Ideally, the width of each band should be roughly one critical band. In practice, there isn't much much point in having a single frequency bin per critical band, so although the ear has roughly 25 critical bands, we only use about 15-20 in CELT. Here's an example using 256-sample MDCTs (128 output samples) and 15 bands. The band boundaries are:

{0, 2, 4, 6, 8, 12, 16, 20, 24, 28, 36, 44, 52, 68, 84, 116, 128}

Using this, band number 0 includes samples 0 and 1, while band number 14 includes samples 84 to 115. The remaining samples (116-127) are just discarded because they are outside the ear's range (a 44.1 kHz sampling rate is assumed here).

Now, the first thing we need to do is actually encode the energy of each band in an efficient way. The ear is more sensitive to lower frequencies, so these will need to be encoded with better resolution. Of course, we use the log (dB) domain and add a small value (equivalent to -10 dB) just to prevent overflows when taking the log. In this example, we use a quantisation interval of 0.75 dB for the lowest band, increasing linearly to 4.25 dB for the highest band. Doing naive quantisation/encoding over a fixed range would require a prohobitive number of bits (>100 bits per frame) and is thus not an option. Measuring the ideal entropy (assuming a perfect probability model for the data) for same speech and music samples gives us an average of 71 bits per frame. That's still expensive, considering we're going to encode around 200 frames per second.

The only way to further reduce the number of bits used for energy quantisation is to eliminate redundancy. Energy usually doesn't vary that much from one frame to the next, so we can use a time-domain predictor of the form P(z) = 1 - alpha*z^-1. That means we remove from the current energy alpha times the previous energy (we're already in log domain). Here's what the entropy per frame looks like (as a function of alpha) if we use that predictor:

That's already much better. As we increase the prediction coefficient (alpha) from 0 to 1, we can reduce the entropy from 71 bits down to around 45 bits, a 26 bits improvement. Unfortunately, using alpha=1 for prediction isn't practical because it would mean that any transmission error (e.g. lost packet) would propagate through time with no attenuation (even 0.95 would take too long). A value of alpha around 0.7 would be a nice tradeoff between redundancy reduction and limited error propagation. That's 52 bits per frame. However, we're not done yet eliminating redundancy. There's still a correlation across the bands in the same frame. This time, we can use any predictor we like because a frame either arrives completely or it doesn't. So we use a second predictor Q(z) = (1 - z^-1)/(1 - beta*z^-1). With that second predictor, the entropy goes down again:

With alpha=0.7 and beta=0.5, we have just under 44 bits of entropy. Much better than the 71 bits we started from and even better than only the first predictor with alpha=1. Of course, that entropy value is optimistic because it assumes a perfect probability model and because it assumes that prediction isn't degraded by quantisation.

For encoding, it's not very practical to use the actual probability model because would require storing the probability for each value of each band (and for each bit-rate if we change the resolution). However, it turns out that the distributions are somewhere between a Gaussian distribution and a Laplacian distribution. Although actually closer to being Gaussian, we use a Laplacian model because it reduces the spikes in bit-rate (a Gaussian would significantly underestimate the probability of extreme values). Despite the rough approximation, the average actual encoding rate for all 15 bands is 46 bits per frame. That's just 2 bits worse than the theoretical best case using that predictor. Not bad at all.

I've also played with the DCT (for intra-frame redundancy) without getting better results, mainly because it's harder to control the error in each band. Still, there may be better ways that what I've done so far to reduce the bit-rate for the energy. I'm open to ideas/suggestions on that.

**Updated:**Fixed the definition of the example bands.