Carrier Frequencies
As long as the bandwidth of a signal fits in a passband, it can be transmitted without interfering with adjacent passbands. But how is it possible to take a particular voice signal, occupying the passband 300 to 3300 Hz, and transmit the same information in the passband of, for instance, 64,000 to 68,000 Hz?
The trick to moving a signal from one passband to another is to apply a trigonometric theorem concerning the multiplication of two sine waves with frequencies A and B, respectively. The theorem states that the two sine waves multiplied together form a composite that contains two frequencies: A + B, and A - B.
One of these sine waves is given a “well known” frequency (e.g., 107.9 MHz on your FM dial). This sine wave will be varied (modulated) in proportion to another sine wave (the information) so that it can “carry” the information. Hence, the term carrier wave, or carrier frequency.
Suppose we want to take a particular signal, a sine wave with frequency 1,000 Hz, and send the same information in a particular passband, at the frequency 67,000 Hz. If we multiply a “carrier” sine wave with frequency 68,000 Hz by our signal (1,000 Hz), we get a composite signal containing two frequencies: 67,000 and 69,000 Hz. Filter out the upper frequency, and we have our desired frequency.
Moreover, the same circuitry can be used to recover the original signal at the other end of the transmission medium. If we multiply our 67,000 Hz signal by a 68,000 Hz carrier and filter out the higher resulting frequency, the result is our original 1,000 Hz signal!
In the telephone system the signal to be transmitted is not merely a pure sine wave, but a complex signal containing a range of frequencies. Nevertheless, the same principle applies (since Fourier assures us that the voice signal can be represented as a sum of sine waves). Thus, to transmit a voice signal (original passband 300 to 3300 Hz) in the 64 to 68 kHz passband, we multiply the signal by a 68,000 Hz carrier. Verify by applying the formula that the composite signal has two “sidebands”: 64,700 to 67,700 Hz, and 68,300 to 71,300 Hz. The upper sideband is filtered out, leaving the lower sideband in the desired range.
Carrier Modulation Techniques
In amplitude modulation (AM), the amplitude of the carrier is modulated, resulting in the composite shown on the accompanying visual. On the visual, it is easy to see the input signal is now represented as the different amplitudes of the carrier wave. Connect the tops of the AM wave to see the original input signal.
In frequency modulation (FM), the frequency of the carrier is modulated in proportion to the amplitude of the signal. This is not as easy to see, but with some imagination it is still possible. Remember that higher frequencies have shorter wavelengths and the sine wave in the visual is more tightly packed. This represents the high amplitude of the input signal. During the portion of the input signal that has lower amplitude, the FM signal represents this as the more loosely packed (i.e., lower frequency) sine wave.
Quadrature Amplitude Modulation
In quadrature amplitude modulation (QAM) a modulator creates 16 different signals by introducing both phase and amplitude modulation. By introducing 12 possible phase shifts and two possible amplitudes, we obtain the 16 different signal types required for 4 bits per baud signaling.
Another way of viewing QAM is to observe that it modulates the amplitudes of two waves in “quadrature” (i.e., 90° out of phase), using four different amplitudes for each of the two waves. If we label the four amplitudes as ±A1 through A4, the 16 different signal types are obtained by using all possible pairs as amplitudes (e.g., A1 sin (Ft) + A1 cos (Ft), A1 sin (Ft) + A2 cos (Ft)... will create the necessary signal types). We leave it as an exercise for the interested reader to find values for the amplitude parameters that create the illustrated “constellation.”
Although our example uses four different amplitudes of the sine and cosine components (a four-level system), other systems may use greater (or fewer) amplitude levels. The greater the number of levels, the greater the number of constellation points and correspondingly greater numbers of bits per signal. Like all multibit encoding systems, QAM is limited to the number of levels that can be discerned at the receiver in the face of noise.
