28 0y the dashed line the value of j0 1 is the

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Unformatted text preview: ) is 1 ra frequency deviation is 2 Hz. According to Equation 1.52 this sign 0.8 ∞ s s(t) = Ac 0.n=−∞ Jn (1) cos(2π (20 + 2n)t). The spectrum of th 6 ermined by inspection of Equation 1.53. Each cosinusoidal comp 0.4 e a pair of delta functions located at f = ±(20 + 2n) with amplit 0.2 lue of Jn (1). The resulting line spectrum is shown in Figure 1 10. 20. 30. 40. - .2 his example0the amplitudes of the sidebands are smaller than t igure 1.29 -0.ould be compared with Figure 1.28. The amplitud sh 4 0. n e spectrum-ca6 be determined by inspecting the values of Jn (β ) -8 Figure 1.28 0.y the dashed line. The value of J0 (1) is the amplitu b F gtr 1 29: L u s o m f a g - io du a t s g al wi h 20 u c r o a d 2 H uh mponent;iodueaei.nvalineepectfruJ1o(1n)legmvelstedhienamtplitHz daerier f nthe zfirst sideb m l t g tone. k phase deviation of an angle-modulated signal happens to corr r B f he ( f o d l t mo dul t an le th u t d s r al p l o have a a r er one of thteum oetsseslinusuinacly ionsa,ted)hegn-modelaceorigneswilonntding ccroimponent w BW 2(∆fmax + W ). When the mo dulating signal has finite bandwidth W can usually be taken to be β = 1.0 1.0 0.8 |S (f )| 0.6 0.4 0.2 0.0 38 −15 −10 −5 0 5 10 15 f −fc CHAPTER 1. COMMUNICATION SIGNALS AND SYSTEMS fm Figure 1.30: Spectrum of a carrier angle-mo dulated by a single tone with mo dulation β = 2.4 index β = 1. Note that the frequency axis is the normalized frequency difference −f from the carrier fr1q0 ency, i.e. ffm c . e. u 0.8 the highest frequency that is contained in the signal. If the mo dulating signal has infinite ba(f )| idth0.6 en W is chosen so that it contains most of the power in the th |S ndw signal. 0.4 Notice that when the peak frequency deviation is much smaller than the highest 0.2 frequency in m(t), i.e., if ∆fmax W , then Carson’s Rule reduces to 0.0 −15 −10 B W 5 2W 0 − 5 10 15 (1.56) f −fc mo dulfmion at When this approximation is valid, the is called “narrowband” angle mo dulation, and the bandwidth is essentially the same as for AM or DSB. On the Figure 1.31: Spectrum of a carrier angle-mo dulated by a single tone with mo dulation index β = 2.4. β = 7.0 Figure 1.31: Spectrum of a carrier angle-mo dulated by a single tone with mo dulation β = 7.0 index β = 2.4. 1.0 0.8 |S (f )| |S (f )| β = 7.0 10 0..6 08 0..4 06 0..2 04 0..0 −15 0.2 −10 −5 0 5 10 15 f −fc fm 0.0 −15 −10 −5 0 5 10 15 Figure 1.32: Spectrum of a carrier angle-mo dulaf −dcby a single tone with mo dulation te f fm index β = 7. Figure 1.32: Spectrum of a carrier angle-mo dulated by a single tone with mo dulation β = 20.0 index β = 7. 1.0 0.8 |S (f )| |S (f )| β = 20.0 10 0..6 08 0..4 06 0..2 04 0..0 −25 −20 −15 −10 −5 0.2 0.0 0 5 10 15 20 25 f −fc fm −25 −20 −15 −10 −5 0 5 10 15 20 25 Figure 1.33: Spectrum of a carrier angle-mo dulaf −dcby a single tone with mo dulation te f fm index β = 20. Figure 1.33: Spectrum of a carrier angle-mo dulated by a single tone with mo dulation 64 CHAPTER 2. RECEIVERS sufficient to demo dulate the signal. The blo ck diagram of such a tuned-detector receiver is shown in Figure 2.1. The tuned-detector is a passive circuit, and the audio power available si (t) = A(1 + m(t)) cos(ωct + θ)+noise Demodulator so (t) = A(1 + m(t))+noise Figure 2.1: Top: blo ck diagram of a tuned-detector receiver. Bottom: typical implementation. This type of receiver is also known as a “crystal set”, because early implementations of the dio de junction were implemented with the point-contact junction between a wire and crystal such as galena (lead sulfide) or carborundum (silicon carbide). 66 CHAPTER 2. RECEIVERS F (ω ) so (ω ) G si (ω ) Demodulator fc s o (ω ) s i (ω ) = GF (ω ) 1−AGF (ω ) A Figure 2.3: A regenerative receiver employing a regenerative amplifier in front of a demo dulator. frequency is GF (ωo )/(1 − AGF (ωo )), which can be made arbitrarily large if AGF (ωo ) is allowed to approach 1 (from below!). The positive feedback that is built into the regenerative amplifier is precisely what is necessary to build an oscillator. Note that if the quantity AGF (ωo ) is allowed to become equal to one, the transfer function becomes infinite, which by a comfortable margin. Leakage from the output back to the input provides a feedback path, which can lead to oscillation. In early receivers, the internal feedback within the trio de vacuum tubes used as amplifiers provided enough reverse coupling to cause oscillation unless the forward gain was kept small. A technique called “neutralization” was developed to cancel the internal feedback of the active devices, and the TRF receiver employing neutralization became known as the neutro dyne receiver. The trio de-based neutro dyne was followed by a second generation TRF receiver based on the tetro de vacuum tube, which had almost no internal feedback and did not require neutralization. The tetro de-based TRF receiver was available through the early 1930s. Demodulator fc fc fc Figure 2.2: Tuned-radio-frequen...
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This test prep was uploaded on 03/13/2014 for the course ECE 453 taught by Professor Staff during the Spring '08 term at University of Illinois, Urbana Champaign.

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