How does Pre-emphasis mitigate finite precision effects?

Question

"The digitised speech signal $s(n)$ is put through a low order digital system (usually first order FIR filter) to spectrally flatten the signal and make it less susceptible to finite precision effects later in the signal processing"

This is from Fundamentals of Speech Recognition (Rabiner) material.

Gathered three reason but I could not reason out any of them.

1. Pre-emphasis compensates for the natural energy decay in the higher frequencies of speech.

How does just boosting the higher frequency components relative to its lower counterparts mitigate finite precision effects?

2. By emphasizing higher frequency components the signal to noise ratio is improved.

Doesn't noise occur usually at the high frequencies? Then shouldn't it rather decrease SNR?

3. This rapid reduction in energy from low to higher frequencies leads to practical problems in implementations.

If we would implement a discrete Fourier transform with fixed-point arithmetic, then the accuracy would be very different in different parts of the spectrum.

Typically the spectrum at 6kHz is 15dB lower than at 0Hz. On a linear scale 15dB corresponds to a factor of 6. In other words, on a 16-bit CPU, if we use the full range of a signed 15-bit representation for the lowest frequencies, than we use effectively only 12-bit range for frequency components at 6kHz.

If higher frequencies are of lower magnitude, as in the example we will be using lower number of bits to represent it; unused bits being redundant. But why does this matter or even lead to finite precision effects ?

Answer 1

The basic idea here is to apply pre-emphasis, then do the fixed point processing (or quantization) and then apply the inverse emphasis to restore the original spectrum.

The noise that is introduce by fixed point processing is generally white (i.e. equal energy at all frequencies). If you want constant signal to noise ratio at all frequencies, the signal needs to be white as well. But speech isn't white so you apply pre-emphasis to make it white.

The key here is the inverse emphasis filter applied (explicitly or implicitly) after the processing. This will restore the original spectrum of the speech AND will knock down the noise at high frequencies.