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Unformatted text preview: e on any of M discrete values, in which case we have an M − ary
data sequence and log2 M bits are transmitted each T seconds. In general, the number of
information bits transmitted per second will be the signaling rate (T −1 ) times the number
of bits represented by each pulse that is transmitted (log2 M ).
Pulseshapes that are commonly used to generate the message signal include the rectangular pulse (Figure 1.8a):
1 , −1 ≤
2 t
T < 1
2 F (m(t)) = M (f ) = M (f ) = ∞ m(t)e−j 2πf t dt −∞ ∞ −∞
[
an p(t − nT )]e−j 2πf t dt M (f ) = P (f ) n an e−j 2πf nT n < M (f ) >= P (f ) < 
2 2
n an e−j 2πf nT 2 > If the pulses, p(t), have bandwidth W then the message signal,
m(t), will also have bandwidth W . CHAPTER 1. COMMUNICATION SIGNALS AND SYSTEM
rectangular pulse rectangular pulse spectrum 1.0 1.0 p(t) P (f ) 0.5 0.0 −5 −4 −3 −2 −1 0 1
t/ T (a) 2 3 4 5 0.0
−5 −4 −3 −2 −1 0 1
fT halfcosine pulse 2 3 4 5 3 4 5 halfcosine pulse spectrum 1.0 1.0 p(t) P (f ) 0.5 0.0 (b) −5 −4 −3 −2 −1 0 1
t/ T 2 3 β = 0.1, raised cosine pulse 4 5 0.0
−5 −4 −3 −2 −1 0 1
fT 2 β = 0.1, raised cosine spectrum β = 0.1, raised cosine pulse β = 0.1, raised cosine spectrum
1.0 1.0
p(t) P (f )
0.5 0.0 −5 −4 −3 −2 −1 0 1
t/ T (c) 2 3 4 5 0.0
−2 β = 0.5, raised cosine pulse −1 0
fT 1 2 β = 0.5, raised cosine spectrum
1.0 1.0
p(t) P (f )
0.5 0.0 −5 −4 −3 −2 −1 0 1
t/ T (d) 2 3 4 5 0.0
−2 β = 0.9, raised cosine pulse −1 0
fT 1 2 β = 0.9, raised cosine spectrum
1.0 1.0
p(t) P (f )
0.5 0.0 (e) −5 −4 −3 −2 −1 0 1
t/ T 2 3 4 5 0.0
−2 −1 0
fT 1 2 Figure 1.8: Pulse shapes (and their spectra) suitable for use in digital communications sys P (f ) = 1
πT ( 2 + f ) + πT ( 1 − f )
2 o ccupies signiﬁcantly less bandwidth than the rectangular pulse.
A very useful family of pulses with ﬁnite bandwidth is obtained by deﬁning the pulse such
that its Fourier transform transitions from a constant (ﬂat) central region to zero through a
smo oth transition region having a raised cosine shape. These pulses are called raisedcosine
pulses (Figure 1.8(ce)). The Fourier transform of the raisedcosine pulse is 1
T
f T  ≤ 2 (1 − β ) T
π
1
1
1
P (f ) =
(1.6)
2 {1 + cos[ β (f T  − 2 (1 − β )]}
2 (1 − β ) < f T  ≤ 2 (1 + β ) , 1
0
f T  > 2 (1 + β ) 1.4. MESSAGE SIGNALS
1.4. MESSAGE SIGNALS 21
21 where 0 ≤ β ≤ 1. The dimensionless parameter β controls the width of the raisedcosine
whersit0o≤ rβg≤ n.. Thhetimeensmailn spulse amaperoβ tcontross dce swidtpuofe tihe raisedcosine
tran e i n e io 1 T e dim do ion e s par sh et e f he rai l e th o ine h ls s
transition region. The timedomain pulse shape of the raisedcosine pulse is
sin(π t/T ) cos(π β t/T )
p(t) = sin(π t/T ) cos(π β t/T ) 2 .
(1.7)
π t/T 1 − (2β t/T ) .
p(t) =
(1.7)
π t/T 1 − (2β t/T )2
The bandwidth of these pulses is W = 21 (1 + β ). The parameter β is called the fractional
The bandwidth of these pulses is W = 21T (1 + β ). The parameter β is called the fractional
T
excess bandwidth and it is often expressed as a percentage. The minimum possible bandwidth
excess bandwidth and it is often expressed a1 a percentage. The minimum possible bandwidth
s
results when β = 0 which results in W = 2T . In this case, the spectrum becomes rectangular
1
results when β = 0 which results in W = 2T . In this case, the spectrum becomes rectangular
and the pulse shape is a sinc function with relatively large sidelobes that extend over many
and the pulse shape is a sinc function with relatively large sidelobes that extend over many
signaling intervals on either side of the center of the pulse. For 0 < β ≤ 1, the amplitude
signaling intervals on either side of the center of the pulse. For 0 < β ≤ 1, the amplitude
spectrum exhibits a gradual transition to zero. As β increases, the sidelobes are increasingly
spectrum exhibits a gradual transition to zero. As β increases, the sidelobes are increasingly
damped and the timedomain pulse becomes more compact and extends over fewer signaling
damped and the timedomain pulse becomes more compact and extends over fewer signaling
intervals.
intervals.
It is important to point out that the spectrum of the chosen pulse will determine the
It is important to point out that the spectrum of the chosen pulse will determine the
shape of the spectrum of the message signal m(t) and hence the bandwidth o ccupied by the
shape of the spectrum of the message signal m(t) and hence the bandwidth o ccupied by the
message signal. Let us assume that the summation in equation 1.5 is ﬁnite and consists of ∞ Message signal m(t) = an p(t − nT )
n=−∞ Raised − cosine (spectrum) pulse β=1 t
sin( πt ) cos(πβ T )
T
p(t) =
πt
t
1 − 4β 2 ( T )2
T β = .5 β = 0.0 2.0 m(t) 1.0
0.0
−1.0
−2.0
0 40 80 120 160 200 t/ T
2 < M (f )
E 2 > 3.0
2.0
1.0
0.0
0.00 0.25 0.50
fT 0.75 1.00 Figure 1...
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