Single Sideband Modulation

As we saw in Chapter 5, multiplying two real sinusoids together results in a signal with two new components at the sum and difference of the original frequencies. If we carry out the same operation with complex sinusoids, we get only one new resultant frequency; this is one result of the greater mathematical simplicity of complex sinusoids as compared to real ones. If we multiply a complex sinusoid $1, Z, {Z^2}, \ldots$ with another one, $1, W, {W^2}, \ldots$ the result is $1, WZ, {{(WZ)}^2}, \ldots$, which is another complex sinusoid whose frequency, $\angle(ZW)$, is the sum of the two original frequencies.

In general, since complex sinusoids have simpler properties than real ones, it is often useful to be able to convert from real sinusoids to complex ones. In other words, from the real sinusoid:

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

(with a spectral peak of amplitude $a/2$ and frequency $\omega$) we would like a way of computing the complex sinusoid:

$$X[n] = a \left ( \cos (\omega n) + i \sin (\omega n) \right )$$

so that

$$x[n] = \mathrm{re} (X[n])$$

We would like a linear process for doing this, so that superpositions of sinusoids get treated as if their components were dealt with separately.

Of course we could equally well have chosen the complex sinusoid with frequency $-\omega$:

$$X'[n] = a \left ( \cos (\omega n) - i \sin (\omega n) \right )$$

and in fact $x[n]$ is just half the sum of the two. In essence we need a filter that will pass through positive frequencies (actually frequencies between 0 and $\pi$, corresponding to values of $Z$ on the top half of the complex unit circle) from negative values (from $-\pi$ to 0, or equivalently, from $\pi$ to $2\pi$--the bottom half of the unit circle).

One can design such a filter by designing a low-pass filter with cutoff frequency $\pi /2$, and then performing a rotation by $\pi /2$ radians using the technique of Section 8.3.4. However, it turns out to be easier to do it using two specially designed networks of all-pass filters with real coefficients.

Calling the transfer functions of the two filters $H_1$ and $H_2$, we design the filters so that

$$\angle({H_1}(Z)) - \angle({H_2}(Z)) \approx \left \{ \be... ...pi} \\ -\pi/2 & {-\pi < \angle(Z) < 0} \end{array} \right .$$

or in other words,

$${H_1}(Z) \approx i {H_2}(Z) , \; 0 < \angle(Z) < \pi$$

$${H_1}(Z) \approx -i {H_2}(Z) , \; -\pi < \angle(Z) < 0$$

Then for any incoming real-valued signal $x[n]$ we simply form a complex number $a[n] + i b[n]$ where $a[n]$ is the output of the first filter and $b[n]$ is the output of the second. Any complex sinusoidal component of $x[n]$ (call it $Z^n$) will be transformed to

$${H_1}(Z) + i {H_2}(Z) \approx \left \{ \begin{array}{ll} ... ...ngle(Z) < \pi} \\ 0 & \mbox{otherwise} \end{array} \right .$$

Having started with a real-valued signal, whose energy is split equally into positive and negative frequencies, we end up with a complex-valued one with only positive frequencies.