Databases Reference
In-Depth Information
F I GU R E 13 . 13
Sinan image coded at 0.5 bits per pixel using the JPEG algorithm.
13.7 Application to Audio Compression
__
The
MDCT
As mentioned in the previous section, the use of the block-based transform has the unfortunate
effect of causing distortion at the block boundaries at low rates. A number of techniques that
use overlapping blocks have been developed over the years [
198
]. One that has gained wide
acceptance in audio compression is a transform based on the discrete cosine transform called
the modified discrete cosine transform (MDCT). It is used in almost all popular audio coding
standards from MP3 and AAC to Ogg Vorbis.
The MDCT used in these algorithms uses 50% overlap. That is, each block overlaps half
of the previous block and half of the next block of data. Consequently, each audio sample is
part of two blocks. If we were to keep all the frequency coefficients, we would end up with
twice as many coefficients as samples. Reducing the number of frequency coefficients results
in the introduction of distortion in the inverse transform. The distortion is referred to as time
domain aliasing [
199
]. The reason for the name is evident if we consider that the distortion is
being introduced by subsampling in the frequency domain. Recall that sampling at less than the
Nyquist frequency in the time domain leads to an overlap of replicas of the frequency spectrum,
or frequency aliasing. The lapped transforms are successful because they are constructed in
such a way that while the inverse transform of each block results in time-domain aliasing, the
aliasing in consecutive blocks cancels each other out.
Consider the scenario shown in Figure
13.15
. Let's look at the coding for block
i
and
block
i
1. The inverse transform of the coefficients resulting from both these blocks results
in the audio samples in the subblock
q
. We assume that the blocksize is
N
, and, therefore, the
subblock size is
N
+
/
2. The forward transform can be represented by an
N
/
2
×
N
matrix
P
.
