Oversampling Series II: Fourier transform and signal-to-noise ratio

The Fourier transform makes people look at the problem from the time domain to the frequency domain , One more dimension . The fast Fourier transform algorithm popularizes the application of Fourier transform in engineering field , In the fields of scientific computing and digital signal processing , Discrete Fourier transform （DFT） It is still one of the most effective tools .

For example, the following figure shows an amplitude of 1、 The frequency is 2Hz Sine wave and its discrete Fourier transform result .

Signal-to-noise ratio （SNR） Is the ratio of signal to noise , It is one of the important indexes to measure the performance of communication or simulation systems , And Fourier transform is inextricably linked . In many cases , We use Fourier transform to evaluate the signal-to-noise ratio , If the evaluation method is wrong , It is difficult to achieve the desired results , It often backfires .

solve SNR The process of , We use ‘ assessment SNR’ To describe , That means we can't calculate exactly SNR, Can only be evaluated , And so it is .

assessment SNR The method can be divided into time domain and frequency domain . We aim to analyze a set of discrete sample points , See how to evaluate SNR, And its mistakes .

Time domain estimation SNR

Xs(n) Is a signal sequence ,Xn(n) Is the noise sequence , Then signal X(n)=Xs(n)+Xn(n), Is a set of discrete sequences with noise , Evaluate in time domain X(n) The SNR formula of is as follows ：

Its meaning is to obtain discrete signals respectively 、 Noise power and , Calculate the ratio of the two . Here's a premise , We need to separate the signal from the noise , Then we can solve ,

The problem, however, is that , For a given discrete time series , It is difficult to separate signal and noise completely , So time domain evaluation SNR There are limitations , And it's not intuitive , So we usually solve it in the frequency domain .

Frequency domain estimation SNR

In the frequency domain SNR The calculation principle is very close to the time domain , Or is the ratio of signal power to noise power only . The simplest way is in the spectrum X(m) Set the threshold on , Above the threshold is the signal , Below the threshold is noise . In this way, there will be estimation accuracy problems caused by threshold setting , At the same time, there will be more or less noise superimposed in the signal frequency band , Calculate in frequency domain SNR It is also an approximation .

SNR With dB As a unit ,SNR(dB)=10*log10(SNR).

We often use Matlab assessment SNR,Matlab Is a very powerful mathematical tool , It integrates SNR Calculation function , If the application is incorrect , The error will be very large , Unable to get expected results , Examples are as follows .

The following figure shows an amplitude of 1、 The frequency is 2Hz Sine wave and its discrete Fourier transform result .

use Matlab SNR The result of function calculation is as follows , Blue is the signal component of interest , Orange is noise . This frequency domain diagram can represent SNR, The lower the orange noise value 、 The lower the , It indicates that the higher the signal-to-noise ratio 、 The better . Even if there is a little noise in the red circle , In the evaluation of single frequency component signals SNR when , This is still a very effective means .

Suppose a signal x(n)=sin(4pi*t)+0.5*sin(18pi*t), As shown in the figure below, an amplitude of 1、 The frequency is 2Hz The sine wave superposition amplitude of is 0.5、 The frequency is 9Hz The result of sine wave .

use Matlab SNR The result of function calculation is as follows , Blue is the signal component of interest , Orange is noise ,matlab Discarded frequency 9Hz、 Less energetic ingredients , Only calculated 2Hz Signal to noise ratio of signal .

So if our signals of interest are complex , It can't be used directly SNR The function calculates directly .

So we must understand DFT And SNR Based on the relationship , Use... Correctly matlab To get what you expect SNR result .

In oversampling , Signal-to-noise ratio 、ADC Effective digits 、 Oversampling rates are inextricably linked , After understanding the basic concepts , We understand the principle of oversampling step by step .