Adding "realistic" noise to signals of different amplitudes

Question

I have a question regarding basic signal processing, which I have to start using since I have started studying fault isolation. The thing is, I have three computed signals, which are gathered in the matrix $\mathbf X \in \Re^{p \times n}$, where $p$ is the signal length and $n=3$ is the number of signals. The three signals should emulate some real measurements in different sensors, for instance,

$$\mathbf x_i = A_i \sin(\omega t), \quad \text{with } A_1 \neq A_2 \neq A_3.$$

Now, I want to add some white Gaussian noise to these signals, but then I started thinking; how can I do this somewhat "realistically" when the three signals have different amplitudes? My basic understanding is that the amplitude of the noise in real measurements is independent of the amplitudes of the signals, hence implying that my signal-to-noise ratio will not be the same for the three signals (since the variance of the noise is taken as constant). Is this correct?

What I have done at the moment is simply to form $\mathbf Y = \text{vec}\left(\mathbf X\right) \in \Re^{pn \times 1}$ (with $\text{vec}$ being the vectorization operator), calculated the variance of $\mathbf Y$, and then added noise as a fraction of this variance (with zero mean). The reason for this is that I often come across the phrase "we added this percentage of noise to the signals", and I don't suspect that one adds this to each signal seperately according to the variance of each signal.

Sorry for the long post. I hope you get my point and can answer my questions.

James

Answer 1

I am assuming we're talking about discrete measurements here, i.e. $n$ instead of $t$ with $n = 0, \cdots, p-1$. If all you want is adding some realistic noise to your three-component matrix $\mathbf X$, you could do this in the case of white Gaussian noise $\mathbf W_{p\times n}$. Using the subscript $\sigma$ for noisy data:

1. Same noise in all three channels: \begin{align} \mathbf X_\sigma &= \mathbf X + \mathbf W\tag{$p\times n$ matrices}\\ \Rightarrow\mathbf x_{i\sigma} &= \mathbf x_i + \mathbf w\tag{$p\times 1$ vectors}\\ \Rightarrow x_{i\sigma}[n] &= A_i \sin(\omega n)+ w[n] \quad\text{ with } w \sim \mathcal N\left(0, \sigma^2\right) \end{align}

   Even though you're adding the same noise in all three channels, i.e. $w_1=w_2=w_3=w$, the SNR of the channels will be different (directly proportional the amplitudes $A_i$) unless the three different sensors are capturing exactly identical (real copies) measurements.

2. Different noise values in the three channels: \begin{align} \mathbf X_\sigma &= \mathbf X + \mathbf W\tag{$p\times n$ matrices}\\ \Rightarrow\mathbf x_{i\sigma} &= \mathbf x_i + \mathbf w_i\tag{$p\times 1$ vectors}\\ \Rightarrow x_{i\sigma}[n] &= A_i \sin(\omega n)+ w_i[n] \quad\text{ with } w_i \sim \mathcal N\left(0, \sigma_i^2\right) \end{align} Your SNR of each channel $i$ will be different as the value will be directly proportional to the amplitudes $A_i$ and inversely proportional the channel's variance $\sigma_i^2$.

3. If you were having one physical quantity in all three signals (a three-axis sensor for instance), you could compute the norm for each value of $n$ and work with one signal measurement, the magnitude of size $p\times 1$. Like: $$\left|x[n]\right|=\sqrt{\sum_{i=1}^3 x_i^2[n]}$$ Then you would be looking at one $p\times 1$ signal component versus one $p\times 1$ noise component.

Answer 2

I would do it this way in Matlab:

stanDevNoise_i=sqrt((Ai^2)/(2*EbNo));
noise_i=stanDevNoise*(randn(length(xi),1)+1i*randn(length(xi),1));

The reason I put (Ai^2)/2 is because you want to put there your signal power. The noise is just a random number with the same length as your signal (p, in your case).