## Single Sideband (SSB) Modulation 3

To tie the previous two posts together, LSSB and USSB are given by the following spectrum and time domain expressions. So what does this expression give? It provides the basis for the construction of the block diagram level SSB transmitter.

This means that the message signal m(t) gets sent to two separate paths. The top path multiplies m(t) by (1/2)*A_c*cos(2*pi*f_c*t). The lower branch takes the message and passes it through a Hilbert transformer to generate ^m(t). ^m(t) is then multiplied by (1/2)*A_c*sin(2*pi*f_c*t). The resulting outputs of each of the multipliers (also known as frequency mixers) is fed to a summing junction. If LSSB is desired, then the sine term is added to the cosine term. If USSB is desired, then the sine term is subtracted from the cosine term.

Why do all of this when the most intuitive thing to do would have been to put a low-pass filter in front of a DSB-SC transmitter (for LSSB) or a high-pass filter in front of a DSB-SC transmitter (for USSB)? The reason is that the filter requirements are not practical and trying to build something that approaches the ideal low-pass/high-pass filter means a lot of components will be necessary.

Of course, as with most things in communication systems, there is a trade-off for using the first technique. Though the filters have to be basically ideal in the latter case, in the former case, a wideband phase shifter is necessary in the phasing method of SSB transmission. The Hilbert transformer has to phase shift every frequency in the bandwidth of the message signal by -90 degrees. The Hilbert transformer, luckily, operates near the baseband and so it's not impractical to build such a Hilbert transformer.

According to the IIT Delhi lecture previously linked, in practice both are used and both have their pros and cons. Realizing a -90 phase shifter at a single frequency is easy theoretically. An ideal capacitor has an impedance that is such that the capacitor output is precisely -90 degrees out of phase with the input.

Since SSB is essentially DSB-SC with one sideband chopped off, coherent demodulation can be done in the same way that it was done with DSB-SC: a product modulator followed by a low-pass filter recovers the message -- exploiting the fact that cosine squared can equivalently be written in terms of a constant and a cosine of twice the frequency (just filter out the high-frequency term and recover the message). This assumes that there is phase coherence between the transmitter and the receiver of course. This is enough theory for now. I will post more later.