在Python中向信号添加噪声

问题:在Python中向信号添加噪声

我想在我正在Python中模拟的100 bin信号中添加一些随机噪声-使其更加真实。

在基本级别上,我的第一个想法是逐个bin,只在特定范围内生成一个随机数,然后从信号中进行相加或相减。

我希望(因为这是python),可能有更聪明的方式通过numpy或其他方式来执行此操作。(我认为,理想情况下,从高斯分布中提取并添加到每个仓中的数字也会更好。)

预先感谢您的任何答复。


我只是在计划代码的阶段,所以没有什么可展示的。我只是在想可能会有更复杂的方式来产生噪音。

在输出方面,如果我有以下值的10个bin:

Bin 1:1 Bin 2:4 Bin 3:9 Bin 4:16 Bin 5:25 Bin 6:25 Bin 7:16 Bin 8:9 Bin 9:4 Bin 10:1

我只是想知道是否存在一个预定义的函数,该函数可能会增加噪音,使我产生类似以下内容:

Bin 1:1.13 Bin 2:4.21 Bin 3:8.79 Bin 4:16.08 Bin 5:24.97 Bin 6:25.14 Bin 7:16.22 Bin 8:8.90 Bin 9:4.02 Bin 10:0.91

如果没有,我将逐个逐个添加一个从高斯分布中选择的数字。

谢谢。


这实际上是我正在模拟的射电望远镜发出的信号。我希望最终能够选择仿真的信噪比。

I want to add some random noise to some 100 bin signal that I am simulating in Python – to make it more realistic.

On a basic level, my first thought was to go bin by bin and just generate a random number between a certain range and add or subtract this from the signal.

I was hoping (as this is python) that there might a more intelligent way to do this via numpy or something. (I suppose that ideally a number drawn from a gaussian distribution and added to each bin would be better also.)

Thank you in advance of any replies.


I’m just at the stage of planning my code, so I don’t have anything to show. I was just thinking that there might be a more sophisticated way of generating the noise.

In terms out output, if I had 10 bins with the following values:

Bin 1: 1 Bin 2: 4 Bin 3: 9 Bin 4: 16 Bin 5: 25 Bin 6: 25 Bin 7: 16 Bin 8: 9 Bin 9: 4 Bin 10: 1

I just wondered if there was a pre-defined function that could add noise to give me something like:

Bin 1: 1.13 Bin 2: 4.21 Bin 3: 8.79 Bin 4: 16.08 Bin 5: 24.97 Bin 6: 25.14 Bin 7: 16.22 Bin 8: 8.90 Bin 9: 4.02 Bin 10: 0.91

If not, I will just go bin-by-bin and add a number selected from a gaussian distribution to each one.

Thank you.


It’s actually a signal from a radio telescope that I am simulating. I want to be able to eventually choose the signal to noise ratio of my simulation.


回答 0

您可以生成噪声阵列,并将其添加到信号中

import numpy as np

noise = np.random.normal(0,1,100)

# 0 is the mean of the normal distribution you are choosing from
# 1 is the standard deviation of the normal distribution
# 100 is the number of elements you get in array noise

You can generate a noise array, and add it to your signal

import numpy as np

noise = np.random.normal(0,1,100)

# 0 is the mean of the normal distribution you are choosing from
# 1 is the standard deviation of the normal distribution
# 100 is the number of elements you get in array noise

回答 1

…对于那些像我这样的人,他们的学习曲线还很早,

import numpy as np
pure = np.linspace(-1, 1, 100)
noise = np.random.normal(0, 1, 100)
signal = pure + noise

… And for those who – like me – are very early in their numpy learning curve,

import numpy as np
pure = np.linspace(-1, 1, 100)
noise = np.random.normal(0, 1, 100)
signal = pure + noise

回答 2

对于那些试图在SNR和numpy生成的普通随机变量之间建立联系的人:

[1] ,重要的是要记住,P是平均功率

或以dB为单位:
[2]

在这种情况下,我们已经有一个信号,并且我们想产生噪声以提供所需的SNR。

虽然噪音可能会有不同的味道取决于你模拟什么,一个良好的开端(尤其是这个射电望远镜的例子)是加性高斯白噪声(AWGN) 。如先前的回答所述,要对AWGN建模,您需要在原始信号中添加零均值高斯随机变量。该随机变量的方差将影响平均噪声功率。

对于高斯随机变量X,平均功率(也称为第二)为
[3]。

因此,对于白噪声,平均功率等于方差

在python中进行建模时,您可以
1.根据所需的SNR和一组现有测量值计算方差,如果您希望测量值具有相当一致的幅度值,则可以使用该方法。
2.或者,您可以将噪声功率设置为已知水平,以匹配接收器噪声。可以通过将望远镜对准自由空间并计算平均功率来测量接收器噪声。

无论哪种方式,重要的是要确保将噪声添加到信号中,并在线性空间而不是dB​​单位中求平均值。

这是一些生成信号并绘制电压,功率(瓦)和功率(dB)的代码:

# Signal Generation
# matplotlib inline

import numpy as np
import matplotlib.pyplot as plt

t = np.linspace(1, 100, 1000)
x_volts = 10*np.sin(t/(2*np.pi))
plt.subplot(3,1,1)
plt.plot(t, x_volts)
plt.title('Signal')
plt.ylabel('Voltage (V)')
plt.xlabel('Time (s)')
plt.show()

x_watts = x_volts ** 2
plt.subplot(3,1,2)
plt.plot(t, x_watts)
plt.title('Signal Power')
plt.ylabel('Power (W)')
plt.xlabel('Time (s)')
plt.show()

x_db = 10 * np.log10(x_watts)
plt.subplot(3,1,3)
plt.plot(t, x_db)
plt.title('Signal Power in dB')
plt.ylabel('Power (dB)')
plt.xlabel('Time (s)')
plt.show()

这是根据所需的SNR添加AWGN的示例:

# Adding noise using target SNR

# Set a target SNR
target_snr_db = 20
# Calculate signal power and convert to dB 
sig_avg_watts = np.mean(x_watts)
sig_avg_db = 10 * np.log10(sig_avg_watts)
# Calculate noise according to [2] then convert to watts
noise_avg_db = sig_avg_db - target_snr_db
noise_avg_watts = 10 ** (noise_avg_db / 10)
# Generate an sample of white noise
mean_noise = 0
noise_volts = np.random.normal(mean_noise, np.sqrt(noise_avg_watts), len(x_watts))
# Noise up the original signal
y_volts = x_volts + noise_volts

# Plot signal with noise
plt.subplot(2,1,1)
plt.plot(t, y_volts)
plt.title('Signal with noise')
plt.ylabel('Voltage (V)')
plt.xlabel('Time (s)')
plt.show()
# Plot in dB
y_watts = y_volts ** 2
y_db = 10 * np.log10(y_watts)
plt.subplot(2,1,2)
plt.plot(t, 10* np.log10(y_volts**2))
plt.title('Signal with noise (dB)')
plt.ylabel('Power (dB)')
plt.xlabel('Time (s)')
plt.show()

这是一个基于已知噪声功率添加AWGN的示例:

# Adding noise using a target noise power

# Set a target channel noise power to something very noisy
target_noise_db = 10

# Convert to linear Watt units
target_noise_watts = 10 ** (target_noise_db / 10)

# Generate noise samples
mean_noise = 0
noise_volts = np.random.normal(mean_noise, np.sqrt(target_noise_watts), len(x_watts))

# Noise up the original signal (again) and plot
y_volts = x_volts + noise_volts

# Plot signal with noise
plt.subplot(2,1,1)
plt.plot(t, y_volts)
plt.title('Signal with noise')
plt.ylabel('Voltage (V)')
plt.xlabel('Time (s)')
plt.show()
# Plot in dB
y_watts = y_volts ** 2
y_db = 10 * np.log10(y_watts)
plt.subplot(2,1,2)
plt.plot(t, 10* np.log10(y_volts**2))
plt.title('Signal with noise')
plt.ylabel('Power (dB)')
plt.xlabel('Time (s)')
plt.show()

For those trying to make the connection between SNR and a normal random variable generated by numpy:

[1] , where it’s important to keep in mind that P is average power.

Or in dB:
[2]

In this case, we already have a signal and we want to generate noise to give us a desired SNR.

While noise can come in different flavors depending on what you are modeling, a good start (especially for this radio telescope example) is Additive White Gaussian Noise (AWGN). As stated in the previous answers, to model AWGN you need to add a zero-mean gaussian random variable to your original signal. The variance of that random variable will affect the average noise power.

For a Gaussian random variable X, the average power , also known as the second moment, is
[3]

So for white noise, and the average power is then equal to the variance .

When modeling this in python, you can either
1. Calculate variance based on a desired SNR and a set of existing measurements, which would work if you expect your measurements to have fairly consistent amplitude values.
2. Alternatively, you could set noise power to a known level to match something like receiver noise. Receiver noise could be measured by pointing the telescope into free space and calculating average power.

Either way, it’s important to make sure that you add noise to your signal and take averages in the linear space and not in dB units.

Here’s some code to generate a signal and plot voltage, power in Watts, and power in dB:

# Signal Generation
# matplotlib inline

import numpy as np
import matplotlib.pyplot as plt

t = np.linspace(1, 100, 1000)
x_volts = 10*np.sin(t/(2*np.pi))
plt.subplot(3,1,1)
plt.plot(t, x_volts)
plt.title('Signal')
plt.ylabel('Voltage (V)')
plt.xlabel('Time (s)')
plt.show()

x_watts = x_volts ** 2
plt.subplot(3,1,2)
plt.plot(t, x_watts)
plt.title('Signal Power')
plt.ylabel('Power (W)')
plt.xlabel('Time (s)')
plt.show()

x_db = 10 * np.log10(x_watts)
plt.subplot(3,1,3)
plt.plot(t, x_db)
plt.title('Signal Power in dB')
plt.ylabel('Power (dB)')
plt.xlabel('Time (s)')
plt.show()

Here’s an example for adding AWGN based on a desired SNR:

# Adding noise using target SNR

# Set a target SNR
target_snr_db = 20
# Calculate signal power and convert to dB 
sig_avg_watts = np.mean(x_watts)
sig_avg_db = 10 * np.log10(sig_avg_watts)
# Calculate noise according to [2] then convert to watts
noise_avg_db = sig_avg_db - target_snr_db
noise_avg_watts = 10 ** (noise_avg_db / 10)
# Generate an sample of white noise
mean_noise = 0
noise_volts = np.random.normal(mean_noise, np.sqrt(noise_avg_watts), len(x_watts))
# Noise up the original signal
y_volts = x_volts + noise_volts

# Plot signal with noise
plt.subplot(2,1,1)
plt.plot(t, y_volts)
plt.title('Signal with noise')
plt.ylabel('Voltage (V)')
plt.xlabel('Time (s)')
plt.show()
# Plot in dB
y_watts = y_volts ** 2
y_db = 10 * np.log10(y_watts)
plt.subplot(2,1,2)
plt.plot(t, 10* np.log10(y_volts**2))
plt.title('Signal with noise (dB)')
plt.ylabel('Power (dB)')
plt.xlabel('Time (s)')
plt.show()

And here’s an example for adding AWGN based on a known noise power:

# Adding noise using a target noise power

# Set a target channel noise power to something very noisy
target_noise_db = 10

# Convert to linear Watt units
target_noise_watts = 10 ** (target_noise_db / 10)

# Generate noise samples
mean_noise = 0
noise_volts = np.random.normal(mean_noise, np.sqrt(target_noise_watts), len(x_watts))

# Noise up the original signal (again) and plot
y_volts = x_volts + noise_volts

# Plot signal with noise
plt.subplot(2,1,1)
plt.plot(t, y_volts)
plt.title('Signal with noise')
plt.ylabel('Voltage (V)')
plt.xlabel('Time (s)')
plt.show()
# Plot in dB
y_watts = y_volts ** 2
y_db = 10 * np.log10(y_watts)
plt.subplot(2,1,2)
plt.plot(t, 10* np.log10(y_volts**2))
plt.title('Signal with noise')
plt.ylabel('Power (dB)')
plt.xlabel('Time (s)')
plt.show()


回答 3

对于那些想为熊猫数据框甚至是numpy ndarray中加载的多维数据集添加噪声的人,下面是一个示例:

import pandas as pd
# create a sample dataset with dimension (2,2)
# in your case you need to replace this with 
# clean_signal = pd.read_csv("your_data.csv")   
clean_signal = pd.DataFrame([[1,2],[3,4]], columns=list('AB'), dtype=float) 
print(clean_signal)
"""
print output: 
    A    B
0  1.0  2.0
1  3.0  4.0
"""
import numpy as np 
mu, sigma = 0, 0.1 
# creating a noise with the same dimension as the dataset (2,2) 
noise = np.random.normal(mu, sigma, [2,2]) 
print(noise)

"""
print output: 
array([[-0.11114313,  0.25927152],
       [ 0.06701506, -0.09364186]])
"""
signal = clean_signal + noise
print(signal)
"""
print output: 
          A         B
0  0.888857  2.259272
1  3.067015  3.906358
""" 

For those who want to add noise to a multi-dimensional dataset loaded within a pandas dataframe or even a numpy ndarray, here’s an example:

import pandas as pd
# create a sample dataset with dimension (2,2)
# in your case you need to replace this with 
# clean_signal = pd.read_csv("your_data.csv")   
clean_signal = pd.DataFrame([[1,2],[3,4]], columns=list('AB'), dtype=float) 
print(clean_signal)
"""
print output: 
    A    B
0  1.0  2.0
1  3.0  4.0
"""
import numpy as np 
mu, sigma = 0, 0.1 
# creating a noise with the same dimension as the dataset (2,2) 
noise = np.random.normal(mu, sigma, [2,2]) 
print(noise)

"""
print output: 
array([[-0.11114313,  0.25927152],
       [ 0.06701506, -0.09364186]])
"""
signal = clean_signal + noise
print(signal)
"""
print output: 
          A         B
0  0.888857  2.259272
1  3.067015  3.906358
""" 

回答 4

以上真棒答案。最近,我需要生成模拟数据,而这正是我所使用的。分享以防万一,对其他人也有帮助,

import logging
__name__ = "DataSimulator"
logging.basicConfig(level=logging.INFO)
logger = logging.getLogger(__name__)

import numpy as np
import pandas as pd

def generate_simulated_data(add_anomalies:bool=True, random_state:int=42):
    rnd_state = np.random.RandomState(random_state)
    time = np.linspace(0, 200, num=2000)
    pure = 20*np.sin(time/(2*np.pi))

    # concatenate on the second axis; this will allow us to mix different data 
    # distribution
    data = np.c_[pure]
    mu = np.mean(data)
    sd = np.std(data)
    logger.info(f"Data shape : {data.shape}. mu: {mu} with sd: {sd}")
    data_df = pd.DataFrame(data, columns=['Value'])
    data_df['Index'] = data_df.index.values

    # Adding gaussian jitter
    jitter = 0.3*rnd_state.normal(mu, sd, size=data_df.shape[0])
    data_df['with_jitter'] = data_df['Value'] + jitter

    index_further_away = None
    if add_anomalies:
        # As per the 68-95-99.7 rule(also known as the empirical rule) mu+-2*sd 
        # covers 95.4% of the dataset.
        # Since, anomalies are considered to be rare and typically within the 
        # 5-10% of the data; this filtering
        # technique might work 
        #for us(https://en.wikipedia.org/wiki/68%E2%80%9395%E2%80%9399.7_rule)
        indexes_furhter_away = np.where(np.abs(data_df['with_jitter']) > (mu + 
         2*sd))[0]
        logger.info(f"Number of points further away : 
        {len(indexes_furhter_away)}. Indexes: {indexes_furhter_away}")
        # Generate a point uniformly and embed it into the dataset
        random = rnd_state.uniform(0, 5, 1)
        data_df.loc[indexes_furhter_away, 'with_jitter'] +=  
        random*data_df.loc[indexes_furhter_away, 'with_jitter']
    return data_df, indexes_furhter_away

Awesome answers above. I recently had a need to generate simulated data and this is what I landed up using. Sharing in-case helpful to others as well,

import logging
__name__ = "DataSimulator"
logging.basicConfig(level=logging.INFO)
logger = logging.getLogger(__name__)

import numpy as np
import pandas as pd

def generate_simulated_data(add_anomalies:bool=True, random_state:int=42):
    rnd_state = np.random.RandomState(random_state)
    time = np.linspace(0, 200, num=2000)
    pure = 20*np.sin(time/(2*np.pi))

    # concatenate on the second axis; this will allow us to mix different data 
    # distribution
    data = np.c_[pure]
    mu = np.mean(data)
    sd = np.std(data)
    logger.info(f"Data shape : {data.shape}. mu: {mu} with sd: {sd}")
    data_df = pd.DataFrame(data, columns=['Value'])
    data_df['Index'] = data_df.index.values

    # Adding gaussian jitter
    jitter = 0.3*rnd_state.normal(mu, sd, size=data_df.shape[0])
    data_df['with_jitter'] = data_df['Value'] + jitter

    index_further_away = None
    if add_anomalies:
        # As per the 68-95-99.7 rule(also known as the empirical rule) mu+-2*sd 
        # covers 95.4% of the dataset.
        # Since, anomalies are considered to be rare and typically within the 
        # 5-10% of the data; this filtering
        # technique might work 
        #for us(https://en.wikipedia.org/wiki/68%E2%80%9395%E2%80%9399.7_rule)
        indexes_furhter_away = np.where(np.abs(data_df['with_jitter']) > (mu + 
         2*sd))[0]
        logger.info(f"Number of points further away : 
        {len(indexes_furhter_away)}. Indexes: {indexes_furhter_away}")
        # Generate a point uniformly and embed it into the dataset
        random = rnd_state.uniform(0, 5, 1)
        data_df.loc[indexes_furhter_away, 'with_jitter'] +=  
        random*data_df.loc[indexes_furhter_away, 'with_jitter']
    return data_df, indexes_furhter_away

回答 5

AWGN类似于Matlab函数

def awgn(sinal):
    regsnr=54
    sigpower=sum([math.pow(abs(sinal[i]),2) for i in range(len(sinal))])
    sigpower=sigpower/len(sinal)
    noisepower=sigpower/(math.pow(10,regsnr/10))
    noise=math.sqrt(noisepower)*(np.random.uniform(-1,1,size=len(sinal)))
    return noise

AWGN Similar to Matlab Function

def awgn(sinal):
    regsnr=54
    sigpower=sum([math.pow(abs(sinal[i]),2) for i in range(len(sinal))])
    sigpower=sigpower/len(sinal)
    noisepower=sigpower/(math.pow(10,regsnr/10))
    noise=math.sqrt(noisepower)*(np.random.uniform(-1,1,size=len(sinal)))
    return noise

回答 6

在现实生活中,您希望模拟具有白噪声的信号。您应该在信号中添加具有高斯正态分布的随机点。如果我们谈论的设备灵敏度以单位/ SQRT(Hz)给出,那么您需要设计出与该设备的标准偏差。在这里,我提供了为您执行此操作的函数“ white_noise”,其余的代码将进行演示,并检查其是否应执行的操作。

%matplotlib inline
import numpy as np
import matplotlib.pyplot as plt
from scipy import signal

"""
parameters: 
rhp - spectral noise density unit/SQRT(Hz)
sr  - sample rate
n   - no of points
mu  - mean value, optional

returns:
n points of noise signal with spectral noise density of rho
"""
def white_noise(rho, sr, n, mu=0):
    sigma = rho * np.sqrt(sr/2)
    noise = np.random.normal(mu, sigma, n)
    return noise

rho = 1 
sr = 1000
n = 1000
period = n/sr
time = np.linspace(0, period, n)
signal_pure = 100*np.sin(2*np.pi*13*time)
noise = white_noise(rho, sr, n)
signal_with_noise = signal_pure + noise

f, psd = signal.periodogram(signal_with_noise, sr)

print("Mean spectral noise density = ",np.sqrt(np.mean(psd[50:])), "arb.u/SQRT(Hz)")

plt.plot(time, signal_with_noise)
plt.plot(time, signal_pure)
plt.xlabel("time (s)")
plt.ylabel("signal (arb.u.)")
plt.show()

plt.semilogy(f[1:], np.sqrt(psd[1:]))
plt.xlabel("frequency (Hz)")
plt.ylabel("psd (arb.u./SQRT(Hz))")
#plt.axvline(13, ls="dashed", color="g")
plt.axhline(rho, ls="dashed", color="r")
plt.show()

In real life you wish to simulate a signal with white noise. You should add to your signal random points that have Normal Gaussian distribution. If we speak about a device that have sensitivity given in unit/SQRT(Hz) then you need to devise standard deviation of your points from it. Here I give function “white_noise” that does this for you, an the rest of a code is demonstration and check if it does what it should.

%matplotlib inline
import numpy as np
import matplotlib.pyplot as plt
from scipy import signal

"""
parameters: 
rhp - spectral noise density unit/SQRT(Hz)
sr  - sample rate
n   - no of points
mu  - mean value, optional

returns:
n points of noise signal with spectral noise density of rho
"""
def white_noise(rho, sr, n, mu=0):
    sigma = rho * np.sqrt(sr/2)
    noise = np.random.normal(mu, sigma, n)
    return noise

rho = 1 
sr = 1000
n = 1000
period = n/sr
time = np.linspace(0, period, n)
signal_pure = 100*np.sin(2*np.pi*13*time)
noise = white_noise(rho, sr, n)
signal_with_noise = signal_pure + noise

f, psd = signal.periodogram(signal_with_noise, sr)

print("Mean spectral noise density = ",np.sqrt(np.mean(psd[50:])), "arb.u/SQRT(Hz)")

plt.plot(time, signal_with_noise)
plt.plot(time, signal_pure)
plt.xlabel("time (s)")
plt.ylabel("signal (arb.u.)")
plt.show()

plt.semilogy(f[1:], np.sqrt(psd[1:]))
plt.xlabel("frequency (Hz)")
plt.ylabel("psd (arb.u./SQRT(Hz))")
#plt.axvline(13, ls="dashed", color="g")
plt.axhline(rho, ls="dashed", color="r")
plt.show()