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Appendix I: Digitization noise - can adding noise really help?

Digitization noise, also called quantization noise, is an artifact caused by the rounding or truncation of numbers to a fixed number of figures. It can originate in the analog-to-digital converter that converts an analog signal to a digital one, or in the circuitry or software involved in transmitting the digital signal to a computer, or even in the process of transferring the data from one program to another, as in copying and pasting data to and from a spreadsheet.  The result is a series of non-random steps of equal height. The frequency distribution is white, because of the sharpness of the steps, as you can see by observing the power spectrum.

            


The figure on the left, top panel, shows the effect of integer digitization on a sine wave with an amplitude of +/- 10. Ensemble averaging, which is usually the most effective of noise reduction techniques, does not reduce this type of noise (bottom panel) because it is non-random.

Interestingly, if additional random noise is present in the signal, then ensemble averaging becomes effective in reducing
both the random noise and the digitization noise. In essence, the added noise randomizes the digitization, allowing it to be reduced by ensemble averaging. Moreover, if there is insufficient random noise already in the signal, it can be beneficial to add additional noise artificially! The script RoundingError.m illustrates this effect, as shown the animation on the right, which shows the digitized sine wave with gradually increasing amounts of added random noise in line 8 (generated by the randn.m function) followed by ensemble averaging of 100 repeats (in lines 17-20). Look closely at the waveform in this animation as it changes in response to the random noise addition shown in the title. You can clearly see how the noise starts out mostly quantization noise but then quickly decreases as small but increasing amounts of random noise are added before the ensemble averaging step, then eventually increases as too much noise is added. The optimum standard deviation of random noise is about 0.36 times the quantization size, as you can demonstrate by adding lesser or greater amounts via the variable Noise in line 6 of this script. Note that this works only for ensemble averaged signals where the noise is added before the quantification.


  An
audible example of this idea is illustrated by the Matlab/Octave script DigitizedSpeech.m, which starts with an audio recording of the spoken phrase "Testing, one, two, three", previously recorded at 44000 Hz and saved in WAV format (TestingOneTwoThree.wav) and in .mat format (testing123.mat), rounds off the amplitude data progressively to 8 bits (256 steps; sound link), shown on the left, 4 bits (16 steps; sound link), and 1 bit (2 steps; sound link), and then the 2-step case again with random white noise added before the rounding (2 steps + noise; sound link), plots the waveforms and plays the resulting sounds, demonstrating both the degrading effect of rounding and the remarkable improvement caused by adding noise. (Click on these sound links to hear the sounds on your computer). Although the computer program in this case does not perform an explicit ensemble averaging operation as does RoundingError.m, it's likely that the neurons of the hearing center of your brain provide that function by virtue of their response time and memory effect.

This page is part of "A Pragmatic Introduction to Signal Processing", created and maintained by Prof. Tom O'Haver , Department of Chemistry and Biochemistry, The University of Maryland at College Park. Comments, suggestions and questions should be directed to Prof. O'Haver at toh@umd.edu. Updated July, 2022.