## Spectrum of Frequency Shift Keying waveforms

#### The spectrum of FSK depends on a number of factors, including whether the transition between symbol states has a continuous phase or discontinuous phase, whether the data waveform driving the modulator (typically a VCO) is shaped by filtering, and the frequency separation between symbol states. A good insight into the whole topic is given by Lucky et al (1968).

#### Only two special cases will be considered here – binary FSK with a frequency separation of 1/Tb and no pulse shaping (this is often called Sunde's FSK), and binary Continuous Phase FSK with frequency separation of 0.5 x 1/Tb and no pulse shaping (usually called Minimum Shift Keying).

### Power spectral density for Sunde's FSK

####

#### where

####

#### The characteristic of Sunde's FSK is that it has two discrete components in the spectrum at the two symbol frequencies. These are of great benefit when recovering a carrier component for coherent detection.

### Power spectral density for Minimum Shift Keying (MSK)

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#### Unlike Sunde's FSK, this spectrum has no discrete components, and a much narrower main lobe as would be expected with the narrower frequency separation between symbols. Comparing this result with the power spectral density of QPSK, given by:
we can see that the side-lobe energy for MSK falls off much more quickly than for QPSK, with MSK having a slightly wider main lobe. As both formats offer a potential bandwidth efficiency approaching 2 bits/second/Hz, MSK is often the preferred modulation choice in systems where the constant envelope property of the FSK family is important. Where filtered QPSK with its associated amplitude fluctuations can be tolerated, this will give the ultimate minimum bandwidth solution.