as you can see on page 314 the derivation of the ln[f etc.] equation is done via the matched filter that i have also provided below. i would like to know how this equation is used to derive the ln[f etc.] from its previous probability functions, and how it then finally becomes in the form of ||a-b||<||a-b||

any help would be greatly appreciated. thanks

matched filter 3.2.2

maximum likelihood 1

maximum likelihood 2


The formula you need, which I don't see in the scanned pages, is the N-dimensional Gaussian distribution of the received vector given that the signal $s_m$ was transmitted:

$$f(\vec{r}|s_m)=\frac{1}{(\pi N_0)^{N/2}}\exp\left[-\frac{\sum_{j=1}^N(r_j-x_{mj})^2}{N_0}\right]\tag{1}$$

The maximum likelihood receiver seeks to maximize this conditional probability by choosing the appropriate $s_m$. Since the logarithm is a monotonic function, you can as well maximize the logarithm of (1):

$$\log(f(\vec{r}|s_m))=-\frac{N}{2}\log(\pi N_0)-\frac{1}{N_0}\sum_{j=1}^N(r_j-x_{mj})^2=\\=-\frac{N}{2}\log(\pi N_0)-\frac{1}{N_0}||\vec{r}-s_m||^2\tag{2}$$

Since the first term in (2) is constant (i.e. independent of $s_m$), maximizing (2) is equivalent to minimizing the distance $||\vec{r}-s_m||^2$. If $||\vec{r}-s_m||^2$ is minimal, then obviously also $||\vec{r}-s_m||$ is minimal. So we arrive at the ML decision rule given in your book, i.e. choose the signal $s_m$ for which the distance to the received signal is a minimum.

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