Bandwidth

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

$$F(t) = \int_{-\infty}^\infty C(\omega)\,\cos(\omega\,t)\,d\omega + \int_{-\infty}^{\infty} S(\omega)\,\sin(\omega\,t)\,d\omega,$$ (8.58)

where $C(\omega)$ and $S(\omega)$ 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

$$C(\omega) =\frac{1}{2\pi}\int_{-\infty}^\infty F(t)\,\cos(\omega\,t)\,dt,$$ (8.59)

$$S(\omega) =\frac{1}{2\pi}\int_{-\infty}^\infty F(t)\,\sin(\omega\,t)\,dt.$$ (8.60)

These equations make it manifest that $C(-\omega)=C(\omega)$, and $S(-\omega)=-S(\omega)$. Moreover, it is apparent that if $F(t)$ is an even function of $t$ then $S(\omega)=0$, but if it is an odd function then $C(\omega)=0$.

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, $\omega_0$, 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:

$$A(t) = A_0 + \sum_{n>0} A_n\,\cos(\omega_n\,t-\phi_n),$$ (8.61)

where $A(t)-A_0$ 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 $A_0$ 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, $\omega_n$, 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, $\omega_n\ll \omega_0$ for all $n>0$. Furthermore, the modulation amplitudes $A_n$ are all generally smaller than the carrier amplitude $A_0$.

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

$$\psi(t) = A(t)\,\cos(\omega_0\,t)$$

$$=A_0\,\cos(\omega_0\,t)+\sum_{n>0} A_n\,\cos(\omega_n\,t-\phi_n)\,\cos(\omega_0\,t),$$ (8.62)

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

$$\psi(t) =A_0\,\cos(\omega_0\,t)+\frac{1}{2}\sum_{n>0}A_n\,\cos[(\omega_0+... ..._n)\,t-\phi_n)]+ \frac{1}{2}\sum_{n>0}A_n\,\cos[(\omega_0-\omega_n)\,t+\phi_n)]$$

$$=A_0\,\cos(\omega_0\,t)+\frac{1}{2}\sum_{n>0}A_n\,\cos\phi_n\,\cos[(\omega_0+\omega_n)\,t]$$

$$~~~~+\frac{1}{2}\sum_{n>0}A_n\,\sin\phi_n\,\sin[(\omega_0+\omega_n)\,t]+\frac{1}{2}\sum_{n>0}A_n\,\cos\phi_n\,\cos[(\omega_0-\omega_n)\,t]$$

$$~~~-\frac{1}{2}\sum_{n>0}A_n\,\sin\phi_n\,\sin[(\omega_0-\omega_n)\,t].$$ (8.63)

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

$$C(\omega) = \frac{1}{2\pi}\int_{-\infty}^\infty \psi(t)\,\cos(\omega\,t)\,dt,$$ (8.64)

$$S(\omega) = \frac{1}{2\pi}\int_{-\infty}^\infty \psi(t)\,\sin(\omega\,t)\,dt,$$ (8.65)

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

$$C(\omega>0) = \frac{1}{2}\,A_0\,\delta(\omega-\omega_0) + \frac{1}{4}\sum_{n>... ...eft[\delta(\omega-\omega_0-\omega_n) + \delta(\omega-\omega_0+\omega_n)\right],$$ (8.66)

$$S(\omega>0) = \frac{1}{4}\sum_{n>0} A_n\,\sin\phi_n\left[\delta(\omega-\omega_0-\omega_n) - \delta(\omega-\omega_0+\omega_n)\right].$$ (8.67)

Here, we have only shown the positive frequency components of $C(\omega)$ and $S(\omega)$, because we know that $C(-\omega)=C(\omega)$ and $S(-\omega)=-S(\omega)$.

Figure 8.4: Frequency spectrum of an AM radio signal.

The AM frequency spectrum specified in Equations (8.66) and (8.67) is shown, somewhat schematically, in Figure 8.4. The spectrum consists of a series of delta function spikes. The largest spike corresponds to the carrier frequency, $\omega_0$. 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 $\omega_0+\omega_n$, whereas the lower sidebands correspond to the frequencies $\omega_0-\omega_n$. Thus, in order for an AM radio signal to carry all of the information present in audible sound, for which the appropriate modulation frequencies, $\omega_n$, range from about 0 Hz to about 20 kHz, the signal would have to consist of a superposition of sinusoidal oscillations with frequencies that range from the carrier frequency minus 20 kHz to the carrier frequency plus 20 kHz. In other words, the signal would have to occupy a range of frequencies from $\omega_0-\omega_N$ to $\omega_0+\omega_N$, where $\omega_N$ 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 40 kHz. In fact, commercial AM radio signals are only allowed to broadcast a bandwidth of 10 kHz, in order to maximize the number of available stations. (Two different stations obviously cannot broadcast in frequency ranges that overlap.) This means that commercial AM radio can only carry audible information in the range 0 to about 5 kHz. This is perfectly adequate for ordinary speech, but only barely adequate for music.