next up previous contents index
Next: Time shifts and phase Up: Complex numbers Previous: Complex numbers   Contents   Index

Sinusoids as geometric series

Recall the formula for a (real-valued) sinusoid from page [*]:

\begin{displaymath}
x[n] = a \cos (\omega n + \phi )
\end{displaymath}

This is a sequence of cosines of angles (called phases) which increase arithmetically with the sample number $n$. The cosines are all adjusted by the factor $a$. We can now re-write this as the real part of a much simpler and easier to manipulate sequence of complex numbers, by using the properties of their arguments and magnitudes.

Suppose that our complex number $Z$ happens to have magnitude one, so that it can be written as:

\begin{displaymath}
Z = \cos(\omega) + i \sin(\omega)
\end{displaymath}

Then for any integer $n$, the number $Z^n$ must have magnitude one as well (because magnitudes multiply) and argument $n\theta$ (because arguments add). So,

\begin{displaymath}
{Z^n} = \cos(n\omega) + i \sin(n \omega)
\end{displaymath}

This is also true for negative values of $n$, so for example,

\begin{displaymath}
{1 \over Z} = {Z^{-1}} = cos(\omega) - i \sin(\omega)
\end{displaymath}

Figure 7.2 shows graphically how the powers of $Z$ wrap around the unit circle, which is the set of all complex numbers of magnitude one. They form a geometric sequence:

\begin{displaymath}
\ldots, {Z^0}, {Z^1}, {Z^2}, \ldots
\end{displaymath}

and taking the real part of each term we get a real sinusoid with initial phase zero and amplitude one:

\begin{displaymath}
\ldots, \cos(0), \cos(\omega), \cos(2 \omega), \ldots
\end{displaymath}

The sequence of complex numbers is much easier to manipulate algebraically than the sequence of cosines.

Figure 7.2: The powers of a complex number $Z$ with $\vert Z\vert=1$, and the same sequence multiplied by a constant $A$.
\begin{figure}\psfig{file=figs/fig07.02.ps}\end{figure}

Furthermore, suppose we multiply the elements of the sequence by some (complex) constant $A$ with magnitude $a$ and argument $\phi$. This gives

\begin{displaymath}
\ldots, A, AZ, A{Z^2}, \ldots
\end{displaymath}

The magnitudes are all $a$ and the argument of the $n$th term is $\phi + n \omega$, so the sequence is equal to

\begin{displaymath}
a \cdot [\cos(\phi) + i \sin(\phi)],
\end{displaymath}


\begin{displaymath}
a \cdot [\cos(\omega + \phi) + i \sin(\omega + \phi)],
\end{displaymath}


\begin{displaymath}
\cdots , a \cdot [\cos(n \omega + \phi) + i \sin(n \omega + \phi)], \ldots
\end{displaymath}

and so the real part is just the real-valued sinusoid:

\begin{displaymath}
\mathrm{re}(A{Z^n}) = a \cdot [\cos(n \omega + \phi)]
\end{displaymath}

The complex amplitude $A$ encodes both the amplitude (equal to its magnitude $a$) and the inital phase (its argument $\phi$); the unit-magnitude complex number $Z$ controls the frequency which is just its argument $\omega $.

Figure 7.2 also shows the sequence $A, AZ, A{Z^2}, \ldots$; in effect this is the same sequence as $1, Z, {Z^2}, \ldots$, but amplified and rotated according to the amplitude and initial phase. In a complex sinusoid of this form, $A$ is called the complex amplitude.

Using complex numbers to represent the amplitudes and phases of sinusoids can clarify manipulations that otherwise might seem unmotivated. For instance, in Section [*] we looked at the sum of two sinusoids with the same frequency. In the language of this chapter, we let the two sinusoids be written as:

\begin{displaymath}
X[n] = A {Z^n} , \ Y[n] = B {Z^n}
\end{displaymath}

where $A$ and $B$ encode the phases and amplitudes of the two signals. The sum is then equal to:

\begin{displaymath}
X[n] + Y[n] = (A+B) {Z^n}
\end{displaymath}

which is a sinusoid whose amplitude equals $\vert A+B\vert$ and whose phase equals $\angle(A+B)$. This is clearly a much easier way to manipulate amplitudes and phases than using series of sines and cosines. Eventually, of course, we will take the real part of the result; this can usually be left to the very last step of the calculation.


next up previous contents index
Next: Time shifts and phase Up: Complex numbers Previous: Complex numbers   Contents   Index
Miller Puckette 2006-03-03