STEMS Formulas

ISE = $\int_T |x(t)-\hat{x}(t)|^2 \, dt$

Orthogonality principle:
$\int_T [x(t)-\hat{x}(t)]\phi_k^* dt = 0$ for all $k$ .

A pair of functions $\phi_l(t)$ and $\phi_k(t)$ are said to be orthogonal over the interval $T$ if they satisfy:

$\int_T \phi_l(t)\phi_k^*(t) \, dt = K_k, k = l; 0, k \neq l,$

where $K_k$ is some nonzero constant.

If the basis functions are orthogonal, the equations are decoupled, and we have

$c_k = \frac{1}{K_k} \int_T x(t) \phi_k^*(t) \, dt.$

Coefficients for pulse trains:
$c_0 = A \frac{\tau}{T_0}$
$c_k = A \frac{\tau}{T_0} \frac{\sin(k\pi\frac{\tau}{T_0})}{k\pi\frac{\tau}{T_0}}.$

where $\tau$ = pulse width, $T_0$ = period, $A$ = amplitude.

Problems used in this document: