The discussion so far has assumed that perfect anti-aliasing and reconstruction filters are used. Perfect filters are not available, of course, and because designers must use devices with finite slope and rejection, aliasing can still occur. It’s not easy to specify anti-aliasing filters, particularly the amount of stopband rejection needed. The amount of aliasing resulting would depend on, among other things, the amount of out-of-band energy in the input signal. Very little is known about the energy in typical source material outside the audible range. As a further complication, an out-of-band signal will be attenuated by the response of the anti-aliasing filter to that frequency, but the residual signal will then alias, and the reconstruction filter will reject it according to its attenuation at the new frequency to which it has aliased. To take the opposite extreme, if a microphone were used which had no response at all above the audio band, no anti-aliasing filter would be needed.
It could be argued that the reconstruction filter is unnecessary, since all the images are outside the range of human hearing. However, the slightest non-linearity in subsequent stages would result in gross intermodulation distortion. Most transistorized audio power amplifiers become grossly non-linear when fed with signals far beyond the audio band. It’s this non-linearity which enables amplifiers to demodulate strong radio transmissions. The simple solution is to curtail the response of power amplifiers somewhat beyond the audio band so that they become immune to passing taxis and refrigerator thermostats. This is seldom done in Hi-Fi amplifiers because of the mistaken belief that response far beyond the audio band is needed for high fidelity. The truth of the belief is academic as all known recorded or broadcast music sources, whether analog or digital, are band-limited. As a result there is nothing to which a power amplifier of excess bandwidth can respond except RF interference and inadequately suppressed images from digital sources. The possibility of damage to tweeters and beating with the bias systems of analog tape recorders must also be considered.
Consequently a reconstruction filter is a practical requirement. It would, however, be acceptable to bypass one of the filters involved in a copy from one digital machine to another via the analog domain, although a digital transfer is, of course, to be preferred.
Every signal which has been through the digital domain has passed through both an anti-aliasing filter and a reconstruction filter. These filters must be carefully designed in order to prevent artifacts, particularly those due to lack of phase linearity as they may be audible. The nature of the filters used has a great bearing on the subjective quality of the system. Entire books have been written about analog filters, so they will only be treated briefly here.
FGR. 8 and 9 show the terminology used to describe the common elliptic low-pass filter. These filters are popular because they can be realized with fewer components than other filters of similar response. It’s a characteristic of these elliptic filters that there are ripples in the passband and stopband. Lagadec and Stockham7 found that filters with passband ripple cause dispersion: the output signal is smeared in time and, on toneburst signals, pre-echoes can be detected. In much equipment the anti-aliasing filter and the reconstruction filter will have the same specification, so that the passband ripple is doubled with a corresponding increase in dispersion. Sometimes slightly different filters are used to reduce the effect.
It’s difficult to produce an analog filter with low distortion. Passive filters using inductors suffer non-linearity at high levels due to the B/H curve of the cores. It seems a shame to go to such great lengths to remove the non-linearity of magnetic tape from a recording using digital techniques only to pass the signal through magnetic inductors in the filters. Active filters can simulate inductors which are linear using op amp techniques, but they tend to suffer non-linearity at high frequencies where the falling open-loop gain reduces the effect of feedback. Active filters can also contribute noise, but this is not necessarily a bad thing in controlled amounts, since it can act as a dither source.
FGR. 8 The important features and terminology of low-pass filters used for
anti-aliasing and reconstruction.
FGR. 9 (a) Circuit of typical nine-pole elliptic passive filter with frequency
response in (b) shown magnified in the region of cut-off in (c). Note phase
response in (d) beginning to change at only 1 kHz, and group delay in (e),
which require compensation for quality applications. Note that in the presence
of out-of-band signals, aliasing might only be 60 dB down. A 13-pole filter
manages in excess of 80 dB, but phase response is worse.
It’s instructive to examine the phase response of such filters. Since a sharp cut-off is generally achieved by cascading many filter sections which cut at a similar frequency, the phase responses of these sections will accumulate. The phase may start to leave linearity at only a few kiloHertz, and near the cut-off frequency the phase may have completed several revolutions. As stated, these phase errors can be audible and phase equalization is necessary. An advantage of linear phase filters is that ringing is minimized, and there is less possibility of clipping on transients.
It’s possible to construct a ripple-free phase-linear filter with the required stopband rejection, but it’s expensive in terms of design effort and component complexity, and it might drift out of specification as components age. The money may be better spent in avoiding the need for such a filter. Much effort can be saved in analog filter design by using oversampling. Strictly, oversampling means no more than that a higher sampling rate is used than is required by sampling theory. In the loose sense an 'oversampling convertor' generally implies that some combination of high sampling rate and various other techniques has been applied.
Oversampling is treated in depth in a later section of this section. The audible superiority and economy of oversampling convertors has led them to be almost universal. Accordingly the treatment of oversampling in this volume is more prominent than that of filter design.
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FGR. 10 At normal speed, the reconstruction filter correctly prevents images
entering the baseband, as at (a). When speed is reduced, the sampling rate
falls, and a fixed filter will allow part of the lower sideband of the sampling
frequency to pass. If the sampling rate of the machine is raised, but the filter
characteristic remains the same, the problem can be avoided, as at (c).
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