1. Fourier Series Orthogonal Basis Functions The ‘orthogonal basis functions’ are a set of functions of time, $$\phi(t)$$, such that the followings holds over some specified time interval $$T$$,
$$\forall k,\forall m,k \neq m, \int_{T} \overline{\phi_k(t)} \phi_m(t) =0$$
Triangle Fourier Series the ‘OBF’ of triangle fourier series are as follows
$${cos(nw_0t),sin(nw_0t)}$$
which satisfy that
$$\int_Tcos(nw_0t)sin(mw_0t)dt = 0$$
$$\int_Tcos(nw_0t)cos(mw_0t)dt = \int_Tsin(nw_0t)sin(mw_0t)dt = \begin{cases} \frac{T}{2},m=n\ 0,m\neq n \end{cases}$$
$$\int_T1dt=T$$
Therefore, we can express a periodic signal with fundamental period T as a sum of sinusoids,
1. Moving Average(MA) filter N-points MA filter is defined as
$$y[n]=\sum_{i=0}^{N-1}\frac{1}{N}x[n-i]$$
where N is a positive integer.
For example, in the 3-points MA filter, the output is
$$y[n]=\frac{1}{3}x[n]+\frac{1}{3}x[n-1]+\frac{1}{3}x[n-2]$$
As a general class of system, the input/output relationship is defined as
$$y[n] = \sum_{i=0}^{n}w_ix[n-i],n\ge 0$$
it turns out that any ‘causal linear time-invariant’ DT system with the input x[n] equal to 0 for all n < 0 can be expressed as above.
1.Periodic Continuous-Time Signals given two periodic CT signals
$$\forall t, x_1(t+T_1) = x_1(t)$$
and
$$\forall t, x_2(t+T_2) = x_2(t)$$
if the sum of them is periodic, then
$$\forall t, x_1(t+T)+x_2(t+T)=x_1(t)+x_2(t)$$
it is satisfied if and only if
$$\exists p,q \in N^*, T=pT_1=qT_2$$
in other word, p and q are coprime
$$\frac{T_1}{T_2}=\frac{p}{q}\in Q$$
2. Sifting Property note that $$\forall \lambda \neq t_1, \delta(\lambda-t_1)=0$$
it follows that
$$\int_{-\infty}^{+\infty}f(\lambda)\delta(\lambda-t_1)d\lambda=\int_{t_1-\epsilon}^{t_1+\epsilon}f(\lambda)\delta(\lambda-t_1)d\lambda=f(t_1)$$
where $$f(t)$$ is continuous at $$t=t_1$$