adding noise to a signal in python

Question

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

User · Answer

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()

User · Answer

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

User · Answer

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

User · Answer

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

User · Answer

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

User · Answer

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

User · Answer

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     gt   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

[python] adding noise to a signal in python

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