Bandwidth

(755) |

where and are the temporal cosine and sine Fourier transforms of the waveform, respectively. By analogy with Equations (707)-(709), we can invert the above expression to give

(756) | ||

(757) |

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 500kHz and 1600kHz. 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:

(758) |

where represents the information being transmitted. Typically, this information is speech or music which 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 above 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

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

(759) |

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

(760) |

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

(761) | ||

(762) |

by making use of Equation (727)-(729). It follows that

(763) | ||

(764) |

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

The AM frequency spectrum specified in Equations (763) and (764)
is shown, somewhat schematically, in Figure 53. The spectrum consists of
a series of delta function spikes. The largest spike corresponds to the carrier frequency,
. However, this spike carries no information. Indeed, the signal information
is carried in so-called *sideband frequencies* which are equally spaced on either side of the
carrier frequency. The upper sidebands correspond to the frequencies
, whereas the lower sidebands correspond to the
frequencies
. Thus, in order for an AM radio signal
to carry all of the information present in audible sound, for which the appropriate modulation frequencies,
, range
from about 0 Hz to about 20kHz, the signal would have to consist of a superposition
of sinusoidal oscillations with frequencies that range from the carrier frequency
minus 20kHz to the carrier frequency plus 20kHz. In other words, the signal would have to occupy a
range of frequencies from
to
,
where
is the largest modulation frequency. This is an important result.
An AM radio signal that only consists of a single frequency, such as the carrier frequency, transmits
no information. Only a signal that occupies a finite range of frequencies, centered
on the carrier frequency, is capable of transmitting useful information. The difference between the highest and the lowest frequency components of an AM radio signal, which
is twice the maximum modulation frequency,
is called the *bandwidth* of the signal. Thus, to transmit all of the information
present in audible sound an AM signal would need to have a bandwidth of 40kHz.
In fact, commercial AM radio signals are only allowed to broadcast a bandwidth of
10kHz, in order to maximize the number of available stations. (Two
different stations cannot broadcast in frequency ranges that overlap.) This
means that commercial AM radio can only carry audible information in the
range 0 to about 5kHz. This is perfectly adequate for ordinary speech,
but only barely adequate for music.

Let us now consider how we might transmit a digital signal over AM radio. Suppose that each data ``bit'' in the signal takes the form of a Gaussian envelope, of characteristic duration , superimposed on a carrier wave whose frequency is : that is,

Let us assume that . In other words, the period of the carrier wave is much less than the duration of the bit. Figure 54 illustrates a digital bit calculated for .

The sine Fourier transform of the signal (765) is zero by symmetry. However, its cosine Fourier transform takes the form

(766) |

A comparison with Equations (712)-(715) reveals that

(767) |

where

(768) |

In other words, the Fourier transform of the signal takes the form of a Gaussian in -space, which is centered on the carrier frequency, , and is of characteristic width . Thus, the bandwidth of the signal is of order . The shorter the signal duration, the higher the bandwidth. This is a general rule. A signal of full width at half maximum temporal duration generally has a Fourier transform of full width at half maximum bandwidth , so that

(769) |

This can also be written

(770) |

where is the bandwidth in hertz. The above result is known as the

An old-fashioned black and white TV screen consists of a rectangular grid
of black and white spots scanned by an electron beam. A given spot is ``white'' if the phosphorescent TV
screen was recently (i.e., within about
th of a second) struck by the
electron beam at that location. The spot separation is about
mm. A typical
screen is
, and thus has 500 lines with
500 spots per line, or
spots. Each spot is renewed every
th of
a second. (Every other horizontal line is skipped during a given traversal of the
electron beam over the screen. The skipped lines are renewed on the next
traversal. This technique is known as *interlacing*. Consequently, a given region of the screen, that includes many horizontal lines,
has a flicker rate of 60 Hz.) Thus, the rate at which the instructions ``turn on" and
``turn off'' must be sent to the electron beam is
or
times a second. The transmitted TV signal must therefore
have about
on-off instruction blips per second. If temporal overlap is to be avoided, each blip can be no
longer than
seconds in duration.
Thus, the required bandwidth is
. The carrier wave frequencies used for conventional broadcast TV lie in the
so-called VHF band, and range
from about 55 to 210 MHz. Our previous discussion of AM radio might lead us
to think that the 10MHz bandwidth represents the combined extents of an upper and a lower
sideband of modulation frequencies. In practice, the carrier wave and one
of the sidebands are suppressed. That is, they are filtered out, and never applied
to the antenna. However, they are regenerated in the receiver from the information
contained in the single sideband that is broadcast. This technique, which
is called *single sideband transmission*, halves the bandwidth
requirement to about
. (Incidentally, the lower sideband carries the same
information as the upper one, and thus can be used to completely regenerate the upper sideband, and vice versa.) Thus, between
and
there is room for about 30 TV channels, each using a 5MHz bandwidth. (Actually,
there are far fewer TV channels than this in the VHF band, because part of this
band is reserved for FM radio, air traffic control, air navigation beacons, marine communications, etc.)