6.1 Amplitude Modulation, Double Sideband, and Single Sideband (SSB)

6.1 Amplitude Modulation, Double Sideband, and Single Sideband (SSB)

Modulation is the fundamental process by which information—whether human speech, telegraphy, digital telemetry, or video—is imprinted onto a high-frequency radio frequency (RF) carrier wave suitable for electromagnetic radiation. In the early decades of radio communications, Amplitude Modulation (AM) served as the primary medium for long-distance voice transmission. Although largely superseded in modern amateur communications by Single Sideband (SSB) and narrowband digital modes, the mathematical principles governing amplitude-modulated spectra, sideband power distributions, and non-linear mixing remain foundational to amateur radio engineering.

---

1. Mathematical Fundamentals of Amplitude Modulation

The Time-Domain Waveform

In an amplitude-modulated system, the instantaneous peak voltage of an RF carrier wave is varied linearly in sympathy with the instantaneous amplitude of an information-bearing modulating baseband signal (such as an audio tone or human speech).

Let the unmodulated sinusoidal carrier voltage $v_c(t)$ be defined by:

$$v_c(t) = V_c \sin(\omega_c t)$$

where $V_c$ is the unmodulated carrier peak amplitude and $\omega_c = 2\pi f_c$ is the carrier angular frequency in radians per second.

Let a single-tone sinusoidal modulating baseband signal $v_m(t)$ be defined by:

$$v_m(t) = V_m \sin(\omega_m t)$$

where $V_m$ is the peak amplitude of the modulating signal and $\omega_m = 2\pi f_m$ is the modulating angular frequency ($f_c \gg f_m$).

For conventional AM without envelope inversion, the RF envelope $V_{\text{env}}(t)$ follows the modulating voltage:

$$V_{\text{env}}(t) = V_c + v_m(t) = V_c + V_m \sin(\omega_m t)$$

The complete instantaneous voltage of the amplitude-modulated wave $v(t)$ is expressed by multiplying the instantaneous envelope by the carrier sinusoid:

$$v(t) = [V_c + V_m \sin(\omega_m t)] \sin(\omega_c t)$$

Factoring out the unmodulated carrier amplitude $V_c$:

$$v(t) = V_c [1 + m \sin(\omega_m t)] \sin(\omega_c t)$$

where $m$ is the modulation index (also called the modulation factor or depth of modulation).

Frequency Spectrum and Sideband Generation

Applying the trigonometric product-to-sum identity $\sin(A)\sin(B) = \frac{1}{2}[\cos(A - B) - \cos(A + B)]$ expands the time-domain equation into three distinct frequency components:

$$v(t) = V_c \sin(\omega_c t) + \frac{m V_c}{2} \cos[(\omega_c - \omega_m)t] - \frac{m V_c}{2} \cos[(\omega_c + \omega_m)t]$$

This expansion reveals that an amplitude-modulated signal does not simply vary in amplitude at the single carrier frequency; rather, it consists of precisely three discrete spectral lines:

1. The Carrier Component: $V_c \sin(\omega_c t)$, remaining constant in amplitude ($V_c$) and frequency ($f_c$), completely independent of the presence or amplitude of modulation.
2. The Lower Sideband (LSB): $\frac{m V_c}{2} \cos[2\pi (f_c - f_m)t]$, located at the difference frequency $f_c - f_m$.
3. The Upper Sideband (USB): $-\frac{m V_c}{2} \cos[2\pi (f_c + f_m)t]$, located at the sum frequency $f_c + f_m$.

Each sideband has a peak amplitude of $\frac{m V_c}{2}$. Under 100% modulation ($m = 1$), each sideband reaches exactly half the voltage of the unmodulated carrier ($V_c / 2$).

Amplitude
   ^
   |               Carrier (fc)
   |                    |
Vc |                    |
   |         LSB        |        USB
   |       (fc - fm)    |      (fc + fm)
Vc/2          |         |         |
   |          |         |         |
   +----------+---------+---------+---------> Frequency (f)
              |<-- fm ->|<-- fm ->|
              |<------ B = 2*fm ----->|

AM Bandwidth Requirements

When complex baseband signals such as speech (containing a spectrum of frequencies from $f_{m,\text{min}}$ to $f_{m,\text{max}}$) modulate the carrier, the sidebands expand into continuous spectral bands flanking the carrier. The total occupied RF transmission bandwidth ($B$) is strictly twice the highest modulating frequency:

$$B = 2 f_{m,\text{max}}$$

For standard communications-quality speech restricted to an audio bandwidth of $300\text{ Hz}$ to $3000\text{ Hz}$ ($f_{m,\text{max}} = 3\text{ kHz}$), the resulting AM signal requires a total RF channel bandwidth of:

$$B = 2 \times 3\text{ kHz} = 6\text{ kHz}$$

---

2. Modulation Index and Overmodulation Dynamics

Defining the Modulation Index ($m$)

The modulation index quantifies the degree of carrier envelope excursion relative to the quiescent carrier level:

$$m = \frac{V_m}{V_c}$$

When expressed as a percentage, it is termed the percentage of modulation ($M$):

$$M = m \times 100\%$$

For undistorted single-tone AM with $0\le m\le1$, the maximum and minimum RF envelope voltages are:

$$V_{\text{max}} = V_c + V_m$$ $$V_{\text{min}} = V_c - V_m$$

Adding and subtracting these two relationships yields:

$$V_{\text{max}} + V_{\text{min}} = 2 V_c \implies V_c = \frac{V_{\text{max}} + V_{\text{min}}}{2}$$ $$V_{\text{max}} - V_{\text{min}} = 2 V_m \implies V_m = \frac{V_{\text{max}} - V_{\text{min}}}{2}$$

Dividing $V_m$ by $V_c$ gives the oscilloscope formula for this non-overmodulated single-tone case:

$$m = \frac{V_{\text{max}} - V_{\text{min}}}{V_{\text{max}} + V_{\text{min}}}$$

Operating Regimes: Undermodulation, Complete Modulation, and Overmodulation

For the ideal waveform, distinguish the signed modulation factor from the nonnegative RF envelope:

$$a(t)=V_c[1+m\sin(\omega_mt)],\qquad V_{\text{env}}(t)=|a(t)|$$

At the negative audio peak, $a(t)/V_c$ equals $0.5$, $0$ and $-0.5$ for $m=0.5$, $1$ and $1.5$ respectively. In the last case the carrier reverses phase by $180^\circ$ while the envelope folds upward. This is not a finite interval of zero RF output in the ideal model.

Consequences of Overmodulation in Amateur Transmitters

An envelope detector follows $|a(t)|$, so it cannot recover the original audio faithfully when the signed factor becomes negative. The ideal multiplier waveform still contains the carrier and the two single-tone sidebands; phase reversal alone does not create extra transmitted spectral lines.

A practical high-level AM transmitter may instead reach cutoff during negative peaks and clip the envelope. That nonlinear clipping creates additional harmonics of the audio, which appear as unwanted sidebands around the carrier. They are not restricted to odd harmonics. Excessive microphone gain or inadequate modulator headroom can therefore cause distorted audio and adjacent-channel splatter. Keep the symmetrical sine-wave test within $100\%$ modulation and check the actual envelope and spectrum.

---

3. Power Distribution in AM Signals

Mathematical Power Derivation

To understand the fundamental power inefficiency of full-carrier amplitude modulation, consider an AM signal developed across a pure load resistance $R$ (such as a $50\ \Omega$ antenna system).

The unmodulated carrier power ($P_c$) is the average power dissipated by the carrier alone:

$$P_c = \frac{V_{c,\text{rms}}^2}{R} = \frac{(V_c / \sqrt{2})^2}{R} = \frac{V_c^2}{2R}$$

Each sideband has a peak voltage of $\frac{m V_c}{2}$, corresponding to an RMS voltage of:

$$V_{\text{sb},\text{rms}} = \frac{m V_c / 2}{\sqrt{2}} = \frac{m V_c}{2\sqrt{2}}$$

The power contained in each individual sideband ($P_{\text{LSB}}$ and $P_{\text{USB}}$) is:

$$P_{\text{LSB}} = P_{\text{USB}} = \frac{V_{\text{sb},\text{rms}}^2}{R} = \frac{(m V_c / 2\sqrt{2})^2}{R} = \frac{m^2 V_c^2}{8R} = \frac{m^2}{4} \left( \frac{V_c^2}{2R} \right) = \frac{m^2}{4} P_c$$

Because the carrier, lower sideband, and upper sideband are mutually orthogonal in frequency, total average power ($P_t$) is the direct scalar sum of their individual powers:

$$P_t = P_c + P_{\text{LSB}} + P_{\text{USB}} = P_c + \frac{m^2}{4} P_c + \frac{m^2}{4} P_c = P_c \left( 1 + \frac{m^2}{2} \right)$$

The total power contained in both intelligence-bearing sidebands ($P_{\text{SB}}$) is:

$$P_{\text{SB}} = P_{\text{LSB}} + P_{\text{USB}} = \frac{m^2}{2} P_c$$

Power Distribution at 100% Modulation ($m = 1.0$)

Substituting $m = 1.0$ into the power equations reveals the dramatic imbalance between carrier and sideband dissipation:

$$P_t = P_c \left( 1 + \frac{1^2}{2} \right) = 1.5 \cdot P_c$$ $$P_{\text{SB}} = \frac{1^2}{2} P_c = 0.5 \cdot P_c$$ $$P_{\text{LSB}} = P_{\text{USB}} = \frac{1^2}{4} P_c = 0.25 \cdot P_c$$

Expressing each component as a percentage of total transmitted power ($P_t$):

• Carrier Power Fraction:

$$\frac{P_c}{P_t} = \frac{P_c}{1.5 P_c} = \frac{2}{3} \approx 66.67\%$$

• Total Sideband Power Fraction:

$$\frac{P_{\text{SB}}}{P_t} = \frac{0.5 P_c}{1.5 P_c} = \frac{1}{3} \approx 33.33\%$$

• Individual Sideband Power Fraction (each):

$$\frac{P_{\text{LSB}}}{P_t} = \frac{P_{\text{USB}}}{P_t} = \frac{0.25 P_c}{1.5 P_c} = \frac{1}{6} \approx 16.67\%$$

Core Engineering Insight: At 100% modulation, exactly two-thirds ($66.7\%$) of the total RF power radiated by the antenna is locked in the carrier wave. The carrier contains zero information—its frequency, amplitude, and phase never change with modulation. Furthermore, the remaining one-third ($33.3\%$) of the power is split equally between two redundant sidebands. Thus, a single sideband carries only $16.7\%$ of total transmitter power.

---

4. Worked Calculation: AM Power and Modulation Depth

Worked Example 6.1: Full AM Power and Voltage Analysis

An amateur AM transmitter delivers an unmodulated carrier power of $100\text{ W}$ into a matched $50.0\ \Omega$ dummy load. The transmitter is subsequently modulated by a pure $1000\text{ Hz}$ sine wave.

1. Determine the unmodulated carrier peak voltage ($V_c$) and peak-to-peak voltage ($V_{\text{p-p}}$):

$$V_{c,\text{rms}} = \sqrt{P_c \cdot R} = \sqrt{100\text{ W} \times 50.0\ \Omega} = \sqrt{5000} \approx 70.71\text{ V}$$ $$V_c = \sqrt{2} \cdot V_{c,\text{rms}} = \sqrt{2} \times 70.71\text{ V} = 100.0\text{ V}$$ $$V_{\text{p-p(unmod)}} = 2 \cdot V_c = 200.0\text{ V}$$

1. Calculate the total power, sideband power, and envelope peak values at $m = 0.70$ (70% modulation):

$$P_t = P_c \left( 1 + \frac{m^2}{2} \right) = 100\text{ W} \times \left( 1 + \frac{0.70^2}{2} \right) = 100 \times (1 + 0.245) = 124.5\text{ W}$$ $$P_{\text{SB}} = P_t - P_c = 124.5\text{ W} - 100.0\text{ W} = 24.5\text{ W}$$ $$P_{\text{LSB}} = P_{\text{USB}} = \frac{24.5\text{ W}}{2} = 12.25\text{ W}$$

Envelope peak voltage:

$$V_{\text{max}} = V_c (1 + m) = 100\text{ V} \times (1 + 0.70) = 170.0\text{ V}$$ $$V_{\text{min}} = V_c (1 - m) = 100\text{ V} \times (1 - 0.70) = 30.0\text{ V}$$

Peak Envelope Power (PEP):

$$\text{PEP} = \frac{(V_{\text{max}} / \sqrt{2})^2}{R} = \frac{V_{\text{max}}^2}{2R} = \frac{170.0^2}{2 \times 50.0} = \frac{28900}{100} = 289.0\text{ W}$$

1. Calculate the operating parameters at $m = 1.0$ (100% modulation):

$$P_t = 100\text{ W} \times \left( 1 + \frac{1.0^2}{2} \right) = 150.0\text{ W}$$ $$P_{\text{LSB}} = P_{\text{USB}} = \frac{0.25 \times 100\text{ W}}{1} = 25.0\text{ W}$$ $$V_{\text{max}} = 100\text{ V} \times (1 + 1.0) = 200.0\text{ V}$$ $$V_{\text{min}} = 100\text{ V} \times (1 - 1.0) = 0.0\text{ V}$$ $$\text{PEP} = \frac{V_{\text{max}}^2}{2R} = \frac{200.0^2}{100} = 400.0\text{ W}$$

Notice that at 100% modulation, the Peak Envelope Power (PEP) is precisely four times the unmodulated carrier power ($400\text{ W}$ vs $100\text{ W}$), and the peak voltage doubles ($200\text{ V}$ vs $100\text{ V}$). An amplifier designed for AM service must be capable of delivering this $4\times$ peak power without saturating.

---

5. Double Sideband Suppressed Carrier (DSB-SC)

Because the carrier in an AM signal consumes $66.7\%$ of total power while carrying no information, an obvious engineering optimization is to suppress the carrier completely prior to transmission. This is known as Double Sideband Suppressed Carrier (DSB-SC).

Balanced Modulator Operation

A balanced modulator multiplies the audio modulating signal $v_m(t)$ directly with the RF carrier $v_c(t)$ without adding the carrier offset:

$$v_{\text{dsb}}(t) = v_m(t) \cdot v_c(t) = [V_m \sin(\omega_m t)] \cdot [\sin(\omega_c t)]$$

Applying the product-to-sum identity:

$$v_{\text{dsb}}(t) = \frac{V_m}{2} \cos[(\omega_c - \omega_m)t] - \frac{V_m}{2} \cos[(\omega_c + \omega_m)t]$$

The carrier component $V_c \sin(\omega_c t)$ is completely absent from the output. In practical balanced modulators—such as a diode ring double-balanced mixer or active Gilbert cell transistor array—the carrier is balanced out by symmetrical circuit topology, typically achieving $40\text{ dB}$ to $50\text{ dB}$ of carrier suppression.

Audio In ------->| Balanced    |-------> DSB-SC Output (LSB + USB)
                 | Modulator   |         (Carrier is suppressed)
Carrier In ----->| (Diode Ring)|

While DSB-SC eliminates the carrier power waste (100% of transmitted power resides in the sidebands), it still occupies an RF bandwidth of $B = 2 f_{m,\text{max}}$ ($6\text{ kHz}$ for voice), because both upper and lower sidebands are transmitted.

---

6. Single Sideband Suppressed Carrier (SSB)

The Single Sideband Concept

Because the upper and lower sidebands are exact mirror images of each other, conveying identical intelligence, transmitting both sidebands is redundant. Single Sideband Suppressed Carrier (SSB) eliminates both the carrier wave and one of the two sidebands, transmitting exclusively the Upper Sideband (USB) or the Lower Sideband (LSB).

Bandwidth, Noise and Power Comparisons

1. Bandwidth Conservation: The SSB bandwidth equals the span of the transmitted audio band:

$$B_{\text{SSB}}=f_{m,\text{max}}-f_{m,\text{min}}$$

Audio from $300$ to $3000\text{ Hz}$ occupies $2.7\text{ kHz}$ in one sideband, compared with the full $6\text{ kHz}$ span of conventional AM. Speech passbands around $2.4$ to $2.8\text{ kHz}$ are common.

1. Receiver Noise Bandwidth: For white thermal noise, $N=kTB$, where $B$ is equivalent noise bandwidth. Reducing it from $6.0$ to $2.4\text{ kHz}$ changes admitted noise power by:

$$\Delta N=10\log_{10}(2.4/6.0)\approx-3.98\text{ dB}$$

This is a noise-power comparison. Recovered audio SNR also depends on signal power and demodulation; both AM sidebands contribute to the detected audio.

1. Use the Same Power Reference: Compare ideal signals at $100\text{ W PEP}$, using one audio test tone. At $100\%$ AM modulation, $\text{PEP}=4P_c$, so $P_c=25\text{ W}$, average total power is $37.5\text{ W}$, and each sideband contains $6.25\text{ W}$. A single-tone SSB signal has constant envelope and $100\text{ W}$ average power, all in its one sideband.

For speech, SSB average power is lower than PEP and depends on the waveform and processing. These allocations explain the value of suppressing the carrier and redundant sideband; they do not establish a universal communications gain by adding a sideband-power ratio to a separate noise-bandwidth ratio.

Amateur Radio Sideband Operating Conventions

For SSB voice, the usual amateur operating conventions are:

• Lower Sideband (LSB): Utilised on HF bands below $10\text{ MHz}$:
  • $160\text{ Metres}$ ($1.8\text{ MHz}$)
  • $80\text{ Metres}$ ($3.5\text{ MHz}$)
  • $40\text{ Metres}$ ($7.0\text{ MHz}$)
• Upper Sideband (USB): Utilised on HF bands at and above $10\text{ MHz}$, as well as all VHF, UHF, and microwave bands:
  • $30\text{ Metres}$ ($10.1\text{ MHz}$): Australia also has a domestic SSB voice arrangement. Follow the current WIA band plan for its segment and sideband recommendations; the band is not digital/CW only.
  • $20\text{ Metres}$ ($14.0\text{ MHz}$)
  • $17\text{ Metres}$ ($18.068\text{ MHz}$)
  • $15\text{ Metres}$ ($21.0\text{ MHz}$)
  • $12\text{ Metres}$ ($24.89\text{ MHz}$)
  • $10\text{ Metres}$ ($28.0\text{ MHz}$)
  • $6\text{ Metres}$ ($50\text{ MHz}$), $2\text{ Metres}$ ($144\text{ MHz}$), $70\text{ cm}$ ($432\text{ MHz}$), and higher.

Frequency Translation and Sideband Inversion: Track each spectral component through the mixer. A USB signal at $9\text{ MHz}+f_a$ mixed with a $5\text{ MHz}$ LO gives the sum $14\text{ MHz}+f_a$, still USB. Selecting the difference with a $5.2\text{ MHz}$ LO gives $3.8\text{ MHz}+f_a$, also USB. In contrast, selecting LO minus input with a $12.8\text{ MHz}$ LO gives $3.8\text{ MHz}-f_a$, which is LSB. A difference-frequency output does not automatically imply inversion; which frequency is subtracted matters.

---

7. Generation of SSB Signals

Two classical architectures exist for eliminating the unwanted sideband and carrier: the Filter Method and the Phasing Method.

The Filter Method

The filter method is the most widely implemented analog architecture in commercial and amateur transceivers:

                +-------------------+        +----------------------+        +--------------+
Audio In ------>| Balanced Modulator|------->| Crystal Lattice or   |------->| Mixer &      |---> SSB
                | (Diode Ring/IC)   | DSB-SC | Mechanical Filter    |  SSB   | Linear Power |     Out
Carrier Osc --->|                   |        | (SSB Passband)       |        | Amplifier    |
                +-------------------+        +----------------------+        +--------------+

1. Balanced Modulation: Audio ($300 - 3000\text{ Hz}$) and an IF carrier (e.g., $9.0\text{ MHz}$ or $455\text{ kHz}$) feed a balanced modulator (diode ring or active Gilbert cell). The carrier is cancelled by balance, leaving a DSB-SC signal.
2. Sideband Selection: A crystal or mechanical bandpass filter passes the wanted sideband while rejecting the carrier region and opposite sideband. For example, $9.0003$ to $9.0030\text{ MHz}$ passes $300$–$3000\text{ Hz}$ USB audio and is $2.7\text{ kHz}$ wide. Shape factor is commonly $BW_{60\text{dB}}/BW_{6\text{dB}}$: a low value such as $1.2:1$, close to unity, indicates steep skirts. Actual rejection follows the filter response.
3. Frequency Translation: The resulting low-level SSB signal at the IF is mixed with a local oscillator to translate it to the desired HF operating band, followed by linear (Class A and Class AB) power amplifiers. Non-linear amplifiers (such as Class C) cannot be used because they would destroy the envelope and recreate the missing sideband and carrier through intermodulation.

The Phasing Method

The phasing method eliminates the unwanted sideband through constructive and destructive phase cancellation:

Audio I = Vm*cos(wm*t) ---> [Multiplier1] ---> v1 ---+
RF I = Vc*cos(wc*t) ------> [            ]           |
                                                     +--> v1-v2: USB
Audio Q = Vm*sin(wm*t) ---> [Multiplier2] ---> v2 ---+    v1+v2: LSB
RF Q = Vc*sin(wc*t) ------> [            ]

The two audio inputs come from a quadrature audio network; the two RF inputs come from the same carrier oscillator with quadrature outputs. The combiner selects addition or subtraction as shown.

• The audio signal is split into two components having an identical amplitude but a constant $90^\circ$ phase difference across the entire audio band ($I$ for In-phase, $Q$ for Quadrature):

$$v_{m1}(t) = V_m \cos(\omega_m t), \quad v_{m2}(t) = V_m \sin(\omega_m t)$$

• The RF carrier is split into two components in quadrature ($90^\circ$ apart):

$$v_{c1}(t) = V_c \cos(\omega_c t), \quad v_{c2}(t) = V_c \sin(\omega_c t)$$

• Modulator 1 output:

$$v_1(t) = v_{m1}(t) \cdot v_{c1}(t) = V_m V_c \cos(\omega_m t) \cos(\omega_c t) = \frac{V_m V_c}{2} [\cos((\omega_c - \omega_m)t) + \cos((\omega_c + \omega_m)t)]$$

• Modulator 2 output:

$$v_2(t) = v_{m2}(t) \cdot v_{c2}(t) = V_m V_c \sin(\omega_m t) \sin(\omega_c t) = \frac{V_m V_c}{2} [\cos((\omega_c - \omega_m)t) - \cos((\omega_c + \omega_m)t)]$$

• Subtracting $v_2(t)$ from $v_1(t)$:

$$v_{\text{out}}(t) = v_1(t) - v_2(t) = V_m V_c \cos[(\omega_c + \omega_m)t]$$

The lower sideband is completely cancelled ($180^\circ$ anti-phase), yielding pure Upper Sideband (USB). Summing the two outputs instead yields Lower Sideband (LSB).

Modern Digital Implementation: A digital Hilbert transformer forms quadrature audio components for the same cancellation process. Accurate phase and amplitude balance improves unwanted-sideband rejection. Finite filters and any analogue I/Q paths still affect the result; carrier leakage requires separate balance or offset correction. Arithmetic word length alone does not specify RF carrier or sideband suppression.

---

8. Demodulation of AM and SSB Signals

AM Demodulation: The Diode Envelope Detector

Full-carrier AM possesses a visible carrier envelope that can be demodulated using an inexpensive passive diode envelope detector:

RF In -----[ >| ]------+-------+-------> Audio Out
             Diode     |       |
                      [R]     [C]
                       |       |
                      GND     GND

1. Rectification: The semiconductor diode conducts only on positive half-cycles of the RF waveform, stripping the negative envelope.
2. Filtering: The parallel resistor-capacitor ($RC$) low-pass filter smooths out the high-frequency RF carrier ripple while allowing the audio-frequency variations of the envelope to pass unimpeded.
3. Time Constant Selection: The $RC$ time constant ($\tau = RC$) must be carefully engineered:

$$\frac{1}{f_c} \ll RC \ll \frac{1}{f_{m,\text{max}}}$$

• If $RC$ is too small, the capacitor discharges too quickly between RF cycles, leaving significant RF carrier ripple on the audio line.
• If $RC$ is too large, the capacitor cannot discharge rapidly enough to track steep downward excursions of the audio signal. The diode is reverse-biased and cut off, resulting in diagonal clipping distortion.

SSB Demodulation: The Product Detector

Because an SSB signal lacks a carrier wave, its envelope does not mirror the original audio modulating waveform. Passing an SSB signal through a diode envelope detector produces only garbled, distorted, and completely unintelligible noise.

To recover the original audio, the missing carrier must be artificially regenerated and mixed with the incoming SSB signal inside a Product Detector (which is fundamentally a balanced mixer):

Received SSB Signal (fc + fm) --->| Product  |----[ Low-Pass ]----> Demodulated
                                  | Detector |     Filter          Audio (fm)
Local BFO / CIO (fc) ------------>| (Mixer)  |

1. Carrier Insertion: A highly stable local oscillator—termed the Beat Frequency Oscillator (BFO) or Carrier Insertion Oscillator (CIO)—generates an unmodulated CW signal at the precise frequency of the original suppressed carrier ($f_c$).
2. Multiplication (Mixing): The incoming USB signal at $f_c + f_m$ is multiplied with the BFO carrier at $f_c$ in the product detector:

$$v_{\text{out}}(t) = [V_s \cos(2\pi (f_c + f_m)t)] \cdot [V_b \cos(2\pi f_c t)]$$ $$v_{\text{out}}(t) = \frac{V_s V_b}{2} [\cos(2\pi (2f_c + f_m)t) + \cos(2\pi f_m t)]$$

1. Low-Pass Filtering: The sum component at double the carrier frequency ($2f_c + f_m$) is discarded by a simple audio low-pass filter, leaving the pure difference term: the original baseband audio frequency $f_m$.
2. Frequency and Phase Accuracy: The BFO must match the original carrier frequency within $\pm 20\text{ Hz}$ to $\pm 50\text{ Hz}$. A tuning error of $100\text{ Hz}$ makes human voices sound unnatural ("Donald Duck" pitch elevation if BFO is low, or deep bass growl if BFO is high), and an error exceeding $200\text{ Hz}$ impairs intelligibility.

---

9. Comprehensive Comparison: AM, DSB-SC, and SSB