Statistics 简明教程

Statistics - Signal to Noise Ratio

信噪比（SNR）是科学和工程中用来分析目标信号电平与背景噪声电平的一种度量。它被定义为信号能量与噪声功率之比，通常以分贝为单位表示。大于 1:1（大于 0 dB）的比率表示有信号成分高于噪声成分。虽然 SNR 通常用于电信号，但它也可以应用于任何类型的信号（例如，冰芯中的同位素水平或细胞之间的生化运动）。

Sign to-commotion proportion (contracted SNR) is a measure utilized as a part of science and designing that analyzes the level of a coveted sign to the level of foundation clamor. It is characterized as the proportion of sign energy to the clamor power, regularly communicated in decibels. A proportion higher than 1:1 (more prominent than 0 dB) shows more flag than clamor. While SNR is regularly cited for electrical signs, it can be connected to any type of sign, (for example, isotope levels in an ice center or biochemical motioning between cells).

信噪比定义为信号（有意义的信息）的功率与背景噪声（无用信息）的功率之比：

Signal-to-noise ratio is defined as the ratio of the power of a signal (meaningful information) and the power of background noise (unwanted signal):

如果信号和噪声的方差已知，并且信号为零：

If the variance of the signal and noise are known, and the signal is zero:

如果信号和噪声在相同的阻抗下进行测量，那么可以通过计算幅度比的平方来获得 SNR：

If the signal and the noise are measured across the same impedance, then the SNR can be obtained by calculating the square of the amplitude ratio:

其中 A 是均方根 (RMS) 幅度（例如，RMS 电压）。

Where A is root mean square (RMS) amplitude (for example, RMS voltage).

Decibels

由于许多信号具有非常大的动态范围，因此信号通常使用对数分贝 (dB) 刻度表示。根据分贝的定义，信号和噪声可以用分贝 (dB) 表示为

Because many signals have a very wide dynamic range, signals are often expressed using the logarithmic decibel scale. Based upon the definition of decibel, signal and noise may be expressed in decibels (dB) as

和

and

类似地，SNR 可以用分贝表示为

In a similar manner, SNR may be expressed in decibels as

使用 SNR 的定义

Using the definition of SNR

使用对数的商规则

Using the quotient rule for logarithms

将分贝中的 SNR、信号和噪声定义代入上述方程，当信号和噪声也以分贝表示时，产生了一个计算信噪比的重要公式：

Substituting the definitions of SNR, signal, and noise in decibels into the above equation results in an important formula for calculating the signal to noise ratio in decibels, when the signal and noise are also in decibels:

在上述公式中，P 的单位是功率单位，例如瓦特或毫瓦，而信噪比是一个纯数字。

In the above formula, P is measured in units of power, such as Watts or mill watts, and signal-to-noise ratio is a pure number.

但是，当信号和噪声以幅度的单位伏特或安培测量时，必须对其进行平方才能与功率成正比，如下所示：

However, when the signal and noise are measured in Volts or Amperes, which are measures of amplitudes, they must be squared to be proportionate to power as shown below:

Example

Problem Statement:

Problem Statement:

计算以 48 kHz 采样的 2.5 kHz 正弦波的 SNR。添加标准差为 0.001 的白噪声。将随机数生成器设置为默认设置以获得可重复的结果。

Compute the SNR of a 2.5 kHz sinusoid sampled at 48 kHz. Add white noise with standard deviation 0.001. Set the random number generator to the default settings for reproducible results.

Solution:

Solution: