Complex Exponential

Discrete-Time Oscillatory Heartbeat

The complex exponential describes in compact form an oscillatory behavior with a given frequency and an initial phase. A disrete-time oscillatory heartbeat with amplitude \(A\), frequency \(w\), and an initial phase \(\phi\) can be described as:

\[ x(n) = A \exp \left\{ j(wn + \phi) \right\} = A \left[\cos\left(wn + \phi \right) + j \sin(wn + \phi) \right] \]

Complex Exponential Generating Machine

Rotation can be achieved by multiplication by a complex exponential to a point on a complex plane. This gives a basis to a complex exponential generating machine:

\[ \begin{align} x(n) &= \exp\left\{j (wn + \phi) \right\} \\ x(n + 1) &= \exp\left\{ j(w (n + 1) + \phi) \right\} = \exp \left\{j w \right\} x(n). \end{align} \]

In discrete time, not every sinusoid is periodic. The complex exponential \(\exp \left\{ jwn \right\}\) is periodic in \(n\) iff \(w = \frac{M}{N} 2 \pi\), \(M, N \in \mathbb{N}\), i.e., the frequency is a rational multiple of \(2\pi\):

\[ \begin{align} x(n) &= x(n + N) \\ \exp\left\{j (wn + \phi) \right\} &= \exp\left\{ j(w (n + N) + \phi) \right\} \\ \exp\left\{ jwn \right\} \exp\left\{j \phi \right\} &= \exp\left\{jwn \right\} \exp\left\{jwN \right\} \exp\left\{j \phi \right\} \\ \exp\left\{jwN \right\} &= 1 \\ wN &= 2 M \phi, \ M \in \mathbb{Z} \\ w &= \frac{M}{N} 2 \pi. \end{align} \]

Aliasing

The natural property of a complex exponential is that it has \(2\pi\) periodicity. In discrete time, it puts a limit on how fast we can go around the unit circle with the discrete time signal.