# Amplitude modulation

Amplitude modulation (AM) is a technique used in electronic communication, most commonly for transmitting information via a radio carrier wave. AM works by varying the strength of the transmitted signal in relation to the information being sent. For example, changes in signal strength may be used to specify the sounds to be reproduced by a loudspeaker, or the light intensity of television pixels. Contrast this with frequency modulation, in which the frequency is varied, and phase modulation, in which the phase is varied in accordance to the modulating signal.

In the mid-1870s, a form of amplitude modulation—initially called "undulatory currents"—was the first method to successfully produce quality audio over telephone lines. Beginning with Reginald Fessenden's audio demonstrations in 1906, it was also the original method used for audio radio transmissions, and remains in use today by many forms of communication—"AM" is often used to refer to the mediumwave broadcast band (see AM radio).

Fig 1: An audio signal (top) may be carried by an AM or FM radio wave.

## Forms of amplitude modulation

In radio communication, a continuous wave radio-frequency signal (a sinusoidal carrier wave) has its amplitude modulated by an audio waveform before transmission. The audio waveform modifies the amplitude of the carrier wave and determines the envelope of the waveform. In the frequency domain, amplitude modulation produces a signal with power concentrated at the carrier frequency and two adjacent sidebands. Each sideband is equal in bandwidth to that of the modulating signal, and is a mirror image of the other. Amplitude modulation resulting in two sidebands and a carrier is called "double-sideband amplitude modulation" (DSB-AM).

Amplitude modulation is inefficient in power usage; at least two-thirds of the power is concentrated in the carrier signal. The carrier signal contains none of the original information being transmitted (voice, video, data, etc.). However, it does contain information about the frequency, phase and amplitude needed to demodulate the received signal most simply and effectively. In some communications systems, lower total cost can be achieved by eliminating some of the carrier, thereby lowering electrical power usage even though this requires greater receiver complexity and cost. If some carrier is retained (reduced-carrier transmission, or DSB-RC) receivers can be designed to recover the frequency, phase, and amplitude information from this "pilot" carrier and use it in the demodulation process. If the carrier is eliminated (Double-sideband suppressed-carrier transmissionor DSB-SC) the receiver must provide a substitute carrier, with inevitable loss of fidelity. Completely suppressing both the carrier and one of the sidebands produces single-sideband modulation, widely used in amateur radio and other communications applications. SSB occupies less than half the spectrum of AM so it also has greatly improved bandwidth efficiency. In AM broadcasting, where there are many receivers for each transmitter, the full carrier is provided to allow reception with inexpensive receivers. The broadcaster absorbs the extra power cost to greatly increase potential audience.

A simple form of AM, often used for digital communications, is on-off keying: a type of amplitude-shift keying in which binary data is represented by the presence or absence of a carrier. This is used by radio amateurs to transmit Morse code and is known as continuous wave (CW) operation.

### ITU designations

In 1982, the International Telecommunication Union (ITU) designated the types of amplitude modulation:

Designation Description
A3E double-sideband a full-carrier - the basic Amplitude modulation scheme
R3E single-sideband reduced-carrier
H3E single-sideband full-carrier
J3E single-sideband suppressed-carrier
B8E independent-sideband emission
C3F vestigial-sideband

## Example: double-sideband AM

Left part: Modulation Voltage as a function of time. Right part: Spectrum of the amplitude modulated carrier
Fig 2: Double-sided spectra of baseband and AM signals.

A carrier wave is modeled as a sine wave:

$c(t) = A\cdot \sin(\omega_c t + \phi_c),\,$

in which the frequency in Hz is given by:

$\omega_c / (2\pi).\,$

The constants $A\,$ and $\phi_c\,$ represent the carrier amplitude and initial phase, and are introduced for generality. For simplicity, their respective values can be set to 1 and 0.

Let m(t) represent an arbitrary waveform that is the message to be transmitted, e.g., a simple audio tone of form:

$m(t) = M\cdot \cos(\omega_m t + \phi).\,$

where constant M represent the largest magnitude, and the frequency is:

$\omega_m / (2\pi).\,$

It is assumed that  $\omega_m \ll \omega_c\,$  and that  $\min[ m(t) ] = -M.\,$

Amplitude modulation is formed by the product:

 $y(t)\,$ $= [1 + m(t)]\cdot c(t),\,$ $= A\cdot[1 + M\cdot \cos(\omega_m t + \phi)]\cdot \sin(\omega_c t).$

$A\,$ represents the carrier amplitude, which is a constant that demonstrates the modulation index. The values A=1 and M=0.5 produce y (t), depicted by the top graph (labelled "50% Modulation") in Figure 4. Using trigonometric identities, y(t) can be written in the form

$y(t) = A\cdot \sin(\omega_c t) + \begin{matrix}\frac{AM}{2} \end{matrix} \left[\sin((\omega_c + \omega_m) t + \phi) + \sin((\omega_c - \omega_m) t - \phi)\right].\,$

Therefore, the modulated signal has three components: a carrier wave and two sinusoidal waves (known as sidebands), whose frequencies are slightly above and below  $\omega_c.\,$

### Spectrum

For more general forms of m(t), trigonometry is not sufficient; however, if the top trace of Figure 2 depicts the frequency of m(t) the bottom trace depicts the modulated carrier. It has two components: one at a positive frequency (centered on $+\omega_c$) and one at a negative frequency (centered on $-\omega_c$). Each component contains the two sidebands and a narrow segment in between, representing energy at the carrier frequency. Since the negative frequency is a mathematical artifact, examining the positive frequency demonstrates that an AM signal's spectrum consists of its original (two-sided) spectrum, shifted to the carrier frequency. Figure 2 is a result of computing the Fourier transform of:   $[A + m(t)]\cdot \sin(\omega_c t),\,$ using the following transform pairs:

\begin{align} m(t) \quad \stackrel{\mathcal{F}}{\Longleftrightarrow}&\quad M(\omega) \\ \sin(\omega_c t) \quad \stackrel{\mathcal{F}}{\Longleftrightarrow}&\quad i \pi \cdot [\delta(\omega +\omega_c)-\delta(\omega-\omega_c)] \\ A\cdot \sin(\omega_c t) \quad \stackrel{\mathcal{F}}{\Longleftrightarrow}&\quad i \pi A \cdot [\delta(\omega +\omega_c)-\delta(\omega-\omega_c)] \\ m(t)\cdot A\sin(\omega_c t) \quad \stackrel{\mathcal{F}}{\Longleftrightarrow}& \frac{A}{2\pi}\cdot \{M(\omega)\} * \{i \pi \cdot [\delta(\omega +\omega_c)-\delta(\omega-\omega_c)]\} \\ =& \frac{iA}{2}\cdot [M(\omega +\omega_c) - M(\omega -\omega_c)] \end{align}
Fig 3: The spectrogram of an AM broadcast shows its two sidebands (green), separated by the carrier signal (red).

### Power and spectrum efficiency

In terms of positive frequencies, the transmission bandwidth of AM is twice the signal's original (baseband) bandwidth; both the positive and negative sidebands are shifted up to the carrier frequency. Thus, double-sideband AM (DSB-AM) is spectrally inefficient because the same spectral information is transmitted twice, and fewer radio stations can be accommodated in a given broadcast band than if only one replica of the original signal's spectrum were transmitted. The suppression methods described above may be understood in terms of Figure 2. With the carrier suppressed, there would be no energy at the center of a group; with a sideband suppressed, the "group" would have the same bandwidth as the positive frequencies of $M(\omega).\,$  The transmitter-power efficiency of DSB-AM depends on the type of receiver used. For the most inexpensive type of AM receiver, the carrier is needed to provide undistorted reception, thus 100% of the power is useful. With a single sideband suppressed carrier (SSB-SC) capable receiver, only 16.7% of the transmitted power is useful, since 66.6% of the power is wasted in the carrier and 16.7% in the unused sideband. DSB-SC systems have had very limited application but would theoretically use 33.3% of the transmitted signal.

## Modulation index

The AM modulation index is the measure of the amplitude variation surrounding an unmodulated carrier. As with other modulation indices, in AM this quantity (also called "modulation depth") indicates how much the modulation varies around its "original" level. For AM, it relates to variations in carrier amplitude and is defined as:

$h = \frac{\mathrm{peak\ value\ of\ } m(t)}{A} = \frac{M}{A},$   where $M\,$ and $A\,$ are the message amplitude and carrier amplitude, respectively.

So if $h=0.5$, carrier amplitude varies by 50% above (and below) its unmodulated level; for $h=1.0$, it varies by 100%. To avoid distortion, modulation depth must not exceed 100 percent. Transmitter systems will usually incorporate a limiter circuit (such as a vogad) to ensure this. However, AM demodulators can be designed to detect the inversion (or 180-degree phase reversal) that occurs when modulation exceeds 100 percent; they automatically correct for this defect.[citation needed] Variations of a modulated signal with percentages of modulation are shown below. In each image, the maximum amplitude is higher than in the previous image (note that the scale changes from one image to the next).

Fig 4: Modulation depth

## Modulation methods

Anode (plate) modulation. A tetrode's plate and screen grid voltage is modulated via an audio transformer. The resistor R1 sets the grid bias; both the input and output are tuned circuits with inductive coupling.

Modulation circuit designs may be classified as low- or high-level (depending on whether they modulate in a low-power domain—followed by amplification for transmission—or in the high-power domain of the transmitted signal).[1]

### Low-level generation

In modern radio systems, modulated signals are generated via digital signal processing (DSP). With DSP many types of AM are possible with software control (including DSB with carrier, SSB suppressed-carrier and independent sideband, or ISB). Calculated digital samples are converted to voltages with a digital to analog converter, typically at a frequency less than the desired RF-output frequency. The analog signal must then be shifted in frequency and linearly amplified to the desired frequency and power level (linear amplification must be used to prevent modulation distortion).[2] This low-level method for AM is used in many Amateur Radio transceivers.[3]

AM may also be generated at a low level, using analog methods described in the next section.

### High-level generation

High-power AM transmitters (such as those used for AM broadcasting) are based on high-efficiency class-D and class-E power amplifier stages, modulated by varying the supply voltage.[4]

Older designs (for broadcast and amateur radio) also generate AM by controlling the gain of the transmitter’s final amplifier (generally class-C, for efficiency). The following types are for vacuum tube transmitters (but similar options are available with transistors):[5]

• Plate modulation: In plate modulation, the plate voltage of the RF amplifier is modulated with the audio signal. The audio power requirement is 50 percent of the RF-carrier power.
• Heising (constant-current) modulation: RF amplifier plate voltage is fed through a “choke” (high-value inductor). The AM modulation tube plate is fed through the same inductor, so the modulator tube diverts current from the RF amplifier. The choke acts as a constant current source in the audio range. This system has a low power efficiency.
• Control grid modulation: The operating bias and gain of the final RF amplifier can be controlled by varying the voltage of the control grid. This method requires little audio power, but care must be taken to reduce distortion.
• Clamp tube (screen grid) modulation: The screen-grid bias may be controlled through a “clamp tube”, which reduces voltage according to the modulation signal. It is difficult to approach 100-percent modulation while maintaining low distortion with this system.
• Doherty modulation: One tube provides the power under carrier conditions and another operates only for positive modulation peaks. Overall efficiency is good, and distortion is low.
• Outphasing modulation: Two tubes are operated in parallel, but partially out of phase with each other. As they are differentially phase modulated their combined amplitude is greater or smaller. Efficiency is good and distortion low when properly adjusted.
• Pulse width modulation(PWM) or Pulse duration modulation (PDM): A highly efficient high voltage power supply is applied to the tube plate. This system was pioneered by Hilmer Swanson and has a number of variations, all of which achieve high efficiency and sound quality.

## Demodulation methods

The simplest form of AM demodulator consists of a diode which is configured to act as envelope detector. Another type of demodulator, the product detector, can provide better-quality demodulation with additional circuit complexity.