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Analog Modulation Part II

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Analog Modulation Part II PiPrinc ip les of CiiCommunications Chapter 3: Analog Modulation (continued) Textbook: Ch 3, Ch 4.1-4.4 1 33A3.3 Ang le Mo du la tion Angle modulation is either phase or frequency of the carrier is varied according to the message signal The general form of an angle modulated wave is s()tA=+cc cos2[ πθ ftt ()] where fc = carrier freq, θ(t) is the time-varying phase and varied by the message m(t) The instantaneous frequency of s(t) is 1()dtθ ft()=+ f ic2π dt 2009/2010 Meixia Tao @ SJTU 14 RttifFMdPMilRepresentation of FM and PM signals For phase modulation (PM), we have θ ()tkmt= p () where kp = phase deviation constant For frequency modulation (FM), we have 1 d where k = frequency ft()−= f kmt () = θ () t f icf 2π dt deviation constant The phase of FM is t θ ()tkmd= 2πττ ( ) f ∫0 2009/2010 Meixia Tao @ SJTU 15 Distinguishing Features of PM and FM No perfect regularity in spacing of zero crossing Zero crossings refer to the time instants at which a waveform changes between negative and positive values Constant envelop, i.e. amplitude of s(t) is constant RltiRelations hibthip between PM an dFMd FM ∫ m(t)dt FM wave m(t) Phase integrator modulator A cos 2πf t + k m(t)dt C [ c p ∫ ] AC cos(2πf ct) Discuss the d m(t) properties of FM only m(t) dt Frequency PM wave differentiator modulator AC cos[2πf ct + 2πk f m(t)] AC cos(2πf ct) 2009/2010 Meixia Tao @ SJTU 16 ElSiidlMdltiExample: Sinusoidal Modulation Sinusoid modulating wave m(t) FM wave d m(t) dt PM wave 2009/2010 Meixia Tao @ SJTU 17 ElSMdltiExample: Square Modulation Square modulating wave m(t) FM wave PM wave 2009/2010 Meixia Tao @ SJTU 18 FM by a Sinusoidal Signal Consider a sinusoidal modulating wave m(t) = Am cos(2πfmt) Instantaneous frequency of resulting FM wave is fi (t) = fc + k f Am cos(2πfmt) = fc + Δf cos(2πfmt) where Δ f = k f A m is called the frequency deviation, proportional to the amplitude of m(t), and independent of fm. Hence, the carrier phase is t Δf θπτ()t =−= 2( ff ( )) d τ sin(2 πf t ) ∫0 ic m fm = βπsin(2ftm ) Where β = Δ f f m is called the modulation index 2009/2010 Meixia Tao @ SJTU 19 Example Problem: a siidldltiinusoidal modulating wave o flitd5Vdf amplitude 5V and frequency 1kHz is applied to a frequency modulator. The frequency sensitivity is 40Hz/V. The carrier frequency is 100kHz. Calculate (a) the fditid(b)thdltiidfrequency deviation, and (b) the modulation index Solution: Frequency deviation Δf = k f Am = 40×5 = 200Hz Δf 200 Modulation index β = = = 0.2 fm 1000 2009/2010 Meixia Tao @ SJTU 20 Spectrum Analysis of Sinusoidal FM Wave The FM wave for sinusoidal modulation is s(t) = Ac cos[2πfct + β sin(2πfmt)] = Ac cos[β sin(2πfmt)]cos(2πfct) − Ac sin[β sin(2πfmt)]sin(2πfct) In-phase component Quadrature-phase component sI (t) = Ac cos[β si(in(2πfmt)] sQ (t) = Ac sin[β sin(2πfmt)] Hence, the comppplex envelop of FM wave is ~ jβ sin(2πfmt) s (t) = sI (t) + jsQ (t) = Ace ~ s (t) retains complete information about s(t) j[2πfct+β sin(2πfmt)] ~ j2πfct s(t) = Re{Ace }= Re[s (t)e ] 2009/2010 Meixia Tao @ SJTU 21 ~ s (t) is periodic, can be expanded in Fourier series as ∞ ~ j2πnfmt s (t) = c e ~ jβ sin(2πfmt) ∑ n s (t) = Ace where n=−∞ 1/(2 fm ) ~ − j2πnfmt cn = fm s (t)e dt ∫−1/(2 f ) m 1/(2 fm ) j[β si(in(2πfmt)−2πnfmt] = fm Ac e dt ∫−1/(2 f ) m Let x = 2πfmt π Ac cn =−exp⎡ j (β sin xnxdx)⎤ 2π ∫−π ⎣ ⎦ n-th order Bessel function of the first kind Jn(β) is defined as 1 π Jjn ()ββ=− exp⎡ ( sinxnxdx)⎤ 2π ∫−π ⎣⎦ Hence, cn = Ac J n (β ) 2009/2010 Meixia Tao @ SJTU 22 ~s (t) Substituting cn into c(t) = Ac J n (β ) ∞ ~ s (t) = Ac ∑ J n (β )exp()j2πnfmt n=−∞∞ Hence, FM wave in time domain can be represented by ⎧⎫∞ s()tA=+cn Re⎨⎬∑ J (β )exp[ jf 2π ( cmnf )t] ⎩⎭n=−∞ ∞ =+AJcn∑ ()β cos[2( π fff cmnf )t] n=−∞ In frequency-domain, we have ∞ Ac S( f ) = ∑ J n (β )[δ ( f − fc − nfm ) +δ ( f + fc + nfm )] 2 c n=−∞ 2009/2010 Meixia Tao @ SJTU 23 Property 1:Narrowband FM (for small β≤0.3 ) – Approximations J (β ) ≈1 ? In what ways do a 0 conventinal AM wave and J1(β ) ≈ β 2 a narrow bdband FM wave J n (β ) ≈ 0, n >1 differ from each other – Substitutingg() above into s(t) βA s(t) ≈ A cos(2πf t) + c cos[]2π ( f + f )t c c 2 c m βA − c cos[]2π ( f − f )t 2 c m J n (β ) → 0 as β → ∞ ⎧ J n (β ), n even J −n (β ) = ⎨ ⎩− J n (β ), n odd 10/4/2004 2009/2010 Meixia Tao @ SJTU 24 Property 2: Wideband FM (for large β>1 ) In theory, s(t) contains a carrier and an infinite number of side- frequency components, with no approximations made Property 3: Constant average power The envelop of FM wave is constant, so the average power is also constant, 2 P = Ac / 2 ∞ The averaggpe power is also g iven b y s(t) = Ac ∑ J n (β )cos[2π ( fc + nfm )] n=−∞ 2 2 Ac 2 Ac P = ∑ J n (β ) = 2 n 2 ∞ 2 ∑ J n (β ) =1 n=−∞ 2009/2010 Meixia Tao @ SJTU 25 Example Goal:toinvestigatehowtheamplitude: to investigate how the amplitude Am, and frequency fm, of a sinusoidal modulating wave affect the spectrum of FM wave Fixed fm and varying Am ⇒ frequency deviation Δf = kfAm and modulation index β =Δf/fm are varied 1.0 1.0 β = 5 β =1 fc fc 2Δf 2Δf Increasing Am increases the number of harmonics in the bandwidth 2009/2010 Meixia Tao @ SJTU 26 Fixed Am and varying fm ⇒ Δf is fixed, but β is varied 1.0 1.0 β = 5 β =1 fc fc 2Δf 2Δf Increasing fm decreases the number of harmonics but attht the same time increases the spac ing be tween the harmonics. 2009/2010 Meixia Tao @ SJTU 27 Effective Bandwidth of FW Waves Theoretically, FM bandwidth = infinite In practice, for a single tone FM wave, when β is large, BisB is only slightly greater than the total frequency excursion 2Δf. when β is small, the spectrum is effectively limited to [fc - fm, fc + fm] C’RlCarson’s Rule approxitifilimation for single-tdltitone modulating wave of frequency fm Bff≈ 222(1)Δ+mm = +β f 2009/2010 Meixia Tao @ SJTU 28 99% bandwidth approximation The separation between the two frequencies beyond which none of the side-frequencies is greater than 1% of the unmodulated carrier amplitude i.eB ≈ 2nmax fm where nmax is the max n that satisfies J n (β ) > 0.01 β 0.1 0.3 0.5 1.0 2.0 5.0 10 20 30 2nmax 24 4 6 816285070 2009/2010 Meixia Tao @ SJTU 29 A universal curve for evaluating the 99% bandwidth As β increases, the bandwidth occupied by the significant side- frequencies drops toward that over which the carrier frequency actually deviates, i.e. B become less affected by β 20 2 0.2 2 2009/2010 Meixia Tao @ SJTU 30 FM by an Arbitrary Message Consider an arbitrary m(t) with highest freq component W Define deviation ratio D = Δf / W, where Δ= f kmtf max ( ) D ⇔ β and W ⇔ fm Carson’s rule applies as BfWWD≈Δ+2221 =( + ) Carson’s rule somewhat underestimate the FM bandwidth requirement, while universal curve yields a somewhat conservative result Assess FM bandwidth between the bounds given by Carson’s rule and the universal curve 2009/2010 Meixia Tao @ SJTU 31 ElExample IthAithiIn north America, the maximum va lfflue of frequency deviation Δf is fixed at 75kHz for commercial FM broadcasting by ratio. If we take the modulation frequency W = 15kHz, which is typically the maximum audio frequency of interest in FM transmission, the corresponding value of the deviation ratio is D = 75/15 = 5 Using Carson’ ssrule rule, the approximate value of the transmission bandwidth of the FM wave is B = 2 (()75+15) = 180kHz Using universal curve, B=32B = 3.2 Δf=32x75=240kHzf = 3.2 x 75 = 240kHz 2009/2010 Meixia Tao @ SJTU 32 EiExercise 4 Assuming that mt () = 10sinc10 ( t ) , determine the transmission bandwidth of an FM modulated signal with k f = 4000 2009/2010 Meixia Tao @ SJTU 33 GtifFMGeneration of FM waves Direct approach Design an oscillator whose frequency changes with the input voltage => voltage-controlled oscillator (VCO) Indirect approach First generate a narrowband FM signal first and then change it to a wideband single Due to the similarityyg, of conventional AM signals, the generation of a narrowband FM signal is straightforward. 2009/2010 Meixia Tao @ SJTU 34 Generation of Narrow-band FM Consider a narrow band FM wave s1(t) = A1 cos[2πf1t +φ1(t)] t where f1 = carrier frequency φ1(t) = 2πk1 m(τ )dτ ∫0 k1 = frequency sensitivity Giv en φ1(t) <<1 w ith β ≤ 0. 3, w e may u se ⎧cos[φ1(t)]≈1 ⎨ ⎩sin[φ1(t)]≈ φ1(t) Correspondingly, we may approximate s1(t) as s1(t) = A1 cos(2πf1t)− A1 sin(2πf1t)φ1(t) t = A1 cos()2πf1t − 2πk1 A1 sin ()2πf1t m(τ )dτ ∫0 Narrow-band FW wave 2009/2010 Meixia Tao @ SJTU 35 Narrow-band freqqyuency modulator m(t) Product - integrator Modulator + Narrow-band + FM wave s1(t) A1 sin(2πf1t) Carrier wave -900 phase shifter A1 cos(2πf1t) Next, pass s1(t) through a frequency multiplier, which consists of a non-linear device and a bandpass filter.
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