back on a CD player.

There are a number of different reasons for this, some of which (like the reaction of various electronic components like capacitors) will be left for later; there's just simply too much to cover.

What I'd like to cover is the linking between Square Waves and distortion, specifically in two areas: Gain and Quantisation.

Gain Management
If you've ever used a mixing console (or pretty much anything that shows you the audio levels in the standard green-orange-red format) you've probably kept on turning up the volume until you've got some pretty red lights blinking all over the place. If you ask any experienced engineer they'll tell you that this means that they are "saturating" the console's amplifiers, leading to "some nice compression" and/or distortion.

But what's actually happening here?

At the heart of every bit of audio equipment is a humble little circuit; the Operational Amplifier (Op-Amp). This simple circuit has a never-ending number of variations and uses, but essentially all we need to know is that it takes an input signal and some power, and outputs the same signal but with more volts.

Your stock-standard op-amp drawings. 

Okay, once again I can feel that this picture means nothing to you.  Never fear; let me explain.
 The top picture is how most of us think of amplifiers: you put a signal (+ and -)  into a circuit and you get some kind of output (shown here as the line out the right hand side of the Amp).

In fact, it's perfectly acceptable to only show the +ve side of the input.
Essentially what this image shows is what we mentioned above; you put a small signal into the amp, and you get a bigger signal out. What actually goes on inside the triangle is inconsequential.

However, the bottom picture gives us a little more information. Here we still see our inputs and outputs (U+, U- and U0). There's no difference there. But we now also see two new variables; Ucc and Uee. These represent the "power" that we are supplying to the amplifier.

In normal operation, the input signal is small enough so that the output (U0) is less than the total power that you are feeding to the op-amp (Ucc and Uee).

But if your input is too big, you can get to a situation where U0 needs to be bigger than Ucc. Since the op-amp has run out of power it stops amplifying and simply outputs the maximum power; i.e. Ucc.

Two examples of clipping; we're interested in the lower example

If you look at the image above, you'll see what I mean. The dashed "threshold" lines indicate Ucc and Uee; in other words, the maximum and minimum possible voltages of the output, U0. For the moment, we're only going to look at the lower example.

You'll notice that as soon as the output reaches the maximum (Ucc) or minimum (Uee), the waveform flattens out, no matter what the input signal looks like.

You'll also notice that this looks exactly like a square wave.

If you have a look at the previous article on Wave Theory, you'll note that a Square Wave contains the original frequency as well as a number of  harmonic frequencies. If we were to look at the above waveform on a spectrum analyser you would initially see a single peak (the input frequency). As the input increases beyond the Ucc threshold, you would start to see the single peak get shorter and more harmonic peaks growing up, as if out of nowhere.

As the clipped wave starts to look more like a square wave, you get more harmonics

These harmonics are the "distortion" that we are hearing. 

Since no amplifier circuit is perfect, you'll always have a bit of this Harmonic Distortion cropping up along the way. It's one of the measurements of amplifier power; Total Harmonic Distortion (or THD), and is expressed as a percentage. The percentage refers to the amount of power that is "lost" in the harmonic peaks. In short, a lower number is better - the number is telling you how much "false" information you're hearing due to the amplifier circuit.

Essentially, you can "fake" an amplifier's power rating by increasing the THD tolerance. An amplifier than can deliver 100W at 0.1% THD might be able to deliver 300W at 5% THD. That's because you're increasing the output of the amplifier, and also increasing the harmonics that are creeping into your original signal.

Is that a bad thing?

Distortion isn't necessarily a bad thing. Rock 'n' Rollers have been using fuzz pedals and distortion generators for decades. As with all sound-related things, everyone has their own opinion. But you also need to make sure that you're getting the right gear for the job. A 300W 10% THD amplifier might be perfect for your car stereo, but you might want a 300W 0.1% THD amplifier for your Hi-Fi speakers in your living room.

Quantisation Noise

The other, very similar form of distortion that I'd like to discuss here is Quantisation Noise.

As we saw in my PCM post, digital audio effectively segments up an analogue signal into a bunch of "stepped" signals.

A 4-bit quantisation (grey) of a sine wave (red)

Hopefully, it won't take you much imagination to realise that the grey, "digital" signal will have a number of harmonic frequencies tied up within its square-wave-like structure.

If not, then scroll back up and have a look at the last image showing the build-up of harmonics in the Square Wave example again!

Of course, just as we explained in the PCM article, the above representation is a gross simplification of a digital signal; even a 16-bit signal has millions of steps, making the harsh corners a little more manageable.

That being said, a digital recording will always have a certain amount of distortion, or "Quantisation Noise," inherent in the system. In low bit-rate recordings or transmissions (e.g. bad digital radio), this comes through as a watery-garbling of the audio signal.

Here is an example of an 8-bit recording, and then the noise generated as this is reduced in bit-depth to:

4 Bits (like the above image)

2 Bits

But does it matter?

I've just had a look at a couple of Analogue-to-digital converters in 16 bit terms. These list the THD of the entire converter (including the op-amps and the rest of the circuitry) at about -100dB. That equates to a THD of about 0.0009%. 

Whilst there is a lot of complaints about the "sound" of digital recordings (and yes, I do admit that there are differences between the two), the chances are that the THD of the amplifier is many, many times greater than the harmonic distortion that is caused by the "digital-ness" of the recording.

Pulse-Code Modulation - Music to our ears

Okay, so we've just spent a lot of time looking at how data moves around a network. I hope that you were all with me for the ride. If not then please feel free to comment on any of the posts and I'll answer your post.

For the next couple of weeks I'd like to start looking at the data that you actually put into those packets.

One of the easiest places to start looking at digital signals is the humble Pulse-code Modulation, or PCM, method of encoding analogue information into a digital signal.

Analogue vs Digital
No, I'm not going to get into the "aesthetic" differences between analogue and digital, save to say that the only instrument that you can trust is your own ears. If it sounds better to you, then it sounds better to you.

I would like, however, to clarify something quickly. An "Analogue" signal is a proportionate signal with no real limitation. The local air pressure around a microphone or a signal can be measured by a device and turned into an electrical signal that is proportionate to the pressure. The higher the pressure, the higher the voltage. The voltage is an "Analogy" of the pressure.
This is what we refer to as an "Analogue" signal. Most simple electronic devices will process and run on analogue signals.
At some point, everything is an "Analogue" signal; the pressure changes that reach your ear drums or the light changes that reach your retinas are "Analogue".

A "Digital" signal, however, is somehow encoded so that it is no longer proportionate to the original signal. Through some kind of electronic process the Analogue signal is broken down into symbols of some kind that are readable only by devices that use that format. Before we can interact with them again then they need to be converted into an Analogue signal again. This process is called "Encoding" (Analogue to Digital) and "Decoding" (Digital to Analogue). Combine "enCOder" and "DECoder" and you get CODEC... but we'll get to those later.

Pulse-Code Modulation
Anyone who's ever Google'd "Digital Audio" will have seen a picture similar to this one:

Let's assume that we are looking at the Encode (Analogue to Digital) side of things (although the process is exactly the same in reverse).
The red line is our input signal; a standard sine-wave. This could be anything;  an audio signal, the number of people that like or dislike the current Prime Minister... it doesn't matter. We have a signal that is changing as time goes on.

The analogue signal is continuous and unbroken.

Pulse-Code Modulation sets a value (shown above as 0-15) for each equivalent amplitude. To convert the signal into a digital one, we record the value of the analogue signal at the start of each of the time divisions shown along the bottom access. This process is called "Sampling" - you are taking a sample of the Analogue signal at each of the time divisions.

You'll note that on the image above you can see a difference between the continuous Analogue (Red) signal and the Digital (Grey) one. It looks like a lot, right? In fact, the small differences in images like the above are one of the main arguments used by Analogue supporters. However, there is something missing from this picture...

Bit Rate
The picture above gives you a pretty good look at what you'd see in a phone-line; a 4-bit system. A bit is a single binary "symbol". One bit gives two states; two bits gives four, three bits give eight and four bits give sixteen states.
The number of "bits" that a digital signal contains is referred to as the Bit Depth. In basic terms, the higher the bit depth, the better the quality.

Even your most basic audio (CD-quality) has a bit depth of 16 bits; or about 65 thousand different states. The Human ear isn't really able to detect that kind of resolution; it would be like trying to look at the millimetre markings on a ruler that was 10 meters away.

Still, there are higher bit depths; standard digital audio (AES/EBU-3, which I will cover in a future article) runs at 20 bits (1,048,576 states) or at 24 bits (16,777,216 states). At this point, you're pretty much splitting hairs with a 2000-pound bomb...

But there is another factor that affects the quality of the sound; the rate at which the audio is sampled.
As a general rule, you want to take a sample at twice the frequency of the highest frequency you want to hear. The label on a new-born baby reads 20Hz-20,000Hz, although by the time you've used an MP3 player and gone to a concert or two you will be lucky if you can hear above about 17,000Hz.
Thus, the main sampling rate used in digital audio is 44.1kHz (CD-quality). "Professionals" will use 48kHz, or even go as high as 96kHz. Once again; at that level you are recording detail that humans just can't perceive. It's like taking a photo in ultraviolet; it might look brilliant, but there is no way for us to see the result.





I would like to continue this article, but in the interest of keeping things concise I will hold off for the time being. Next article I will look at a couple of the strange effects of PCM, and how we avoid them.
But for now, I must away. Until next time.