# In general the number of information bits transmitted

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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 ). Pulse-shapes 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 half-cosine pulse 2 3 4 5 3 4 5 half-cosine 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 raised-cosine pulses (Figure 1.8(c-e)). The Fourier transform of the raised-cosine 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 raised-cosine whersit0o≤ rβg≤ n.. Thhetime-ensmailn spulse amaperoβ tcontross d-ce swidtpuofe tihe raised-cosine 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 time-domain pulse shape of the raised-cosine 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 time-domain pulse becomes more compact and extends over fewer signaling damped and the time-domain 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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