# 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.

## 1 Answer

Your formula/method for computing MSE between estimated and known inputs looks good to me. For symbol error rate you could use something like a Hamming distance which simply counts the number of times the estimated symbol is different from the actual symbol. In your 10 symbol example the error rate is 4/10 i.e 40%.

In Matlab you can do something like:

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.

• Thank you for your answer. I want to know what is the difference between mse and bit error rate since both of them seems to be doing the same thing which is reporting the error. In communications the simulation based bit error rate or symbol error rate is often compared with theoretical error rate and then converted to log dB scale. Based on your answer it seems it is not necessary to report the symbol error rate in dB but it can be done in percentage as well. Can you please elaborate on this. Thank you – Srishti M Dec 29 '17 at 7:22
• "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. – Atul Ingle Dec 29 '17 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. – Atul Ingle Dec 30 '17 at 17:57