Digital Signal Processing Reference
In-Depth Information
To ease the requirements on the edge steepness of the band-limiting filters, the
sampling frequency can be chosen accordingly higher than the cutoff frequency. Tak-
ing the highest frequency audible to the human ear and the requirement of doubling
this frequency given by the sampling theorem into consideration, one arrives at the
typical value of 44.1 kHz as used in CD audio. For speech digitisation, lower values
of 16 kHz (broad-band telephony) or even 8 kHz (narrow-band 'standard' telephony)
are typically chosen.
In addition to the time discretisation by sampling, the continuous analogue values
need to be discretised to digital (binary) values [
5
]. The word length
w
of the binary
number is usually limited (mostly 16 bit as in CD audio, or 8 bit as in narrow-band
telephone speech). In the case of binary representation the number
Q
of quantisation
steps is:
2
w
Q
=
.
(6.6)
This limited number of steps results in a quantisation error which is the devia-
tion between the original value and its quantised counter part. This error leads to
quantisation noise. For linear quantisation, i.e., in case of equally sized quantisation
intervals, this quantisation noise can be estimated in terms of signal-to-noise ratio
(SNR)
r
q
as:
10
lg
P
S
20
lg
2
w
r
q
=
P
N
20
lgQ
=
[
dB
]
,
(6.7)
where
P
S
is is the standard power of the preferred signal, and
P
N
is the according
power of the unwanted noise. For longer word lengths this means [
2
]:
r
q
6dB/bit
.
(6.8)
Better values can be reached by adapting the quantisation steps to the signal
characteristics such as by the ITU's A-law as primarily used in Europe or the
-law
as primarily used in Northern America and Japan for telephony in the ITU G.711
standard for Pulse Code Modulation (
http://www.itu.int/rec/T-REC-G.711/e
)
.
μ
6.1.2 Short Time Analysis
Audio signals change over time, i.e., they are time variant [
6
]. Thus, the main para-
meters are also time-dependent. However, one can make the assumption that these
parameters change relatively slower than the signal frequencies. The evolution over
time of the parameters are thus new signals which are, however, sampled with a
considerably lower frequency as compared to the original signal [
7
]. Their sampling
frequency will be referred to as parameter sampling frequency in the ongoing in
contrast to the signal sampling frequency as was introduced above.
The short time analysis considers the signal in a given short interval within which
the audio signal is considered to be stationary [
6
]. To this end, a weighting of the
