In Continuous Time, we use the \emph{Dirac Delta} function. (Technically not a function, and the height is actually infinite)

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%\centering
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%\caption{ The Dirac Delta }
%\label{ fig:dirac }
%\end{figure}

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In Discrete Time, we use the \emph{Kronecker Delta} function.

%\begin{figure}[htp]
%\centering
%\includegraphics{images/kronecker.ps}
%\caption{ The Unit Impulse, a.k.a Kronecker Delta }
%\label{ fig:kronecker }
%\end{figure}

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The Kronecker Delta function is used in the linear combination:
$$ \sum_{m \in M} x(m) \delta(n-m)  $$

Another signal that we use to decompose signals is the \emph{Complex Exponential}.

This signal is given by

$$ e^{i \theta t};  t \in \R$$ 
$$ e^{i \theta n};  n \in \Z$$ 


Signals to Decompose Signals

Week 2

$$x(t) = e^{it}$$

Think of this as the position of a particle on the complex plane. The derivative is defined in the same way, $\frac{d}{dt}e^{st} = se^{st}$ where $s \in \C$, but we will also introduce some new notation. The derivative with respect to time is defined as

$$\dot{x}(t)=\frac{d}{dt}x(t)=ie^{it}$$

At $t = 0$, the \emph{velocity vector} $\dot{x}(0)=i$, and the \emph{position vector} $x(0)=1 + 0i$. The relation between $\dot{x}(t)$ and $x(t)$ is orthogonal, that is
$$\forall t, \dot{x}(t) \perp x(t). $$

$z=e^{i \theta t}$ is on the unit circle, that is $\left| e^{i \theta t}\right| = 1$ for all $t$. $z$ must stay on the unit circle, if the magnitude of $z$ gets smaller over time, this implies a velocity vector $\dot{x}(t)$ pointing inward, which contradicts the orthogonality of the vectors $x(t)$ and $\dot{x}(t)$.


Exponentials as Vectors


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Leonhard Euler demonstrated a relationship between the base of the natural logarithm, sine, cosine, and the imaginary unit, which is now coined \emph{Euler's Formula}: 

\begin{nicebox}
$$ e^{i \theta} = \cos \theta + i \sin \theta $$
\end{nicebox}

%\begin{figure}[htp]
%\centering
%\includegraphics{images/euler/eulers.ps}
%\caption{ A visual look at Euler's Formula }
%\label{ fig:eulers }
%\end{figure}

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Euler's formula can be derived by looking at the complex exponential on the unit circle. The horizontal component of $e^{i \theta}$ is $ \cos \theta$, and the vertical component is $i\sin \theta$.

The formula can also be derived using infinite series.
\begin{align*}
e^\theta &= \sum \limits_{n = 0}^{\infty} \frac{\theta^n}{n!} = 1 + \theta + \frac{\theta^2}{2!} + \frac{\theta^3}{3!} + \frac{\theta^4}{4!} + \cdots \\
\sin \theta &= \sum \limits_{n = 0}^{\infty} \frac{(-1)^n\theta^{2n+1}}{(2n+1)!} = \theta - \frac{\theta^3}{3!} + \frac{\theta^5}{5!} + \cdots \\
\cos \theta &= \sum \limits_{n = 0}^{\infty} \frac{(-1)^n\theta^{2n}}{(2n)!} = 1 - \frac{\theta^2}{2!} + \frac{\theta^4}{4!} + \cdots
\end{align*}
Given the above formulae, we can derive $e^{i\theta}$ in terms of $\sin \theta$ and $\cos \theta$:
\begin{align*}
e^{i\theta} &= \sum \limits_{n = 0}^{\infty} \frac{(i\theta)^n}{n!} = 1 + i\theta + \frac{i^2\theta^2}{2!} + \frac{i^3\theta^3}{3!} + \frac{i^4\theta^4}{4!} + \cdots \\
e^{i\theta} &= \sum \limits_{n = 0}^{\infty} \frac{(i\theta)^n}{n!} = 1 + i\theta - \frac{\theta^2}{2!} - \frac{i\theta^3}{3!} + \frac{\theta^4}{4!} + \frac{i\theta^5}{5!} -\frac{\theta^6}{6!} - \cdots \\
e^{i\theta} &= \sum \limits_{n = 0}^{\infty} \frac{(i\theta)^n}{n!} = \left[ 1 - \frac{\theta^2}{2!} + \frac{\theta^4}{4!} - \cdots \right] + \left[  i\theta - \frac{i\theta^3}{3!} + \frac{i\theta^5}{5!} - \cdots\right] \\
e^{i\theta} &= \sum \limits_{n = 0}^{\infty} \frac{(i\theta)^n}{n!} = \sum \limits_{n = 0}^{\infty} \frac{(-1)^n\theta^{2n}}{(2n)!} +  i\sum \limits_{n = 0}^{\infty} \frac{(-1)^n\theta^{2n+1}}{(2n+1)!} \\
e^{i \theta} &= \cos \theta + i \sin \theta
\end{align*}

We can also use this formula to come up with useful identities:

For $\cos \theta$:
\begin{align*}
e^{-i \theta} &= \cos (\theta) + i \sin(- \theta) \\
e^{-i \theta} &= \cos \theta - i \sin \theta \quad \mbox{(since cosine is even and sine is odd)} \\
e^{i \theta} + e^{-i \theta} &= (\cos \theta + i \sin \theta) +  (\cos \theta - i \sin \theta) \\
e^{i \theta} + e^{-i \theta} &= 2\cos \theta \\
\frac{e^{i \theta} + e^{-i \theta}}{2} &= \cos \theta
\end{align*}

This demonstrates that a cosine wave defined by $\cos \omega_0t = \frac{1}{2}e^{i\omega_0t} + \frac{1}{2}e^{-i\omega_0t}$ has two frequencies, namely $\omega_0$ and $-\omega_0$. The magnitude represents how much each frequency contributes. The same goes for the sine wave:

For $\sin \theta$:
\begin{align*}
e^{i \theta} - e^{-i \theta} &= (\cos \theta + i \sin \theta) -  (\cos \theta - i \sin \theta) \\
e^{i \theta} - e^{-i \theta} &= 2i\sin \theta \\
\frac{e^{i \theta} - e^{-i \theta}}{2i} &= \sin \theta
\end{align*}

For example, the A major triad defined by $x(t) = sin(\omega_0t) + sin(\omega_1t) + sin(\omega_2t)$, where
\begin{align*}
\omega_0 &= 2\pi\cdot 440 = \mbox{A} \\
\omega_1 &= 2\pi \cdot 554 = \mbox{C\#} \\
\omega_2 &= 2\pi \cdot 554 \cdot (\sqrt[12]{2})^3 = \mbox{E}
\end{align*}

This implies that there exists 6 frequencies in the A major triad!

Eulers Formula

The complex exponential can be written in the form
$$e^{i \omega t},$$
where $\omega$ represents the \emph{radial frequency}, and is defined by
$$\omega = 2\pi f,$$ 
where $f$ is measured in \emph{cycles per second} (\emph{Hz}), and $2\pi$ is measured in \emph{radians per cycle}. This gives us a measure for $\omega$ of
$$2\pi\frac{\mbox{radians}}{\mbox{cycle}} \cdot f\frac{\mbox{cycles}}{\mbox{second}} = \omega\frac{\mbox{radians}}{\mbox{second}}$$



Radial Frequency

If we view the complex exponential as a particle traveling around the unit circle, it does not have to be traveling around at a constant rate. For instance, we can represent $\theta$ as a function. If $\theta(t) = \omega_0t + \phi_0$, the particle goes around at a constant rate. We call $\theta(t)$ the \emph{instantaneous angle}, and $\omega(t)$ the \emph{instantaneous frequency} at time $t$:

$$ \omega(t) = \dot{\theta}(t) = \frac{d\theta}{dt} $$

The prototype of a \emph{pure frequency} is $e^{i \omega_0t}$, and represents a phasor going around the unit circle at a constant rate $\omega \frac{\mbox{rad}}{\mbox{sec}}$. If $z = e^{i \theta(t)}$ and $\theta(t) = \omega_0t + \phi_0$, then $\phi_0$ represents the \emph{phase shift}, or initial phase offset.

2.2 The Complex Exponential