0.4 Analog (de)modulation  (Page 4/9)

As illustrated in [link] , the received signal can be demodulated by mixingwith a cosine that has the same frequency and phase as the modulating cosine, and the original message canthen be recovered by low pass filtering. But, as a practical matter,the frequency and phase of the modulating cosine (located at the transmitter)can never be known exactly at the receiver.

Suppose that the frequency of the modulator is $f_c$ but that the frequency at the receiver is $f_c+\gamma$ , for some small $\gamma$ . Similarly, suppose that the phase of the modulator is 0but that the phase at the receiver is $\Phi$ . [link] (b) shows this downconverter, which can be described by

and

where LPF represents a lowpass filtering of the demodulated signal $x\left(t\right)$ in an attempt to recover the message. Thus, the downconversion describedin [link] acknowledges that the receiver's local oscillator may not have thesame frequency or phase as the transmitter's local oscillator.In practice, accurate a priori information is available for carrier frequency, but (relative) phase could be anything, since it dependson the distance between the transmitter and receiver as well as when the transmission begins.Because the frequencies are high, the wavelengths are small and even small motions canchange the phase significantly.

The remainder of this section investigates what happens when the frequency and phase are not known exactly,that is, when either $\gamma$ or $\Phi$ (or both) are nonzero. Using the frequency shift property of Fourier transforms [link] on $x\left(t\right)$ in [link] produces the Fourier transform $X\left(f\right)$

If there is no frequency offset (i.e., if $\gamma =0$ ), then

Because $\cos \left(x\right)=(1/2)(e^{jx}+e^{-jx})$ from [link] , this can be rewritten

Thus, a perfect lowpass filtering of $x\left(t\right)$ with cutoff below $2f_c$ removes the high frequency portions of the signal near $\pm 2f_c$ to produce

The factor $\cos \left(\Phi \right)$ attenuates the received signal (except for the special case when $\Phi =0\pm 2\pi k$ for integers $k$ ). If $\Phi$ were sufficiently close to $0\pm 2\pi k$ for some integer $k$ , then this would be tolerable. But there is no way to know the relative phase, and hence $\cos \left(\Phi \right)$ can assume any possible value within $[-1,1]$ . The worst case occurs as $\Phi$ approaches $\pm \pi /2$ , when the message is attenuated to zero!A scheme for carrier phase synchronization, which automatically tries toalign the phase of the cosine at the receiver with the phase at the transmitter is vital. This is discussedin detail in [link] .