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The concept of modulation is not at all new, it has been around since the early days of radio.

Modulation, especially in the context of RF applications, refers to the mixing of two sinusoidal signals. One signal is the message signal and contains the information to be modulated. It usually consists of a band-limited spectrum of sinusoids (such as music). The other signal is the carrier signal and is generally a pure tone (a sinusoid of a single frequency). The frequency of the carrier is referred to as the carrier frequency and will be denoted by the symbol f_c (or ω_c for radian frequency) throughout this text. Typically, f_c is much higher in frequency than the highest frequency component of the message signal.

The Basics

The concept of modulation comes from the trigonometric identity:

If we assume that the message signal is a pure tone of frequency, f_m, then the message can be mathematically represented as cos(2πf_mt). The same assumption can be made about the carrier signal, thereby expressing it as cos(2πf_ct). The “pure tone” assumption makes the mathematics much more tractable. However, it is important to keep in mind that the message signal is rarely a pure tone.

Typically, it is composed of time variations in amplitude, frequency, phase, or any combination thereof. Even the carrier need not necessarily be a pure sinusoid. Applications exist in which the carrier signal is a square wave with a fundamental frequency, f_c. The harmonics of f_c inherent in the square wave are dealt with by filtering the modulated signal.

The mixing process mentioned earlier can be thought of as a multiplication operation. Therefore, the trigonometric identity above may be employed to represent the mixing process as follows:

Thus, the mixing of the message and carrier results in a transformation of the frequency of the message. The message frequency is translated from its original frequency to two new frequencies — one greater than the carrier (f_c + f_m), and one less than the carrier (f_c - f_m), the upper and lower side bands, respectively. Furthermore, the translated signal undergoes a 6 dB loss (50 percent reduction) as dictated by the factor of ½ appearing on the right-hand side of the equality.

The form of modulation just described is referred to as “double sideband modulation,” because the message is translated to a frequency range above and below the carrier frequency. Another form of modulation, known as single sideband modulation, can be used to eliminate either the upper or lower sideband. One method of performing single sideband modulation is to employ a quadrature modulator. A quadrature modulator mixes the message with two carriers. Both carriers operate at the same frequency, but are shifted in phase by 90 degrees relative to one another (hence the “quadrature” term). This simply means that the two carriers can be expressed as cos(2πf_ct) and sin(2πf_ct). The message, too, is modified to consist of two separate signals: the original and a 90 degree phase-shifted version of the original. The original is mixed with the cosine component of the carrier and the phase-shifted version is mixed with the sine component of the carrier. These two modifications result in the implementation of the single sideband function. Trigonometrically, this can be expressed as:

Note that the right-hand side of the equation reduces to cos(x - y), the lower sideband, only. In the above equation, x is the carrier and y is the message. Incidentally, changing the sign of the operator on the left-hand side of the equation results in only the upper sideband appearing on the right hand side.

In figure 1 the functional representation of a single and double sideband modulator are shown along with the associated frequency spectra. However, the message is shown as a band-limited spectrum rather than a pure tone, which better represents a real-world application. Each constituent frequency in the message is translated to one or both sides of the carrier, as shown.