Amplitude modulation

Amplitude modulation encodes a message by varying the multiplying wave amplitude of a carrier wave. Conventional AM adds a positive constant before multiplication, allowing a simple envelope detector to recover the message within its limits. Multiplication translates the message's frequency spectrum into a band around the carrier, adapting a slow message to a faster channel. This is an application of the modulation property of the Fourier transform.^[ref]

For an original single-message-frequency voltage model, chooseHere $t$ is time, $A$ is the peak voltage amplitude of the unmodulated carrier wave, ${\omega }_c$ is its angular frequency, $\Omega$ is the message angular frequency, and $m$ is the dimensionless AM modulation index. The message shape is $\cos (\Omega t)$; $m$ measures how deeply it changes the multiplying factor. Define the signed factor $q(t)=1+m\cos (\Omega t)$. The nonnegative wave envelope of the representation $z(t)=Aq(t)e^{\mathrm{i}{\omega }_{c}t}$ isThe instantaneous voltage $s(t)$ crosses zero every carrier cycle, even when $E(t)$ is large. Its value at one time, its wave envelope, and its carrier-component amplitude are three different quantities. The narrow-envelope intuition additionally needs $\Omega \ll {\omega }_c$; the algebraic model only assumes $\Omega <{\omega }_c$.

The beauty is that two views reveal different parts of the same operation. In time, the message appears to ride on the oscillations. In the frequency spectrum, amplitude modulation sidebands show it as a pair of translated components. An envelope detector reads their combined shape without needing the carrier's wave phase; coherent demodulation instead uses a synchronized reference and preserves the signed factor.

In the single-sinusoid AM voltage model, $m$ is the peak fractional excursion of the signed multiplying factor from its mean value $1$. At $m=0$ there is only the carrier wave; at $m=0.5$ its multiplying factor ranges from $0.5$ to $1.5$; at $m=1$ it just touches zero. For $0\le m\le 1$, the wave envelope extrema areThe last equality follows by adding and subtracting the first two. An oscilloscope estimate uses local carrier peaks near the message extrema and needs enough carrier cycles to approximate those bounds. Consistent local peak-to-peak amplitude readings give the same ratio because their common factor of two cancels. Mixing a peak reading with a peak-to-peak reading does not.

For $A=2\mathrm{V}$ and $m=0.5$, the wave envelope ranges from $1$ to $3\mathrm{V}$, giving $(3-1)/(3+1)=0.5$. These are model values. At $m=1.4$, the signed factor has minimum $-0.4$, but the nonnegative wave envelope reaches zero at its sign changes; the extrema formula would misleadingly report $1$. It cannot measure the true depth of overmodulation from the folded envelope alone.

Apply the product-to-sum formula to the voltage model:The carrier wave component has peak voltage amplitude $A$; each AM sideband has peak voltage amplitude $Am/2$. These are temporal frequency spectrum components, unlike the spatial sidebands used in pattern stability. The separation $\Omega >0$ makes the three positive angular frequencies distinct, and $\Omega <{\omega }_c$ keeps the lower one positive. The span between sidebands is $2\Omega$, or $2f_m$ in hertz when $f_m=\Omega /(2\pi )$; this is the single-tone span, not a universal signal bandwidth for arbitrary messages.

A one-sided cosine-amplitude display assigns $A$ to the carrier line. A two-sided complex-exponential expansion instead assigns $A/2$ at each of $\pm {\omega }_c$, and $Am/4$ at each of the four sideband locations. The signal is identical; confusing the conventions creates a factor-of-two error. An ideal infinite-duration sinusoid gives a spectral line. A finite recording and finite analyser resolution broaden the displayed peak.

For a voltage applied across an ideal resistor of electrical resistance $R>0$, a component of peak voltage $B$ has root mean square voltage $B/\sqrt{2}$ and mean power $B^2/(2R)$ by Ohm's law. ThusThe proof of additivity is to average $s(t{)}^2/R$ over a long time: cross-products of distinct frequencies have zero mean. Consequently the whole signal's root mean square voltage is $A\sqrt{(1+m^2/2)/2}$, not simply its maximum wave envelope divided by $\sqrt{2}$. The carrier-component $A/\sqrt{2}$ remains unchanged as $m$ varies.

With $A=2\mathrm{V}$, $R=50\Omega$ and $m=0.5$, the carrier supplies $40\mathrm{mW}$ and each $0.5\mathrm{V}$ peak sideband supplies $2.5\mathrm{mW}$, giving $45\mathrm{mW}$ total. At $m=1.4$ each sideband is $1.4\mathrm{V}$ peak and supplies $19.6\mathrm{mW}$, giving $79.2\mathrm{mW}$ total. Neither case changes the carrier-component power.

The fraction of mean power in the message-bearing sidebands is $m^2/(2+m^2)$. It is $1/9$ at $m=0.5$ and $1/3$ at $m=1$. Conventional AM spends the remaining power on a carrier that enables an inexpensive envelope detector. Increasing $m$ beyond $1$ increases this fraction in the linear model but breaks ordinary envelope recovery. Removing the carrier gives suppressed-carrier amplitude modulation and changes the receiver requirement.

For this single-tone AM model, overmodulation means $m>1$. The signed factor $q(t)$ becomes negative whenever $\cos (\Omega t)<-1/m$. On such intervals,The sign reversal is a $\pi$ radian shift of the carrier's wave phase. At each zero of $q$, the wave envelope vanishes and that phase is undefined. The waveform itself is continuous: the formula has not clipped it or created extra spectral lines.

At $m=1.4=7/5$, sign changes occur at message phases $\Omega t=\arccos (-5/7)$ and $2\pi -\arccos (-5/7)$ within each cycle. At the trough $\Omega t=\pi$, the desired signed factor is $-0.4$, whereas its nonnegative wave envelope is $0.4A$. Ideal envelope recovery followed by subtraction of $A$ would give $-0.6A$ there, rather than the desired message voltage $-1.4A$. This is loss of polarity in the detector, not loss of the message from the full modulated signal. Actual transmitter clipping is a separate signal distortion that can generate additional spectral components.

An envelope detector extracts the slowly varying nonnegative wave envelope of a rapid carrier wave. In a simple electrical version a diode charges a capacitor near positive carrier peaks and an attached resistor discharges it between peaks. When $0\le m<1$, ideal envelope extraction gives $E(t)=A[1+m\cos (\Omega t)]$; removing the constant $A$ leaves $Am\cos (\Omega t)$. This permits recovery without a synchronized carrier reference. Carrier injection thus trades transmitted power for simplicity.^[ref]

A working peak detector needs margins as well as $m<1$: $E_{\min }=A(1-m)$ should comfortably exceed the diode's forward-voltage scale and noise, and the time constant $\tau =RC$ must suppress carrier ripple while following a falling wave envelope. The coarse scale window isA time constant that is too short leaves carrier ripple; one that is too long makes the discharging capacitor fall more slowly than the envelope. In the ideal zero-drop, continuous-tracking approximation, following a falling $E$ requires $E/\tau \ge -E^{\prime }$. For $0<m<1$,Differentiating the ratio locates its maximum at $\cos \theta =-m$. This original slope check explains why operation near $m=1$ is demanding even before overmodulation: the upper bound tends to zero while the need to smooth carrier cycles persists. It is an approximate circuit criterion, not a guarantee in the presence of diode drops, loading or noise. At $m=0$ there is no varying message to track; at $m=1$ ideal envelope extraction remains algebraically possible, but the simple detector has no positive trough margin.

For $m>1$, even an ideal noiseless envelope detector returns $A|q(t)|$. Removing its mean or its original offset cannot restore the discarded sign. A more elaborate receiver can use the retained wave phase through coherent demodulation.

Coherent demodulation recovers a message by multiplying the received signal by a reference carrier wave of the same frequency and known relative wave phase, then passing the result through a low-pass filter. For the ideal AM model and reference $2\cos ({\omega }_{c}t+\phi )$,A unity-gain ideal low-pass filter passing $0$ and $\Omega$ but rejecting $2{\omega }_c$ and $2{\omega }_c\pm \Omega$ leaves $Aq(t)\cos \phi$. Such a separating cutoff exists because $\Omega <2{\omega }_c-\Omega$. At $\phi =0$, subtraction of $A$ recovers $Am\cos (\Omega t)$ exactly for any finite $m\ge 0$, including overmodulation, provided the transmitter and channel preserve this linear waveform. This is the signed factor, not its absolute value.

A constant phase error scales the recovered message by $\cos \phi$: a $\pi /3$ radian error halves it, and a $\pi /2$ error removes it. A known error with nonzero cosine can be compensated in gain; a frequency mismatch makes the relative wave phase drift and no longer gives constant attenuation. The lecture illustrates synchronous multiplication, filtering and phase error.^[ref]

Rice University's ELEC 332: In the Lab makes the model experimentally testable. Preparation resets the spectrum analyzer and observes an unmodulated generator output on it and an oscilloscope. A Wavetek function generator supplies external modulation. The specified $10\mathrm{MHz}$ carrier and $10\mathrm{kHz}$ message separate three peaks. Marker readouts give frequency and level; narrowing resolution bandwidth shows how the instrument broadens a line.^[ref]

Part 3 uses an SA602 analog multiplier module. The message generator also triggers the oscilloscope; multiplier output feeds both displays. A no-offset module supplies a suppressed-carrier control. Replacing it with the offset-adding module restores the carrier; varying levels tests signal distortion. Finally a radio receives the conventional AM transmission, and varying message frequency checks that the heard tone belongs to this source. These are prescribed observations, not reported measurements.^[ref]

Calibration matters: verify generator voltage at the connected input, distinguish peak from peak-to-peak amplitude and root mean square, and check analyser units and loading before comparing heights with $Am/2$ or power with $A^2{m}^2/(8R)$. The unmodulated carrier is a reference control. Limited resolution can merge sidebands; generator harmonics can add peaks. Rice explicitly demonstrates both limitations in Part 1.^[ref]

A complementary University of Kansas procedure, A.2 - Amplitude Modulation, calibrates the added carrier and the suppressed-carrier product separately to $4\mathrm{V}$ peak-to-peak before recombining them. It holds the carrier fixed, varies modulation depth, reads envelope extrema and carrier-to-sideband power differences, then connects the demodulator and adjusts its tunable low-pass filter for recovery, comparing the output with the message. These are setup targets and comparison instructions, not measured results.^[ref]

The inference combines time and frequency spectrum readouts: correct sideband spacing tests frequency translation, relative heights test modulation depth, and the recovered message tests the detector. Hardware clipping, receiver filtering and noise can spoil recovery even at $m<1$. Conversely failure of an envelope detector at $m>1$ does not demonstrate failure of a synchronized multiplier receiver. The two diagrams above illustrate the stipulated linear model only.