### Local oscillator error

#### For any bandpass modulation process it is necessary to generate sinewaves in both transmitter and receiver systems, ideally with precisely the same frequency and phase. This begs the question: Is it possible to implement two or more sinewave generators with perfect frequency and phase accuracy under realistic operating conditions (temperature variation, supply variation, ageing, and so on)?

#### A reasonably cost-effective crystal oscillator (<$10) may have the stablity of 1 ppm (part per million) over a given temperature range. This means that for a telephone modem with a carrier of say 2 kHz, the oscillators at each end of the link could have an error of +/–1.10–6 x 2.103 = 0.002 Hz. If we could ensure that both transmit and receiver carrier oscillators begin with the same phase, then we could expect the phase error between them to reach 360o after 1 / 0.004 = 250 seconds, and 90o, giving zero output, after 75 seconds. These figures suggest that, providing an initial phase correction can be achieved, near phase-coherent detection can be ensured for a few seconds without further phase correction being required. |

#### If we now consider the case of a cellular radio modem operating with a carrier of 1 GHz, then the oscillator frequency error for a 1-ppm source is +/–1000 Hz. Here, it is clear that simply achieving a correct starting phase will not allow us to ensure adequate coherency for more than a few microseconds. In this application, it is necessary to find a method of correcting the receiver carrier oscillator frequency and phase to match that of the transmitter. This process is termed carrier recovery.