Upsampling vs. Oversampling for Digital Audio
Vast amounts of marketing efforts are placed on touting the latest and greatest technological advancements in the realm of home audio. We are all aware of the over-inflated, and often baseless claims that companies tend to make when advertising their new products. The latest A/V receiver and A/V Processor offerings are currently marketing super high sampling rates and wide bit words for processing digital audio signals. The ability to upsample to these extreme rates is a main advertising point for many A/V receivers. So does upsampling to higher rates really provide better sound? The following will briefly try to go into the details behind why upsampling is used and if it really is the answer to better sound.
Basics of Sampling - Oversampling and Upsampling
To start with, we must review the system as a whole and look at the terminology. Two main parts of the whole system we are going to consider are upsampling and oversampling. In the purely mathematical context, they are similar operations. When practically implemented though, oversampling refers to using a higher sampling rate than needed to run the A/D or D/A converter thus increasing the rate of the signal. Upsampling is on the other hand a rate conversion from one rate to another arbitrary rate. Oversampling in the ADC has been around for quite a bit of time, while upsampling of audio that results in a simple rate conversion is relatively newer. The figure below shows an example of oversampling. We assume that we can sufficiently sample the analog waveform at twice the bandwidth of the signal as to prevent any aliasing in each case. In one case we are sampling at some frequency, f s and in the other case we are at twice the same sampling rate. The resulting data samples are shown to the right in the figure. In both cases we have assumed that we are at twice the Nyquist rate for the signal to prevent any aliasing.
Figure 1 : Oversampling Operation
A rate conversion in the middle of the digital processing, or upsampling, looks slightly different depending on how it is specifically implemented. The simplest of these implementations entails zero stuffing the original sample stream to increase the sample rate. Other implementations may create the additional samples by taking some weighted averaging of the samples in the original rate. In almost all cases, the upsampling process also includes an interpolation filter to get rid of the images of the original signal.
A simple block diagram of a processing chain is shown below. The Analysis filter sits prior to the ADC and isolates our signal of interest before we sample it. Oversampling is performed at the ADC and then the signal is sent to the digital processing chain that does the filtering and any DSP operations. The upsampler (if any) sits somewhere between the DSP and filtering. Just like the ADC, the DAC also oversamples the signal. Oversampling at the DAC and ADC will make the design of all our filters much simpler as we will see.
Figure 2 : Basic Signal Processing Chain
Effects on Frequency Response
To get a better idea of what is going on with our original signal when we upsample or oversample, we need to look at it in the frequency domain. That will allow us to see what benefits we gain by going to a higher sample rate. The figure below shows the result of taking our standard audio spectrum that ranges from 20Hz to 20kHz and then increasing its rate by 8x.
Figure 3 : Oversampling/Upsampling Effects on Frequency
Whether we upsample or oversample, the effect on the spectrum of our audio signal is similar. Instead of our signal of interest occupying almost our entire bandwidth all the way up to fs/2 (22.05 kHz), it now only occupies 1/8 th of that. This allows us the use of a very simple analysis filter at the head of our processing chain, the internal digital filters, or the reconstruction filter at the end of the chain. Both the processes of upsampling and oversampling give us this benefit. Without increasing the sample rate, we would need to design a very sharp filter that would have to cutoff at just past 20kHz and be 80-100dB down at 22kHz. Such a filter is not only very difficult and expensive to implement, but may sacrifice some of the audible spectrum in its rolloff. If we examine the spectrum at the increased rate, we can see that the filter can roll off gently well past 22kHz and as long as it is down in the cutoff region at 176.4kHz, the image created by the sampling process will easily be removed. The analog filter after the D/A converter is responsible for removing the audio signal's image as well as the frequency spurs caused by the DACs integration steps. An analog filter with a smooth roll off will have nicer phase characteristics as well.
To demonstrate these benefits, let's take a look at two analog filters: one that must operate at Nyquist, and one that can operate at 64x Nyquist. The filter that operates at Nyquist, must have a very sharp cutoff and a higher order. The figure below shows a 10 th order Bessel lowpass filter. We can see that the filter rolloff is very sharp and the corresponding phase response is nonlinear towards the higher frequencies. Such variations in phase are undesirable in our audio signal. The plots below are a function of radians rather than hertz and are in logarithmic scale. On the radians scale, our audio signal occupies 125.66 rad/s up to 138544 rad/s, which roughly corresponds to 10^2 and 10^5 on the plots.
Figure 4 : Analog Filter at Nyquist
Now lets look at a filter that operates on our bandlimited audio signal at 64x Nyquist. This filter is a 3 rd order Bessel lowpass filter that cuts off well after our audio spectrum. The rolloff is much gentler, but the phase response is notably better. It is linear over almost the entire audio spectrum, which extends from 0 rad/s up to 138544 rad/s.
Figure 5 : Analog Filter at 64x Nyquist
The phase response of the analog reconstruction filter after the DAC is a function of the type of filter used and how much oversampling the DAC uses. A higher oversampling will allow for a more linear phase response over the audio spectrum for a given analog filter structure. The DACs oversampling to a higher rate allows for a reasonable analog filter design that gives us linear phase. The key point is that the oversampling in the DAC and the oversampling in the ADC are both important parts of the processing that have been used for a very long time.
Upsampling would give us the same benefits in frequency response that we have gone over, however we can achieve the same effects by sufficiently oversampling our signal both at the DAC and ADC. Upsampling has no effect on our digital filter design problem since all our digital filters are FIR (finite impulse response) and all have linear phase. By sufficiently oversampling at the ADC, we can design a very simple, linear phase, digital filter that has no problems with our audio signal. There has been much misinformation surrounding upsampling and many claims have been made that state that upsampling is necessary to allow for such a desirable digital filter. However, it is the oversampling at the ADC takes care of this, not upsampling. To demonstrate, lets take our audio signal that has been oversampled at 8x Nyquist and design a digital filter for it.
Below we have a symmetric digital FIR filter. The plot below is in normalized frequency where 1 = 176.4kHz. Our audio signal extends from approximately .00011338 up to .125. From this we can see that the passband of our filter is smooth over this frequency range and that the phase response is linear. At just past 22kHz, the response of the filter is down only 6 db and falls off to below -120dB soon after.
Figure 6 : Digital Filter at 8x Nyquist
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