# The energy spectrum of each short segment was

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Unformatted text preview: .9: Top: A message signal pro duced using a random data sequence and raisedcosine pulses with β = 0.5. Bottom: Normalized average energy spectrum of m(t) computed numerically using the following pro cedure: a message signal of length 32768T was generated using a random sequence of bits. The long message signal was divided into short segments of length approximately 100T . The energy spectrum of each short segment was computed, and all spectra were averaged. The average spectrum was normalized so that the area under the spectrum is 1.0. Note that the shape of the energy spectrum exhibits the raised-cosine shape of the pulses. Since β = 0.5, the bandwidth of the spectrum is W T = 1 (1 + 0.5) = 0.75. 2 Digital Television (DTV) message signal: ￿ m(t) = an p(t − nT ) n an ∈ (−3.5, −2.5, −1.5, −0.5, 0.5, 1.5, 2.5, 3.5) 8 possible pulse amplitudes: log2 (8) = 3 bits/pulse Signaling rate T −1 = 10.762238 Mpulses/sec, or 3 bits/pulse ×10.762238 ￿ 32.387 Mbits/sec. About 40 percent of the bits are used to provide forward error correction, so that actual data rate is approximately 0.60×32.387 = 19.4 Mbits/sec. Pulses, p(t), are square-root raised-cosine with β = 0.0575 (5.75% excess bandwidth). Pulse bandwidth (and message signal bandwidth) is W = 21 (1 + β ) = 5.69 MHz. T 1.5. LINEAR MODULATION 15 2.0 1.0 m(t) 0.0 −1.0 −2.0 0 3.0 2.0 1.0 s(t) 0.0 −1.0 −2.0 −3.0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 t/ T 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 t/ T DSB 0 1 Figure 1.9: Top: message signal representing 20 information bits generated using raised-cosine pulses with β = 0.1. Bottom: DSB signal, s(t), normalized such that < s2 (t) >= 1. The DSB/SC signal, s(t), shown in the lower panel has been normalized to have a mean-square value of 1.0. Thus, if this signal represents the voltage developed lope ( maxm(t)(t)|) + 1) > 0 for all t. The signal s (t) was then normalized to have (| m a mean-square of 1.0. The normalized signal, s(t), is shown in the bottom plot of Figure 1.17. 2.0 1.0 m(t) 0.0 −1.0 −2.0 0 3.0 2.0 1.0 s(t) 0.0 −1.0 −2.0 −3.0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 t/ T 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 t/ T DSB, full carrier 0 1 2 3 4 5 Figure 1.17: Top: message signal representing 20 information bits generated using raised-cosine pulses with β = 0.1. Bottom: DSB with full-carrier signal, s(t), normalized such that < s2 (t) >= 1. Compare the bottom plot with the corresponding plot in Figure 1.9. Figure 1.18 illustrates how the envelope detector demo dulates a full-carrier DSB signal. The upper plot shows the result of passing s(t) from Figure 1.17 through a full-wave rectiﬁed s(t) 2.0 1.0 0.0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 t/ T 8 9 10 11 12 13 14 15 16 17 18 19 20 t/ T after low-pass ﬁlter 2.0 1.0 0.0 0 1 2 3 4 5 6 7 Figure 1.18: Upper plot shows |s(t)|, which is the result of passing s(t) through a full-wave rectiﬁer. The lower plot was obtained by passing the signal shown in the upper plot through a low-pass ﬁlter with cutoﬀ frequency larger than the bandwidth of m(t) (W) and smaller than fc − W . The transient at the beginning of the lower plot is the start-up transient of the digital lowpass ﬁlter used to pro duce the simulated signal. 2.0 1.0 m(t) 0.0 −1.0 −2.0 2.0 1.0 m(t) 0.0 ˆ −1.0 −2.0 2.0 1.0 s(t) 0.0 −1.0 −2.0 2.0 1.0 s(t) 0.0 −1.0 −2.0 2.0 1.0 s(t) 0.0 −1.0 −2.0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 t/T 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 t/T 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 t/T 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 t/T 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 t/T DSB-sc 0 1 USB 0 1 LSB 0 1 2.0 1.0 m(t) 0.0 −1.0 −2.0 3.0 2.0 1.0 s(t) 0.0 −1.0 −2.0 −3.0 3.0 2.0 1.0 s(t) 0.0 −1.0 −2.0 −3.0 3.0 2.0 1.0 s(t) 0.0 −1.0 −2.0 −3.0 3.0 2.0 1.0 s(t) 0.0 −1.0 −2.0 −3.0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 t/T DSB-full carrier 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 t/T 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 t/T 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 t/T 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 t/T DSB-sc 0 1 USB 0 1 LSB 0 1 2.0 1.0 m(t) 0.0 −1.0 −2.0 3.0 2.0 1.0 s (t) 0.0 −1.0 −2.0 −3.0 3.0 2.0 1.0 s (t) 0.0 −1.0 −2.0 −3.0 3.0 2.0 1.0 s (t) 0.0 −1.0 −2.0 −3.0 3.0 2.0 1.0 s (t) 0.0 −1.0 −2.0 −3.0 0 1 2 3 4 5 6 7 8 DSB-full carrier: s(t) cos(ωc t) 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 t/T 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 t/T 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 t/T 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 t/T DSB-sc: s(t) cos(ωc t) 0 1 2 3 4 5 USB: s(t) cos(ωc t) 0 1 2 3 4 5 LSB: s(t) cos(ωc t) 0 1 2 3 4 5 9 10 11 12 13 14 15 16 17 18 19 20 t/T 2.0 1.0 m(t) 0.0 −1.0 −2.0 2.0 1.0 s (t) 0.0 −1.0 −2.0 2.0 1.0 s (t) 0.0 −1.0 −2.0 2.0 1.0 s (t) 0.0 −1.0 −2.0 2.0 1.0 s (t) 0.0 −1.0 −2.0 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 t/T DSB-full carrier: LPF[s(t) cos(ωc t)] 1 2 3 4 5 6 7 8 DSB-sc: LPF[s(t...
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