#03 Real-message redundancy, Hilbert convention, VSB recovery and AM-family tradeoffs

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

1. 1. Use real-message conjugate symmetry correctly

   

   For a real baseband message extending to W and carrier above W, double sideband occupies:

   $\displaystyle B_{\mathrm{DSB}}=2W$

   Assume a real message with zero DC and no content beyond the band edge W; state the receiver conditions before discarding a sideband.

   One sideband can represent this real message; two independent complex message channels need not be redundant.

   Real-message Fourier symmetry uses complex conjugation, not identical phases:

   $\displaystyle M\left(-f\right)=M^{*}\left(f\right)$

   The two baseband magnitude spectra are mirrored:

   $\displaystyle |M\left(-f\right)|=|M\left(f\right)|$

   The positive-frequency upper-sideband band edges are:

   $\displaystyle f_{c}\le f\le f_{c}+W$

   Keeping one positive-RF sideband still requires its negative-frequency conjugate image for a real transmitted waveform.

   Narration transcript

   In the previous lesson, both conventional AM and DSB-SC still used two sidebands around the carrier. That raises an important engineering question. Do we truly need both of them? For a real valued message, the upper and lower sidebands are mirrored copies in frequency. They are shifted to opposite sides of the carrier, but they do not carry independent new information. So if bandwidth is precious, transmitting both sidebands can feel wasteful. That observation is the starting point for single sideband and vestigial sideband modulation.

2. 2. Retain one recoverable sideband with a stated convention

   

   Let h be the Hilbert transform of m; keep the upper sideband at the original DSB sideband amplitude:

   $\displaystyle s_{U}\left(t\right)=\left(\frac{1}{2}\right)\left[m\left(t\right)\cos \left(\omega _{c} t\right)-h\left(t\right)\sin \left(\omega _{c} t\right)\right]$

   The lower-sideband alternative uses the opposite sign:

   $\displaystyle s_{L}\left(t\right)=\left(\frac{1}{2}\right)\left[m\left(t\right)\cos \left(\omega _{c} t\right)+h\left(t\right)\sin \left(\omega _{c} t\right)\right]$

   For a positive-frequency test tone, the stated Hilbert convention gives:

   $\displaystyle m\left(t\right)=\cos \left(\omega _{m} t\right), h\left(t\right)=\sin \left(\omega _{m} t\right)$

   The ideal occupied width of one sideband is:

   $\displaystyle B_{\mathrm{SSB}}=W$

   With equal retained-sideband amplitudes and no carrier, removing one equal-power copy gives:

   $\displaystyle P_{\mathrm{SSB}}=\frac{P_{\mathrm{DSB}}}{2}$

   SSB can fit more narrow voice channels in a given RF allocation; practical transition bands and guard spacing still matter.

   For the upper-sideband model, matched frequency and normalized product detection with phase error give:

   $\displaystyle y_{U}\left(t\right)=m\left(t\right)\cos \left(\theta \right)+h\left(t\right)\sin \left(\theta \right)$

   Narration transcript

   Single sideband, or SSB, keeps only one of those mirrored sidebands and discards the other. In practice, the carrier is usually suppressed as well. The result is powerful. The occupied bandwidth drops from about two times the message bandwidth to about one times the message bandwidth. At the same time, power is no longer spent on a duplicated spectral copy. That is why SSB became so important in narrowband voice radio and long distance communication. Its price is higher implementation difficulty, because generating and receiving a clean single sideband is more demanding than simple AM.

3. 3. Trade a finite vestige for realizable filter transition

   

   Let the residual opposite-sideband width be B_v, between zero and W:

   $\displaystyle B_{\mathrm{VSB}}=W+B_{v}$

   Filtering one sideband sharply becomes difficult when useful message components approach the carrier arbitrarily closely.

   The retained vestige provides a finite transition region; correct gain and phase response are still required.

   For real RF filter response H, a distortionless coherent baseband condition across the message band is:

   $\displaystyle H\left(f_{c}+f\right)+H\left(f-f_{c}\right)=K e^{-j2\pi f\tau }$

   With input m times the carrier and product detector using twice the matched carrier, this condition gives:

   $\displaystyle y\left(t\right)=\left(\frac{K}{2}\right)m\left(t-\tau \right)$

   Historical analog television is a VSB example; envelope detection additionally needs suitable residual carrier and controlled quadrature distortion.

   Narration transcript

   Vestigial sideband, or VSB, is the practical compromise. Pure SSB is efficient, but near the carrier frequency it can be difficult to realize the ideal sharp filter that removes exactly one sideband and nothing else. This becomes especially awkward when very low message frequencies matter. VSB solves that problem by keeping one sideband almost fully while leaving a small vestige of the other. That means the bandwidth is slightly larger than SSB, but filtering and demodulation become much more practical. This is why VSB became attractive in wideband broadcast systems such as television video transmission.

4. 4. Compare bandwidth, transmitted power and receiver conditions

   

   Compare occupied bandwidth, transmitted spectral power and receiver complexity as separate quantities.

   For non-overmodulated single-tone conventional AM, the fraction of signal power in both sidebands is:

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

   DSB-SC removes a separate carrier term only for a zero-DC message; both sidebands still occupy the same band.

   SSB uses the least bandwidth among these four methods, but frequency error shifts recovered tones and phase error mixes the Hilbert pair.

   For an intermediate vestige the ideal band ordering is:

   $\displaystyle W<B_{\mathrm{VSB}}<2W$

   VSB trades some bandwidth saving for practical filtering; its carrier level, equalization and detector determine the power and reception tradeoffs.

   Narration transcript

   Now we can compare the AM family with a clearer engineering lens. Conventional AM is easiest to receive because the carrier supports simple envelope detection, but it is poor in power efficiency. DSB-SC removes the wasted carrier power, yet still pays for two sidebands. SSB is the most bandwidth efficient member of this family and also avoids duplicated sideband power, but it demands better oscillators, filtering, and tuning. VSB sits between SSB and DSB styles. It gives back a little efficiency so that practical filters and receivers become easier to build.

5. 5. Choose a method under explicit application constraints

   

   Choose a method from explicit bandwidth, carrier-power and receiver requirements; there is no universal winner.

   Example assumption: the real message extends from near DC to a 3 kHz band edge, with a 0.5 kHz vestige allowed.

   Conventional AM with a well-behaved envelope uses this width in kHz:

   $\displaystyle B_{\mathrm{AM}}=6$

   DSB-SC saves carrier power but keeps the same width in kHz:

   $\displaystyle B_{\mathrm{DSB}}=6$

   Keeping one sideband gives this ideal width in kHz:

   $\displaystyle B_{\mathrm{SSB}}=3$

   One sideband plus the chosen vestige gives this width in kHz:

   $\displaystyle B_{\mathrm{VSB}}=3.5$

   Next: put information into frequency or phase; the linear AM-family bandwidth formulas do not automatically carry over.

   Narration transcript

   So the big idea is this. AM design is a tradeoff between bandwidth, power, and receiver simplicity. If simplicity matters most, conventional AM is attractive. If carrier power should be removed but bandwidth can stay the same, DSB-SC is useful. If spectrum is scarce, SSB becomes very attractive. If a pure SSB filter is too difficult, VSB offers a practical compromise. In the next lesson, we leave amplitude based modulation and ask what changes when the information is placed into frequency or phase instead.

Source video: Analog Communication #03 | Single Sideband (SSB), Vestigial Sideband (VSB) and AM Tradeoffs (3:35)