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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 ± 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 = 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 = 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 | W ( f ) | of a message w ( t ) is shown in [link] (a), and the spectrum after upconversion isshown in part (b); this is the transmitted signal s ( t ) . At the receiver, s ( t ) is sampled, which can be modelled as a multiplication with a train of delta functions in time

y ( t ) = s ( t ) n = - δ ( t - n T s ) ,

where T s is the sample period.

Spectra in a sampling downconverter such as Figure 6-7. The (bandlimited analog) signal W(f) shown in (a) is upconverted to the transmitted signal in (b). Directly sampling this (at a rate equal to f_s=f_c/2 results in the spectrum shown in the bottom plot.
Spectra in a sampling downconverter such as [link] . The (bandlimited analog) signal W ( f ) shown in (a) is upconverted to the transmitted signal in (b).Directly sampling this (at a rate equal to f s = f c 2 ) results in the spectrum shown in the bottom plot.

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

Y ( f ) = 1 T s n = - S ( f - n f s ) ,

where f s = 1 / T s . The magnitude spectrum of Y ( f ) is illustrated in [link] (c) for the particular choice f s = f c / 2 (and T s = 2 / f c ) with B < f c 4 = 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 M 6 receiver project in Chapter  [link] .

System diagram showing how sampling can be used to downconvert a signal. The spectra corresponding to w(t), s(t) and y(t) are shown in Figure 6-6. The output of the LPF contains only the “M” shaped portion nearest zero.
System diagram showing how sampling can be used to downconvert a signal. The spectra corresponding to w ( t ) , s ( t ) and y ( t ) are shown in [link] . The output of the LPF contains only the “M” shaped portion nearest zero.

Create a simulation of a sampling-based modulator that takes a signal with bandwidth 100 Hzand transforms it into the “same” signal centered at 5000 Hz. Be careful; there are two“sampling rates” in this problem. One reflects the assumed sampling rate for the modulation and theother represents the sampling rate that is used in M atlab to represent a “continuous time” signal. You may wish to reuse code from sine100hzsamp.m . What choices have you made for these two sampling rates?

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Source:  OpenStax, Software receiver design. OpenStax CNX. Aug 13, 2013 Download for free at http://cnx.org/content/col11510/1.3
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