Thursday, August 23, 2012

Basic Audio EQs

In my last post, I looked at why it's usually better to do EQ (or filtering) in the time domain than the frequency domain as far as audio is concerned, but I didn't spend much time explaining how you might implement a time-domain EQ. That's what I'm going to do now.

The theory behind time-domain filters could fill a book. Instead of trying to cram you full of theory we'll just skip ahead to what you need to know to do it. I'll assume you already have some idea of what a filter is.

Audio EQ Cookbook

The Audio EQ Cookbook by Robert Bristow-Johnson is a great, albeit very terse, description of how to build basic audio EQs. These EQs can be described as second order digital filters, sometimes called "biquads"because the equation that describes them contains two quadratics. In audio, we sometimes use other kinds of filters, but second order filters are a real workhorse. First order filters don't do much: they generally just allow us to adjust the overall balance of high and low frequencies. This can be useful in "tone control" circuits, like you might find on some stereos and guitars, but not much else. Second order filters give us more control -- we can "dial in" a specific frequency, or increase or decrease frequencies above and below a certain threshold, with a fair degree of accuracy, for example. If we need even more control than a second order filter offers, we can often simply take several second order filters and place them in series to simulate the effect of a single higher order filter.

Notice I said series, though. Don't try putting these filters in parallel, because they not only alter the frequency response, but also the phase response, so when you put them in parallel you might get unexpected results. For example, if you take a so-called all-pass filter and put it in parallel with no filter, the result will not be a flat frequency response, even though you've combined the output of two signals that have the same frequency response as the original signal.

Using the Audio EQ Cookbook, we can design a peaking, high-pass, low-pass, band-pass, notch (or band-stop), or shelving filter. These are the basic filters used in audio. We can even design that crazy all-pass filter I mentioned which actually does come in handy if you are building a phaser. (It has other uses, too, but that's for another post.)

Bell Filter

Let's design a "bell", or "peaking" filter using RBJ's cookbook. Most other filters in the cookbook are either similar to the bell or simpler, so once you understand the bell, you're golden. To start with, you will need to know the sample rate of the audio going into and coming out of your filter, and the center frequency of your filter. The center frequency, in the case of the bell filter, is the frequency that "most affected" by your filter. You will also want to define the width of the filter, which can be done in a number of ways usually with some variation on "Q" or "quality factor" and "bandwidth". RBJ's filters define bandwidth in octaves, and you want to be careful that you don't extend the top of the bandwidth above the Niquist frequency (or 1/2 the sample rate), or your filter won't work. We also need to know how much of our center frequency to add in dB (if we want to remove, we just use a negative value, and for no change, we set that to 0).

Fs = Sample Rate
F0 = Center Frequency (always less than Fs/2)
BW = Bandwidth in octaves
g = gain in dB

Great! Now we are ready to begin our calculations. First, RJB suggests calculating some intermediate values:

A = 10^(g/40)
w0 = 2*pi*f0/Fs c = cos(w0) s = sin(w0) alpha = s*sinh( ln(2)/2 * BW * w0/s )

This is a great chance to use that hyperbolic sin button on your scientific calculator that, until now, has only been collecting dust. Now that we've done that, we can finally calculate the filter coefficients, which we use when actually processing data:

b0 = 1 + alpha*A b1 = -2*c b2 = 1 - alpha*A a0 = 1 + alpha/A a1 = -2*c a2 = 1 - alpha/A

Generally speaking, we want to "normalize" these coefficients, so that a0 = 1. We can do this by dividing each coefficient by a0. Do this in advance or the electrical engineers will laugh at you:

b0 /= a0 b1 /= a0 b2 /= a0 a1 /= a0 a2 /= a0

Now, in pseudocode, here's how we process our data, one sample at a time using a "process" function that looks something like this:

number xmem1, xmem2, ymem1, ymem2;

void reset() {
xmem1 = xmem2 = ymem1 = ymem2 = 0;
number process( number x ) {
number y = b0*x + b1*xmem1 + b2*xmem2 - a1*ymem1 - a2*ymem2;

xmem2 = xmem1;
xmem1 = x;
ymem2 = ymem1;
ymem1 = y;

return y;

You'll probably have some kind of loop that your process function goes in, since it will get called once for each audio sample.

There's actually more than one way to implement the process function given that particular set of coefficients. This implementation is called "Direct Form I" and happens to work pretty darn well most of the time. "Direct form II" has some admirers, but those people are either suffering from graduate-school-induced trauma or actually have some very good reason for doing what they are doing that in all likelihood does not apply to you. There are of course other implementations, but DFI is a good place to start.

You may have noticed that the output of the filter, y, is stored and used as an input to future iterations. The filter is therefore "recursive". This has several implications:

  • The filter is fairly sensitive to errors in the recursive values and coefficients. Because of this, we need to take care of what happens with the error in our y values. In practice, on computers, we usually just need to use a high resolution floating point value (ie double precision) to store these (on fixed point hardware, it is often another matter).
  • Another issue is that you can't just blindly set the values of your coefficients, or your filter may become unstable. Fortunately, the coefficients that come out of RJB's equations always result in stable filters, but don't go messing around. For example, you might be tempted to interpolate coefficients from one set of values to another to simulate a filter sweep. Resist this temptation or you will unleash the numerical fury of hell! The values in between will be "unstable" meaning that your output will run off to infinity. Madness, delirium, vomiting and broken speakers are often the unfortunate casualties.
  • On some platforms you will have to deal with something called "denormal" numbers. This is a major pain in the ass, I'm sorry to say. Basically it means our performance will be between 10 and 100 times worse than it should be because the CPU is busy calculating tiny numbers you don't care about. This is one of the rare cases where I would advocate optimizing before you measure a problem because sometimes your code moves around and it comes up and it's very hard to trace this issue. In this case, the easiest solution is probably to do something like this (imagine we are in C for a moment):

#DEFINE IS_DENORMAL(f) (((*(unsigned int *)&(f))&0x7f800000) == 0)
float xmem1, xmem2, ymem1, ymem2;

void reset() {
xmem1 = xmem2 = ymem1 = ymem2 = 0;
float process( float x ) {
number y = b0*x + b1*xmem1 + b2*xmem2 - a1*ymem1 - a2*ymem2;

if( IS_DENORMAL( y ) )
y = 0;

xmem2 = xmem1;
xmem1 = x;
ymem2 = ymem1;
ymem1 = y;

return y;

Okay, happy filtering!

Wednesday, August 8, 2012

Why EQ Is Done In the Time Domain

In my last post, I discussed how various audio processing may be best done in the frequency or time domain. Specifically, I suggested that EQ, which is a filter that alters the frequency balance of a signal, is best done in the time domain, not the frequency domain. (See my next post if you want to learn how to implement a time-domain filter.)

If this seems counter intuitive to you, rest assured you are not alone. I've been following the "audio" and "FFT" tags (among others) on stack overflow and it's clear that many people attempt to implement EQs in the frequency domain, only to find that they run into a variety of problems.

Frequency Domain Filters

Let's say you want to eliminate or reduce high frequencies from your signal. This is called a "low-pass" filter, or, less commonly, a "high-cut" filter. In the frequency domain, high frequencies get "sorted" into designated "bins", where you can manipulate them or even set them to zero. This seems like an ideal way to do low-pass filtering, but lets explore the process to see why it might not work out so well.

Our first attempt at a low-pass filter, implemented with the FFT might look something like this:
  • loop on audio input
  • if enough audio is received, perform FFT, which gives us audio in the frequency domain
    • in frequency domain, perform manipulations we want. In the case of eliminating high frequencies, we set the bins representing high frequencies to 0.
    • perform inverse FFT, to get audio back in time domain
    • output that chunk of audio

But there are quite a few problems with that approach:
  • We must wait for a chunk of audio before we can even begin processing, which means that we will incur latency in our processing. The higher quality filter we want, the more audio we need to wait for. If the input buffer size does not match the FFT size, extra buffering needs to be done.
  • The FFT, though efficient compared to the DFT (which is the FFT without the "fast" part), performs worse than linear time, and we need to do both the FFT and it's inverse, which is computationally similar. EQing with the FFT is therefore generally very inefficient compared to comparable time-domain filters.
  • Because our output chunk has been processed in the frequency domain independent of samples in neighboring chunks, the audio in neighboring chunks may not be continuous. One solution is to process the entire file as one chunk (which only works for offline, rather than real-time processing, and is computationally expensive). The better solution is the OLA or Overlap Add method but this involves complexity that many people miss when implementing a filter this way.
  • Filters implemented via FFT, as well as time-domain filters implemented via IFFT, often do not perform the way people expect. For example, many people expect that if they set all values in bins above a certain frequency to 0, then all frequencies above the given frequency will be eliminated. This is not the case. Instead, frequency responses at the bin values will be 0, but the frequency response between those values is free to fluctuate -- and it does fluctuate, often greatly. This fluctuation is called "ripple." There are techniques for reducing ripple but they are complex, and they don't eliminate ripple. Note that, in general, frequencies across the entire spectrum are subject to ripple, so even just manipulating a small frequency band many create ripple across the entire frequency spectrum.
  • FFT filters suffer from so-called "pre-echo", where the sounds can be heard before the main sound hits. In and of itself, this isn't really a problem, but sounds are "smeared" so badly by many designs, that many in the audio world feel that these filters can effect the impact of transients and stereo imaging if not implemented and used correctly.
So it's clear that FFT filters may not be right, or if they are, they involve much more complexity than many people first realize.

As a side note, one case where it might be worth all that work is a special case of so-called FIR filters (also sometimes called "Linear phase" filters). These are used sometimes in audio production and in other cases. In audio, they are usually used only in mastering because of their high latency and computational cost, but even then, many engineers don't like them (while others swear by them). FIR filters are best implemented in the time domain, as well, until the number of "taps"in the filter becomes enormous, which it sometimes does, and it actually becomes more efficient to implement using an FFT with OLA. FIR filters suffer from many of the problems mentioned above including pre-echo, high computational cost and latency, but they do have some acoustical properties that make them desirable in some applications.

Time Domain Filters

Let's try removing high frequencies in the time domain instead. In the time domain, high frequencies are represented by the parts of the signal that change quickly, and low frequencies are represented as the parts that change slowly. One simple way to remove high frequencies, then, would be to use a moving average filter:

y(n) = { x(n) + x(n-1) + .... + x(n-M) } / (M+1)

where x(i) is your input sample at time i, and y(i) is your output sample at time i. No FFT required for that (This is not the best filter for removing high frequencies -- in fact we can do WAY better -- but it is my favorite way to illustrate the point. The moving average filter is not uncommon in economics, image processing and other fields partly for this reason.). Several advantages are immediately obvious, and some are not so obvious:
  • Each input sample can be processed one at a time to produce one output sample without having to chunk or wait for more audio. Therefore, there are also no continuity issues and minimal latency.
  • It is extremely efficient, with only a few multiplies, adds and memory stores/retrievals required per sample.
  • These filters can be designed to closely mimic analog filters.
A major disadvantage is that it is not immediately obvious how to design a high-quality filter in the time domain. In fact, it can take some serious math to do so. It's also worth noting that many time-domain filters, like frequency domain filters, also suffer from ripple, but for many design methods, this ripple is well defined and can be limited in various ways.

In the end, the general rule is that for a given performance, you can get much better results with the time-domain than the frequency domain.

Saturday, August 4, 2012

When to (not) use the FFT

In the last post I discussed one use for the FFT: pitch tracking. I also mentioned that there were better ways to do pitch tracking. Indeed, aside from improvements on that method, you could also use entirely different methods that don't rely on the FFT at all.

The FFT transforms data into the "frequency domain", or, if your data is broken down into chunks, the FFT transforms it into the "time-frequency domain," which we often still think of as the frequency domain. However, the most basic "domain" you can work in is usually the "time domain." In the time domain, audio is represented as sequence of amplitude values. You may know this as "PCM" audio. This is what's usually stored in WAVs and AIFs, and when we access audio devices like soundcards, this is the most natural way to transfer data. It turns out we can also do a whole lot of processing and analysis in the time domain as well.

ProcessTime DomainFrequency Domain
Pitch ShiftingOkayOkay
Pitch TrackingOkayOkay
Guitar effects
SR ConversionYes!No!
Panning, Mixing, etcYes!No!
Table 1: Recommendations for Audio Processing in the Time Domain vs. the Frequency Domain

Wow, so impulse reverb is really the only thing on that list you need an FFT for? Actually even that can be done in the time domain, it's just much more efficient in the frequency domain (so much so that it might be considered impossible in the time domain).

You might wonder how to adjust the frequency balance of a signal, which is what an EQ does, in the time domain rather than the frequency domain. Well, you can do it in the frequency domain, but you are asking for trouble. I'll talk about this in my next post.