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MIT 6 003 - Modulation

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6.003: Signals and Systems Lecture 23 December 3, 200916.003: Signals and SystemsModulationDecember 3, 2009Communications SystemsToday we will look at applications of signals and systems in com-munication systems.Example: Transmit voice via telephone wires (copper)mic amp telephone wire amp speakerWorks well: basis of local land-based telephones.Wireless CommunicationIn cellular communication systems, signals are transmitted via elec-tromagnetic (E/M) waves.mic amp E/M wave amp speakerFor efficient transmission and reception, antenna length should be≈ one-quarter of the wavelength.Telephone-quality speech contains frequencies from 200 to 3000 Hz.How long should the antenna be?Check YourselfFor efficient transmission and reception, the antenna lengthshould be comparable to one-quarter of the wavelength.Telephone-quality speech contains frequencies between 200 Hzand 3000 Hz.How long should the antenna be?0. 4 m 1. 40 m 2. 400 m3. 4 km 4. 40 km 5. 400 kmCheck YourselfWhat frequency E/M wave is well matched to an antennawith a length of 4 cm (about 1.5 inches)?0. 200 kHz 1. 2 MHz 2. 20 MHz3. 200 MHz 4. 2 GHz 5. 20 GHzWireless CommunicationSpeech is not well matched to the wireless medium.Many applications require the use of signals that are not wellmatched to the required media.signal applicationsaudio telephone, radio, phonograph, CD, cell phone, MP3video television, cinema, HDTV, DVDinternet coax, twisted pair, cable TV, DSL, optical fiber, E/MWe can often modify the signals to obtain a better match.Today we will introduce simple matching strategies based onmodulation.6.003: Signals and Systems Lecture 23 December 3, 20092Check YourselfConstruct a signal Y that codes the audio frequency informationin X using frequency components near 2 GHz.ω|X(jω)|ωωc|Y (jω)|Determine an expression for Y in terms of X.1. y(t) = x(t) ejωct2. y(t) = x(t) ∗ ejωct3. y(t) = x(t) cos(ωct) 4. y(t) = x(t) ∗ cos(ωct)5. none of the aboveAmplitude ModulationMultiplying a signal by a sinusoidal carrier signal is called amplitudemodulation (AM). AM shifts the frequency components of X by ±ωc.×x(t) y(t)cos ωctω|X(jω)|ωωc−ωcωωc−ωc|Y (jω)|Amplitude ModulationMultiplying a signal by a sinusoidal carrier signal is called amplitudemodulation. The signal “modulates” the amplitude of the carrier.×x(t) y(t)cos ωcttx(t) cos ωcttx(t)tcos ωctAmplitude ModulationHow could you recover x(t) from y(t)?×x(t) y(t)cos ωctFrequency-Division MultiplexingMultiple transmitters can co-exist, as long as the frequencies thatthey transmit do not overlap.x1(t)x2(t)x3(t)z1(t)z2(t) z(t)y(t)z3(t)cosw1tcosw2tcoswctcosw3tLPFFrequency-Division MultiplexingMultiple transmitters simply sum (to first order).x1(t)x2(t)x3(t)z1(t)z2(t) z(t)y(t)z3(t)cosw1tcosw2tcoswctcosw3tLPF6.003: Signals and Systems Lecture 23 December 3, 20093Frequency-Division MultiplexingThe receiver can select the transmitter of interest by choosing thecorresponding demodulation frequency.Z(jw)X3(jw)wwwww1w2w3Broadcast Radio“Broadcast” radio was championed by David Sarnoff, who previouslyworked at Marconi Wireless Telegraphy Company (point-to-point).• envisioned “radio music boxes”• analogous to newspaper, but at speed of light• receiver must be cheap (as with newsprint)• transmitter can be expensive (as with printing press)Sarnoff (left) and Marconi (right)Check YourselfThe problem with making an inexpensive radio receiver is thatyou must know the carrier signal exactly!z(t)x(t) y(t)cos(wct) cos(wct+f)L P FWhat happens if there is a phase shift φ between the signalused to modulate and that used to demodulate?AM with CarrierOne way to synchronize the sender and receiver is to send the carrieralong with the message.× +Cx(t) z(t)cos ωctz(t) = x(t) cos ωct + C cos ωct = (x(t) + C) cos ωcttz(t)x(t) + CAdding carrier is equivalent to shifting the DC value of x(t).If we shift the DC value sufficiently, the message is easy to decode:it is just the envelope (minus the DC shift).Inexpensive Radio ReceiverIf the carrier frequency is much greater than the highest frequencyin the message, AM with carrier can be demodulated with a peakdetector.z(t)z(t)y(t)RCty(t)In AM radio, the highest frequency in the message is 5 kHz and thecarrier frequency is between 500 kHz and 1500 kHz.This circuit is simple and inexpensive.But there is a problem.Inexpensive Radio ReceiverAM with carrier requires more power to transmit the carrier than totransmit the message!x(t)xpxrmsxp> 35xrmstSpeech sounds have high crest factors (peak value divided by rmsvalue). The DC offset C must be larger than xpfor simple envelopedetection to work.The power needed to transmit the carrier can be 352≈ 1000× thatneeded to transmit the message.Okay for broadcast radio (WBZ: 50 kwatts).Not for point-to-point (cell phone batteries wouldn’t last long!).6.003: Signals and Systems Lecture 23 December 3, 20094Superheterodyne ReceiverEdwin Howard Armstrong invented the superheterodyne receiver,which made broadcast AM practical.Edwin Howard Armstrong also invented andpatented the “regenerative” (positive feed-back) circuit for amplifying radio signals(while he was a junior at Columbia Univer-sity). He also invented wide-band FM.Amplitude, Phase, and Frequency ModulationThere are many ways to recode a signal to a different frequencyband. Here are three.Amplitude Modulation (AM): y1(t) = x(t) cos ωctPhase Modulation (PM): y2(t) = cos(ωct + kx(t))Frequency Modulation (FM): y3(t) = cosωct + kRt−∞x(τ)dτFrequency ModulationIn FM, the instantaneous frequency of the carrier is modulated bythe signal.FM: y3(t) = cosωct + kZt−∞x(τ)dτ| {z }φ(t)ωi(t) = ωc+ddtφ(t) = ωc+ kx(t)Frequency ModulationCompare AM to FM for x(t) = cos ωmt.AM: y1(t) = (cos ωmt + 1.1) cos ωcttFM: y3(t) = cos(ωct + m sin ωmt)tAdvantages of FM:• constant power• no need to transmit carrier (unless DC important)• bandwidth?Frequency ModulationEarly investigators thought that narrowband FM could have arbitrar-ily narrow bandwidth, allowing more channels than AM. Wrong!y3(t) = cosωct + kZt−∞x(τ)dτ= cos ωct coskZt−∞x(τ)dτ− sin ωct sinkZt−∞x(τ)dτIf k → 0 thencoskZt−∞x(τ)dτ→ 1sinkZt−∞x(τ)dτ→ kZt−∞x(τ)dτy3(t) ≈ cos ωct − sin ωctkZt−∞x(τ)dτBandwidth of narrowband FM is the same as that of AM!(integration does not change bandwidth)Frequency ModulationWideband FM is useful


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MIT 6 003 - Modulation

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