0.5 Sampling with automatic gain control  (Page 4/19)

Downconversion via sampling

The processes of modulation and demodulation, which shift the frequencies of a signal,can be accomplished by mixing with a cosine wave that has a frequency equal to the amountof the desired shift, as was demonstrated repeatedly throughout Chapter  [link] . But this is not the only way.Since sampling creates a collection of replicas of the spectrum of a waveform, it changes the frequencies ofthe signal.

When the message signal is analog and bandlimited to $\pm B$ , sampling can be used as a step in the demodulation process. Suppose that the signal is transmitted with a carrierat frequency $f_c$ . Direct sampling of this signal creates a collection of replicas, one near DC.This procedure is shown in [link] for $f_s=\frac{f_c}{2}$ , though beware: when $f_s$ and $f_c$ are not simply related, the replica may not land exactly at DC.

This demodulation-by-sampling is diagrammed in [link] (with $f_s=\frac{f_c}{n}$ , where $n$ is a small positive integer), and can be thought of as an alternative tomixing with a cosine (that must be synchronized in frequency and phase with the transmitter oscillator).The magnitude spectrum $\left|W\right(f\left)\right|$ of a message $w\left(t\right)$ is shown in [link] (a), and the spectrum after upconversion isshown in part (b); this is the transmitted signal $s\left(t\right)$ . At the receiver, $s\left(t\right)$ is sampled, which can be modelled as a multiplication with a train of delta functions in time

where $T_s$ is the sample period.

Using [link] , this can be transformed into the frequency domain as

where $f_s=1/T_s$ . The magnitude spectrum of $Y\left(f\right)$ is illustrated in [link] (c) for the particular choice $f_s=f_c/2$ (and $T_s=2/f_c$ ) with $B<\frac{f_c}{4}=\frac{f_s}{2}$ .

There are three ways that the sampling can proceed:

1. sample faster than the Nyquist rate of the IF frequency,
2. sample slower than the Nyquist rate of the IF frequency, and then downconvert the replica closest to DC, and
3. sample so that one of the replicas is directly centered at DC.

The first is a direct imitation of the analog situation where no aliasing will occur. This may be expensivebecause of the high sample rates required to achieve Nyquist sampling. The third is the situation depicted in Figures [link] and [link] , which permit downconversion to baseband without an additional oscillator.This may be sensitive to small deviations in frequency (for instance, when $f_s$ is not exactly $f_c/2$ ). The middle method downconverts part of the wayby sampling and part of the way by mixing with a cosine. The middle method is used inthe ${\mathcal{M}}^6$ receiver project in Chapter  [link] .