Waves and modulation

In physics, mathematics, and related fields, a wave is a disturbance of a field in which a physical attribute oscillates repeatedly at each point or propagates from each point to neighboring points, or seems to move through space.

The phase velocity of electromagnetic radiation may – under certain circumstances (for example anomalous dispersion) – exceed the speed of light in a vacuum, but this does not indicate any superluminal information or energy transfer.

## Superposition principle

The **superposition principle**, also known as **superposition property**, states that, for all linear systems, the net response caused by two or more stimuli is the sum of the responses that would have been caused by each stimulus individually. So that if input *A* produces response *X* and input *B* produces response *Y* then input (*A* + *B*) produces response (*X* + *Y*).

The importance of linear systems is that they are easier to analyze mathematically; there is a large body of mathematical techniques, frequency domain linear transform methods such as Fourier, Laplace transforms, and linear operator theory, that are applicable. Because physical systems are generally only approximately linear, the superposition principle is only an approximation of the true physical behaviour.

## Wave interference

In physics, **interference** is a phenomenon in which two waves superpose to form a resultant wave of greater, lower, or the same amplitude. Constructive and destructive interference result from the interaction of waves that are correlated or coherent with each other, either because they come from the same source or because they have the same or nearly the same frequency. Interference effects can be observed with all types of waves, for example, light, radio, acoustic, surface water waves, gravity waves, or matter waves. The resulting images or graphs are called **interferograms**.

The principle of superposition of waves states that when two or more propagating waves of same type are incident on the same point, the resultant amplitude at that point is equal to the vector sum of the amplitudes of the individual waves. If a crest of a wave meets a crest of another wave of the same frequency at the same point, then the amplitude is the sum of the individual amplitudes—this is constructive interference. If a crest of one wave meets a trough of another wave, then the amplitude is equal to the difference in the individual amplitudes—this is known as destructive interference.

Constructive interference occurs when the phase difference between the waves is an **even multiple** of π (180°) , whereas destructive interference occurs when the difference is an **odd multiple** of π. If the difference between the phases is intermediate between these two extremes, then the magnitude of the displacement of the summed waves lies between the minimum and maximum values.

Consider, for example, what happens when two identical stones are dropped into a still pool of water at different locations. Each stone generates a circular wave propagating outwards from the point where the stone was dropped. When the two waves overlap, the net displacement at a particular point is the sum of the displacements of the individual waves. At some points, these will be in phase, and will produce a maximum displacement. In other places, the waves will be in anti-phase, and there will be no net displacement at these points. Thus, parts of the surface will be stationary—these are seen in the figure above and to the right as stationary blue-green lines radiating from the centre.

Interference of light is a common phenomenon that can be explained classically by the superposition of waves, however a deeper understanding of light interference requires knowledge of wave-particle duality of light which is due to quantum mechanics. Prime examples of light interference are the famous double-slit experiment, laser speckle, anti-reflective coatings and interferometers. Traditionally the classical wave model is taught as a basis for understanding optical interference, based on the Huygens–Fresnel principle.

## Huygens–Fresnel principle

The Huygens–Fresnel principle states that every point on a wavefront is itself the source of spherical wavelets. The sum of these spherical wavelets forms the wavefront.

## Amplitude modulation

Amplitude modulation (AM) is a modulation technique used in electronic communication, most commonly for transmitting information via a radio carrier wave. In amplitude modulation, the amplitude (signal strength) of the carrier wave is varied in proportion to that of the message signal being transmitted.

The transmitted wave is the carrier wave multiplied by the message signal. At the receiving station, the message signal is extracted from the modulated carrier by demodulation.

## Heterodyne

**Heterodyning** is a signal processing technique that creates new frequencies by combining or mixing two frequencies. Heterodyning is used to shift one frequency range into another, new one, and is also involved in the processes of modulation and demodulation.

The two frequencies are combined in a nonlinear signal-processing device such as a vacuum tube, transistor, or diode, usually called a *mixer*.

In the most common application, two signals at frequencies *f*1 and *f*2 are mixed, creating two new signals, one at the sum *f*1 + *f*2 of the two frequencies, and the other at the difference *f*1 - *f*2. These frequencies are called *heterodynes*.

Typically only one of the new frequencies is desired, and the other signal is filtered out of the output of the mixer. Heterodyne frequencies are related to the phenomenon of "beats" in acoustics.

A major application of the heterodyne process is in the superheterodyne radio receiver circuit, which is used in virtually all modern radio receivers.

Heterodyning is based on the trigonometric identity:

sin
⁡

θ

1

sin
⁡

θ

2

=

1
2

cos
⁡
(

θ

1

−

θ

2

)
−

1
2

cos
⁡
(

θ

1

θ

2

)

{\displaystyle \sin \theta \_{1}\sin \theta \_{2}={\frac {1}{2}}\cos(\theta \_{1}-\theta \_{2})-{\frac {1}{2}}\cos(\theta \_{1}+\theta \_{2})}

The product on the left hand side represents the multiplication ("mixing") of a sine wave with another sine wave. The right hand side shows that the resulting signal is the difference of two sinusoidal terms, one at the sum of the two original frequencies, and one at the difference, which can be considered to be separate signals.

Using this trigonometric identity, the result of multiplying two sine wave signals

sin
⁡
(
2
π

f

1

t
)

{\displaystyle \sin(2\pi f\_{1}t)\,}

 and

sin
⁡
(
2
π

f

2

t
)

{\displaystyle \sin(2\pi f\_{2}t)\,}

 at different frequencies

f

1

{\displaystyle f\_{1}}

 and

f

2

{\displaystyle f\_{2}}

 can be calculated:

sin
⁡
(
2
π

f

1

t
)
sin
⁡
(
2
π

f

2

t
# )

1
2

cos
⁡
[
2
π
(

f

1

−

f

2

)
t
]
−

1
2

cos
⁡
[
2
π
(

f

1

f

2

)
t
]

{\displaystyle \sin(2\pi f\_{1}t)\sin(2\pi f\_{2}t)={\frac {1}{2}}\cos[2\pi (f\_{1}-f\_{2})t]-{\frac {1}{2}}\cos[2\pi (f\_{1}+f\_{2})t]\,}

The result is the sum of two sinusoidal signals, one at the sum *f*1 + *f*2 and one at the difference *f*1 - *f*2 of the original frequencies

The two signals combine in a device called a *mixer*. As seen in the previous section, an ideal mixer would be a device that multiplies the two signals.

Most circuit elements in communications circuits are designed to be linear. This means they obey the superposition principle; if *F*(*v*) is the output of a linear element with an input of *v*:

F
(

v

1

v

2

# )
F
(

v

1

)
F
(

v

2

)

{\displaystyle F(v\_{1}+v\_{2})=F(v\_{1})+F(v\_{2})\,}

So if two sine wave signals at frequencies *f*1 and *f*2 are applied to a linear device, the output is simply the sum of the outputs when the two signals are applied separately with no product terms. Thus, the function *F* must be nonlinear to create mixer products.

A perfect multiplier only produces mixer products at the sum and difference frequencies (*f*1 ± *f*2), but more general nonlinear functions produce higher order mixer products: *n*⋅*f*1 + *m*⋅*f*2 for integers *n* and *m*.

## Analog multiplier

In electronics, an **analog multiplier** is a device which takes two analog signals and produces an output which is their product. Such circuits can be used to implement related functions such as *squares* (apply same signal to both inputs), and *square roots*.

If one input of an analog multiplier is held at a steady state voltage, a signal at the second input will be scaled in proportion to the level on the fixed input. In this case the analog multiplier may be considered to be a voltage controlled amplifier. Although analog multipliers are often used for such applications, voltage-controlled amplifiers are not necessarily true analog multipliers. The two inputs are not symmetrical.

By contrast, in what is generally considered to be a *true* analog multiplier, the two signal inputs have identical characteristics.

Analog multiplication can be accomplished by using the Hall Effect.

The Gilbert cell is a circuit whose output current is a 4 quadrant multiplication of its two differential inputs.

Although analog multiplier circuits are very similar to operational amplifiers, they are far more susceptible to noise and offset voltage-related problems as these errors may become multiplied. When dealing with high frequency signals, phase-related problems may be quite complex. For this reason, manufacturing wide-range general-purpose analog multipliers is far more difficult than ordinary operational amplifiers. This means they have a relatively high cost and so they are generally used only for circuits where they are indispensable.