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Note 24.
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Amplitude Modulation. Sidebands
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Let us take amplitude modulation as an example. In amplitude modulation, the amplitude of the high-frequency carrier wave is varied or modulated in accordance with audio, video, or other type of signal information to be transmitted. The general principle of amplitude modulation, AM, will be clear from a study of Fig. 30-A, which let us now concentrate on, as follows.
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Fig. 30-A
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To the left, in the gure, is shown a portion of an unmodulated high-frequency carrier wave. (For convenience in drawing, the carrier is shown as a triangular wave, but it will actually be a sinusoidal wave.) Note that the peak value of the unmodulated carrier wave is denoted by Vc . Now suppose the carrier is AMPLITUDE MODULATED by, say, a sinusoidal wave of audio frequency voltage. The appearance of the resulting wave would then be such as shown to the right in the gure. The dashed line connecting the peaks of the carrier wave is called the envelope of the wave. For distortionless modulation the envelope must have the same waveshape as the modulating voltage. Now let Va be the peak value of the modulated component of the carrier, as indicated in the gure. Then the ratio of Va to Vc is called the modulation factor and is denoted by m; thus Va m Vc 14-A
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In the discussion here we ll assume a sinusoidal carrier wave and sinusoidal modulating signal, and use the notation !c carrier frequency in radians=second; and !a frequency of modulating signal; rad=sec: hence vc Vc sin !c t instantaneous value of unmodulated carrier; and va Va sin !a t instantaneous value of modulation envelope
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or, by eq. (14-A) va mVc sin !a t
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After a detailed consideration of the above we come to the conclusion that the instantaneous value v of the resulting amplitude-modulated wave must be equal to v Vc 1 m sin !a t sin !c t Note that this equation satis es the requirement that if m 0 (the condition of no modulation), then all that is left is the unmodulated carrier, Vc sin !c t. Next, upon multiplying as indicated, the equation becomes v Vc sin !c t mVc sin !c t sin !a t Now, in the trigonometrical identity for sin x sin y (see note 25), set x !c t and y !a t. Upon doing this, the last equation becomes the very important result that v Vc sin !c t mVc mVc cos !c !a t cos !c !a t 2 2 15-A
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The equation brings out the important fact that, when a high-frequency sinusoidal carrier of frequency !c 2fc is amplitude-modulated by a single sinusoidal signal of frequency !a 2fa the resulting amplitude-modulated wave is composed of three component sinusoidal waves, thus 1. 2. 3. the CARRIER wave, of frequency !c 2fc , the LOWER SIDEBAND wave, of frequency (!c !a 2 fc fa , the UPPER SIDEBAND wave, of frequency !c !a 2 fc fa .
For several practical reasons the carrier must be a high-frequency ( radio-frequency ) wave, much higher in frequency than the frequency of the modulating signal; that is, fc must be much higher than fa . Thus the sidebands fc fa ) are also high-frequency waves, centered around the carrier wave. If the carrier is being modulated by a non-sinusoidal wave, as would normally be the case, then a pair of sidebands exists for each harmonic of the modulating wave. Thus the carrier wave is at the center of a cluster of high-frequency sideband waves, as indicated in Fig. 31-A, in which Vc is the rms value of the carrier wave. A brief but fundamental discussion follows the gure.
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