Sinusoids as geometric series

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

$$x[n] = a \cos(\omega n + \phi)$$

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:

$$Z = \cos(\omega) + i sin(\omega)$$

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,

$${Z^n} = cos(n\omega) + i \sin(n \omega)$$

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

$${1 \over Z} = {Z^{-1}} = cos(n\omega) - i \sin(n \omega)$$

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:

$$\cdots, {Z^0}, {Z^1}, {Z^2}, \cdots$$

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

$$\cdots, \cos(0), \cos(\omega), \cos(2 \omega), \cdots$$

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 $A$.

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

$$\cdots, A, AZ, A{Z^2}, \cdots$$

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

$$a \cdot [\cos(\phi) + i \sin(\phi)],$$

$$a \cdot [\cos(\omega + \phi) + i \sin(\omega + \phi)],$$

$$\cdots , a \cdot [\cos(n \omega + \phi) + i \sin(n \omega + \phi)], \cdots$$

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

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

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 $A, AZ, A{Z^2}, \cdots$; in effect this is the same sequence as $1, Z, {Z^2}, \cdots$, 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 1.6 we considered the sum of two sinusoids with the same frequency. In the language of this chapter, we let the two sinusoids be written as:

$$X[n] = A {Z^n} , \ Y[n] = B {Z^n}$$

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

$$X[n] + Y[n] = (A+B) {Z^n}$$

which is a sinusoid whose amplitude equals $\vert A+B\vert$ and whose phase equals $\mathrm{arg}(A+B)$. 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.