This is a review of signal processing.
Let $N \in \setZ^+$ and $m,n \in [0, N-1]$ $$ f(m) : \setZ \rightarrow \setC $$ periodicity $$ f(m + N) = f(m) $$ This is a vectorspace. Impulse Function $$ \delta(m) \equiv \begin{cases} 0 \quad\text{if}\quad m \neq 0 \\ 1 \quad\text{if}\quad m = 0 \end{cases} $$ $$ f(m) = \sum_{m'=0}^{N-1} \delta(m - m') f(m') $$ Unit Step Function $$ u(m) \equiv \begin{cases} 0 \quad\text{if}\quad m < 0 \\ 1 \quad\text{if}\quad m \geq 0 \end{cases} $$ $$ \delta(m) = u(m) - u(m - 1) $$ Standard Inner Product $$ \inner{f}{g} \equiv \sum_{m=0}^{N-1} \conj{f(m)} g(m) $$ DFT Basis Function $N \in \setZ^+$ and $m,n \in [0, N-1]$ (Unitary convention) $$ b_n(m) = \frac{1}{\sqrt{N}} e^{i \frac{2 \pi n}{N} m} $$ $$ \inner{b_n}{b_{n'}} = \delta_{nn'} $$ where $$ \delta_{nn'} = \begin{cases} 0 \quad\text{if}\quad n \neq n' \\ 1 \quad\text{if}\quad n = n'\end{cases} $$
$$
\inner{b_n}{b_{n'}} = \sum_{m=0}^{N-1} \conj{\bb{\frac{1}{\sqrt{N}} e^{i \frac{2 \pi n}{N} m}}} \bb{\frac{1}{\sqrt{N}} e^{i \frac{2 \pi n'}{N} m}}
$$
or
$$
\inner{b_n}{b_{n'}} = \frac{1}{N} \sum_{m=0}^{N-1} e^{i \frac{2 \pi [n' - n]}{N} m}
$$
if $n = n'$, then $[n' - n] = 0$ and thus all exponentials become $e^0 = 1$ so
$$
\inner{b_n}{b_{n}} = \frac{1}{N} \sum_{m=0}^{N-1} 1 = \frac{1}{N} N = 1
$$
otherwise, let $n \neq n'$. to evaluate this sum, a lemma is needed.
Lemma. Let $f:\setZ \rightarrow \setC$ and $A, b, c \in \setC$
$$
f(m) = A b^{c m}
$$
$$
S = \sum_{m=0}^{N-1} A b^{c m}
$$
$$
S = A \bb{1 + b^c + b^{c 2} + \dots + b^{c [N - 2]} + b^{c [N - 1]}}
$$
$$
b^c S = A \bb{b^c + b^{c 2} + b^{c 3} + \dots + b^{c [N - 1]} + b^{c N}}
$$
$$
S - b^c S = A \bb{1 - b^{c N}}
$$
$$
S = A \frac{1 - b^{c N}}{1 - b^c}
$$
Apply this by letting $A = 1/N, b = e, c = i 2 \pi [n' - n] / N$. Then
$$
\inner{b_n}{b_{n'}}
= \frac{1}{N} \frac{1 - e^{i 2 \pi [n' - n]}}{1 - e^{i \frac{2 \pi [n' - n]}{N}}}
= \frac{1}{N} \frac{1 - 1}{1 - e^{i \frac{2 \pi [n' - n]}{N}}}
= 0
$$
which follows because the denominator is non-zero and $e^{i 2 \pi [n' - n]} = 1$ due to the argument being an integer multiple of $2 \pi$.
Combining these cases, you get
$$
\inner{b_n}{b_{n'}} = \delta_{nn'}
$$
$$ f(m) = \sum_{n=0}^{N-1} b_n(m) c_n $$ $$ c_n = \inner{b_n}{f} $$
$$
\inner{b_n}{f}
= \inner{b_n}{\sum_{n'=0}^{N-1} b_{n'} c_{n'}}
= \sum_{n'=0}^{N-1} \inner{b_n}{b_{n'}} c_{n'}
= \sum_{n'=0}^{N-1} \delta_{nn'} c_{n'}
= c_n
$$
Linearity $$ T(f(m) a + g(m) b) = T(f(m)) a + T(g(m)) b $$ Shift Invariance (often called time/space invariant) $$ g(m) = T(f(m')) $$ $$ g(m - m_0) = T(f(m' - m_0)) $$ Let $T$ be a linear shift invariant function. Then $$ T(f(m)) = T\pp{\sum_{m'=0}^{N-1} \delta(m - m') f(m')} = \sum_{m'=0}^{N-1} T(\delta(m - m')) f(m') = \sum_{m'=0}^{N-1} h(m - m') f(m') $$ where $h = T(\delta)$.
Convolution $$ [\conv{f}{g}](m) \equiv \sum_{m'=0}^{N-1} f(m - m') g(m') $$
Properties
Linearity $$ \conv{f}{[g + h]} = \conv{f}{g} + \conv{f}{h} $$ $$ \conv{f}{[a g + b h]} = \sum_{m'=0}^{N-1} f(m - m') [a g(m') + b h(m')] = a \sum_{m'} f(m - m') g(m') + b \sum_{m'} f(m - m') h(m') = a [\conv{f}{g}] + b [\conv{f}{h}] $$ Shift Invariance $$ \conv{f}{g} = [\conv{f}{g}](m + m_0) $$ Associativity $$ \conv{[\conv{f}{g}]}{h} = \conv{f}{[\conv{g}{h}]} $$ $$ [\conv{[\conv{f}{g}]}{h}](m) = \sum_{m'=0}^{N-1} \bb{\sum_{m''=0}^{N-1} f(m - m' - m'') g(m'')} h(m') = \ = \conv{f}{[\conv{g}{h}]} $$ $$ d_{m_0} (m) = \delta(m - m_0) $$ $$ [\conv{d_{m_0}}{f}](m) = f(m - m_0) $$ $$ [\conv{f}{g}] = \sum_{m=0}^{N-1} f([m - m_0] - m') g(m') = $$ Commutivity $$ \conv{f}{g} = \conv{g}{f} $$
$$
[\conv{f}{g}](m) = \sum_{m'=0}^{N-1} f(m - m') g(m')
$$
Let $m - m' = m''$. Then
$$
[\conv{f}{g}](m) = \sum_{m''=0}^{N-1} f(m'') g(m - m'')
$$
Eigenfunctions of LTI systems
$$ q_\omega(m) = A e^{i \omega m} $$ $$ [\conv{q_\omega}{h}](m) = \sum_{m'=0}^{N-1} A e^{i \omega [m - m']} h(m') = A e^{i \omega m} \sum_{m'=0}^{N-1} e^{-i \omega m'} h(m') = q_\omega \lambda_\omega $$ where $$ \lambda_\omega = \sum_{m'=0}^{N-1} e^{-i \omega m'} h(m') $$ is the associated eigenvalue of eigenvector $q_\omega$ of the linear function of convolution by $h$. $$ [\conv{h}{}] q_\omega = \lambda_\omega q_\omega $$