6.20 Entropy

212 CHAPTER 6. INFORMATION COMMUNICATION

From our work in Fourier series, we know that this signal’s spectrum contains odd-harmonics of the fundamental, which here equals 1 . Thus, strictly speaking, the signal’s bandwidth is infinite. In practical2T terms, we use the 90%-power bandwidth to assess the effective range of frequencies consumed by the signal. The first and third harmonics contain that fraction of the total power, meaning that the effective bandwidth of our baseband signal is 3 or, expressing this quantity in terms of the datarate, 3R . Thus, a digital2T 2 communications signal requires more bandwidth than the datarate: a 1 Mbps baseband system requires a bandwidth of at least 1.5 MHz. Listen carefully when someone describes the transmission bandwidth of digital communication systems: Did they say “megabits” or “megahertz?”

Exercise 6.15 (Solution on p. 256.)

Show that indeed the first and third harmonics contain 90% of the transmitted power. If the receiver uses a front-end filter of bandwidth 3 , what is the total harmonic distortion of the received signal?2T

Exercise 6.16

What is the 90% transmission bandwidth of the modulated signal set?

(Solution on p. 256.)

6.15 Frequency Shift Keying18

In frequency-shift keying(FSK), the bit affects the frequency of a carrier sinusoid.

s0 (t) = ApT (t) sin (2πf0t) s1 (t) = ApT (t) sin (2πf1t)

A s0(t)

A

tT s1(t)

T t

(6.41)

Figure 6.11

The frequencies f0, f1 are usually harmonically related to the bit interval. In the depicted example,

f0 = 3 and f1 = 4 . As can be seen from the transmitted signal for our example bit stream (Figure 6.12),

T T the transitions at bit interval boundaries are smoother than those of BPSK.

To determine the bandwidth required by this signal set, we again consider the alternating bit stream. Think of it as two signals added together: The first comprised of the signal s0 (t), the zero signal, s0 (t), x(t) “0” “1” “1” “0”

A

t

T 2T 3T 4T

Figure 6.12: This plot shows the FSK waveform for same bitstream used in the BPSK example (Figure 6.8).