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
Is squaring a signal different than multiplying it by itself 


No, it is not different.  Squaring the signal is multiplying it by itself, sample by sample.

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.