# Sideband and the Frequency Domain

## What is Sideband?

Whenever a carrier is modulated by an information signal, new signals at different frequencies are generated as part of the process. These new frequencies, which are called side frequencies, or sidebands, occur in the frequency spectrum directly above and directly below the carrier frequency. More specifically, the side bands occur at frequencies that are the sum and difference of the carrier and modulating frequencies. When signals of more than one frequency make up a waveform, it is often better to show the AM signal in the frequency domain rather than in the time domain.

## Sideband Calculations

When only a single-frequency sine wave modulating signal is used, the modulation process generates two side bands. If the modulating signal is a complex wave, such as voice or video, a whole range of frequencies modulate the carrier, and thus a whole range of side bands are generated. The upper side band fUSB and lower sideband fLSB are computed as

where fc is the carrier frequency and fm is the modulating frequency. The existence of sidebands can be demonstrated mathematically, starting with the equation for an AM signal described previously:

By using the trigonometric identity that says that the product of two sine waves is and substituting this identity into the expression a modulated wave, the instantaneous amplitude of the signal becomes

where the first term is the carrier; the second term, containing the difference fc 2 fm, is the lower sideband; and the third term, containing the sum fc 1 fm, is the upper sideband. For example, assume that a 400-Hz tone modulates a 300-kHz carrier. The upper and lower sidebands are

Observing an AM signal on an oscilloscope, you can see the amplitude variations of the carrier with respect to time. This time-domain display gives no obvious or outward indication of the existence of the sidebands, although the modulation process does indeed produce them, as the equation above shows. An AM signal is really a composite signal formed from several components: the carrier sine wave is added to the upper and lower sidebands, as the equation indicates. This is illustrated graphically in Fig. 3-6.

Adding these signals together algebraically at every instantaneous point along the time axis and plotting the result yield the AM wave shown in the fi gure. It is a sine wave at the carrier frequency whose amplitude varies as determined by the modulating signal.

### Frequency-Domain Representation of AM

Another method of showing the sideband signals is to plot the carrier and sideband amplitudes with respect to frequency, as in Fig. 3-7. Here the horizontal axis represents frequency, and the vertical axis represents the amplitudes of the signals. The signals may be voltage, current, or power amplitudes and may be given in peak or rms values. A plot of signal amplitude versus frequency is referred to as a frequency-domain display. A test instrument known as a spectrum analyzer is used to display the frequency domain of a signal. Fig. 3-8 shows the relationship between the time- and frequency-domain displays of an AM signal. The time and frequency axes are perpendicular to each other. The amplitudes shown in the frequency-domain display are the peak values of the carrier and sideband sine waves. Whenever the modulating signal is more complex than a single sine wave tone, multiple upper and lower sidebands are produced by the AM process. For example, avoice signal consists of many sine wave components of different frequencies mixed together. Recall that voice frequencies occur in the 300- to 3000-Hz range.

Therefore, voice signals produce a range of frequencies above and below the carrier frequency, as shown in Fig. 3-9. These sidebands take up spectrum space. The total bandwidth of an AM signal is calculated by computing the maximum and minimum sideband frequencies. This is done by finding the sum and difference of the carrier frequency and maximum modulating frequency (3000 Hz, or 3 kHz, in Fig. 3-9). For example, if the carrier frequency is 2.8 MHz (2800 kHz), then the maximum and minimum sideband frequencies are

The total bandwidth is simply the difference between the upper and lower sideband frequencies:

As it turns out, the bandwidth of an AM signal is twice the highest frequency in the modulating signal: BW 5 2fm, where fm is the maximum modulating frequency. In the case of a voice signal whose maximum frequency is 3 kHz, the total bandwidth is simply

Example 3-2 A standard AM broadcast station is allowed to transmit modulating frequencies up to 5 kHz. If the AM station is transmitting on a frequency of 980 kHz, compute the maximum and minimum upper and lower sidebands and the total bandwidth occupied by the AM station.

As Example 3-2 indicates, an AM broadcast station has a total bandwidth of 10 kHz. In addition, AM broadcast stations are spaced every 10 kHz across the spectrum from 540 to 1600 kHz. This is illustrated in Fig. 3-10. The sidebands from the fi rst AM broadcast frequency extend down to 535 kHz and up to 545 kHz, forming a 10-kHz channel for the signal. The highest channel frequency is 1600 kHz, with sideband extending from 1595 up to 1605 kHz. There are a total of 107 10-kHz-wide channels for AM radio stations.

## Pulse Modulation

When complex signals such as pulses or rectangular waves modulate a carrier, a broad spectrum of side bands are produced. According to Fourier theory, complex signals such as square waves, triangular waves, sawtooth waves, and distorted sine waves are simply made up of a fundamental sine wave and numerous harmonic signals at different amplitudes. Assume that a carrier is amplitude-modulated by a square wave that is made up of a fundamental sine wave and all odd harmonics. A modulating square wave will produce sidebands at frequencies based upon the fundamental sine wave as well as at the third, fi fth, seventh, etc., harmonics, resulting in a frequency-domain plot like that shown in Fig. 3-11. As can be seen, pulses generate extremely wide-bandwidth signals. In order for a square wave to be transmitted and faithfully received without distortion or degradation, all the most signifi cant sidebands must be passed by the antennas and the transmitting and receiving circuits. Fig. 3-12 shows the AM wave resulting when a square wave modulates a sine wave carrier. In Fig. 3-12(a), the percentage of modulation is 50; in Fig. 3-12(b), it is 100. In this case, when the square wave goes negative, it drives the carrier amplitude to zero. Amplitude modulation by square waves or rectangular binary pulses is referred to as amplitude-shift keying (ASK). ASK is used in some types of data communication when binary information is to be transmitted. Another crude type of amplitude modulation can be achieved by simply turning the carrier off and on. An example is the transmitting of Morse code by using dots and dashes.

A dot is a short burst of carrier, and a dash is a longer burst of carrier. Fig. 3-13 shows the transmission of the letter P, which is dot-dash-dash-dot (pronounced “dit-dah-dahdit”). The time duration of a dash is three times the length of a dot, and the spacing between dots and dashes is one dot time. Code transmissions such as this are usually called continuous-wave (CW) transmissions. This kind of transmission is also referred to as ON/OFF keying (OOK). Despite the fact that only the carrier is being transmitted, sidebands are generated by such ON/OFF signals. The sidebands result from the frequency or repetition rate of the pulses themselves plus their harmonics. As indicated earlier, the distortion of an analog signal by overmodulation also generates harmonics. For example, the spectrum produced by a 500-Hz sine wave modulating a carrier of 1 MHz is shown in Fig. 3-14(a). The total bandwidth of the signal is 1 kHz. However, if the modulating signal is distorted, the second, third, fourth, and higher harmonics are generated. These harmonics also modulate the carrier, producing many more sidebands, as illustrated in Fig. 3-14(b). Assume that the distortion is such that the harmonic amplitudes beyond the fourth harmonic are insignificant (usually less than 1 percent); then the total bandwidth of the resulting signal is about 4 kHz instead of the 1-kHz bandwidth that would result without overmodulation and distortion. The harmonics can overlap into adjacent channels, where other signals may be present and interfere with them. Such harmonic sideband inter ference is sometimes called splatter because of the way it sounds at the receiver. Overmodulation and splatter are easily eliminated simply by reducing the level of the modulating signal by using gain control or in some cases by using amplitude-limiting or compression circuits.