# Is there a rule of thumb for selecting the variance of the input?

Considering an estimation problem, where I want to estimate the unknown input $x$ using the known channel parameters. This estimation problem can be solved by Least Squares.

Thus, the model is $$\hat{x}[n] = \mathbf{A}^T\mathbf{x}_{n} + w[n]$$ where $\mathbf{A} = [a_0,a_1,\ldots,a_{L-1}]^T \in \mathbb{R}^{1 \times L}$ represents the channel's impulse response, and $\mathbf{x}_{n} = [x[n], x[n-1], x[n-2],\ldots,x[n-L+1]^T$ and $L$ is the order. The model is an FIR filter and $w[n]$ is the measurement noise which is $w \sim N(0,\sigma^2_w)$.

After the estimation, I want to see the performance using Mean square error defined as $MSE = \frac{1}{NT}\sum_{n=1}^Ne[n]^2$ where $T$ is the number of independent runs, that is for every SNR, I generate $T = 10$ different channel parameters at random and for each $T$ I perform estimation.

The error is $e[n] = x[n] - \hat{x}[n]$.

The graph to plot would be SNR on $X$ axis vs. $MSE$ on $Y$ axis. SNR is defined as $\frac{\sigma^2_x}{\sigma^2_w}$.

But I don't know what is the value of the numerator, $\sigma^2_x$. I cannot understand how to plot the curve. Can somebody please explain (with an illustration / graph would be very useful) what is the procedure to plot the graph, what is the value of variance of the input, $\sigma^2_x$?

Please correct me where ever I have done mistake (especially, in the formula ).

• Your formula for SNR needs to be amended. Perhaps you can define signal power (the numerator) as the mean of the sum of squares of the elements of $x[n]$ and use that instead of $\sigma_x^2$. – Atul Ingle Jul 14 '17 at 0:58

You can define your model as $$\hat{x}[n] = \frac{1}{||\mathbf{x}_{n}||}\mathbf{A}^T\mathbf{x}_{n} +\frac{1}{\sqrt{\mathsf{SNR}}} w[n]$$ In this way, you define a range for $\mathsf{SNR}$ in dB (e.g. between 0 to 10 dB), then translate it to natural (non-dB) scale, and then $\frac{1}{\sqrt{\mathsf{SNR}}}$ gives you the noise power. The signal power is normalized to one which makes it easy.

Here, $w\sim \mathcal{N}(0,1)$ is a standard Gaussian RV and you can simulate it as w = randn;. In fact, $\frac{1}{\sqrt{\mathsf{SNR}}}w=w'\sim \mathcal{N}(0,\sigma^2_{w'})$, where $\sigma^2_{w'}$ has the right proportion wrt the SNR.

• signal power is normalized to one in my answer. The exact value is not so important since SNR is a ratio. You can also use awgn as well. Since you specify measured it uses the right values. – msm Jul 14 '17 at 19:58
• Please read the answer carefully. a SNR of dB should first be converted to non-dB scale which becomes 1. The noise variable is standard Gaussian and when is scaled by inverse square root of SNR the result becomes a Gaussian RV with the appropriate variance. We only care about the ratio when it comes to SNR. Hence, the exact values are not important (as long as there is no other constraints in your specific problem). – msm Jul 15 '17 at 4:17