Spectral effects of multiplying a signal by itself?

This has to do with creating the synchronization circuit for MSK in a DSP as described in "Minimum Shift Keying" by Subbarayan Pasupathy published in IEEE Communications Magazine.

At the beginning of the circuit he shows signal passing through a "squarer". The explanation reads:

"Although the MSK signal $s(t)$ has no discrete components which can be used for synchronization, it produces string discrete spectral components at $2f_+$ and $2f_-$ when passed through the squarer. (The squarer, in effect, doubles the modulation index and produces an FSK signal with $\delta_f = 1/T$, known as Sunde's FSK. Sunde's FSK has 50 percent of its total power in the line components at the two transmitter frequencies.)"

$s(t)$ consists of a signal containing components at $\pm 300 \mathrm{Hz}$ around the carrier frequency.

I'm confused by this because when I multiply the signal by itself it doesn't seem to produce a signal with components at twice the offset frequencies. Is squaring a signal different than multiplying it by itself (sample by sample, not convolution with itself)?

Thank you

• Are you using a high-enough sampling frequency? In your simulation, you'll need a sampling frequency that is more than double the maximum frequency that can ever appear in the system.
– MBaz
Oct 9 '14 at 21:05

In the frequency domain this is equivalent to convolving the original frequency spectrum with itself. The convolution "spreads" the signal out. If the signal is at $\pm300$ Hz around a carrier frequency $f_c$, then you should see energy at $\pm(2f_c + 600)$ Hz, $\pm2f_c$ Hz, $\pm(2f_c - 600)$ Hz, $0$ Hz, and $\pm600$ Hz.