"The reason digital radio is so reliable is because it employs a smart receiver. Inside each digital radio receiver there is a tiny Computer: a computer capable of sorting through the myriad of reflected and atmospherically distorted transmissions and reconstructing a solid, usable signal for the set to process." from http://radioworks.cbc.ca/radio/digital-radio/drri.html (2/2/03)
Telecommunications technologies using electromagnetic transmission surround us: television images flicker, radios chatter, cell phones (and telephones) ring, allowing us to see and hear each other anywhere on the planet. Email and the Internet link us via our computers, and a large variety of common devices such as CDs, DVDs, and hard disks augment the traditional pencil and paper storage and transmittal of information. People have always wished to communicate over long distances: to speak with someone in another country, to watch a distant sporting event, to listen to music performed in another place or another time, to send and receive data remotely using a personal computer. In order to implement these desires, a signal (a sound wave, a signal from a TV camera, or a sequence of computer bits) needs to be encoded, stored, transmitted, received, and decoded. Why? Consider the problem of voice or music transmission. Sending sound directly is futile because sound waves dissipate very quickly in air. But if the sound is first transformed into electromagnetic waves, then they can be beamed over great distances very efficiently. Similarly, the TV signal and computer data can be transformed into electromagnetic waves.
2.1 ELECTROMAGNETIC TRANSMISSION OF ANALOG WAVEFORMS
There are some experimental (physical) facts that cause transmission systems to be constructed as they are. First, for efficient wireless broadcasting of electromagnetic energy, an antenna needs to be longer than about 1/10 of a wavelength of the frequency being transmitted. The antenna at the receiver should also be proportionally sized.
The wavelength A and the frequency / of a sinusoid are inversely proportional. For an electrical signal travelling at the speed of light c (= 3 x 10® meters/second), the relationship between wavelength and frequency is
For instance, if the frequency of an electromagnetic wave is / = 10 KHz, then the length of each wave is
Efficient transmission requires an antenna longer than 0.1 A, which is 3 km! Sinusoids in the speech band would require even larger antennas. Fortunately, there is an alternative to building mammoth antennas. The frequencies in the signal can be translated (shifted, up-converted, or modulated) to a much higher frequency called the carrier frequency, where the antenna requirements are easier to meet. For instance,
• AM Radio: / « 600 - 1500 KHz A « 500 m - 200 m=> 0.1 A > 20 m
• VHF (TV): / « 30 - 300 MHz => A « 10 m - 1 m => 0.1 A > 0.1 m
• UHF (TV): / « 0.3 - 3 GHz => A « 1 m -0.1 m=> 0.1 A > 0.01 m
• Cell phones (US): / « 824 - 894 MHz => A « 0.36 - 0.33 m 0.1 A > 0.03 m
• PCS: / « 1.8 - 1.9 GHz => A « 0.167 - 0.158 m => 0.1 A > 0.015 m GSM (Europe): / « 890 - 960 MHz => A « 0.337 - 0.313 m=>0.1A> 0.03
• LEO satellites: / « 1.6 GHz => A « 0.188 m=>0.1A> 0.0188 m
A second experimental fact is that electromagnetic waves in the atmosphere exhibit different behaviors depending on the frequency of the waves:
• Below 2 MHz, electromagnetic waves follow the contour of the earth. This is why short wave (and other) radio can sometimes be heard hundreds or thousands of miles from their source.
• Between 2 and 30 MHz, sky-wave propagation occurs with multiple bounces from refractive atmospheric layer.
• Above 30 MHz, line-of-sight propagation occurs with straight line travel between two terrestrial towers or through the atmosphere to satellites.
• Above 30 MHz, atmospheric scattering also occurs, which can be exploited for long distance terrestrial communication.
Manmade media in wired systems also exhibit frequency dependent behavior. In the phone system, due to its original goal of carrying voice signals, severe attenuation occurs above 4 KHz.
The notion of frequency is central to the process of long distance communications. Because of its role as a carrier (the AM/UHF/VHF/PCS bands mentioned above) and its role in specifying the bandwidth (the range of frequencies occupied by a given signal), it is important to have tools with which to easily measure the frequency content in a signal. The tool of choice for this job is the Fourier transform (and its discrete counterparts, the DFT and the FFT1). Fourier transforms are useful in assessing energy or power at particular frequencies. The Fourier transform of a signal w(t) is defined as
- oo where j = \J — 1 and / is given in Hz (i.e. cycles/sec or 1/sec).
Speaking mathematically, W(f) is a function of the frequency /. Thus for each /, W(f) is a complex number and so can be plotted in several ways. For instance, it is possible to plot the real part of W(f) as a function of / and to plot the imaginary part of W(f) as a function of /. Alternatively, it is possible to plot the real part of W(f) versus the imaginary part of W(f). The most common plots of the Fourier transform of a signal are done in two parts: the first graph shows the magnitude |VF(/)| versus / (this is called the magnitude spectrum) and second graph shows the phase angle of W(f) versus / (this is called the phase spectrum). Often, just the magnitude is plotted, though this inevitably leaves out information. The relationship between the Fourier transform and the DFT is discussed in considerable detail in Appendix D, and a table of useful properties appears in Appendix A.
If, at any particular frequency /o, the magnitude spectrum is strictly positive (|VF(/o)| > 0), then the frequency /o is said to be present in w(t). The set of all frequencies that are present in the signal is the frequency content, and if the frequency content consists of all frequencies below some given p, then the signal is said to be bandlimited to p. Some bandlimited signals are:
• Telephone quality speech: maximum frequency ~ 4 KHz
• Audible music: maximum frequency ~ 20 KHz
But real world signals are never completely bandlimited, and there is almost always some energy at every frequency. Several alternative definitions of bandwidth are in common use, which try to capture the idea that "most of" the energy is contained in a specified frequency region. Usually, these are applied across positive frequencies, with the presumption that the underlying signals are real valued (and hence have symmetric spectra).
1. Absolute bandwidth is _/*2 — /i where the spectrum is zero outside the interval /i < / < ¡2 along the positive frequency axis.
2. 3-dB (or half-power) bandwidth is _/*2 — /i where, for frequencies outside /i < f < f2, \H(f) \ is never greater than l/i/2 times its maximum value.
1 These are the discrete Fourier transform, which is a computer implementation of the Fourier transform, and the fast Fourier transform which is a slick, computationally efficient method of calculating the DFT.
3. Null-to-null (or zero-crossing) bandwidth is fo — fi where fo is first null in \H(f) \ above /o and, for bandpass systems, /i is the first null in the envelope below /o where /o is frequency of maximum |_ff(/)|. For baseband systems, /i is usually zero.
4. Power bandwidth is fo — fi where fi<f<fi defines the frequency band in which 99% of the total power resides. Occupied bandwidth is such that 0.5% of power is above fo and 0.5% below /i.
These definitions are illustrated in Figure 2.1.
FIGURE 2.1: Various ways to define bandwidth.
Bandwidth refers to the frequency content of a signal. Since the frequency response of a linear filter is the transform of the impulse response, it can also be used to talk about the bandwidth of a linear system or filter.
Suppose that the signal w(t) contains important information that must be transmitted. There are many kinds of operations that can be applied to w(t). Linear operations are those for which superposition applies, but linear operations cannot augment the frequency content of a signal - no sine wave can appear at the output of a linear operation if it was not already present in the input.
Thus the process of modulation (or upconversion), which requires a change of frequencies, must be a nonlinear operation. One useful nonlinearity is multiplication; consider the product of the message waveform w(t) with a cosine wave s(t) = w(t) cos(27r/0i) (2.2)
where fo is called the carrier frequency. The Fourier transform can now be used to show that this multiplication shifts all frequencies present in the message by exactly fo Hz.
Using one of Euler's identities (A.2)
the spectrum (or frequency content) of the signal s(t) can be calculated using the definition of the Fourier transform given in (2.1). In complete detail, this is
I (ej27r/ot e-j2nf0t^
ej2nJ0t e-j2nf0t\
e-'Wdt r w(t) (e-rMf-fo)t + e-rMf+f0)t\ dt J — oo
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