4.5. Powers and waves

Now that we’ve seen how complex numbers behave when multiplied together, we might also wonder what happens when a complex number is multiplied by itself repeatedly. Of specific importance in signal processing is the sequence

\[ z^0, z^1, z^2, \cdots, z^n, \cdots \]

If we pick an arbitrary point in this sequence, \(z^n\) for some \(n\geq 0\), we can express the corresponding term in polar form as follows

\[ z^n = {\left(r\cdot e^{\mathsf{j}\theta}\right)}^n = r^n \cdot e^{\mathsf{j} \cdot n \theta}, \]

where \(z = r\cdot e^{\mathsf{j}\theta}\) and the second equality follows from the product rule for exponents.

From this, we can see that the \(n\)th step in the sequence will place the point at angle \(n\cdot \theta\), and radius \(r^n\).

If \(r < 1\), this sequence will have decreasing radius, and it will spiral into the origin. If \(r > 1\), the sequence will have increasing radius, and it will spiral out. If \(r=1\), each element will have the same radius since \(1^n = 1\): the sequence will therefore loop around the circle indefinitely.

If \(0 \leq \theta \leq \pi\), the spiral will wind counter-clockwise, and otherwise it will wind clockwise, as demonstrated below. Larger angles (closer to \(\pm\pi\)) rotate faster than smaller angles (closer to \(0\)).

Once Loop Reflect

Fig. 4.10 Raising a complex number \(z\) to a sequence of powers \(n=0, 1, 2, \dots\) produces an inward spiral (if magnitude is less than 1); an outward spiral (if magnitude is greater than 1), or an orbit around the unit circle (if magnitude is exactly 1).

If we take just the real (or imaginary) part of \(z^n\) for the sequence of values of \(n\), we’ll see a sinusoid with decaying (\(r < 1\)), increasing (\(r>1\)), or stable (\(r=1\)) amplitude, as illustrated below for the real part (top) and imaginary part (bottom).

Fig. 4.11 The real (top) and imaginary (bottom) components of \(z^n\) as \(n\) grows, for different choices of \(z\).

From these plots, we can see that any complex number \(z\) can be used to generate what is known as a complex sinusoid by raising it to successive powers \(z^n\). This idea forms the basis of the (discrete) Fourier transform, which we will see in the next chapter.