Digital to Analogue Conversion

1. Digital to Analogue Conversion • Natural signals tend to be analogue • Need to convert to digital

2. Sample and Hold Analogue To digital converter Sampling • Need to hold the signal steady for long enough to enable the A to D converter to generate an output

3. Quantization converts continuous value to a discrete (usually integer) value • Output value is rounded so accuracy lost • Maximum quantization error of ±0.5 lsb • Error is combined with signal as noise

4. Quantization accuracy • Least significant bit determines accuracy. So for a 2 Volt peak to peak signal, an 8 bit converter can accurately represent multiples of 7.81mV but anything in between will be rounded

5. Quantization error Error is at most ±1/2 an lsb, or ±3.905 mV for the 8 bit converter or ±15.25µV in the 16 bit case

6. Quantization error • Relatively small signal changes are subject to severe quantization errors

7. Quantization error creates steps • Steps create distortion which is visible in the frequency domain • Noise shown on dB scale as it is relatively small compared to the signal

8. Sampling theory • Signal needs to be sampled at twice the speed of the fastest change to be captured • Shannon or the Nyquist sampling theorem, (authors of 1940s papers) • Theorem states that a continuous signal can be properly sampled, only if it does not contain frequency components above one-half of the sampling rate

9. Correctly sampled signal • Signal frequency is 0.09 of the sample rate (i.e. sample rate is about 11x signal freq) • e.g. 90Hz signal sampled at 1kHz

10. Sample rate still ok • Signal frequency is 0.31 of the sample rate (i.e. sample rate about 3x signal freq) • 3.2 samples / cycle but freq still preserved

11. Improper Sampling • Signal frequency is 0.95 of the sample rate (i.e. sample rate only slightly higher than signal freq) • Only 1.05 samples per cycle. Produces a 0.05Hz alias signal which is mixed with the original

12. Sidebands • Sampling a signal is effectively multiplication of signals in the time domain • Multiples of the sample frequency are produced as well as sum and difference frequencies (sidebands)

13. No sampling no sidebands • Time domain to frequency domain of an analogue signal

14. Sampled signal produces sidebands

15. Incorrectly sampled signal • Breaching Nyquist causes aliasing with overlapping sidebands

16. Simulated sampling • Using a sample rate of 1kHz, the frequency spectrum with noise was calculated from: • Then modified to illustrate aliasing by changing 300Hz signal to 800Hz:

17. Correct sample rate

18. Aliasing at 200Hz

19. Anti-alias filter • Frequencies higher than those of interest (such as noise) need to be blocked before sampling. Use an analogue low pass filter

21. Analogue Anti-alias Filter Sample and Hold Analogue to digital converter Digital to analogue converter Analogue Reconstruction Filter DSP DSP system • Low pass input filter removes F > 0.5F(S) • Reconstruction filter removes high frequency F(S) multiples

22. Sound Blaster block diagram