Oversampling Series III: quantization error and oversampling rate

The digital world is a mirror image of the analog world , and ADC Is the gate connecting the two worlds . Once all analog signals have passed ADC After discretization , Its amplitude is bound to be distorted , The important reason is ADC Limited resolution , It can only approximate the true amplitude .

We cannot reach the truth , Can only be infinitely close to the truth .

The resolution is ADC One of the important parameters of , It and precision are two different quantities , Precision describes the accuracy of discrete results , The resolution describes ADC The smallest signal that can be resolved , by 1LSB.

In other words , High resolution ADC Can distinguish smaller signals , But the accuracy of the result of the transformation is limited by the accuracy .

One 8bit ADC, Distinguishable 256 Kind level , When the input range is 2.56V when ,1LSB That is to say 10mV. Limited by resolution ,ADC There is an error between the output value and the actual value .

The following figure is a schematic diagram of quantization error , For variations less than 1LSB The signal of ,ADC Is indistinguishable , The error of input and output at this time is the quantization error .

A simplified mathematical model for quantifying noise is as follows ,

e(t)=st, -q/2s < t < +q/2s

According to the input signal 、ADC Relationship between resolution and quantization error , We can derive an important SNR Calculation formula . The detailed derivation process will be replied in the background of the official account ： Oversampling

Here is the classic ADC SNR Calculation formula .

SNR = 6.02N + 1.76dB

DC to fs/2 Bandwidth range

If you use digital filtering to filter out bandwidth BW Other noise components , Then a correction factor is included in the equation

‍ Or writing

BW Is the signal bandwidth ,FS Is the sampling rate ,OSR=Fs/(2*BW) Is the oversampling rate .

When the oversampling rate is increased 4 times , Can improve ADC 1bit The effective resolution of is calculated according to the above formula , The oversampling rate can refer to the previous article ：

Oversampling series a ： Sampling theorem and oversampling rate

Why? “ Every time the oversampling rate increases 4 times , Can improve ADC 1bit The effective resolution of ”？

Take a chestnut ：

When the oversampling rate OSR by 1 when ,

When the oversampling rate OSR by 4 when ,

Contrast formula 1 And the formula 2, Only the red boxes are different , I.e. caused by oversampling SNR Gain and increase resolution N Can be transformed into equivalent .

Extra digits N+：

N+=10log(OSR)/6.02,

When OSR=1,4,16,,,, when ,N+=1,2,3,,,,,

This is what is usually said , Every time the oversampling rate increases 4 times , Can improve 1bit The reason for the resolution .

Is it possible to improve the resolution by increasing the sampling rate ？

It's not true , From formula 2 It can be seen that ,10log(4) Turn into 10log(1) 了 , This process also requires a reduction in sampling , Or pull down , In addition to reducing the amount of data , It can improve the resolution .

How to draw , It's a learning , If you simply average , Often only improve the signal-to-noise ratio , The purpose of increasing the number of significant digits is not achieved , A lot of people will dig pits here .

How to correctly use down pumping to increase the number of significant digits ？ The most classical explanation of quantization error and oversampling rate is spectral density , Limited space , The following articles will be continuously updated .