# Filters and Ringing: Part 6

In this part, I’m going to deviate just a little from something I said at the beginning of this series. To be honest, if I hadn’t admitted this, you probably wouldn’t notice – but I would prefer to keep things clean… The deviation is that, for this part, I’m making a slight change to how Q is defined. This is not serious enough to get into the details of exactly how the definition is different .

Using the slightly-different definition of Q, let’s make a peaking filter with a centre frequency of 1 kHz, a boost of 12 dB and a Q of 2. This will have the response shown below in Figure 1.

Using the same modified definition of Q, let’s also look at the response of a dip filter with the same parameter values, but a gain of -12 dB instead.

If you look at the magnitude responses of these two filters, you’ll see that it looks like they are mirror images of each other. In fact, they are.

If you look at the phase responses of these two filters, you’ll also see that it looks like they are mirror images of each other. In fact, they are.

If you look at their impulse responses, you’ll see that it would be difficult to see that they are related at all… But never mind this.

If I connect the output of the first filter to the input of the second filter, and measure the total throughput of the system, it will look like this:

Just in case you’re suspicious, I didn’t fake this. I actually connected the boost to the dip and sent an impulse through the whole thing and you’re looking at the result. No tricks! (Note that I could have reversed their order with the same total result.)

What you can see here is that the responses of the dip and the boost negate each other. Whatever one does, the other does exactly the opposite.

Generally speaking, we audio geeks use some special words to describe not-very special cases like this.

Often, you’ll hear us talking about a linear system which is a fancy way of saying ‘the effects of this system can be undone’. In this example, the dip filter can ‘undo’ the effect of the boost (and vice versa) therefore both must be linear filters.

Just as often, you’ll hear us talking about time-invariant systems, which just means that they don’t change over time. Because I implemented those two filters using equations done on my computer, if I run the math again tomorrow, I’ll get exactly the same answer. If I test them using an impulse that is quieter or louder, I also get exactly the same responses. (If I had implemented them using resistors and capacitors and transistors or vacuum tubes, I might not get the same answer tomorrow or with a different signal level because of temperature changes, for example. Although now I’m really splitting hairs, just to make a point.)

The reason I said “just as often” is because, normally we use the two terms together as a package deal. So, we ask whether a system (like something as simple as a filter or as complicated as a reverb unit or an upmixing algorithm) is Linear Time-Invariant or LTI. This is an important question because it packs a lot of information in it.

For example, if a reverb unit is LTI, then I can measure it today with an impulse, and I know that it will behave the same tomorrow with lute music or a snare drum. It does the same thing all day, every day, regardless of the input signal or its level. One measurement, and I can go away and analyse it for the rest of the week.

If it’s not LTI, then its characteristics will change for some reason that I don’t necessarily know. Maybe the internal delays are modulating in time, so its response in 10 seconds will be different than it is now. Maybe it has a compressor or a noise gate built in, so it changes its behaviour according to the level of the signal.

If we get back to our (rather simple) peak / dip filter example. We know they’re LTI (because I said so – and you have to trust me). We also know that the dip filter is the opposite of the boost. The question is “how, exactly, did I make this happen?”

The general answer to this question has already been answered – the magnitude and the phase responses are mirror images of each other. Therefore, for any given frequency, one filter boosts by the same amount that the other cuts, and one filter advances in phase by the same amount that the other delays in phase.

The more geeky answer to this question requires that we look at the Z-Plane, which I’ve talked about throughly in another series of postings starting with this one. I’ll repeat myself a little by saying that a Z-Plane representation shows a different way of looking at the ‘ingredients’ in a filter. It contains ‘poles’ that are placed at frequencies that are infinitely boosted, and ‘zeroes’ that are placed at frequencies that are infinitely cut. By carefully placing poles and zeros relative to each other in the Z-Plane, you can decide how the filter will behave for other frequencies between 0 Hz and the Nyquist frequency.

When you design (or analyse) filters this way, there are a couple of basic rules:

The ‘safe zone’ in the Z-Plane is defined by a circle. If you start placing poles outside it, then the filter can become unstable. If a filter is unstable, this means that its ringing can get louder over time instead of decaying.

If you place a pole in exactly the same place as a zero, they cancel each other out, and the total result is as if neither were there.

So, let’s look at our two filters above in their Z-Plane representations.

Admittedly, the resolution of the display in the software that I’m using to show this isn’t great, but if you compare the Z-Plane plots on the left and right, you can see that the zeros (marked with ‘o’) and the poles (‘x’) swap places. Just to make things a little clearer, I moved the centre frequency to 10 kHz and kept the gain and Q values the same. These are shown in Figure 5.

What’s the point of showing you this? The Magnitude and Phase response plots (which, combined, comprise the filters’ Frequency Responses) are ‘just’ descriptions of the behaviour of the filter. They tell you what happens to a signal that goes through them.

The Z-Plane representations show you how the filters are actually implemented.

It’s like the difference between reading a description of how a cake tastes and reading the recipe.

What you can see in the Z-Plane is not only that the responses of the filters negate each other: they’re built to ensure that this is the case. The poles and zeros of one filter cancel the zeros and poles of the other, and vice versa.

There’s one other extra piece of information that you already know. The fact that the poles for any of these filters are inside the circle helps to tell us that they’re stable and therefore LTI. It also tells us something else that we’ll talk about in the next part.

# Filters and Ringing: Part 5

## Phase

There are lots of people in audio who will make some claims about one kind of filter being better than another kind of filter because of something to do with the time response. They’ll throw around words like “minimum phase” or “linear phase” or “apodising” or other names, which sound impressive, but don’t really mean anything to normal people. In fact, in most cases, they don’t even mean anything to abnormal people (a.k.a. audio engineers). They’ll even make some statements about why one is better than the other, with some psychoacoustic claims to back themselves up.

One thing to remember is that these terms are very general headings that each sit on top of a lot of sub-headings. It’s also important to separate these terms from the incorrectly-interchanged terms ‘FIR’ and ‘IIR’ (which stand for ‘Finite Impulse Response’ and ‘Infinite Impulse Response’) which are different descriptions for the same filters. For example, many people say “FIR” when they mean “linear phase”, forgetting that an FIR can be used to create a non-linear phase filter.

In this posting, we’ll start to look at the difference between ‘minimum phase’ and ‘linear phase’ filters, but this requires a little set-up first.

Up to now in this series of postings, we’ve only looked at the filters’ magnitude responses (the gain of the filter vs. frequency) and time responses (or impulse responses). Let’s shift gears a little and think about the phase response instead.

Remember from Part 1, we looked at how an impulse is the result of adding an infinite number of cosine waves that all started at the beginning of time, and will continue until the end of time. Those waves all cancel each other out at all moments in time (forwards and backwards) except for that one instant (which we call Time = 0, also known as NOW) where they all add up to make a click.

What happens when we shift the time alignment? The intuitive answer is that we get something different than a simple click. The more we shift the frequency components in time, the more different we get from a simple click.

However, when we talk about shifting frequency components in time, it doesn’t make sense to actually measure that shift in time. I know that sounds like a stupid thing to say, so I’ll illustrate what I mean…

We saw that if we add a bunch of cosine waves together they start looking like an impulse, as shown in Figure 1.

What happens if I delay all of those individual waves by 0.5 second (or 500 ms)? The result is shown in Figure 2.

It should be pretty obvious that the result in Figure 2 is identical to the result in Figure 1. The only difference is that it’s been shifted in time by 500 ms. The shape of the wave has not changed because we shifted all of the waves together, so their relationship to each other has not changed.

So, if we want to change the shape of the total result, we need to shift the components relative to each other, as shown in Figure 3.

Figure 3 shows the same components with the same amplitudes, but shifted so that they all cross the T=0 point at the 0 line instead of at the maximum (as in Figure 1). This means that I’ve shifted each component individually by 90º, which is a different amount of time (in seconds) for each one. (In other words, I’m summing sine waves instead of cosine waves.) The summed result is quite different, as you can see in the bottom plot.

You can also shift some components differently (measured in phase) as well. For example, take a look at Figure 4. In that one, the first 4 components with the lowest frequencies are cosine waves, and I’ve shifted the 5th component by 90º. As you can see in the bottom plot, just shifting one component can make a large difference.

And it probably goes without saying, but I’ll say it anyway, that if you change the relative levels of the components, you’ll also change their total sum, as shown in Figure 5.

Let’s turn this around (finally…). In the examples above, I was playing with the components’ amplitudes and relative phases to produce different total summed results, even through the frequencies of the components were the same each time.

If we think of this backwards, we can conclude that, if the time response of a filter is NOT a perfect impulse, then it must have done something to the relative levels and/or the relative phases of the collection of infinite frequency components that went through it. Using math (the same Fourier Transform that I mentioned in Part 2) we can take the impulse response and calculate what happened to the components, both in amplitude (the Magnitude Response) and phase (the Phase Response), which together give us the filter’s Frequency Response.

Let’s look at an example: a bandpass filter with a centre frequency of 1 kHz and a Q of 2, shown in Figure 6.

The top and middle plots in Figure 6 should not come as surprises now, so let’s talk about that bottom plot. What is shows us, generally speaking, is that if you send a sinusoidal wave through the bandpass filter at the centre frequency (1 kHz) then the output will have the same phase as the input, since the red line is at 0 degrees at 1 kHz.

If the sinusoidal wave that you send in is above 1 kHz, then the output will be later in phase than the input. This does NOT necessarily mean that it’s delayed in time. We can’t know this because as soon as I said “sinusoidal wave”, this implied that it has no start or stop time – it’s just a sinusoidal tone that has always been there and will always be there. (In order to start or stop it, you need other frequency components.)

Philosophically, this may be difficult to consider – but think of it the same way you you experience seeing Niagara Falls. You really have no first-hand knowledge of when the water started falling or when it will stop – it’s as if it’s always been doing this and it always will – and you just get to see it for a small slice of time in its “infinitely”-long existence.

It’s really important to remember that what we’re looking at in Figure 8 is a phase shift and NOT a time delay (even though it looks like it). Repeat this sentence until you believe it before looking at the next plot.

Figure 9 shows an example of why you have to believe that we’re not talking about a time delay – just a phase shift. As you can see there, in the case of a bandpass filter, if the signal frequency is below the centre frequency, the phase shift is backwards, which looks like the output is ahead of the input. Of course, this is impossible. Bandpass filters are not time machines.

Now go back and look at the bottom plot in Figure 6. You’ll see that frequencies above the centre frequency of the filter (1 kHz) have a phase shift that is below 0º – they’re negative numbers approaching -90º as the frequency increases. Compare this to Figure 8 and you can make the link that a negative phase shift is “later” (in phase, not in time!).

Conversely, lower frequencies have a positive phase shift in Figure 6, which (as can be seen in Figure 9) correspond to a phase shift that moves “earlier”.

Remember that a peak/dip filter is a combination of a bandpass and a throughput. So now let’s look at the phase shift that results when you use one.

Looking at the magnitude response, it should now be fairly easy to see the merging of a throughput (which would be a straight line at 0 dB across all frequencies) and a bandpass (which causes the bump around 1 kHz).

It should be almost as easy to see the merging in the phase response as well. A throughput would have a phase response of 0º at all frequencies – which is why the plot starts at 0º in the very low frequencies and ends at 0º in the very high frequencies (because the bandpass doesn’t have much contribution out there). In the middle, the phase response of the bandpass shows up; so around 1 kHz, the phase responses of Figure 10 and 6 are very similar.

Let’s change the Q and see what happens.

Figure 11 shows the same peaking filter with the Q increased to 10. Notice 5 things (not in any obvious order):

• The bump in the magnitude response is narrower
• The ringing starts at a lower level
• The impulse response is ringing for a lot longer in time
• The deviation from 0º in the phase response has a narrower bandwidth.
• The slope of the phase response at 1 kHz is steeper.

Let’s put some of these together. I’ll take these in a slightly different order, but after reading the paragraphs below, the points above should all interlock.

The bump in the magnitude response is narrower; therefore it has a smaller bandwidth. This should be expected, since Q = Fc/BW, so if we don’t change Fc, then the higher Q goes, the smaller BW gets.

Notice that both the filter in Figure 10 and the filter in Figure 11 have a gain at Fc of 12 dB. However, since the Q is lower in Figure 10, this means that, overall, more frequencies are boosted by more. Consequently, if you have a signal that has all frequencies in it (say, pink noise or Metallica), then the output of Figure 10’s filter will be generally louder than the output of Figure 11’s. Another way to see this is that the level of the start of the ‘tail’ of the impulse response is higher.

There is a direct link between the length of time the filter rings (which you can see in the impulse responses) and the slope of the phase response. The steeper the slope at a given frequency, the longer the filter will ring at that frequency. So, if you only look at the phase response plots, it’s easy to tell which of the two filters will ring for a longer time, and at what frequency. This will come in handy in the next part.

# Filters and Ringing: Part 4

Let’s put together a couple of things that were said in the last postings, which should help to support each other:

A peak or a dip filter is created by adding a bandpass filter to a throughput, as shown in Figure 1.

To change from peak to dip, you switch the polarity of the bandpass portion by making the “gain” negative instead of positive. (In other words, you subtract the bandpass from the throughput instead of adding it). To change the gain of the peak/dip filter, you change the gain of the bandpass portion. To change the Q of the peak/dip, you change the Q of the bandpass.

We also saw at the end of Part 3 that changing the gain does not change the rate of the decay.

This should all come together nicely to make sense for the first of the three points. For example, since the bandpass portion is the part that’s ringing, and since changing the gain of the peak (or dip) is just a matter of changing the gain applied to the bandpass portion, then there is no reason why the decay rate of the ringing should change. It will start at a higher or lower level, but its decay slope will be the same.

## Q vs Time

We also saw at the end of Part 3 that changing the Q will change the slope of the decay inversely proportionally, but that changing the frequency will change the slope of the decay proportionally.

There is a nice little rule-of-thumb that’s used by electrical engineers for measuring the Q of a filter. Let’s say that you can’t (or couldn’t be bothered to take the time to) measure the frequency or magnitude response, and you want to figure out the Q based on the time response only, you can calculate this by looking at its impulse response.

For example, Figure 2 shows the initial part of the impulse response of an unknown filter. I’ve highlighted two points that are reasonably close to the tops of two of the cosine wave cycles. I picked the first one (on the left) and then noted its Y value (Y = 0.027). Then I found a top of another wave that was as close to half that value as I could find. You can see there that it’s 2 cycles later, where Y = 0.0149.

So, you multiply the number of cycles it takes to drop by 50% (in this example, 2 cycles) and multiply that by 4.53, which results in a value of about 9. This is a good estimate of the Q of the filter (which is actually 10, if I measure it using the -3 dB points in the magnitude response).

Note that it doesn’t matter which cycle I chose to get the first value, since the rate of decay is the same through the entire time response of the filter. In other words, if I chose the 3rd cycle to do the first measurement, I would have found that the 5th cycle is about 50% lower because it’s also 2 cycles later.

It also doesn’t matter whether we’re talking about peaks or dips, since, as we already know, from a perspective of the individual building blocks of the filter, these are the same thing.

## So what?

Of course, most normal people aren’t measuring the time response of filters to calculate the Q. However, this piece of information is good from the opposite perspective: if you know the Q of the filter, you can figure out how fast it’s decaying. For example, a filter with a Q of 2 will take 2 / 4.53 = 0.44 cycles to decay by 50% (or 6 dB). If you know the frequency, then you can then translate that into a decay rate per seconds, because the period in seconds (the total time of one cycle of the wave) = 1 / Fc.

So, if that filter with a Q of 2 has an Fc of 100 Hz, then the period is 1/100 = 0.01 sec, and therefore it will decay by 6 dB (50%) in 0.44 cycles * 0.01 sec/cycle = 0.0044 sec or 4.4 ms.

If the Fc of the filter is 5 kHz, then the the period is 1/5000 = 0.0002 sec, and therefore it will decay by 5 dB in 0.0002 * 0.44 = 0.000088 sec = 88 µsec. (This is roughly equivalent to 2 samples at 48 kHz.)

Another good thing to remember is that Q = Fc / BW where BW is the bandwidth of the response measured between the two -3 dB points. This means, for example, that if Q = 1, then Fc = BW, therefore the bandwidth is about 1 octave. If Q = 2, then the bandwidth is about 1/2 of an octave, if Q = 12 then the bandwidth is about 1 semitone (1/12th of an octave), and so on.

# Filters and Ringing: Part 3

Now we’ve seen that if we have a filter that results in either a peak or a dip in the magnitude response, we’ll also result in the signal ringing in time. We’ve also seen that the frequency of the ringing is the centre frequency of the filter. Now let’s dig a little deeper into the behaviour of that ringing; or, more specifically its decay characteristics.

We’ll repeat the process from Part 2: measure the impulse response of a peaking filter where Fc = 1 kHz, gain = +12 dB, and Q = 2. However, this time I’ll look at the time response with a different scaling. Instead of plotting the linear value over time, I’ll convert each instantaneous value to dB and plot that. This looks like Figure 1.

The important thing to notice here is that, when I plot the instantaneous amplitude in decibels (in other words, on a logarithmic scale), the decay is a straight line with a slope.

Let’s get two things out of the way here. This isn’t really decibels, because decibels requires some time averaging. Also, I’m actually plotting the absolute value of the impulse response in a decibel scale, because if I try to calculate the log of a negative number, things get ugly. This means that the math I’m actually using to create the bottom plot is

20 * log10(abs(signal))

If I draw a line across the tops of the bumps in that plot, I can look at the decay of the filter’s ringing as in Figure 2.

For this filter, the decay rate of the ringing is -1360 dB per second (which is very fast). Let’s change some parameters and see what happens.

If I increase the gain of the filter without changing the Fc or the Q, I get the following:

I could plot lots more of these so that you start to see a pattern, but I’ll jump to the punch lines and you can use the plots above to check that things make sense.

If I have a filter that is using a definition of Q = Fc / BW (where BW is the distance between the -3 dB points down from the maximum), then:

• Changing the gain does not change the rate of the decay (all least, as long as it’s a boost, according to what we’ve seen so far…)
• Changing the Q will change the slope of the decay inversely proportionally if we’re measuring the slope in dB/sec. For example, if I multiply the Q by 2, the ringing decays twice as slowly. If I multiply the Q by 10, the ringing will take 10 times longer to decay to the same level.
• Changing the frequency will change the slope of the decay proportionally if we’re measuring the slope in dB/sec. For example, if I multiply the frequency by 2, the ringing will decay twice as fast.

Let’s talk about the last of these first, since it’s the easiest to understand conceptually. In the plots above, I’m showing the time in seconds. So, the higher the frequency, the more cycles I’m showing in the same plot. However, if I were plotting time in cycles of the cosine wave instead, the slope would be the same regardless of frequency.

In other words, the level of the ringing decays by the same amount per number of cycles of the cosine wave.

This is why, if you count the number of “bumps” in the dB plots in Figure 2 and 5, you’ll see that they are the same number. It takes about 12 cycles to get down to -100 dB, but the shorter the cycles (because the frequency is higher) the faster you get there when measuring in seconds. If the X-axis were not “Time in milliseconds”, but “Time in periods of the centre frequency” instead, then the slopes would be identical in Figures 2 and 5.

# Filters and Ringing: Part 2

## Rocks, Guitars, and Children

If you throw a rock into a pond on a windless day, you’ll see the ripples moving away in an expanding circle from the place where the rock hit. The ripples are places on the water where the water is either higher or lower than where it was before you hit the rock. The water itself only moves up and down, but the waves expand sideways. (You can see this if there is something floating on the water, for example – it bobs up and down as the waves go by.)

A similar thing happens when you pluck a guitar string. The point where your finger plucked is the same as the point where the rock landed in the water, and waves radiate away from that place on the string in two directions (because there are only two directions to travel in on a string: this way and that way). However, when those waves reach the end of the string, they reflect and come back in the opposite direction.

In both cases, the water and the guitar string, the wave has some speed at which it travels. It’s slow enough on the water for you to watch it, but it’s much too fast on a guitar string. In fact, it’s so fast that, when you pluck it, the wave travels to the end of the string, reflects in the opposite direction, hits the other end of the string, reflects again, and gets back to where you plucked it in about 1/82nd of a second if it’s the low E string. Since the wave doesn’t stop there – it keeps going, repeating the back-and-forth journey along the length of the string every 1/82nd of a second, then we hear a note with a fundamental frequency of 82 Hz (82 cycles per second): a low E.

That ringing that happens on the guitar string will happen no matter how you start the movement on it. You could hit the string with a chopstick, you could just thump the side of the guitar with your fist, you could even stand next to the guitar and cough loudly. All of these things will “inject” energy into the string, causing it to move, and the wave starts banging back and forth.

The rate of repetition is dependent on two things: the length of the string and the speed of the wave. The speed of the wave is dependent on two things: the mass of the string (e.g. how heavy is 1 m of it?) and the tension (how tightly is it stretched?) Increase the tension, and you increase the speed of the wave. Decrease the mass and you increase the speed of the wave. Increase the speed of the wave, and the repetition takes less time, so you hear a higher note.

That frequency at which the string will naturally ring is called a resonance. A child on a swing will go back and forth at the same rate (number of times per second) no matter how gently or forcefully you push them – apply energy, and the system will resonate.

Now, let’s think about that push of the child, the rock hitting the water, or the pluck of the guitar string. All of those things are a short injection of energy: a kind of impulse, and the way the child, the water, or the string behaves afterwards is its impulse response – how it responds to that impulse.

But here’s a strange thing to consider. This means that the note (the frequency) that you hear from the guitar string was one of the many frequencies in the initial pluck itself.

So, another way to think of this is that, by plucking the string, you inject a signal with all frequencies in it, and all of those frequencies decay (“die away”) very quickly except for one.

Okay, okay, if we’re going to be pedantic, I should be including not only the fundamental frequency but all of the additional harmonics; typically multiples of that frequency. But we don’t need to complicate things with the truth at the moment…

## What does this have to do with filters?

From a “big picture” point of view, a guitar string is a filter. I feed in some signal (the pluck) and I get out a modified version of that signal (the note ringing). From the same perspective, a filter in an equaliser is the same: I feed in a signal (music) and I get out a modified version of it (the same music, but slightly louder at 1 kHz, for example). What’s interesting is that the two things basically work the same way.

Let’s take the example of the filter at the end of Part 1: a peaking filter with a boost of 12 dB at 1 kHz, with a Q of 2. If I feed in a sine wave (which only contains energy at 1 frequency) at a very low frequency (say, 100 Hz or lower) then the level of the output will equal that of the input. If I do the same with a very high frequency (say, 10 kHz) then the level of the output will also equal that of the input. However, if I feed in a sine wave at 1 kHz, the output will be 4 times louder than the input (+12 dB = 4 time the amplitude because 20*log10(4) = 12-ish).

At some other frequency around 1 kHz, I’ll get a different answer. However, this is a VERY long and tedious way to measure the magnitude response of the filter. Another option is to measure its impulse response.

If I feed the input of the filter with an impulse (which is a sound that contains all frequencies at the same level, as we saw in Part 1), and look at the filters output in time, it might look like this:

Notice that the impulse looks like an impulse at Time = 0, but then something extra happens afterwards – like a guitar string ringing in time. If I zoom in vertically and look at the same plot, it will look like Figure 3.

And if we zoom in horizontally as well, it will look like this.

So, as you can see there, it’s almost as if we kept the impulse, and then just added a cosine wave with a period (a repetition time) of 1 ms, starting at Time = 0 and decaying over time. In fact, that’s exactly what the filter does.

## Time response to Frequency response

The excuse I gave above for sending an impulse through the filter (instead of sine waves) was that this will be a faster way to measure its response. The time response of the filter is already done. We can see that in the figures above. But how do we see the filter’s frequency response? This is done using a clever bit of math called a Fourier Transform, which lets you take a signal in time, and analyse its content by frequency. I won’t explain that here, but if you’re interested in how it works, you can start by reading this.

If I take the total impulse response (also known as a time response measurement) of the filter: in other words, I send in an impulse, I record the output and don’t stop recording until the ringing has decayed to a level low enough that I no longer care (for the purposes of this discussion, at least). Then, I do a Fourier Transform of the recording, I get something like Figure 5.

There is no new information in Figure 5. It’s just a setup for Figures 6 and 7.

Let’s now start slicing up the time response selectively to see what frequencies are contained in the output of the filter at what time. We’ll start by just taking the first and second samples of the impulse at the output, shown in Figure 6.

As you can see in Figure 6, if I remove the ringing that comes after the impulse, then the response of the signal has an almost-flat magnitude response and a gain of about 2 dB or so. This should not come as a surprise, since it’s almost an impulse. The only real difference between the portion that I’ve used and a real impulse is that the second value is not 0. So far so good…

Let’s look at the remainder of the time response. This is shown in Figure 7.

Figure 7 shows something interesting. We see the response of a band-pass filter with a centre frequency of 1 kHz, and a gain of 9 dB, which is the response of the filter after the initial impulse has passed.

## What does this all mean!?

If we leave out one important thing for now, this means that a peaking filter that has a boost of 12 dB, an Fc of 1 kHz and a Q of 2 is actually the sum of two things:

• a through-put with a little gain (about 1 dB)
• a bandpass filter with a gain of about 9 dB

This is, in essence, true. You can create a peaking filter by summing a bandpass filter to a through-put. However, an important point to realise here is that the band pass signal essentially comes after the onset of the signal. In Part 3, we’ll talk about whether this is a problem – or, more accurately, when this might be a problem. For now, however, I’ll throw one more example at you.

Up to now, we’ve only looked at the example of a peaking filter with a boost. What happens when the filter has a cut instead?

Notice that a dip filter also rings in time after the initial impulse, but decays much faster than the equivalent boost. (I’ll have to be a bit more careful about my use of the word “equivalent”, actually – but I’ll straighten that out at the end of the series. To be continued…)

Okay, what’s going on here? A peaking filter with a boost is a through-put plus a bandpass. A dip filter is ALSO a through-put plus a somewhat quieter (sort-of) bandpass. This doesn’t make any sense.

Actually it doesn’t make any sense because there’s a piece of information that I’m leaving out – the phase of the ringing. Notice that, with the peaking filter, the decay portion starts positive and then goes negative initially. With the dip filter, the decay starts negative and goes positive. So, the previous paragraph should have read: “A peaking filter with a boost is a through-put PLUS a bandpass. A dip filter is ALSO a through-put MINUS a somewhat quieter (sort-of) bandpass.”

The phases of the decays of the bandpass portions are opposite for the two filters. Another way to think of this is that the ringing in the dip filter cancels the energy around 1 kHz in the initial impulse, whereas the ringing in the peak filter adds to it.

However, it’s really important to note for now that both filters – the peak and the dip result in ringing in time.

# Filters and Ringing: Part 1

Let’s say that, for some reason, you want to apply an equaliser to an audio signal. It doesn’t matter why you want to do this: maybe you like more bass, maybe you need more treble, maybe you’re trying to reduce the audibility of a room mode. However, one thing that you should know is that, by changing the frequency response of the system, you are also changing its time response.

Now, before we go any farther, do NOT mis-interpret that last sentence to mean that a change in the time response is a bad thing. Maybe the thing you’re trying to fix already has an issue with its time response, and sometimes you have to fight fire with fire.

Before we start talking about filters, let’s talk about what “time response” means. I often work in an especially-built listening room that has acoustical treatments that are specifically designed and implemented to result in a very controlled acoustical behaviour. I often have visitors in there, and one of the things they do to “test the acoustics” is to clap their hands once – and then listen.

On the one hand (ha ha) this is a strange thing to do, because the room is not designed to make the sound of a single hand clap performed at the listening position sound “good” (whatever that means). On the other hand, the test is not completely useless. It’s a “play-toy” version of a very useful test we use to measure a loudspeaker called an impulse response measurement. The clap is an impulsive sound (a short, loud sound) and the question is “how does the thing you’re measuring (a room or a loudspeaker, for example) respond to that impulse?”

So, let’s start by talking about the two important reasons why we use an impulse.

## Time response

If a thing in a room makes a sound, then the sound radiates in all directions and starts meeting objects in its path – things like walls and furniture and you. When that happens, the surface it meets will absorb some amount of energy and reflect the rest, and this is balance of absorbed-to-reflected energy is different at different frequencies. A cat will absorb high frequencies and low frequencies will just pass by it. A large flat wall made of gypsum will reflect high frequencies and absorb whatever frequency it “wants” to vibrate at when you thump it with your fist.

The energy that is absorbed is (eventually) converted to heat: that’s lost. The reflected energy comes back into the room and heads towards another surface – which might be you as well, but probably isn’t unless you’re in a room about the size of an ancient structure known to archeologists as a “phone booth”.

At your location, you only hear the sound that reaches you. The first part of the sound that you hear “immediately” after the thing made the noise, probably travelled a path directly from the source to you. Let’s say that you’re in a large church or an aircraft hangar – the last sound that you hear as it decays to nothing might be 5 seconds (or more!) after the thing made the noise, which means that the sound travelled a total of 5 sec * 344 m/s = 1.72 km bouncing around the church before finally arriving at your position.

So, if I put a loudspeaker that radiates simultaneously in all directions equally at all frequencies (audio geeks call this a point source) somewhere in a room, and I put a microphone that is equally sensitive to all frequencies from all directions (audio geeks call this an omnidirectional microphone) and I send an impulse (a “click”) out of the loudspeaker and record the output of the microphone, I’ll see something like this:

Some things to notice about that plot shown above

• There is some silence before the first sound starts. This is the time it takes for the sound to get from the loudspeaker to the microphone (travelling at about 344 m/s, and with an onset of about 30 ms, this means that the microphone was about 10.3 m away.
• There are some significant spikes in the signal after the first one. These are nice, clean reflections off some surfaces like walls, the floor or the ceiling.
• Mostly, this is a big mess, so it’s difficult to point somewhere else and say something like “that is the reflection off the coffee mug on the table over there, after the sound has already hit the ceiling and two walls on the way” for example…

So, this shows us something about how the room responds to an impulse over time. The nice (theoretical) thing is that this is a plot of what will happen to everything that comes out of the loudspeaker, over time, when captured at the microphone’s position. In other words, if you know the instantaneous sound pressure at any given moment at the output of the point-source loudspeaker, then you can go through time, multiplying that value by each value, moment by moment, in that plot to predict what will come out of the microphone. But this means that the total output of the microphone is all of the sound that came out of the loudspeaker over the 1000 ms plotted there, with each moment individually multiplied by each point on the plot – and all added together.

This may sound complicated, but think of it as a more simple example: When you’re sitting and listening to someone speak in a church, you can hear what that person just said, in addition to the reverberation (reflections) of what they said seconds ago. There is one theory that this is how harmony was invented: choirs in churches noticed that the reverb from the previous note blended nicely with the current note, and so chords were born.

## Frequency Response

There is a second really good reason for using an impulse to test a system. An impulse (in theory) contains all frequencies at the same level. This is a little difficult to wrap ones head around (at least, it took me years to figure out why…) but let me try to explain.

Any sound is the combination of some number of different frequencies, each with some level and some time relationship. This means that, I can start with the “ingredients” and add them together to make the sound I want. If I start with two frequencies: 1 Hz and 2 Hz and add them together, using cosine waves (a cosine wave is the same as a sine wave that starts 90º late), the result is as shown in Figure 2.

Let’s do this again, but increase the number to 5 frequencies: 1 Hz, 2 Hz, 3 Hz, 4 Hz, and 5 Hz.

You may notice that the peak at Time = 0 ms is getting bigger relative to the rest of the result. However, we get the same peak values at Time = -1000 ms and Time = 1000 ms. This is because the frequencies I’m choosing are integer values: 1 Hz, 2 Hz, 3 Hz, and so on. What happens if we use frequencies in between? Say, 0.1 Hz to 10 Hz in steps of 0.1 Hz, thus making 100 cosine waves added together? Now they won’t line up nicely every second, so the result looks like Figure 4.

Let’s get crazy. Figure 5 shows 10,000 cosine waves with frequencies of 0 to 100 Hz in steps of 0.01 Hz.

You may start to notice that the result of adding more and more cosine waves together at different frequencies is starting to look a lot like an impulse. It’s really loud at Time = 0 ms (whenever that is, but typically we think that it’s “now”) and it’s really quiet forever, both in the past and the future.

So, the moral of the story here is that if you click your fingers and make a “perfect” impulse, one philosophical way to think of this is that, at the beginning of time, cosine waves, all of them at different frequencies, started sounding – all of them cancelling each other until that moment when you decided to snap your fingers at Time = 0. Then they all continue until the end of time, cancelling each other out forever…

Or, another way to think of it is simply to say “an impulse contains all frequencies, each with the same amplitude”.

One small point: you may have noticed in Figure 5 that the impulse is getting big. That one added up to 10,001 – and we were just getting started. Theoretically, a real impulse is infinitely short and infinitely loud. However, you don’t want to make that sound because an infinitely loud sound will explode the universe, and that will wreck your analysis… It will at least clip your input.

## Equalisation

Let’s take a simple example of an equaliser. I’ll use an EQ to apply a boost of 12 dB with a centre frequency of 1 kHz and a Q of 2. (Note that “Q” has different definitions. The one I’ll be using here is where the Q = Fc / BW, where BW is the bandwidth in Hz between the -3 dB points relative to the highest magnitude. If you want to dig deeper into this topic, you can start here.) That filter will have a magnitude response that looks like this:

As you can see there, this means that a signal coming into that filter at 20 Hz or 20 kHz will come out at almost exactly the same level. At 1000 Hz, you’ll get 12 dB more at the output than the input. Other frequencies will have other results.

The question is: “how does the filter do that, conceptually speaking?”

That’s what we’ll look at in the next part of this series.

# “High-Res” Audio: Part 13: Wrapping up

As I’ve stated a couple of times through this series, my reason for writing this stuff was not to prove that high res audio is better or worse than normal res audio (whatever that is…). My reason was to highlight some of the advantages and disadvantages associated with LPCM audio at different bit depths and sampling rates. Just as a bullet-point summary of things-to-remember/consider (with some loose grouping):

• “High resolution audio” could mean
• “more than 16 bits per sample”
or
• “a sampling rate higher than 44.1 kHz”
or
• both.
• These two dimensions of the specifications have different implications on the signal

• Doubling the sampling rate only increases your audio bandwidth by 1 octave.
Yes, it’s twice as much information, but that’s only one octave. If you add an extra octave on top of a piano, you don’t get twice as many notes.
• Just because you have more bits per sample doesn’t mean that you are actually getting more resolution.
There are examples out there where a “24-bit recording” is just a 16-bit recording with 8 zeros stuck on the end.
• Just because you have a higher sampling rate doesn’t mean that you are actually getting a recording that was done at that sampling rate.
There are examples out there where, if you do a spectral analysis of a “high-res” recording, you’ll see the cutoff filter of the original 44.1 kHz recording.
• Just because you have a recording done at a higher sampling rate doesn’t mean that the extra information you get is actually useful.

• There are many cases where you want equipment that has higher specifications than your audio signal.
• If you have a volume control after the conversion to analogue, then 93 dB of dynamic range (16 bits, TPDF dithered) might be enough – especially if you listen to music with a limited dynamic range. However, if your volume control is in the digital domain, and you have a speaker that can play loudly, then you’ll probably want more dynamic range, and therefore more bits per sample hitting the DAC.

Like I said, I’m not here to tell you that one thing is better or worse than another thing.

As I said, my intention in writing all of this is to help you to never fall into the trap of assuming that “high resolution audio” is better than “normal resolution audio” in all respects.

More is not necessarily better, sometimes, it’s not even more. Don’t fall victim to misleading advertising.

# “High-Res” Audio: Part 12: Outputs

This series has flipped back and forth between talking about high resolution audio files & sources and the processing that happens in the equipment when you play it. For this posting, we’re going to deal exclusively with the playback side – regardless of the source content.

I work for a company that makes loudspeakers (among other things). All of the loudspeakers we make use digital signal processing instead of resistors, capacitors, and inductors because that’s the best way to do things these days…

Point 1: This means that our volume control is a gain (a multiplier) that’s applied to the digital signal.

We also make surround processors (most of our customers call them “televisions”) that take a multichannel audio input (these days, this is under the flag of “spatial audio”, but that’s just a new name on an old idea) and distribute the signals to multiple loudspeakers. Consequently, all of our loudspeakers have the same “sensitivity”. This is a measurement of how loud the output is for a given input.

Let’s take one loudspeaker model, Beolab 90, as an example. The sensitivity of this loudspeaker is set to be the same as all other Bang & Olufsen loudspeakers. Originally, this was based on an analogue signal, but has since been converted to digital.

Point 2: Specifically, if you send a 0 dB FS signal into a Beolab 90 set to maximum volume, then it will produce a little over 122 dB SPL at 1 m in a free field (theoretically).

Let’s combine points 1 and 2, with a consideration of bit depth on the audio signal.

If you have a DSP-based loudspeaker with a maximum output of 122 dB SPL, and you play a 16-bit audio signal with nothing but TPDF dither, then the noise floor caused by that dither will be 122 – 93 = 29 dB SPL which is pretty loud. Certainly loud enough for a customer to complain about the noise coming from their loudspeaker.

Now, you might say “but no one would play a CD at maximum volume on that loudspeaker” to which I say two things:

1. I do.
The “Banditen Galop” track from Telarc’s disc called “Ein Straussfest” has enough dynamic range that this is not dangerous. You just get very loud, but very short spikes when the gunshots happen.
2. That’s not the point I’m trying to make anyway…

The point I’m trying to make is that, if Beolab 90 (or any other Bang & Olufsen loudspeaker) used 16-bit DACs, then the noise floor would be 29 dB SPL, regardless of the input signal’s bit depth or dynamic range.

So, the only way to ensure that the DAC (or the bit depth of the signal feeding the DAC) isn’t the source of the noise floor from the loudspeaker is to use more than 16 bits at that point in the signal flow. So, we use a 24-bit DAC, which gives us a (theoretical) noise floor of 122 – 141 = -19 dB SPL. Of course, this is just a theoretical number, since there are no DACs with a 141 dB dynamic range (not without doing some very creative cheating, but this wouldn’t be worth it, since we don’t really need 141 dB of dynamic range anyway).

So, there are many cases where a 24-bit DAC is a REALLY good idea, even though you’re only playing 16-bit recordings.

Similarly, you want the processing itself to be running at a higher resolution than your DAC, so that you can control its (the DAC’s) signal (for example, you want to create the dither in the DSP – not hope that the DAC does it for you. This is why you’ll often see digital signal processing running at floating point (typically 32-bit floating point) or fixed point with a wider bit depth than the DAC.

# “High-Res” Audio: Part 11: How high can you go?

If you you get an audiometry test done, you’ll be shown into a small room, about the size of a public bathroom stall. Someone will put a pair of headphones on you, and pass you a small handle with a button. Your instructions are to press the button if you hear a tone. Then the audiometrist will leave the room, closing the door, and you’ll suddenly realise that if there’s any noise in this room, it’s because you’re making it.

Then you hear a beep in your left ear. You press the button. You hear a quieter beep. Press. Quieter beep. Press…. …. …. Beep, press… …. …. …. Beep, press…. New frequency beep, loud again. Press… and so on.

What’s happening here is that you’re presented with a sine tone at some frequency, probably loud enough for you to hear. You press. The tone gets quieter, and you press again. Eventually, the tone is so quiet that you cannot hear it (this is normal) so you don’t press. So, the tone gets louder, and you press. Then it gets quieter again, until you can’t hear it again.

By crossing over that threshold of “can hear” and “can’t hear” a couple of times, the audiometrist finds out whether or not you got lucky… If you bottom out at the same level a couple of times in a row, then that’s your threshold of hearing at that frequency in that ear.

The frequency changes (usually by 1 octave, but sometimes less), and the whole process is repeated.

If you get a full test done, then this is probably done at 9 frequencies (250, 500, 1k, 1.5k, 2k, 3k, 4k, 6k, and 8kHz) in both ears individually – 18 tests in all.

You’ll then be given a sheet of paper, or at least shown a plot of your hearing threshold. Typically, if you have “normal” hearing (whatever that means) your thresholds will all be sitting on a horizontal line marked 0 dB. If you’re “better than normal” then you get a negative score, if you’re “worse than normal” you get a positive score.

What does this mean?

Let’s start over.

If a lot of people do this test, and we only test at 1 kHz, we’ll find out that, after the results are averaged, the group can hear the 1 kHz sine tone when the change in air pressure at the ear entrance is 20 µPa. We’re not going to talk about what this means other than to say that “sound is a change in air pressure over time, and that pressure is measured in pascals, abbreviated Pa”. Needless to say, 20 µPa is pretty quiet, since it’s the quietest sound a group of people can hear at 1 kHz when you take their average.

If you did that test at a much lower frequency, you would find out that people aren’t as good at hearing quiet sounds. In other words, at 100 Hz, the sine tone has to be louder than 20 µPa for people to hear it.

The same is true if you repeated the test at a much higher frequency – say, 10,000 Hz.

If you did this test at a lot of frequencies, then you’d find out that, on average, the threshold of hearing for a human follows the bottom red line of the plot in Figure 1, borrowed from Wikipedia.

That bottom plot shows the threshold of hearing for different frequencies, plotted in dB SPL. Notice that, at 1 kHz, the line is at 0 dB SPL. This is because 0 dB SPL is defined to be the average threshold of hearing of a human at 1 kHz, which is 20 µPa. So, it’s not an accident…

Looking at that plot, you can see that, in order to hear a sine tone at 20 Hz, the tone has got to be more than 70 dB louder (that’s a LOT louder). So, a microphone “sees” a 73 dB SPL, 20 Hz sine tone as being louder than a 0 dB SPL, 1 kHz sine tone – but as far as you’re concerned, they’re both “the quietest sound you can hear” – therefore, they’re the same level.

If we take that threshold of hearing curve, and we play tones at those levels for those frequencies, then you should “just be able to” hear them. So, we’ll call those levels “0 dB” – since it’s the same as what is expected of you.

In other words, the piece of paper you got from the audiometrist tells you how much above or below that red threshold of hearing YOU sit.

Now, let’s back up a bit.

1. I said that, in your test, you only went up to 8 kHz. This is because, above that (and possibly even before that) the headphones might not be trust-worthy, and even a tiny movement (say a couple of millimetres) in the position of the headphones will have a (relatively) big effect on the level at your eardrum. So, rather than get people worried about losing their hearing at 20,000 Hz (when, in fact, they were actually just wearing the headphones 1 mm too far forward), you won’t get tested.
2. Notice how variable that threshold of hearing line is. There are big changes in level over the “audible” frequency range.
3. Remember that the threshold of hearing curve is an AVERAGE of a lot of people. Just like no one has 2.6 children, no one has this exact response. And, if you are some freak of nature and you DO have exactly that response, you don’t for long… we all get old…
4. Notice how that threshold of hearing curve only goes up to about 16 kHz, and above that it says “estimated”. See point #1.

Now, you should know that your ability to hear a sine tone at some frequency is defined as how your ability compares to an expectation based on an average, within a relatively small frequency band: 250 to 8 kHz.

Then you look at a textbook or you read a website that says “humans can hear from 20 Hz to 20 kHz”, which is not enough information to be either true or false… It’s like saying “humans are usually between 0 and 10 m tall” which is also sort of true, but also adequately vague to be potentially worse-than-useless information.

The truth is, unfortunately, much more complicated… However, it’s fair to say that, in order for you to just hear a sine tone at 20 kHz, it would have to be much, much louder than one at 1 kHz. In fact, if I played a 20 kHz sine tone loud enough for you to hear, measured that level, and then played a 1 kHz sine tone for you at the same level, you’d probably punch me – after you had passed out due to the pain, woken up, hunted me down, and found me… (I’d already have run away by then….)

## So what?

We humans like nice, tidy, answers. “It will rain tomorrow” is preferable to “there is a 70 – 80% chance of scattered showers in the afternoon tomorrow”. We even get mad when the information is correct, but we interpret it tidily… For example, we’ll complain about getting rained on in the middle of our hike, when there was only a 10% chance of rain. On the other hand, if there was a 10% chance of winning 1 Million dollars in the lottery, we’d all buy a ticket.

Anyways, once-upon-a-time, when the committee for inventing the compact disc was holding meetings, they said “what should the sampling rate be?” and someone said “at least 40 kHz, because we can hear up to 20 kHz”. (The reason it’s 44100 is related to the fact that the bits were stored as black and white stripes on video tape, and NTSC and PAL come close to meeting each other close to that number, when you look at the numbers of lines per field and frames per second.)

Of course, like any first-generation thing, digital recording equipment wasn’t very good at the start (back around 1980 or so) – so the first DDD recordings that were released on CD sounded… well…. weird. There was quantisation distortion because they hadn’t figured out dither yet, only 12 or 13 of the bit values were working properly on the ADC’s, the anti-aliasing filters were implemented as analogue circuits, so they let some stuff through that aliased, and they rang (“sang along”) with the signal at a high frequency… All of that added up to “weird” – possibly even “bad”. Then, people who had good equipment (high-end turntables or, even better, 1/4″ tape running at 30 ips) listened to this new format, decided it was bad, and that was that.

Some of them asked “why is is bad?” and one answer they came up with was the band limiting… If the system can’t capture or store or play materials above 20 kHz, then it’s useless… Right? Maybe…

Then, instruments were put in front of measurement microphones and spectra were measured – and the proof was in. Trumpets with harmon (wah-wah) mutes, when pointing directly at the microphone, contain harmonics as high as 50 kHz! This must explain why CDs sound bad! Right? Maybe…

Then Rupert Neve did a demo at an AES (Audio Engineering Society) convention where he played people two tones. Both were at 7 kHz, but one was a sine wave and the other was a square wave (at some level). The question was: have a listen and tell me which is which. The results were the same as if everyone was just guessing. (Remember that, in order to make a square wave, you need to add odd harmonics – so the lowest-frequency content difference between a 7 kHz sine wave and a 7 kHz square wave is at 21 kHz.) Proof that we don’t need to go above 20 kHz, right? Maybe…

Some years ago, I took some “high resolution” audio files and measured their spectral content. One particularly interesting result is shown in Figures 2, below.

Look at that spike in the top end – around 20 kHz. What musical instrument makes that sound? The answer is “no musical instrument makes that sound – at least none of the baroque instruments in that recording make that sound. As I wrote back in 2014:

If you’re wondering what it might be, I asked a bunch of smart friends, and the best explanation we can come up with is that it’s noise from a switched-mode power supply that is somehow bleeding into the recording. HOW it’s bleeding into the recording is a potentially interesting question for recording engineers. One possibility is that one of the musicians was charging up a phone in the room where the microphones were – and the mic’s just picked up the noise. Another possibility is that the power supply noise is bleeding electrically into the recording chain – maybe it’s a computer power supply or the sound card and the manufacturer hasn’t thought about isolating this high frequency noise from the audio path. Or, maybe it’s something else.

Interestingly, this is a conflict of two engineers. The designer of the power supply (assuming that’s what it is…) said “I’ll put the switching frequency above 20 kHz so that no one will hear it” and the recording engineer said “I’ll record this at 96 kHz so that people can get the content they’re missing…” The problem is that the content you’re missing is something you don’t want…

Similarly, if you listen to Eric Clapton’s “Unplugged” album with headphones or loudspeakers that have a low-enough low-frequency range, you’ll hear a loud thump, thump, thump going along with the music. This is the sound of someone tapping their foot on a temporary stage floor, shaking a vocal microphone. In my not-very-humble opinion, that should never have made it out to the public release. However, my guess is that the speakers it was mastered on didn’t go low enough… (OR, it was an artistic decision, and I would have done it differently.) Assuming that I’m right, then this is a second example where a “better” system sounds “worse”.

Of course, through all of this, I have assumed that your loudspeakers or headphones can produce the signals that we’re talking about in the direction that you’re sitting in, and that those signals are not being masked by other sounds in the room (like phone chargers singing…) However, to complicate things with reality would just be too far to go today…

## Conclusions?

I don’t have any, but I have some questions and (as usual) some opinions…

• Does a harmon mute on a trumpet produce energy at 50 kHz, if you’re sitting right in front of it?
Yes.
• Do you want to sit right in front of a trumpet with a harmon mute?
Debatable.
• Can a high-res audio recording include the sound of a phone charger?
Yes.
• Do you want to have an expensive recording of a baroque ensemble with obligato phone charger?
Probably not – the charger is not in Buxtehude’s original score as far as I can see.
• Can you hear the difference between a 7 kHz sine and a 7 kHz square wave?
Depends on the speaker / headphone, the listening position, the background noise level, and whether or not you were out clubbing last night. Heads or tails?
• Will you feel better by knowing that your file contains “audio” content above 20 kHz? Probably.
Placebos have been known to work bigger miracles than this. (But don’t forget the stuff I said about sampling rate converters earlier…)