Filter for digital-to-analog converters

Details Filter for digital-to-analog converters Filter for digital-to-analog converters

2000-03-09

2002-01-08

JeanPierre, Peguy (Department: 2819)

Coded data generation or conversion

Analog to or from digital conversion

Differential encoder and/or decoder

C341S155000, C341S144000

Reexamination Certificate

active

06337645

ABSTRACT:

BACKGROUND OF THE INVENTION
1. Field of the Invention
The invention relates generally to improvements in digital-to-analog (D-to-A) conversion, and, more particularly, to improvements in such conversion for audio signals. It is most useful in high quality audio for music reproduction in devices such as Compact Disc (CD) players, DVD players, etc.
2. Description of Related Art
Essentially all modem analog-to-digital (A-to-D) and D-to-A converters used for audio operate at a sample rate higher than the output/input sample rate. These converters are known as over-sampling converters, and they use digital filters to decimate down to the output sample rate in the case of A-to-D converters and to interpolate up from the input sample rate in the case of D-to-A converters. An important reason for this configuration is that filters are necessary to minimize distortions present in both A-to-D and D-to-A conversion processes caused by unwanted frequencies, and digital filters are more stable, more reproducible, and less expensive to implement than analog filters of an equivalent quality.
FIG. 1
a
is a simplified block diagram of an A-to-D converter and
FIG. 1
b
is a simplified block diagram of a corresponding D-to-A converter.
The actual sampling rate for the converters themselves in over-sampling digital converters may be many times higher than the input/output sample rate. The decimation or interpolation process is normally done in several stages, with the last decimation filter and the first interpolation filter normally done with a two-to-one frequency ratio. These filters for the last/first operations have the most effect on the sonic performance of audio converters because their cutoff frequencies are close to the frequencies in the program material.
FIG. 2
a
shows the frequency response of a typical filter used in an A-to-D converter as the final decimation filter.
FIG. 2
b
shows the response of a typical filter used in a D-to-A converter as the first interpolation filter. The Y-axis of the graphs shows the magnitude of the amplitude response of the filters in decibels, and the X-axis shows the frequency as a fraction of the output/input sampling rate, Fs. The reason that we need to examine both the A-to-D and D-to-A filters is that they function as a system in determining the effects of several distortions in the output signal
180
(
FIG. 1
b
).
The frequency 0.5 in the middle of the graphs (
FIGS. 2
a
and
2
b
), the Nyquist frequency, has a special significance. It is important because the sampling theorem states that in a sampled data system, frequencies above one half the sampling frequency cannot be uniquely represented by the sampled data stream, in the case of the A-to-D converter, any frequencies above the Nyquist frequency in the input signal that are not removed by the decimation filter appear as spurious frequencies in the output known as alias frequencies or alias distortion. The ideal decimation filter from an alias distortion perspective would pass all frequencies below 0.5 Fs and no energy above a frequency of 0.5 Fs. Such a filter is not realizable in practice, but practical filters usually try to approximate the ideal response. Any residual frequencies above Nyquist in the original signal fold over or alias into frequencies below Nyquist in the output signal
120
(
FIG. 1
a
), with the relationship that a frequency f in the input
90
becomes Fs-f in the output
120
.
An alias distortion mechanism also exists in D-to-A converters in the interpolation process. The incoming digital signal
122
(
FIG. 1
b
) can be considered to have no frequencies above 0.5 Fs. The first stage of interpolation consists of adding zero value samples in between each of the original samples to double the sample rate and then passing the result through a low pass filter with a frequency response such as in
FIG. 2
b.
The result is that the zero valued samples are replaced by values that are interpolated from the surrounding data.
The distortion arises from the fact that new frequencies are created above the original Nyquist frequency and that these new frequencies correspond to frequencies present in the original signal. In order to analyze the potential impact of this distortion, it is useful to graph the composite frequency response of the decimation/interpolation system. If one takes the frequency response of the A-to-D decimation filter in
FIG. 2
a
and performs the equivalent decimation followed by inserting the zero value samples prior to the interpolation filter, one gets a frequency response shown in
FIG. 3
a.
For each frequency below the Nyquist
200
, a new frequency above Nyquist is created. These have the same relationship, i.e., f_new equals FS-f, that alias products have in the A-to-D case, as can be seen from the symmetry about Nyquist
200
. These new frequencies, f_new, are uniquely represented because the sampling rate is now twice as great.
If one now adds the cascade of the interpolation filter response of
FIG. 2
b,
one gets the composite response shown in FIG.
3
b. The frequencies above 0.5 are signals which were not there in the original signal and are alias distortion products. They fall into two general groups: those corresponding to the stop band of the interpolation filter
220
, and those associated with the transition band behavior of both the decimation and interpolation filters
210
.
Many people consider these distortion products to be of little importance because they are extra signals above the band of interest and are inaudible in the case of a CD or any other system with a sampling rate greater than 40 kHz. If everything in the audio system following the interpolation filter were really linear, this would be true. Unfortunately, the real world is not strictly linear. Non-linearities exist in the D-to-A converter, small signal amplifiers, power amplifiers, loudspeakers, and even human hearing.
In
FIG. 3
b,
the acceptable level of stop band distortion products
220
determine the stop band performance requirement for the interpolation filter. The only way to reduce these distortions is to improve the performance of the interpolation filter stop band rejection.
The transition band distortions at
210
are more limited in frequency range, but they have much higher amplitude and can cause really audible problems in the output of a system. As an example, consider a cymbal crash in music, which generates large amplitude high frequency components. For each component just below Nyquist, there is a corresponding one at a mirror image frequency above Nyquist, and each pair of original frequency and alias frequency will generate a difference component when it encounters a non-linearity later in the system. In the case of a CD system with these filters, these difference components are in the frequency range of 0 to 5 kilohertz, where human hearing is very sensitive and where they are not masked very well by the signal that created them. They result in a “dirty” sound to the cymbals, which is very typical of digital systems.
Transition band distortions
210
result primarily from the transition band behavior of the interpolation filter. The type of filter that is normally used in this position in a system design is called a half-band filter. As can be seen from FIG.
2
b,
it is 6 dB down at Nyquist with considerable response above the 0.5 frequency. It is used in most systems because it is very economical to implement computationally, and because it has good time domain behavior. It is a symmetrical finite impulse response (FIR) filter with linear phase response in which all even order coefficients except the middle one are exactly zero, and therefore, those multiplications do not have to be performed. This type of filter is used on the vast majority of commercial D-to-A converters designed for audio use.
A prior art approach to solving the problems associated with half-band interpolation filters is covered in U.S. Pat. Nos. 5,479,168 and 5,808,574 and related materials. The solution that is optimal from a performance point of view is to use an

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