# Bandwidth

It is possible to Fourier transform in time, as well as in space. Thus, a general temporal waveform can be written as a superposition of sinusoidal waveforms of various different angular frequencies, . In other words,

 (8.58)

where and are the temporal cosine and sine Fourier transforms of the waveform, respectively. By analogy with Equations (8.10)–(8.12), we can invert the previous expression to give

 (8.59) (8.60)

These equations make it manifest that , and . Moreover, it is apparent that if is an even function of then , but if it is an odd function then .

The current flowing in the antenna of an amplitude-modulated (AM) radio transmitter is driven by a voltage signal that oscillates sinusoidally at a frequency, , which is known as the carrier frequency. In commercial (medium wave) AM radio, each station is assigned a single carrier frequency that lies somewhere between about 500 kHz and 1600 kHz. However, the voltage signal fed to the antenna does not have a constant amplitude. Rather, it has a modulated amplitude that can be expressed, somewhat schematically, as a Fourier series:

 (8.61)

where represents the information being transmitted. Typically, this information is speech or music that is picked up by a microphone, and converted into an electrical signal. The constant amplitude is present even when the transmitter is sending no information. The remaining terms in the previous expression are due to the signal picked up by the microphone. The modulation frequencies, , are thus the frequencies of audible sound waves. In other words, they are so-called audio frequencies lying between about 20 Hz and 20 kHz. This implies that the modulation frequencies are much smaller than the carrier frequency; that is, for all . Furthermore, the modulation amplitudes are all generally smaller than the carrier amplitude .

The signal transmitted by an AM station, and received by an AM receiver, is an amplitude-modulated sinusoidal oscillation of the form

 (8.62)

which, with the help of some standard trigonometric identities (see Appendix B), can also be written

 (8.63)

We can calculate the cosine and sine Fourier transforms of the signal,

 (8.64) (8.65)

by making use of Equation (8.30)–(8.32). It follows that

 (8.66) (8.67)

Here, we have only shown the positive frequency components of and , because we know that and .