0.4 Analog (de)modulation  (Page 5/9)

To continue the investigation, suppose that the carrier phase offset is zero, (i.e.,  $\Phi =0$ ), but that the frequency offset $\gamma$ is not. Then the spectrum of $x\left(t\right)$ from [link] is

and the lowpass filtering of $x\left(t\right)$ produces

This is shown in [link] . Recognizing this spectrum as a frequency shiftedversion of $w\left(t\right)$ , it can be translated back into the time domain using [link] to give

Instead of recovering the message $w\left(t\right)$ , the frequency offset causes the receiver to recover a low frequencyamplitude modulated version of it. This is bad with even a small carrier frequency offset.While $\cos \left(\Phi \right)$ in [link] is a fixed scaling, $\cos \left(2\pi \gamma t\right)$ in [link] is a time-varying scaling that will alternately recover $m\left(t\right)$ (when $\cos \left(2\pi \gamma t\right)\approx 1$ ) and make recovery impossible (when $\cos \left(2\pi \gamma t\right)\approx 0$ ). Transmitters are typically expected to maintain suitableaccuracy to a nominal carrier frequency setting known to the receiver. Ways of automatically tracking (inevitable) smallfrequency deviations are discussed at length in [link] .

The following code AM.m generates a message $w\left(t\right)$ and modulates it with a carrier at frequency $f_c$ . The demodulation is done with a cosine offrequency $f_c+\gamma$ and a phase offset of $\Phi$ . When $\gamma =0$ and $\Phi =0$ , the output (a lowpass version of the demodulated signal)is nearly identical to the original message, except for the inevitable delay caused by the linearfilter. [link] shows four plots: the message $w\left(t\right)$ on top, followed by the upconverted signal $v\left(t\right)=w\left(t\right)cos\left(2\pi f_{c}t\right)$ , followed in turn by the downconverted signal $x\left(t\right)$ . The lowpass filtered versionis shown in the bottom plot; observe that it is nearly identical to the original message, albeit with a slight delay.