Example of Oversampling

1. Example of Oversampling

2. Aims and learning outcomes To demonstrate how oversampling can improve the quality of conversion for acoustic signals. • Students will be able to: • contrast the representations: PCM and DPCM; • describe the components of an oversampling ADC; • describe the advantages of using differential coding and oversampling convertors; • describe noise shaping as a system to reduce S/N ratio; • calculate the improvement in S/N ratio for a simple sigma DPCM convertor

3. Introduction Oversampling is sampling at a higher rate than is strictly needed. It is very popular in quality digital audio equipment. Advantages include less circuitry (cheaper) and better S/N ratio. It allows less expensive circuits to be used, as the sampling frequency is traded off against the number of bits needed to represent the signal. Fundamental to oversampled convertors, is the representation of signals in Differential Post Code Modulation (DPCM), so this is where we will start.

4. What is DPCM?

5. Definitions PCM is the familiar digital representation of an analog signal. It consists of a series of pulses, evenly spaced in time, the pulse heights being proportional to the analog signal at the sample time. DPCM is formed from the difference between adjacent PCM signal values If the PCM signal is PCM x1, x2, x3, .... xn, then the DPCM signal is (x2-x1), (x3-x2) .... (xn - xn-1) DPCM is effectively numerical differentiation of a PCM signal, hence the name.

6. Exercise for students • Three samples of a PCM signal are 0.15, 0.25, 0.57, 0.1, -0.1 volts, the first signal is at t=0 and the time interval is 1x10-4s. Calculate the equivalent DPCM from t=0 to t=4x10-4s. 2) Sketch the DPCM equivalent of the PCM signals shown in two figures to the right. (assume the sampling frequency is 10 kHz.)

7. Answers 1) time (ms) 0 0.1 0.2 0.3 0.4 DPCM (volts) Undefined 0.1 0.32 -0.47 -0.2 2) The middle figure gives a signal like (pk=pk amp. 4V). The right figure gives a DPCM signal of Acos(2Bft) where A is 0.6283 volts.

8. Why and more on DPCM Why DPCM ? Because it gives a better S/N ratio in convertors by using a process of noise shaping – see below. As we are now converting the difference between adjacent samples, rather than the samples themselves, there is no longer a limit to how large a signal the system can represent. The limiting factor is, however, how big a difference between adjacent samples it can represent - so system is limited by its slew rate. To encode large differences between adjacent samples, the convertor can either increase the number of bits that the quantizer produces, or increase the sampling frequency.

9. The signal output from a DPCM falls at 6dB/octave compared to a PCM signal. (it is limited by the slew rate). The noise floor due to quantization error and dither is constant across all frequencies. Obviously, this signal roll off (shown right above) is undesirable, so the signal is integrated (shown below right). This tilts the noise floor, so noise is greater at high frequencies - hence the sigma DPCM converters are known as noise shaping converters. This will be exploited in the oversampling systems we will discuss.

10. Oversampling Figure 1 illustrates that in a digital system we can play off the number of bits against the sampling frequency. If we lower the number of bits representing a signal, we can still get an accurate rendition of the input signal provided we increase the sampling frequency (the number of steps). This fact is key to the use of oversampling converters which are commonly used on quality audio systems..

12. Consider a 4x oversampling system for an audio system designed to work to 20kHz. Non-oversampling system: Analogue signal  anti-aliasing filter 24dB/oct, -3dB point 18kHz  AD convertor  Digital out, fs = 40kHz, 16 bit Oversampling system: Analogue signal  anti-aliasing filter 6dB/oct  AD convertor (14 bit, fs=160kHz)  Digital low pass filter, 24dB/oct, -3dB point 18kHz  Digital filter to decrease sample rate  Digital out, fs = 40kHz, 16 bit

13. Oversampling is used because: • Analogue anti-aliasing filters are expensive due to the high quality analogue components needed. In an oversampling system, the analogue anti-aliasing filter can have a very slow roll off and be very cheap. • When oversampled, the noise floor rises with frequency (see figure above). Consequently, this noise can be filtered out by the digital low pass filter as it is outside the audio frequency range. In this 4x oversampling system the signal to noise ratio is improved by 12dB. This is a process called noise shaping.

14. We can use less circuitry for the convertor and so reduce the cost. As the sampling frequency increases the number of bits we need to hold the same information decreases. By increasing the sampling frequency, we can reduce the word length used in conversion. Disadvantage of oversampling • The data storage is much more inefficient, but with SACD and similar technology oversampled data is now being stored directly (i) because computer storage now cheap (ii) removes up and down sampling stages which are detrimental to quality.

15. As we can trade off sample rate verses number of quantization bits in a DPCM system - see above - there is no reason why we can’t go to the extreme of only having a one bit quantizer. This is delta modulation. The output is simply +1 or 0 depending on whether the previous signal values are less than or greater than the current value. It obviously has to work at an extremely high sample rate. One advantage is that the conversion circuit can be very simple.

16. This produces single bit DPCM information. If we then integrate the output from the delta modulator, we get PCM. Such a convertor is known as a delta-sigma modulator. (shown above) So when considering the effects of sampling in an audio system, the simple considerations of quantisation error, and the simple formulations for dynamic range given before are not correct. Previously, we have assumed that any quantisation error produced stays within the audio band, but in an oversampling system the error is deliberately shifted out of the audio range so that it can be removed by low-pass filtering in the digital domain. If 16 bit convertor can be replaced by a delta-modulator, than the noise in the audio range decreases by a factor of (216 ≡ 48dB!) compared with non-oversampling.

17. MATLAB tasks Quite a few concepts have been introduced so far. Now it is time to show how realistic these are by doing some MATLAB simulations. (Incidentally, real researchers developing convertors use these sought of MATLAB demonstrations). In the course notes, the one-bit convertor was shown (figure above). It may not be immediately obvious that this achieves noise-shaping and so improves signal to noise ratio. The intention of this tutorial exercise is to prove its operation through the MATLAB script oversampling.m. Run and examine the workings at output of the script.

18. There are many things to note, but of primary interest: • Note that a 1 bit signal does really convey frequency information (Figures 2 and 3). This can seem fantastical, but you can see it in the time picture (if you stare at this long enough you can also see a picture by Bridget Riley). • Figure 3 shows the input frequency and all the additional frequencies generated by quantisation. Notice the rising noise floor due to this being a noise shaping convertor. The frequencies of interest are to the far left of the graph, and so the noise >16kHz is going to be lost when we downsample. • Figure 4 shows the input and output signals. Notice they are not identical. The sine wave is distorted by the quantisation – some of this is the effect of dither, the rest is the rounding errors.

19. Figure 5 shows the final spectrum of the converted signal – the noise floor is too high to be useful, higher order integrators are needed. • Listen to the sound. At the end of the script the signal before and after conversion is played. (These are played alternatively 5 times). Notice that you can hear the dither noise on the after conversion signal, but there is no tonal change in the pure tone. Remove the dither from the convertor by commenting out the relevant line. Rerun and listen. Notice that the tonal changes, this is the problem of signal dependent distortion as was found for a non-oversampled case. Look for evidence of distortion in the graphs.

20. Oversampling in DAC Figure 1 Noise shaping in DAC - after Watkinson pg157, Z-1 means delay one sample

21. The process of error interaction leading to noise shaping is also used in DA conversion. The interpolating filter generates 28 bit word length, which is then requantized to 28 bits before being transmitted through a DAC, reconstruction filter and then is the analog output signal. The 14 LSB from the requantizer are delayed by one sample and added to the interpolating digital filter output.

22. The 14 LSBs from the last sample (the quantization error) and 28 bits fr om the present sample interfere. Consequently we no longer have a white noise spectrum but something different. The RH graph was generated aong these principles using a spreadsheet. The bottom line is a triangular dither quantization error spectrum – which is white noise as expected. The top line has been noise shaped by adding the quantization error of the previous sample to the signal before quantization. A rising noise floor can be seen. If this rising noise floor shifts the noise beyond the audible range – which is possible in an oversampling system. This quantization error noise is then lost and inaudible. Better S/N ratio.