Subband Coders

Subband splitting of a signal is often performed with a two-channel uniform filterbank. The filterbank is used several times to achieve finer bandsplitting, in a filter tree:

$$0-20000\ Hz\ \rightarrow \ 0-10000\ Hz, 10000-20000\ Hz$$

$$0-10000\ Hz\ \rightarrow \ 0-5000\ Hz, 5000-10000\ Hz\ ...$$

This way, the coder can split the input data into uniform bands, in each of which a quantizer, adopted to the masking threshold (see section 3), is applied.

The filterbank filterbank consists of four filters, $F_0(\omega)$, $F_1(\omega)$, $G_0(\omega)$ and $G_1(\omega)$. The $F$'s are used to split the input data in a high and a low frequency band. Both bands are then subsampled with a factor of 2. To reconstruct, both bands are upsampled again, and filtered with the $G$'s. Thus, the output $\hat{X}(\omega)$ can be written as

$$\hat{X}(\omega) = \frac{1}{2}(F_0(\omega)G_0(\omega)+F_1(\omega)G_1(\omega))X(\omega)$$

$$+ \frac{1}{2}(F_0(\omega+\pi)_0(\omega)+F_1(\omega+\pi)G_1(\omega)) X(\omega+\pi)\ {\rm (aliasing)}.$$ (1)

To obtain aliasing cancellation, the following is required:

$$G_0(\omega) = F_1(\omega + \pi)$$ (2)

$$-G_1(\omega) = F_0(\omega + \pi).$$ (3)

Optimally,

$$\frac{1}{2}(F_0(\omega)G_0(\omega)+F_1(\omega)G_1(\omega)) = 1, \ \omega\in[0..\pi],$$ (4)

so that perfect reconstruction is achieved. This is, though, often hard to achieve. The filters used in most coders are Quadrature Mirror Filters (QMF), which achieve aliasing cancellation by choosing $F_1$ as the mirror image of $F_0$ (around $\pi/2$):

$$F_1(\omega) = F_0(\omega + \pi) = -G_1(\omega) = G_0(\omega + \pi).$$ (5)

To get good reconstruction without relying on the aliasing cancellation, which cannot be relied on in the presence of quantization noise, the QMF filters have to have a steep pass-to-stop-band transition.