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If ideal low-pass anti-aliasing and anti-image filters are assumed, having a vertical cut-off slope at half the sampling rate, an ideal spectrum shown in FGR. 6(a) is obtained. It was shown in Section 2 that the impulse response of a phase linear ideal low-pass filter is a sin x/x waveform in the time domain, and this is repeated in (b). Such a waveform passes through zero volts periodically. If the cut-off frequency of the filter is one half of the sampling rate, the impulse passes through zero at the sites of all other samples. It can be seen from FGR. 6(c) that at the output of such a filter, the voltage at the centre of a sample is due to that sample alone, since the value of all other samples is zero at that instant. In other words the continuous time output waveform must join up the tops of the input samples. In between the sample instants, the output of the filter is the sum of the contributions from many impulses, and the waveform smoothly joins the tops of the samples. If the time domain is being considered, the anti-image filter of the frequency domain can equally well be called the reconstruction filter. It’s a consequence of the band-limiting of the original anti-aliasing filter that the filtered analog waveform could only travel between the sample points in one way. As the reconstruction filter has the same frequency response, the reconstructed output waveform must be identical to the original band-limited waveform prior to sampling. It follows that sampling need not be audible. The reservations expressed by some journalists about 'hearing the gaps between the samples' clearly have no foundation whatsoever. A rigorous mathematical proof of reconstruction can be found in Betts.
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FGR. 6 If ideal 'brick wall' filters are assumed, the efficient spectrum of
(a) results. An ideal low-pass filter has an impulse response shown in (b).
The impulse passes through zero at intervals equal to the sampling period.
When convolved with a pulse train at the sampling rate, as shown in (c),
the voltage at each sample instant is due to that sample alone as the impulses
from all other samples pass through zero there.
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FGR. 7 As filters with finite slope are needed in practical systems, the sampling
rate is raised slightly beyond twice the highest frequency in the baseband.
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The ideal filter with a vertical 'brick-wall' cut-off slope is difficult to implement. As the slope tends to vertical, the delay caused by the filter goes to infinity: the quality is marvelous but you don't live to hear it. In practice, a filter with a finite slope has to be accepted as shown in FGR. 7. The cut-off slope begins at the edge of the required band, and consequently the sampling rate has to be raised a little to drive aliasing products to an acceptably low level. There is no absolute factor by which the sampling rate must be raised; it depends upon the filters which are available and the level of aliasing products which are acceptable. The latter will depend upon the word length to which the signal will be quantized.
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