Digital_Communications_Part_A

DSBFC
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Double Sideband Full Carrier
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types_dsbfc_map.png

Including a carrier on your modulated signal will be beneficial for the receiver to detect, synchronize and demodulate the transmitted signal properly.

We can augment the DSBSC and add the carrier to the modulated signal.

Block Diagram
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design_icon.png
DSBFC_block_diagram.png

Lab Bench
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lab_icon.png

Using the lab equipment, I connect the output of the previous DSBSC to input 1 of the adder and the carrier to input 2 of the adder , where and are knob gains that control how much of the signal passes to the adder respectively.

Carrier and Baseband Signals
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  • Orange Line: baseband signal
  • Blue Line: carrier signal

Time Domain Waveforms
dsbsc_carrier_baseband_time.png

dsbsc_carrier_baseband_time_zoom.png

Frequency Spectrum
dsbsc_carrier_baseband_spectrum.png

Attention

Since we have the ability to control how much of the signals we add together, we are going to tinker with the knobs and learn different modulated output signals.

Tinker Signal 1
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  • gain
  • gain , to compensate the multiplication factor k = 1/2.
  • Orange Line: baseband signal
  • Blue Line: modulation output

Time Domain Waveform
dsbfc_modulation_baseband_time.png

Frequency Spectrum
dsbfc_modulation_spectrum.png

Note
  • No surprises here. We have the same output as the DSBSC.
  • No carrier on the spectrum and the two sidebands at
  • I have the scale on the y-axis up to 5V so we can compare different spectrums later.

If I move the gain knob up and down, all I am doing is increasing and decreasing the amplitude of the modulated signal both in time and frequency domain.

dsbfc_time_signal_1.gif

dsbfc_spectrum_signal_1.gif

Tinker Signal 2
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  • Orange Line: baseband signal
  • Blue Line: modulation output
  • Keep gain the same.
  • Vary gain from 0 until the upper envelope of looks more or less like the diagram below
    am_envelope_diagram.png

Time Domain Waveform
dsbfc_time_signal_2.gif

Frequency Spectrum
dsbfc_spectrum_signal_2.gif

Note
  • Fun stuff happening now 🙂
  • As I increase the gain, the carrier starts showing up on the spectrum.
  • is initially weird but at the end before the animation stops, we can see on the upper envelope of

Let me shift to the top of so we can visualize the upper envelope of being shaped by .

dsbfc_time_signal_1_shift.gif

Tinker Signal 3
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  • Orange Line: baseband signal
  • Blue Line: modulation output
  • Keep gain the same.
  • Vary gain from previous tinker signal 2 to maximum gain.

Time Domain Waveform
dsbfc_time_signal_3.gif

Frequency Spectrum
dsbfc_spectrum_signal_3.gif

Note
  • From this point on we don't lose the upper or lower envelope of the modulated signal.
  • Increasing all it does is to increase the signal strength of

Quick Note
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there is a mathematical way to know which of these 3 tinker signals, or any other signal in between your modulated signal will be.

The method is called modulation index, it is a ratio that describes the amount of change in amplitude present in AM waveforms. Based on that ratio, signals can be adjusted accordingly so we can recover the signal at the receiver properly.

I am not covering it here so this chapter doesn't get too extensive. Check the modulation index chapter for more details about it.

Time Domain Analysis
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theory_icon.png

  • Baseband:
  • Carrier:

The modulation output is the following:
We can use Desmos (or similar mathematical toolboxes) to validate the mathematical expressions.

Frequency Domain Analysis
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Now it is time to verify the spectrum domain starting from the time domain equation.
If you understood and followed the frequency domain analysis of the DSBSC, you recognize that the part of is equal to the DSBSC modulated signal .
We know the Fourier result of that already:
The part of has a simple transform pair:
The final result:

Frequency Spectrum
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Waveforms software only shows the positive side of the spectrum. With that in mind we can eliminate the frequencies that are negative from () and rewrite the equation as:

Note

If we sketch we get the spectrum below, which complies with what we observed.

  • Full carrier on the modulated signal
  • Two sidebands at frequencies

fourier_spectrum_icon.pngSpectrum
dsbfc_modulation_spectrum_math.png

Info

If you are confirming the magnitudes on the spectrum:

  • Don't forget that the hardware multiplier that I am using divides the output by k = 1/2
  • To accommodate this, I need to divide my spectrum by two
  • gets cancelled out by Waveforms software FFT visualization

Alternative Design
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You can find in literature an alternative block diagram that produces the same output as the DSBFC described previously.

Instead of adding the carrier to the DSBSC modulated signal, we add a DC component to the baseband signal before multiplying it with .

Block Diagram 2
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design_icon.png
DSBFC_AM_block_diagram.png

For this case, it is assumed that and

The modulation output is the following:
where ends up to be the amplitude of the carrier

If we factor out and we get the following equation:
and making we get:
where is called modulation index.

Note

As mentioned on the Quick Note, I will cover modulation index on a different chapter.

Attention

That assumption of instead of can be confusing to accept. We have been using with and all of a sudden we drop that 🤔

However, there is a another subtle hidden assumption that we can uncover with the explanation below that will helps us digest

Let's use the expression that we used during the DSBFC simulation using Desmos.

Desmos equation:

We can remove because that is a constant due to the hardware that I am using to test these equations. On that same hardware the modules responsible to generate and don't have a way to change the amplitudes and , which makes them a constant too.

For simplicity, if we set (it could be any other constant, as long as we make ) we get the following:
which and end up to control the amplitude of and accordingly.

Note
  • This alternative block diagram assumes that and have equals magnitudes from the modules that are generating these signals.
  • Later we can manipulate those magnitudes according to the modulation index

Summary
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  • In DSBFC the carrier signal is present on the AM modulated signal.

  • It is less power efficient method compared with DSBSC because we need to transmit the carrier and the sidebands.

  • For the modulated signal to be similar to tinker signal 2 and 3, the magnitude of the carrier is significantly higher that the sidebands which in turn will require more power to transmit.

  • On the other hand, because there is a carrier, the receiver side or demodulator will be able to synchronize with the carrier easily.

References
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Table Of Contents

DSBFC

Double Sideband Full Carrier

Block Diagram

Lab Bench

Carrier and Baseband Signals

Tinker Signal 1

Tinker Signal 2

Tinker Signal 3

Quick Note

Time Domain Analysis

Frequency Domain Analysis

Frequency Spectrum

Alternative Design

Block Diagram 2

Summary

References