#02 Carrier and sidebands, modulation depth, envelope detection and power efficiency

Below are all the lines written in the notebook together with the full narration transcript.

1. 1. Distinguish conventional AM and suppressed-carrier modulation

   

   Choose a zero-mean normalized single-tone message:

   $\displaystyle m_{n}\left(t\right)=\cos \left(\omega _{m} t\right)$

   Conventional AM, with positive carrier amplitude and nonnegative depth:

   $\displaystyle s_{A}\left(t\right)=A_{c}\left[1+\mu m_{n}\left(t\right)\right]\cos \left(\omega _{c} t\right)$

   For positive modulation sensitivity and a nonnegative signed amplitude, a larger message raises the envelope.

   A smaller message lowers that signed amplitude; after a zero crossing its magnitude is not a faithful message copy.

   The conventional waveform includes this separate carrier term:

   $\displaystyle c\left(t\right)=A_{c} \cos \left(\omega _{c} t\right)$

   Compare the same sidebands with the carrier removed:

   $\displaystyle s_{D}\left(t\right)=A_{c} \mu m_{n}\left(t\right)\cos \left(\omega _{c} t\right)$

   Keeping a carrier enables simple envelope reception under suitable conditions; removing it saves carrier power.

   Narration transcript

   Now that we know why a carrier is useful, we can ask how the message is placed on that carrier. In amplitude modulation, the message controls the carrier amplitude. When the message grows, the carrier envelope expands. When the message falls, that envelope contracts. In conventional AM, a strong carrier remains present all the time, so the envelope is easy to observe. In double sideband suppressed carrier, or DSB-SC, the sidebands still exist but the carrier itself is removed. So both waveforms are closely related, yet one keeps the carrier for simplicity while the other suppresses it for better power efficiency.

2. 2. Locate carrier and sidebands under a stated tone model

   

   For the chosen tone, expand the product:

   $\displaystyle s_{D}\left(t\right)=\left(\frac{A_{c} \mu }{2}\right)\left[\cos \left(\left(\omega _{c}-\omega _{m}\right)t\right)+\cos \left(\left(\omega _{c}+\omega _{m}\right)t\right)\right]$

   The two positive-frequency tone locations are:

   $\displaystyle f_{L}=f_{c}-f_{m}, f_{U}=f_{c}+f_{m}$

   Conventional AM adds a carrier at the centre:

   $\displaystyle s_{A}\left(t\right)=c\left(t\right)+s_{D}\left(t\right)$

   DSB-SC has no independent carrier term when the message has zero DC; it retains both sidebands.

   For a message limited to the band edge W and carrier above W:

   $\displaystyle B=2W$

   Removing only the carrier line does not remove either sideband or halve the occupied band.

   Carrier power for peak voltage amplitude across resistance R:

   $\displaystyle P_{c}=\frac{A_{c}^{2}}{2R}$

   Narration transcript

   The frequency domain makes that relationship clearer. A single tone message creates two sidebands located at the carrier frequency plus and minus the message frequency. Conventional AM contains those two sidebands and also the carrier spike at the center. DSB-SC keeps the same mirrored sidebands but removes that center carrier. So the occupied bandwidth is still twice the message bandwidth in both cases. Suppressing the carrier does not save bandwidth. It mainly saves transmitter power because the carrier no longer burns energy without carrying new information.

3. 3. Keep the signed envelope separate from its magnitude

   

   Separate signed carrier amplitude from its nonnegative magnitude:

   $\displaystyle a\left(t\right)=A_{c}\left[1+\mu \cos \left(\omega _{m} t\right)\right]$

   For the normalized tone the signed amplitude stays positive when:

   $\displaystyle 0\le \mu <1$

   At the ideal non-crossing boundary:

   $\displaystyle \mu =1, a_{\mathrm{min}}=0$

   An overmodulated example crosses zero:

   $\displaystyle \mu =1.3, a_{\mathrm{min}}=-0.3 A_{c}$

   The measured magnitude envelope folds at a sign reversal:

   $\displaystyle E\left(t\right)=|a\left(t\right)|$

   For a non-overmodulated symmetric tone, recover depth from envelope extrema:

   $\displaystyle \mu =\frac{E_{\mathrm{max}}-E_{\mathrm{min}}}{E_{\mathrm{max}}+E_{\mathrm{min}}}$

   The ideal non-crossing limit is not a guarantee of distortion-free practical reception; diode threshold and RC tracking need margin.

   Narration transcript

   For conventional AM, the modulation index tells us how strongly the message changes the envelope. If the modulation index is below one, the envelope stays clean and easy to follow. At exactly one, we are using the full envelope range without yet causing distortion. But if the modulation index rises above one, the envelope folds over itself. That condition is called over-modulation. At that point a simple envelope detector can no longer track the message correctly, because the apparent shape of the envelope is no longer a faithful copy of the original signal. So for basic AM reception, keeping the modulation index at or below one is a practical design rule.

4. 4. Compare envelope and coherent detection with their conditions

   

   Conventional AM can trade transmitted carrier power for a simpler receiver.

   A diode and RC detector must smooth carrier ripple while following the message; carrier and message time scales must be well separated.

   Remove the envelope's DC offset and account for its gain to recover the message.

   A simple magnitude detector loses the sign of a DSB-SC message:

   $\displaystyle E_{D}\left(t\right)=A_{c} \mu |m_{n}\left(t\right)|$

   Use a local oscillator with the correct frequency and phase; a residual synchronization error matters.

   For nonzero modulation gain and ideal normalized coherent detection at matched frequency:

   $\displaystyle y\left(t\right)=m_{n}\left(t\right)\cos \left(\theta \right)$

   A ninety-degree phase error suppresses this recovered signal; carrier removal saves transmitted power, not all receiver or hardware losses.

   The practical comparison includes synchronization, noise, detector distortion and implementation complexity.

   Narration transcript

   This is why conventional AM became so attractive in early radio. If a carrier is transmitted and the envelope is well behaved, a very simple diode and RC filter can recover the message. The receiver only needs to trace the envelope. DSB-SC is different. Because the transmitted carrier is missing, the receiver must recreate a synchronized carrier locally before detection. That usually means coherent or synchronous detection. So DSB-SC is more power efficient at the transmitter, but it asks more from the receiver. This is the classic tradeoff between efficiency and implementation simplicity.

5. 5. Separate carrier-power savings from occupied bandwidth

   

   Compare occupied bandwidth, transmitted carrier power and receiver complexity separately.

   The two sidebands of a real message remain in both conventional AM and DSB-SC.

   For a single sinusoidal tone and a resistive load, total sideband power is:

   $\displaystyle P_{\mathrm{SB}}=\frac{P_{c} \mu ^{2}}{2}$

   The conventional AM sideband-power fraction is:

   $\displaystyle \eta =\frac{\mu ^{2}}{2+\mu ^{2}}$

   At full non-overmodulated single-tone depth:

   $\displaystyle \mu =1, \eta =\frac{1}{3}$

   DSB-SC removes carrier power, but both sidebands still occupy the same band; sideband fraction is not DC-to-RF amplifier efficiency.

   For a real message, one sideband contains recoverable information under the appropriate coherent receiver assumptions.

   Next: single-sideband and vestigial-sideband methods, including their filtering and recovery conditions.

   Narration transcript

   Let us summarize the engineering picture. Both conventional AM and DSB-SC place information into two sidebands around a carrier frequency. Conventional AM keeps the carrier, which makes envelope detection simple but wastes transmitter power. DSB-SC suppresses that carrier, which improves power efficiency but requires coherent detection at the receiver. Neither method removes the duplicated sideband structure, so the bandwidth cost is still there. That naturally leads to the next question. If one sideband already mirrors the other, can we remove one and transmit only what we truly need? That is exactly where single sideband and vestigial sideband modulation enter the story.

Source video: Analog Communication #02 | Double-Sideband Suppressed Carrier & Conventional AM (4:10)