0.1 A telecommunication system  (Page 4/15)

Thus, the process of modulation (or upconversion), which requires a change of frequencies, must be either nonlinear or time varying(or both). One useful way to modulate is with multiplication; consider the product of the message waveform $w\left(t\right)$ with a cosine wave

where $f_0$ is called the carrier frequency. The Fourier transform can now be used to show thatthis multiplication shifts all frequencies present in the message by exactly $f_0$ Hz.

Using one of Euler's identities [link] ,

one can calculate the spectrum (or frequency content) of the signal $s\left(t\right)$ from the definition of the Fourier transform given in [link] . In complete detail, this is

Thus, the spectrum of $s\left(t\right)$ consists of two copies of the spectrum of $w\left(t\right)$ , each shifted in frequency by $f_0$ (one up and one down) and each half as large.This is sometimes called the frequency shifting property of the Fourier transform, and sometimes called the modulation property. [link] shows how the spectra relate. If $w\left(t\right)$ has the magnitude spectrum shown in part (a) (this is shown bandlimited to $f^{†}$ and centered at zero Hz or baseband , though it could be elsewhere),then the magnitude spectrum of $s\left(t\right)$ appears as in part (b). This kind of modulation (or upconversion , or frequency shift), is ideal for translating speech, music, or other low frequency signalsinto much higher frequencies (for instance, $f_0$ might be in the AM or UHF bands)so that they can be transmitted efficiently. It can also be used to convert a high frequency signalback down to baseband when needed, as will be discussed in [link] and in detail in [link] .

Any sine wave is characterized by three parameters: the amplitude, frequency, and phase. Any of these characteristics can be used asthe basis of a modulation scheme: modulating the frequency is familiar from the FM radio, and phase modulation is commonin computer modems. A major example in this book is amplitudemodulation as in [link] , where the message $w\left(t\right)$ is multiplied by a high frequency sinusoid with fixed frequency and phase.Whatever the modulation scheme used, the idea is the same: a sinusoid is used to translate themessage into a form suitable for transmission.

Frequency division multiplexing

When a signal is modulated, the width (in Hertz) of the replicasis the same as the width (in Hertz) of the original signal. This is a direct consequence of [link] . For instance, if the message is bandlimited to $\pm f^{*}$ , and the carrier is $f_c$ , then the modulated signal has energy in the range from $-f^{*}-f_c$ to $+f^{*}-f_c$ and from $-f^{*}+f_c$ to $+f^{*}+f_c$ . If $f^{*}\ll f_c$ , then several messages can be transmitted simultaneously by using differentcarrier frequencies.