Gravitational wave data analysis. More of Signal processing techniques

[saikumar18]

I am using the matched filtering technique to extract the data from a heavy noise background in the process of detection of gravitational waves. I calculate the correlation between the experimental data and a theoretical template.
I have been told that the maximum of the correlation function will be the signal to noise ratio. Just for confirming this, I just took an example.
I generated a sine function (pure sine wave), and then added gaussian white noise(mean=0, variance=1) to it. Now I cross-correlate these two, ie pure sine wave and sine wave added with noise. I used the correlation theorm to calculate it, ie doing an fft and taking the ifft of it. I find that the maximum value turns out to be somewhere between 40 and 65.
Now for checking whether that is the true snr, I tried calculating the snr as
snr=(Amplitude of Signal/ Amplitude of noise)^2;
I calculated the amplitude as the rms value in both the cases(signal and noise). The answer always turned out to be somewhere between 0.38 and 0.65 or around. I am not able to understand my mistake and whether I am correct in checking the snr like this.
For further clarification, I did the same thing with a gaussian signal, and found a similar problem. Can anyone please tell me, where am I going wrong?

 

[Stephen Tashi]

then added gaussian white noise(mean=0, variance=1) to it.

Did you generate the values of the process at discrete time intervals? Did the gaussian random variable you used at each interval have variance = 1? If so, shouldn't you have made it smaller?

 

[saikumar18]

I am using octave for my analysis. What I did was, I defined time variable t from 0 to 10 in steps 0.1. then generated gaussian random values of the same length(101) using the randn function, which gives gaussian random numbers with mean=0 and variance=1 by default. are u saying, i shud make the variance smaller? how is that going to help? and am I following the correct procedure of calculating the snr?
Thank you very much for your reply.

 

[Stephen Tashi]

I defined time variable t from 0 to 10 in steps 0.1. then generated gaussian random values of the same length(101) using the randn function, which gives gaussian random numbers with mean=0 and variance=1 by default. are u saying, i shud make the variance smaller? how is that going to help?

I don't know how it will help but it can't hurt to straighten this point out. I'm not a signal processing guy and I'm looking at what you're doing from the point of view of probability theory. You say that the Gaussian noise has "amplitude 1". What does that mean to you? From the point of view of a stochastic process, the variability of a continuous process based on a Gaussian distribution is isn't determined by the standard deviation of a Gaussian distribution independently of how often (in time) you make a random draw from that that distribution. If you had drawn random values from Gaussian distribution with standard deviation 1 every .001 seconds, you would have a process that is more variable than the process you got by drawing from that distribution every 0.1 seconds. If "amplitude" is to make sense the formula for calculating it must have a "per unit time" consideration in it.

What is the definition of the "amplitude" of Gaussian noise in signal processing? (This isn't a hint, because I really don't know.) When you have a data sample, what calculation do you do to estimate its amplitude?

and am I following the correct procedure of calculating the snr?

Not being a signal processing guy, I can't tell you. If we settle on how to calculate the amplitude of Gaussian noise, then we'll worry about that.

 

[blue_raver22]

I can offer some insights and suggestions regarding your question. First, it is important to note that the matched filtering technique is a widely used and effective method for extracting signals from noisy data. So, using this technique for gravitational wave data analysis is a good approach.

Regarding your question about the maximum of the correlation function being the signal-to-noise ratio (SNR), it is important to understand that this is an approximation and may not always be accurate. The SNR can also be affected by other factors such as the frequency and amplitude of the signal, the duration of the data, and the noise characteristics. So, while the maximum of the correlation function can give you an estimate of the SNR, it may not always be the exact value.

In your example, it is possible that your calculation of the SNR using the amplitude of the signal and noise is not accurate. This is because the amplitude of the noise may not be constant and may vary across different frequencies. Additionally, the rms value may not be the best measure of the amplitude of the signal and noise. It is recommended to use the root mean square of the signal and noise instead of the amplitude.

Furthermore, it is important to note that the SNR is not the only factor to consider in the detection of gravitational waves. Other factors such as the significance level and the false alarm rate also play a crucial role. So, it is important to use a combination of techniques and methods to accurately detect gravitational waves.

In conclusion, while your approach using the matched filtering technique is correct, your calculation of the SNR may not be accurate. It is recommended to use a more robust method for calculating the SNR, such as root mean square, and also consider other factors in the detection of gravitational waves. I hope this helps clarify your doubts and further your research in this field.