0.1 A telecommunication system  (Page 2/15)

Efficient transmission requires an antenna longer than $0.1\lambda$ , 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,upconverted, or modulated) to a much higher frequency called the carrier frequency , where the antenna requirements are easier to meet. For instance,

• AM Radio: $f\approx 600-1500$ KHz $\Rightarrow$ $\lambda$ $\approx$ 500 m $-$ 200 m $\Rightarrow$ 0.1 $\lambda$ $>$ 20 m
• VHF (TV): $f$ $\approx$ 30 $-$ 300 MHz $\Rightarrow$ $\lambda$ $\approx$ 10 m $-$ 1 m $\Rightarrow$ 0.1 $\lambda$ $>$ 0.1 m
• UHF (TV): $f$ $\approx$ 0.3 $-$ 3 GHz $\Rightarrow$ $\lambda$ $\approx$ 1 m $-$ 0.1 m $\Rightarrow$ 0.1 $\lambda$ $>$ 0.01 m
• Cell phones (U.S.): $f$ $\approx$ 824 $-$ 894 MHz $\Rightarrow$ $\lambda$ $\approx$ 0.36 $-$ 0.33 m $\Rightarrow$ 0.1 $\lambda$ $>$ 0.03 m
• PCS: $f$ $\approx$ 1.8 $-$ 1.9 GHz $\Rightarrow$ $\lambda$ $\approx$ 0.167 $-$ 0.158 m $\Rightarrow$ 0.1 $\lambda$ $>$ 0.015 m
• GSM (Europe): $f$ $\approx$ 890 $-$ 960 MHz $\Rightarrow$ $\lambda$ $\approx$ 0.337 $-$ 0.313 m $\Rightarrow$ 0.1 $\lambda$ $>$ 0.03 m
• LEO satellites: $f$ $\approx$ 1.6 GHz $\Rightarrow$ $\lambda$ $\approx$ 0.188 m $\Rightarrow$ 0.1 $\lambda$ $>$ 0.0188 m

Recall that KHz $=$ ${10}^3$ Hz; MHz $=$ ${10}^6$ Hz; GHz $=$ ${10}^9$ Hz.

A second experimental fact is that electromagnetic waves in the atmosphere exhibitdifferent behaviors, depending on the frequency of the waves:

• Below 2 MHz, electromagnetic waves follow the contour of the earth. This is why shortwave (and other) radios can sometimes be heardhundreds or thousands of miles from their source.
• Between 2 and 30 MHz, sky-wave propagation occurs with multiple bounces from refractive atmospheric layers.
• 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.

Humanmade media in wired systems also exhibit frequency dependent behavior. In the phone system, due to its original goal ofcarrying voice signals, severe attenuation occurs above 4 KHz.

The notion of frequency is central to the process of long distance communications. Because of its roleas a carrier (the AM/UHF/VHF/PCS bands mentioned above) and its role in specifying the bandwidth (the range of frequencies occupiedby a given signal), it is important to have tools with which to easily measure the frequency content in a signal. The tool of choicefor this job is the Fourier transform (and its discrete counterparts, the DFT and the FFT) These are the discrete Fourier transform, which is a computer implementation of the Fourier transform, and thefast Fourier transform, which is a slick, computationally efficient method of calculating the DFT. .Fourier transforms are useful in assessingenergy or power at particular frequencies. The Fourier transform of a signal $w\left(t\right)$ is defined as

where $j=\sqrt{-1}$ and $f$ is given in Hz (i.e., cycles/sec or 1/sec).

Speaking mathematically, $W\left(f\right)$ is a function of the frequency $f$ . Thus for each $f$ , $W\left(f\right)$ is a complex number and so can be plotted in several ways. For instance, it is possible to plot thereal part of $W\left(f\right)$ as a function of $f$ and to plot the imaginary part of $W\left(f\right)$ as a function of $f$ . Alternatively, it is possible to plot the real part of $W\left(f\right)$ versus the imaginary part of $W\left(f\right)$ . The most common plots of the Fourier transform of a signal are donein two parts: the first graph shows the magnitude $\left|W\right(f\left)\right|$ versus $f$ (this is called the magnitude spectrum) and second graph shows the phase angle of $W\left(f\right)$ versus $f$ (this is called the phase spectrum).Often, just the magnitude is plotted, though this inevitably leaves out information.