0195136047.pdf

728 COMMUNICATION SYSTEMS

When Equation (15.3.23) is satisfied, the bandwidth of the channel needed to supportsFSK(t)is

about 2ωbrad/s. The transmitted signal in Figure 15.3.19 can be seen to be two unipolar ASK

signals in parallel, one at carrier frequencyω 2 and the other atω 1. The receiver then becomes two

coherent ASK receivers in parallel (see Figure 15.3.14).

The noncoherent FSK version results when each product device and the following inte-

grator in the receiver are replaced by a corresponding matched filter followed by an envelope

detector.

Comparison of Digital Communication Systems

The noise performances of different digital communication systems are usually compared on the

basis of their bit-error probabilitiesPe. Keeping the ratioEb/N 0 as an independent parameter,

whereEbrepresents the average signal energy per bit andN 0 stands for the noise level, the

following expressions are given here with no derivation:

Pe=



































1

2

[

1 +

√

1

2 π(Eb/N 0 )

]

exp

(

−Eb

2 N 0

)

noncoherent ASK

1

2 exp

(

−Eb

2 N 0

)

noncoherent FSK

coherent ASK

1

2 erfc

[√

Eb

2 No

]

coherent FSK

}

unipolar PCM

1

2 exp

(

−Eb

No

)

DPSK

PSK

1

2 erfc

[√

Eb

No

]

polar PCM

}

Manchester PCM

(15.3.24)

where

erfc(x)= 1 −

2

√

π

∫x

0

e−ξ

2

dξ (15.3.25)

is known as thecomplementary error function,which is equal to 1 atx=0 and decreases to 0 as

x→∞; its behavior may be approximated by

erfc(x)∼=

e−x

2

x

√

π

whenx> 2 (15.3.26)

and

erfc(−x)= 2 −erfc(x) (15.3.27)

Figure 15.3.20 gives the plots of Equation (15.3.24). Of the PCM (noncarrier) systems, the

polar and Manchester-formatted systems are superior to the unipolar system, since no energy is

transmitted during half the bits (on average) in the unipolar system. Of the carrier-modulated

systems, coherent systems perform slightly better (by about 1 dB) than their corresponding

noncoherent versions. However, in practice, to avoid having to generate a local carrier in the

receiver, it may be worth the expense of 1 dB more in transmitted power. The PSK system can

be seen to be superior by about 3 dB for a givenPeto both FSK and ASK (which exhibit nearly

the same performance).