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

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

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

22. Sound Blaster block diagram