2. Roadmap
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1. Bandpass Signals and Systems
2. Double-Sideband Amplitude Modulation
3. Modulators and Transmitters
4. Suppressed-Sideband Amplitude Modulation
5. Frequency Conversion and Demodulation
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3. BANDPASS SIGNALS AND SYSTEMS
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• Analog Message Conventions
• Bandpass Signals
• Bandpass Transmission
• Bandwidth
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envelope-and-phase description
where A(t) is the envelope and φ(t) is the phase, both functions of time
The envelope is defined as nonnegative, so that A(t) ≥ 0 . Negative “amplitudes,”
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when they occur, are absorbed in the phase by adding ±180o .
8. DOUBLE-SIDEBAND AMPLITUDE
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MODULATION
• AM Signals and Spectra
• DSB Signals and Spectra
• Tone Modulation and Phasor Analysis
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9. AM Signals and Spectra
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If Ac denotes the unmodulated carrier amplitude, modulation by x(t) produces the
AM signal
modulation
index
The signal’s envelope is
xc(t) has no time-varying phase, its in-phase and quadrature components are
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11. The envelope clearly reproduces the shape of if
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The condition fc >> W ensures that the carrier oscillates rapidly compared to the
time variation of x(t); otherwise, an envelope could not be visualized.
The condition μ ≤ 1 ensures that Ac[ 1 + μx(t) ] does not go negative.
With 100 percent modulation (μ = 1), the envelope varies between Amin = 0
and Amax = 2Ac .
Overmodulation ( μ > 1), causes phase reversals and envelope distortion
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13. Another important consideration is the average transmitted power
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Upon expanding
averages to zero under the
condition fc >> W
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14. The term Pc represents the unmodulated carrier power, since ST = Pc when μ = 0
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the term Psb represents the power per sideband since, when μ ≠ 0, ST consists of
the power in the carrier plus two symmetric sidebands.
The modulation constraint
requires that
Consequently, at least 50 percent (and often close to 2/3) of the total transmitted
power resides in a carrier term that’s independent of and thus conveys no
message information.
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15. DSB Signals and Spectra
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The “wasted” carrier power in amplitude modulation can be eliminated by
setting and suppressing the unmodulated carrier-frequency component. The
resulting modulated wave becomes
which is called double-sideband–suppressed-carrier modulation—or DSB for
short. (The abbreviations DSB–SC and DSSC are also used.)
the DSB spectrum looks like an AM spectrum without the unmodulated carrier
impulses. The transmission bandwidth thus remains unchanged . 15
17. The envelope here takes the shape of |x(t)|, rather than x(t), and the modulated
wave undergoes a phase reversal whenever x(t) crosses zero.
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Full recovery of the message requires knowledge of these phase reversals, and
could not be accomplished by an envelope detector.
Carrier suppression does put all of the average transmitted power into the
information-bearing sidebands.
Practical transmitters also impose a limit on the peak envelope power
We’ll take account of this peak-power limitation by examining the ratio
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DSB conserves power but requires
complicated demodulation circuitry,
whereas AM requires increased power
to permit simple envelope detection.
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19. EXAMPLE Consider a radio transmitter rated for
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Let the modulating signal be a tone with
If the modulation is DSB,
the maximum possible power per sideband equals the
lesser of the two values determined from
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20. If the modulation is AM with μ = 1, then
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To check on the average-power limitation,
Hence, the peak power limit again dominates and the maximum sideband
power is
Since transmission range is proportional to Psb , the AM path length would be 20
only 25 percent of the DSB path length with the same transmitter.
21. Tone Modulation and Phasor
Analysis
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Setting
the tone-modulated DSB waveform
tone-modulated AM wave
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23. EXAMPLE: AM and Phasor Analysis
tone-modulated AM with
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the phasor sum equals the envelope
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24. Suppose a transmission channel completely removes the lower sideband,
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Now the envelope becomes
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from which the envelope distortion can be determined.
25. SUPPRESSED-SIDEBAND AMPLITUDE
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MODULATION
• SSB Signals and Spectra
• SSB Generation
• VSB Signals and Spectra
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28. Rectangular-to-Polar conversion yields
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The lowpass-to-bandpass transformation in the time domain.
The corresponding frequency-domain transformation is
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29. Since we’ll deal only with real bandpass signals, we can keep the hermitian
symmetry, in mind and use the simpler expression
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It’s usually easier to work with the lowpass equivalent spectra related by
which is the lowpass equivalent transfer function.
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31. In particular, after finding , you can take its inverse Fourier transform
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The lowpass-to-bandpass transformation then yields the output signal
Or you can get the output quadrature components or envelope and phase
immediately from
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33. The resulting signal in either case has
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Removing one sideband line leaves only the other line. Hence,
Note that the frequency of a tone-modulated SSB wave is offset from fc by ±fm
and the envelope is a constant proportional to Am.
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Obviously, envelope detection won’t work for SSB.
34. To analyze SSB with an arbitrary message x(t),
we’ll draw upon the fact that the sideband filter is a bandpass system with a
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bandpass DSB input
and a bandpass SSB output
applying the equivalent lowpass method.
Since xbp(t) has no quadrature component, the lowpass equivalent input is simply
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The bandpass filter transfer function for USSB along with the equivalent lowpass
function
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36. The corresponding transfer functions for LSSB are
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Both lowpass transfer functions can be represented by
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37. yields the lowpass equivalent spectrum for either USSB or LSSB, namely
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Now recall that
Finally, we perform the lowpass-to-bandpass transformation
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