Dsb lc,sc

Linear CW Modulation

Roadmap1. Bandpass Signals and Systems2. Double-Sideband Amplitude Modulation3. Modulators and Transmitters4. Suppressed-Sideband Amplitude Modulation5. Frequency Conversion and Demodulation

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BANDPASS SIGNALS AND SYSTEMS

• Analog Message Conventions• Bandpass Signals• Bandpass Transmission• Bandwidth

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Analog Message Conventions

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sinusoidal or tone modulation

Bandpass Signals

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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,” when they occur, are absorbed in the phase by adding ±180o .

envelope-and-phase description

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quadrature-carrier description

quadraturecomponent

in-phase component

DOUBLE-SIDEBAND AMPLITUDE MODULATION

• AM Signals and Spectra• DSB Signals and Spectra• Tone Modulation and Phasor Analysis

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AM Signals and Spectra

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If Ac denotes the unmodulated carrier amplitude, modulation by x(t) produces the AM signal

The signal’s envelope is

modulation index

xc(t) has no time-varying phase, its in-phase and quadrature components are

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The condition fc >> W ensures that the carrier oscillates rapidly compared to thetime 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

The envelope clearly reproduces the shape of if

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12AM transmission bandwidth

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Another important consideration is the average transmitted power

Upon expanding

averages to zero under the condition fc >> W

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The term Pc represents the unmodulated carrier power, since ST = Pc when μ = 0

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 transmittedpower resides in a carrier term that’s independent of and thus conveys nomessage information.

DSB Signals and Spectra

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The “wasted” carrier power in amplitude modulation can be eliminated bysetting and suppressing the unmodulated carrier-frequency component. Theresulting modulated wave becomes

which is called double-sideband–suppressed-carrier modulation—or DSB forshort. (The abbreviations DSB–SC and DSSC are also used.)

the DSB spectrum looks like an AM spectrum without the unmodulated carrierimpulses. The transmission bandwidth thus remains unchanged .

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the DSB envelope and phase are

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The envelope here takes the shape of |x(t)|, rather than x(t), and the modulatedwave undergoes a phase reversal whenever x(t) crosses zero.

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 theinformation-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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EXAMPLE Consider a radio transmitter rated for

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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If the modulation is AM with μ = 1, then

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 only 25 percent of the DSB path length with the same transmitter.

Tone Modulation and Phasor Analysis

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Setting

the tone-modulated DSB waveform

tone-modulated AM wave

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tone-modulated AM with

EXAMPLE: AM and Phasor Analysis

the phasor sum equals the envelope

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Suppose a transmission channel completely removes the lower sideband,

Now the envelope becomes

from which the envelope distortion can be determined.

SUPPRESSED-SIDEBAND AMPLITUDEMODULATION

• SSB Signals and Spectra• SSB Generation• VSB Signals and Spectra

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The in-phase and quadrature functions must be lowpass signals

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the lowpass equivalent spectrum

lowpass equivalent signal

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Rectangular-to-Polar conversion yields

The lowpass-to-bandpass transformation in the time domain.

The corresponding frequency-domain transformation is

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Since we’ll deal only with real bandpass signals, we can keep the hermitiansymmetry, 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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In particular, after finding , you can take its inverse Fourier transform

The lowpass-to-bandpass transformation then yields the output signal

Or you can get the output quadrature components or envelope and phase immediately from

SSB Signals and Spectra

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The resulting signal in either case has

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 ±fmand the envelope is a constant proportional to Am.

Obviously, envelope detection won’t work for SSB.

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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 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 lowpassfunction

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The corresponding transfer functions for LSSB are

Both lowpass transfer functions can be represented by

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yields the lowpass equivalent spectrum for either USSB or LSSB, namely

Now recall that

Finally, we perform the lowpass-to-bandpass transformation

SSB Generation

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VSB Signals and Spectra

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Review Questions

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Problems to Ponder

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Review Questions

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Problems to Ponder

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MODULATORS AND TRANSMITTERS

• Product Modulators• Square-Law and Balanced Modulators• Switching Modulators

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Product Modulators

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Square-Law and Balanced Modulators

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Switching Modulators

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Review Questions

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Problems to Ponder

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SUPPRESSED-SIDEBAND AMPLITUDEMODULATION

• SSB Signals and Spectra• SSB Generation• VSB Signals and Spectra

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SSB Signals and Spectra

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SSB Generation

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VSB Signals and Spectra

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Review Questions

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Problems to Ponder

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FREQUENCY CONVERSION AND DEMODULATION

• Frequency Conversion• Synchronous Detection• Envelope Detection

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Frequency Conversion

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Synchronous Detection

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Envelope Detection

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Review Questions

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Problems to Ponder

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