Preliminary Information

• Continuous (analog). Signals which are continuous in both dimensions (time and amplitude)

$$ f : \R \rightarrow \R \text{ or } \mathbb{C} $$

• Discrete. Signals whose domain is discrete and codomain is continuous

$$ f : \Z \rightarrow \R \text{ or } \mathbb{C} $$

• P-periodic. A discrete-time signal $f$ is $p$-periodic if there exists a $p$ such that the signal repeats itself after $p$ samples

$$ \exists p \in \N \space | \space f(n + p) = f(n), \forall n \in \Z $$

$p$ must be the smallest positive integer that satisfies this equation.

• Period Decomposition. Any $p$-periodic discrete-time signal $f$ can be decomposed into a $p$-dimensional vector with entries being values of the signal at samples $n \in \{0 \space \dots \space p-1\}$

$$ f(n) \rightarrow \mathbf{f} = \begin{bmatrix} f(0) \\ \vdots \\ f(p-1) \end{bmatrix} $$

Note that $n$ does not have to start at $0$, sometimes it is more convenient to choose a different starting sample.

• Kronecker Delta Function. Function which is non-zero if and only if its argument is 0.

$$ \delta(n) = \begin{cases} 1 \hspace{15pt} n=0 \\ 0 \hspace{15pt} n \neq 0\end{cases} \hspace{15pt}\delta : \Z\rightarrow \{0, 1\} $$

Euler’s Formula

Gives the complex exponential its geometric intuition.

$$ e^{i(\omega t + \theta)} = \cos(\omega t + \theta) + i \sin(\omega t + \theta) $$

• Proof

Inverse Euler Formulas

Algebraic rearrangement of Euler’s Formula.

$$ \cos(\omega t + \theta) = \frac{e^{i(\omega t + \theta)} + e^{-i(\omega t + \theta)}}{2} \\ \hspace{0pt} \\ \sin(\omega t + \theta) = \frac{e^{i(\omega t + \theta)} - e^{-i(\omega t + \theta)}}{2i} $$

• Derivation

Discrete Time Complex Exponentials

Family of $p$-periodic signals over a discrete sample domain. These signals are typically only plotted on stem plots if the function value is real for all $n \in \Z$.

$$ x(n) = e^{i \omega _0 n} \hspace{25pt} x : \Z \rightarrow \mathbb{C}, \hspace{5pt} \omega_0 \in \R $$