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This situation is depicted in [link] , where three different messages are representedby the triangular, rectangular, and half-oval spectra, each bandlimited to ± f * . Each of these is modulated by a different carrier( f 1 , f 2 , and f 3 ), which are chosen so that they are further apart than the widthof the messages. In general, as long as the carrier frequencies are separated by more than 2 f * , there will be no overlap in the spectrum of the combined signal. This process ofcombining many different signals together is called multiplexing , and because the frequencies are divided up among the users, the approach of [link] is called frequency division multiplexing (FDM).

Whenever FDM is used, the receiver must separate the signal of interest from all the other signals present.This can be accomplished with a bandpass filter as in [link] , which shows a filter designed to isolate the middle user from the others.

Suppose that two carrier frequencies are separated by 1 KHz. Draw the magnitude spectra if(a) the bandwidth of each message is 200 Hz and (b) the bandwidth of each message is 2 KHz. Comment on theability of the bandpass filter at the receiver to separate the two signals.

Three different upconverted signals are assigned different frequency bands. This is called frequency division multiplexing.
Three different upconverted signals are assigned different frequency bands. This is called frequency divisionmultiplexing.
Separation of a single FDM transmission using a bandpass filter.
Separation of a single FDM transmission using a bandpass filter.

Another kind of multiplexing is called time division multiplexing (TDM), in which two (or more) messages use the samecarrier frequency but at alternating time instants. More complex multiplexing schemes(such as code division multiplexing) overlap the messages in both time and frequency in such a way that they can bedemultiplexed efficiently by appropriate filtering.

Filters that remove frequencies

Each time the signal is modulated, an extra copy (or replica) of the spectrum appears. When multiple modulations are needed(for instance, at the transmitter to convert up to the carrier frequency, and at the receiver to convert back down tothe original frequency of the message), copies of thespectrum may proliferate. There must be a way to remove extra copies in order to isolate the original message.This is one of the things that linear filters do very well.

There are several ways of describing the action of a linear time invariant filter.In the time domain (the most common method of implementation), the filter is characterized by its impulse response(which is defined to be the output of the filter when the input is an impulse function). By linearity, the outputof the filter to any arbitrary input is then the superposition of weighted copies of time shifted version ofthe impulse response, a procedure known as convolution. Since convolution may be difficult tounderstand directly in the time domain, the action of a linear filteris often described in the frequency domain.

Perhaps the most important property of the Fourier transform is the duality between convolution andmultiplication, which says that

  • convolution in time multiplication in frequency, and
  • multiplication in time convolution in frequency.

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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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