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Aperture effect
The reconstruction process of FGR. 6 only operates exactly as shown if the impulses are of negligible duration. In many DACs this is not the case, and many keep the analog output constant for a substantial part of the sample period or even until a different sample value is input. This produces a waveform which is more like a staircase than a pulse train.
The case where the pulses have been extended in width to become equal to the sample period is known as a zero-order-hold system and has a 100 percent aperture ratio. Note that the aperture effect is not apparent in a track-hold system; the holding period is only for the convenience of the quantizer which then outputs a value corresponding to the input voltage at the instant hold mode was entered.
It was shown in Section 3 that whereas pulses of negligible width have a uniform spectrum, which is flat within the audio band, pulses of 100 percent aperture ratio have a sin x/x spectrum which is shown in FGR. 16.
The frequency response falls to a null at the sampling rate, and as a result is about 4 dB down at the edge of the audio band. If the pulse width is stable, the reduction of high frequencies is constant and predictable, and an appropriate equalization circuit can render the overall response flat once more. An alternative is to use resampling which is shown in FGR. 17. Resampling passes the zero-order-hold waveform through a further synchronous sampling stage which consists of an analog switch which closes briefly in the centre of each sample period. The output of the switch will be pulses which are narrower than the original. If, for example, the aperture ratio is reduced to 50 percent of the sample period, the first frequency response null is now at twice the sampling rate, and the loss at the edge of the audio band is reduced. As the figure shows, the frequency response becomes flatter as the aperture ratio falls. The process should not be carried too far, as with very small aperture ratios there is little energy in the pulses and noise can be a problem. A practical limit is around 12.5 percent where the frequency response is virtually ideal.
The term resampling will also be found in descriptions of sampling rate convertors, where it refers to the process of finding samples at new locations to describe the original waveform. The context usually makes it clear which meaning is intended.
FGR. 17 (a) Resampling circuit eliminates transients and reduces aperture
ratio. (b) Response of various aperture ratios.
Quantizing
Quantizing is the process of expressing some infinitely variable quantity by discrete or stepped values. Quantizing turns up in a remarkable number of everyday guises. FGR. 18 shows that an inclined ramp enables infinitely variable height to be achieved, whereas a step-ladder allows only discrete heights to be had. A step-ladder quantizes height.
When accountants round off sums of money to the nearest pound or dollar they are quantizing. Time passes continuously, but the display on a digital clock changes suddenly every minute because the clock is quantizing time.
FGR. 18 An analog parameter is continuous whereas a quantized parameter
is restricted to certain values. Here the sloping side of a ramp can be used
to obtain any height whereas a ladder only allows discrete heights.
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FGR. 19 Quantizing assigns discrete numbers to variable voltages. All voltages
within the same quantizing interval are assigned the same number which causes
a DAC to produce the voltage at the centre of the intervals shown by the dashed
lines in (a).
This is the characteristic of the mid-tread quantizer shown in (b). An alternative system is the mid-riser system shown in (c). Here 0 volts analog falls between two codes and there is no code for zero. Such quantizing cannot be used prior to signal processing because the number is no longer proportional to the voltage. Quantizing error cannot exceed ±1 /2Q as shown in (d).
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In audio the values to be quantized are infinitely variable voltages from an analog source. Strict quantizing is a process which operates in the voltage domain only. For the purpose of studying the quantizing of a single sample, time is assumed to stand still. This is achieved in practice either by the use of a track-hold circuit or the adoption of a quantizer technology which operates before the sampling stage.
FGR. 19(a) shows that the process of quantizing divides the voltage range up into quantizing intervals Q, also referred to as steps S. In applications such as telephony these may advantageously be of differing size, but for digital audio the quantizing intervals are made as identical as possible. If this is done, the binary numbers which result are truly proportional to the original analog voltage, and the digital equivalents of mixing and gain changing can be performed by adding and multiplying sample values. If the quantizing intervals are unequal this cannot be done. When all quantizing intervals are the same, the term uniform quantizing is used. The term linear quantizing will be found, but this is, like military intelligence, a contradiction in terms.
The term LSB (least significant bit) will also be found in place of quantizing interval in some treatments, but this is a poor term because quantizing works in the voltage domain. A bit is not a unit of voltage and can have only two values. In studying quantizing, voltages within a quantizing interval will be discussed, but there is no such thing as a fraction of a bit.
Whatever the exact voltage of the input signal, the quantizer will locate the quantizing interval in which it lies. In what may be considered a separate step, the quantizing interval is then allocated a code value which is typically some form of binary number. The information sent is the number of the quantizing interval in which the input voltage lies.
Whereabouts that voltage lies within the interval is not conveyed, and this mechanism puts a limit on the accuracy of the quantizer. When the number of the quantizing interval is converted back to the analog domain, it will result in a voltage at the centre of the quantizing interval as this minimizes the magnitude of the error between input and output.
The number range is limited by the word-length of the binary numbers used. In a sixteen-bit system, 65 536 different quantizing intervals exist, although the ones at the extreme ends of the range have no outer boundary.
Quantizing error
It’s possible to draw a transfer function for such an ideal quantizer followed by an ideal DAC, and this is also shown in FGR. 19. A transfer function is simply a graph of the output with respect to the input.
In audio, when the term linearity is used, this generally means the straightness of the transfer function. Linearity is a goal in audio, yet it will be seen that an ideal quantizer is anything but linear.
FGR. 19(b) shows the transfer function is somewhat like a staircase, and zero volts analog, corresponding to all zeros digital or muting, is half-way up a quantizing interval, or on the centre of a tread. This is the so-called mid-tread quantizer which is universally used in audio. FGR. 19(c) shows the alternative mid-riser transfer function which causes difficulty in audio because it does not have a code value at muting level and as a result the numerical code value is not proportional to the analog signal voltage.
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FGR. 20 At (a) an arbitrary signal is represented to finite accuracy by PAM
needles whose peaks are at the centre of the quantizing intervals. The errors
caused can be thought of as an unwanted signal (b) added to the original. In
(c) the amplitude of a quantizing error needle will be from -1/2Q to +1/2Q
with equal probability. Note, however, that white noise in analog circuits
generally has Gaussian amplitude distribution, shown in (d).
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Quantizing causes a voltage error in the audio sample which is given by the difference between the actual staircase transfer function and the ideal straight line. This is shown in FGR. 19(d) to be a sawtooth-like function which is periodic in Q. The amplitude cannot exceed ± 1/2Q peak-to-peak unless the input is so large that clipping occurs.
Quantizing error can also be studied in the time domain where it’s better to avoid complicating matters with the aperture effect of the DAC.
For this reason it’s assumed here that output samples are of negligible duration. Then impulses from the DAC can be compared with the original analog waveform and the difference will be impulses representing the quantizing error waveform. This has been done in FGR. 20. The horizontal lines in the drawing are the boundaries between the quantizing intervals, and the curve is the input waveform. The vertical bars are the quantized samples which reach to the centre of the quantizing interval. The quantizing error waveform shown at (b) can be thought of as an unwanted signal which the quantizing process adds to the perfect original. If a very small input signal remains within one quantizing interval, the quantizing error is the signal.
As the transfer function is non-linear, ideal quantizing can cause distortion. As a result practical digital audio devices deliberately use non ideal quantizers to achieve linearity. The quantizing error of an ideal quantizer is a complex function, and it has been researched in great depth.
It’s not intended to go into such depth here. The characteristics of an ideal quantizer will be pursued only far enough to convince the reader that such a device cannot be used in quality audio applications.
As the magnitude of the quantizing error is limited, its effect can be minimized by making the signal larger. This will require more quantizing intervals and more bits to express them. The number of quantizing intervals multiplied by their size gives the quantizing range of the convertor. A signal outside the range will be clipped. Provided that clipping is avoided, the larger the signal, the less will be the effect of the quantizing error.
Where the input signal exercises the whole quantizing range and has a complex waveform (such as from orchestral music), successive samples will have widely varying numerical values and the quantizing error on a given sample will be independent of that on others. In this case the size of the quantizing error will be distributed with equal probability between the limits. FGR. 20(c) shows the resultant uniform probability density.
In this case the unwanted signal added by quantizing is an additive broadband noise uncorrelated with the signal, and it’s appropriate in this case to call it quantizing noise. This is not quite the same as thermal noise which has a Gaussian probability shown in FGR. 20(d) (see Section 3 for a treatment of probability). The difference is of no consequence as in the large signal case the noise is masked by the signal. Under these conditions, a meaningful signal-to-noise ratio can be calculated as follows:
In a system using n-bit words. there will be 2n quantizing intervals. The largest sinusoid which can fit without clipping will have this peak-to peak amplitude. The peak amplitude will be half as great, i.e. 2n-1 Q and the rms amplitude will be this value divided by _2. The quantizing error has an amplitude of 1/2Q peak which is the equivalent of Q/___ 12 rms. The signal-to-noise ratio for the large signal case is then given by:
By way of example, a sixteen-bit system will offer around 98.1 dB SNR.
Whilst the above result is true for a large complex input waveform, treatments which then assume that quantizing error is always noise give results which are at variance with reality. The expression above is only valid if the probability density of the quantizing error is uniform.
Unfortunately at low levels, and particularly with pure or simple waveforms, this is simply not the case.
At low audio levels, quantizing error ceases to be random, and becomes a function of the input waveform and the quantizing structure as FGR. 20 showed. Once an unwanted signal becomes a deterministic function of the wanted signal, it has to be classed as a distortion rather than a noise. Distortion can also be predicted from the non-linearity, or staircase nature, of the transfer function. With a large signal, there are so many steps involved that we must stand well back, and a staircase with 65 000 steps appears to be a slope. With a small signal there are few steps and they can no longer be ignored.
The non-linearity of the transfer function results in distortion, which produces harmonics. Unfortunately these harmonics are generated after the anti-aliasing filter, and so any which exceed half the sampling rate will alias. FGR. 21 shows how this results in anharmonic distortion within the audio band. These anharmonics result in spurious tones known as bird-singing. When the sampling rate is a multiple of the input frequency the result is harmonic distortion. This is shown in FGR. 22.
Where more than one frequency is present in the input, intermodulation distortion occurs, which is known as granulation.
As the input signal is further reduced in level, it may remain within one quantizing interval. The output will be silent because the signal is now the quantizing error. In this condition, low-frequency signals such as air conditioning rumble can shift the input in and out of a quantizing interval so that the quantizing distortion comes and goes, resulting in noise modulation.
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FGR. 21 Quantizing produces distortion after the anti-aliasing filter; thus
the distortion products will fold back to produce anharmonics in the audio
band. Here the fundamental of 15 kHz produces second and third harmonic distortion
at 30 and 45 kHz.
This results in aliased products at 40 - 30 = 10 kHz and 40 - 45 = (-)5 kHz.
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FGR. 22 Mathematically derived quantizing error waveform for sine wave sampled
at a multiple of itself. The numerous autocorrelations between quantizing errors
show that there are harmonics of the signal in the error, and that the error
is not random, but deterministic.
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Needless to say, any one of the above effects would preclude the use of an ideal quantizer for high-quality work. There is little point in studying the adverse effects further as they should be and can be eliminated completely in practical equipment by the use of dither. The importance of correctly dithering a quantizer cannot be emphasized enough, since failure to dither irrevocably distorts the converted signal: there can be no process which will subsequently remove that distortion.
The signal-to-noise ratio derived above has no relevance to practical audio applications as it will be modified by the dither and by any noise shaping used.