# Error plot between known and estimated data

The channel is an FIR model with input $u$. The input takes in values which are symbols from some constellation. Using an equalizer such as the Least Mean Squares (LMS), I estimate the input to the channel from the known noisy observations and the channel coefficients. Let the estimated input to the channel be denoted by $\hat{u}[n]$.

The coefficients of the channel are denoted by $c$ of length $L$. The noisy output of the channel is $y$. Therefore we have, $y[n] = \mathbf{c}^H*\mathbf{u} + v[n] \tag{1}$ for $n = 1,2,\ldots,N$ where $v[n] \sim N(0,\sigma^2)$ is a Gaussian noise. The weights of the equalizer are given by $w$ and the equalizer is of length $2L+1$. The input to the equalizer is $y$. The output of the equalizer is given by $\hat{u}[n] = y*\mathbf{w}$ where $*$ denotes the convolution. My questions are:

1) Mean Square error between the estimated input and known / desired input to the channel for different values of Signal-To-Noise ratio (SNR) --The result of equalization will give us the estimates of the input to the channel. The estimate of the input to the channel is $\hat{u}[n]$. If I need to calculate the error between the estimated input and the known input, I should probably use the formula: $MSE = \frac{1}{T}\frac{\sum_{n=1}^{N}\hat{u}[n] - {u}[n]}{N}$ and $T$ denotes the number of independent trails. As an example, the input can take values from the set $\{1,2,3,4\}$. The first 10 symbols of the data can be $2,1,3,1,1,2,4,1,2,3$. These are the known values. Then to validate the model after equalization,let the estimates input symbols to the FIR channel be $\hat{u} = 1,3,1,1,1,2,4,1,2,2$. The error, $e = (u-\hat{u})^2+(u-\hat{u})^2+(u-\hat{u})^2 + (u-\hat{u})^2 +(u-\hat{u})^2 + ...+(u-\hat{u})^2$ Is this the correct way? I am not using the communication tool-box in Matlab, and building my own code

2) Symbol error rate -- In communications, I have seen the symbol error plot (please refer :http://www.dsplog.com/2008/03/29/comparing-16psk-vs-16qam-for-symbol-error-rate/).

How is Using the same data which is used to obtain the MSE plot between the estimated symbols and known symbols, how can I obtain the SER plot? An example in Matlab will be very helpful.

Thank you.

symbol_error_rate_percent = sum(u_hat ~= u)/length(u) * 100 
You can compute this for $T$ trials and average them, like you did for MSE. Or you could just make a really long sequence of length $10 \times T$ and do it once.
• "both of them seem to be doing the same thing" -> yes both are ways to assess error, but using different measures of what is meant by error. Which one you use depends on your application / what you care about. As for BER, yes it's a percent value and you can plot it on a linear scale. But you can also take it's $\log_{10}$ and plot it on a log scale, which seems to be quite common. I'm not sure if it makes sense to plot BER as dB. Dec 29, 2017 at 16:47
• It is common to plot $\log_{10}(BER)$ of the raw fraction, not percent. Take a look at this image: upload.wikimedia.org/wikipedia/commons/thumb/7/77/… Notice how the Y-axis has powers of 10. You can think of this as a log plot where the Y-axis labels from top to bottom are 0, -2, -4, etc. And you can interpret -2, for example, as BER of 0.01 i.e. 1%. You could in theory plot $\log_{10}$ of the percent values too - that will convey exactly the same information, just that the labels will be offset by 2 because you are multiplying by 100. Dec 30, 2017 at 17:57