ELE636 Communication Systems

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1 ELE636 Communication Systems Chapter 5 : Angle (Exponential) Modulation 1 Phase-locked Loop (PLL) The PLL can be used to track the phase and the frequency of the carrier component of an incoming signal. It can be used (i) synchronous demodulation of AM signals with suppressed carrier; (ii) demodulation of angle-modulated signals. A PLL has three basic component 1. A voltage-controlled oscillator (VCO); whose frequency can be controlled by an external voltage. If the input voltage is e o (t), the output is a sinusoid of frequency ω given by ω(t) =ω c + ce o (t) where c is a constant, ω c is called free-running frequency of the VCO. 2. A multiplier, serving as a phase detector (PD) or a phase comparator 3. A loop filter (LPF) Consider that the instantaneous frequency of the VCO is ω(t) =ω c + ce o (t) If the VCO output is B cos[ω c t + θ o (t)], then its instantaneous frequency is 2

2 ω(t) =ω c + θ o (t), therefore, θ o (t) =ce o (t) (1) where c and B are constants of the PLL. PLL is a typical feedback system. The signal fed back tends to follow the input signal. The difference signal (known as the error) will change the signal fed back until it is close to the input signal. How the PLL works: Let the input to the PLL be A sin(ω c t + θ i ), and let the VCO output be B cos(ω c t + θ o ). The multiplier output is x(t) =AB sin(ω c t + θ i ) cos(ω c t + θ o )= AB 2 [sin(θ i θ o )+sin(2ω c t + θ i + θ o )] The last term is suppressed by the loop filter. Hence, the input to the VCO is where θ e is the phase error (θ i θ o ). e o = AB 2 sin θ e θ e = θ i θ o When the loop is locked, then the frequencies of both the input and the output sinusoids are identical. In this steady state, θ i,θ o and θ e are constant. When the input sinusoid frequency suddenly changes from ω c to ω c + k. Then the 3 incoming signal is A cos[(ω c + k)t + θ i ]=A cos(ω c t + ˆθ i ) ˆθi = kt + θ i The increase in θ i leads e o increases, which, in turn, increases the frequency of the VCO output to match the increase in the input frequency. The VCO thus tracks the frequency and the phase of the incoming signal. Free-running frequency/centre frequency: This is the frequency at which the VCO operates when not locked to an input signal, pre-set by external component (R, L, ) to PLL. Capture range/pull-in/acquisition range: The maximum initial frequency difference between the input signal and the VCO output for which the PLL can acquire lock, denoted as f p. Lock range/tracking/hold-in range: The frequency range in the vicinity of f o over which the PLL, once locked to the input, will remain in lock, denoted as f L. 4

3 More analysis on PLL working and parameter measurement If v i (t) =, PD output =, which implies the control/error voltage to VCO=. VCO continues to operate at its free running frequency f c (also referred to as the centre frequency of the PLL). If v i (t), PD generates a non-zero output since PD acts as a multiplier. v i (t)v o (t) cos(ω c t + φ i )sin(ω o t + φ ) [sin((ω c ω o )t +(φ i φ ))] + [sin((ω c + ω o )t +(φ i + φ ))] If f o f c >ɛ(a threshold), then both components in the product v i (t)v o (t), i.e., the sum and difference terms are eliminated by the loop filter, the input to the VCO is still zero and the PLL does not acquire a lock. If f o f c <ɛ, the sum component will be eliminated by the loop filter. But the difference term will be within the passband of the loop filter and it is applied as a control signal to the VCO. If f c is sufficiently close to f o, the feedback action of the loop causes the VCO to lock to the instantaneous frequency of the input signal. Once in lock, the VCO frequency is identical to that of the input signal. The action of the loop is to cause φ i φ o to take on that value which is required to generate the VCO 5 control voltage necessary to change the frequency of the VCO from its free running frequency to that of the input signal frequency. This action allows the PLL to track any instantaneous frequency changes of the input once lock has been acquired. Parameter measurement: Ve f_in f c f L f c f p f c f c + f p f c + f L f_in from low to high f_in from high to low Figure 1: Diagram to illustrate the capture and lock for PLL. 6

4 Angle Modulation: Basic Concept In general, we define angle modulated waveform by the expression φ(t) =A c cos(θ(t)) where A c is constant, and θ(t) is modulated by m(t). Angle Modulation: either the phase or the frequency of the carrier wave is varied according to the message signal, while the amplitude of the carrier wave is maintained at a constant value. Observe that φ(t) represents a rotating phasor of amplitude (=length) A c and generalized angle θ(t) with instanteneous carrier frequency given by ω i (t) = d dt θ(t) or f i (t) = 1 2π ω i(t) = 1 d 2π dt θ(t) 7 Let us consider some specific cases: Unmodulated carrier θ(t) =ω c t + θ o ω i (t) =ω c,f i = f c where θ o is an arbitrary initial phase term. Hence, θ(t) represents a phasor rotating at a constant angular velocity. Phase Modulation (PM), θ(t) m(t), and assuming θ =without loss of generality, θ(t) =ω c t + k p m(t) ω i (t) =ω c + k p dm(t) dt φ PM (t) =A c cos(ω c t + k p m(t)) = ω c + k p m(t) Frequency modulation (FM): ω i is varied linearly with m(t) (ω i m(t)) θ(t) = ω i (λ)dλ = ω i (t) =ω c + k f m(t) φ FM (t) =A c cos [ω c + k f m(λ)]dλ = ω c t + k f [ ω c t + k f ] m(λ)dλ m(λ)dλ 8

5 m(t) We can consider FM as a special PM where θ m(λ)dλ or equivalently, PM as a special case of FM where ω i (t) m(t). m(t) m(α)dα Phase modulator s_fm(t) Frequency modulator m(t) d/dt m(t) Frequency modulator s_pm(t) Phase modulator Figure 2: Phase and frequency modulation are inseparable. Hence we need to study only one. Properties of the other can be easily deducted. Since FM is the more common one, we will study it first and more in detail. Example 1 Sketch FM and PM waves for the modulating signal m(t). The constant k f and k p are 2π 1 5 and 1π, respectively, and the carrier frequency f c is 1 MHz (Textbook Example5.1, p256). 9 x For FM: ω i = ω c + k f m(t) f i = f c + k f 2π m(t) = m(t) (f i ) min = [m(t)] min = 99.9MHz (f i ) max = [m(t)] max = 1.1MHz the instantaneous frequency increases linearly (decreases) from min to max (or max to min) in each half cycle. For PM: f i = f c + k f m(t) = m(t) 2π (f i ) min = [ m(t)] min = 99.9MHz (f i ) max = [ m(t)] max = 1.1MHz 1

6 Because m(t) switches back and forth from min to max, the carrier frequency switches back and forth from 99.9 to 1.1 MHz each half-cycle. 11 Tone Modulated FM In particular, we want to study the spectrum of an FM waveform. We will start with the simplest case where m(t) is a pure tone m(t) =A m cos(ω m t), then ω i (t) = ω c + k f m(t) =ω c + k f A m cos(ω m t) = ω c + ω cos(ω m t) where ω = k f A m ( f = ω/2π), represent the maximum frequency deviation/departure of ω i from ω c. Observe that which is independent of ω m, we have θ(t) = where β = ω ω m ω i (λ)dλ = ω A m = max[m(t)] = ω c t + β sin(ω m t) [ω c + ω cos(ω m λ)]dλ = ω c t + ω ω m sin(ω m t) = f f m, is called modulation index for FM signal. 12

7 Hence, the FM signal waveform can be written as φ FM (t) = A c cos(θ(t)) = A c cos(ω c t + β sin(ω m t)) = A c cos(ω c t) cos(β sin(ω m t)) A c sin(ω c t)sin(β sin(ω m t)) Case 1: narrow band FM: β is small cos[β sin(x)] 1 sin[β sin(x)] β sin(x) hence, φ FM (t) =A c cos(ω c t) A c β sin(ω c t)sin(ω m t) (2) compare with tone-modulated AM-LC waveform φ AM (t) =A c [1 + µ cos(ω m t)] cos(ω c t)=a c cos(ω c t)+a c µ cos(ω m t) cos(ω c t) Usually the value of β<.3 is taken to be sufficient for the above approximation to be valid. Such signals are referred to as NBFM. Although NBFM and AM have similarities, they are distinctively different methods of modulation. 13 Bandwidth of Angle-modulated Waves We now use power series to obtain signal bandwidth. The FM waveform can be expressed as ) φ FM (t) =A c cos (ω c t + k f m(λ)dλ For general case, let a(t) = m(λ)dλ and we have { } φ FM (t) = A c cos(ω c t + k f a(t)) = Re A c e j[ω ct+k f a(t)] = Re {A } c e j(ωct) e jk f a(t) { [ ]} = Re = A c [ A c e j(ω ct) 1+jk f a(t) k2 f 2! a2 (t)+ cos ω c t k f a(t)sinω c t k2 f 2! a2 (t) cos ω c t + k3 f 3! a3 (t)sinω c t + ] (3) 14

8 where we have applied power series Remember that if e jx =1+jx x2 2! + + jn xn n! + m(t) is bandlimited to B-Hz, a(t) is bandlimited to B-Hz, since integration is linear; a 2 (t) is bandlimited to 2B-Hz, a n (t) is bandlimited to nb-hz, Clearly, the modulated wave φ FM (t) is not band-limited. It has an infinite bandwidth. However, most of the modulated signal power resides in a finite bandwidth. Narrow-band Angle Modulation: when k f is very small (that is, k f a(t) << 1), then all but the first two terms in (3) are negligible, and we have φ FM (t) A c [cos ω c t k f a(t)sinω c t] The bandwidth of φ FM (t) is 2B (bandwidth of m(t) narrow-band FM -NBFM. 15 NBPM case is similarly given by φ PM (t) A c [cos ω c t k p m(t)sinω c t] m(t) Ackfa(t)sin(ωct) Ackpm(t)sin(ωct) DSB SC m(t) a(t) DSB SC modulator NBPM modulator NBFM Ac sin(ωct) Ac sin(ωct) π/2 π/2 ~ ~ Ac cos(ωct) Ac cos(ωct) (b) (a) Figure 3: Narrow-band PM and FM wave generation. Wide-band FM (WBFM) analysis diagram p261 16

9 In general, suppose m(t) is band-limited to B Hz. Cells: staircase approximation to m(t). Each cell has a width 1/2B (Nyquisite rate: f sample =2B). kth cell: starting from time t = t k, constant amplitude= m(t k ), corresponding FM signal: f c + k f m(t k ) with duration 1/2B. Spectrum of the kth cell: sinc function centered at f c + k f m(t k ) with a spread of 2B. Spectrum of ˆm(t): Significant frequency component range (m p = max t [m(t)]): f min = f c k f m p 2B f max = f c + k f m p +2B 17 B FM f max f min =2k f m p +4B = 2( f +2B) (4) where f = k f m p is the maximum frequency deviation. The above FM-BW is an approximation, and not a very good one. In the case of NBFM, our earlier result shows B FM =2B. Therefore, a better estimation of (4) is B FM =2( f + B) =2(k f m p + B) Carson s Rule (5) Define β = f/b as deviation ratio, which controls the amount of modulation. For tone-modulated FM, β is called the modulation index (β = f/f m ). Then B FM =2B(β + 1) 18

10 Spectrum of the Tone Modulated FM To determine the spectrum of φ FM (t) for arbitrary β in a closed form format is not possible. We will still look at the single-tone modulation case m(t) =A m cos(ω m t), then therefore, a(t) = m(λ)dλ = A m ω m sin(ω m t) ( φ FM (t) =A c cos ω c t + k ) f A m sin ω m t = Re { A c e jβ sin ωmt e jω ct } ω m where β = f f m = k f A m ω m is the deviation ratio (or the modulation index in this case). Furthermore, e jβ sin ωmt is periodic with a period 2π/ω m, and can be expanded using FS as e jβ sin ω mt = n D n e jnω mt with D n = ω m 2π π/ωm π/ω m e jβ sin ω mt e jnω mt dt = 1 2π π π e j(β sin x nx) dx = J n (β) (6) 19 which is Bessel function of 1st kind (denoted by J), order n and argument β. Changing variable x = ω m t has also been used in derivation of (5). Therefore, { } φ FM (t) = Re A c J n (β)e j(ω c+nω m )t = A c J n (β) cos(ω c + nω m )t (7) Consequently, the spectrum is n n Φ FM (ω) =A c π n J n (β)[δ(ω (ω c + nω m )) + δ(ω +(ω c + nω m ))] (8) Hence, J n s determine the overall shape of Φ FM. The strength of the nth sideband at ω = ω c + nω m is J n (β) (real valued). Some key observations: For a given β, J n (β) decreases with n, and turns to negligible for n>β+1. Hence, the number of significant sidebands is β +1. The bandwidth is given by which verifies our previous results. B FM =2nf m = 2(β + 1)f m =2( f + B) When β<<1 (NBFM), there is only one significant sideband and the bandwidth B FM =2f m =2B. J n (β) =J n (β) for n even, J n (β) = J n (β) for n odd, J n (β) =( 1) n J n (β). 2

11 for β small (β <.3) (NBFM), J (β) 1 J 1 (β) =β/2 J 1 (β) = β/2 J n (β), n 2 n J n(β) 2 =1 J ns are well tabulated n= n=1 n= 2 n= 3 J n (β) β β = 1 β = 2 β = 3 β = 6 J n (β) n 21 Example NBFM (β <.3) therefore, J (β) 1 J 1 (β) =β/2 J 1 (β) = β/2 J n (β), n 2 φ FM (t) =A c cos(ω c t)+ A cβ 2 cos(ω c + ω m )t A cβ 2 cos(ω c ω m )t as before (NBFM: a carrier + 2 sidebands). WBFM: Observations FM with sinusoidal modulation: infinite number of sidebands, f c ± nf m, n =, 1, 2, Magnitudes of sidebands decrease as f >> f c (n ) For practical purposes the power in φ FM is contained within a finite bandwidth Carrier amplitude = A c J (β) depends on β! Total power in φ FM φ 2 FM = A2 c 2 n J 2 n(β) = A2 c 2 = constant 22

12 But P SB and P c changes as a function of β. Since β = f/f m, we can change β by (i) f=constant, change f m : β = f m more spectral lines are added within 2 ω (ii) f changed, f m =constant β = f more spectral lines at constant spacing are added Example 2: (textbook 5.3) (a) Estimate B FM and B PM for the modulating signal m(t) in Example 1 for k f =2π 1 5 and k p =5π. (b) Repeat the problem if the amplitude of m(t) is doubled. Solution: (a) The peak amplitude of m(t) is unity. hence, m p =1. We now determine the essential bandwidth B of m(t). Since m(t) is a periodic function with fundamental frequency and m(t) can be written as ω = m(t) = n 2π = 14 π f =5kHz 8 π C n cos nω t C n = 2 n n odd 2 n even 23 Since C n decreases rapidly with n. We apply the bandwidth of m(t) as the frequency of the third harmonic, thus B =3 5kHz = 15kHz FM: Method I f = 1 2π k f m p = 1 2π (2π 15 )(1) = 1kHz B FM = 2( f + B) = 23kHz Method II β = f B = 1 ( ) 1 15 B FM =2B(β + 1) = = 23kHz PM: Method I f = 1 2π k pm p = 5kHz B FM = 2( f + B) = 13kHz Method II β = f B = 5 ( ) 5 15 B FM =2B(β + 1) = = 13kHz 24

13 (b) Doubling m(t) doubles its peak value. Hence, m p =2, but the bandwidth is still B = 15kHz. FM: Method I f = 1 2π k f m p = 1 2π (2π 15 )(2) = 2kHz B FM = 2( f + B) = 43kHz Method II β = f B = 2 ( ) 2 15 B FM =2B(β + 1) = = 43kHz PM: Method I f = 1 2π k pm p = 1kHz B FM = 2( f + B) = 23kHz Method II β = f B = 1 ( ) 1 15 B FM =2B(β + 1) = = 23kHz 25 Example 5.5 : An angle-modulated signal with carrier frequency ω c =2π 1 5 is described by the equation φ EM (t) = 1 cos(ω c t + 5 sin 3t + 1 sin 2πt) 1. Find the power of the modulated signal P = 1 2 /2 = Find the frequency deviation f ω i = d dt θ(t) =ω c + 15, cos 3t + 2, π cos 2πt The maximal value of frequency deviation is ω = 15, + 2, π. hence, f = ω/2π = Hz 3. Find the deviation ratio β: since the signal BW is the highest frequency in m(t). We have B = 2π/2π = 1Hz. Therefore, β = f/b = /1 = Find the phase deviation φ: The angle deviation is (5 sin 3t + 1 sin 2πt). φ = 15rad. 5. Estimate the bandwidth of φ EM (t): B EM = 2( f + B) = Hz. 26

14 Generation of FM Waves Generation of NBFM: from earlier analysis ψ FM (t) = A c cos (ω c t + k f when k f is small, ) m(λ)dλ = A c cos [ω c t + k f a(t)] (9) = A c cos(ω c t) cos[(k f a(t)] A c sin[k f a(t)] sin(ω c t) (1) ψ FM (t) = A c cos(ω c t) A c k f a(t) sin(ω c t) m(t) k f a(t) S_FM(t) A c sin(ω ct) π/2 ~ Figure 4: NBFM Generation. 27 Indirect Method of Armstrong: m(t) S NBFM S WBFM For this purpose, we use a Frequency Multiplier. i(t) non linear BPF (n f_c) f c f nf c n f consider the case where o(t) =[a n i n (t)] h BPF o(t) = a 2 i 2 (t) h BPF = a 2 A 2 c cos 2 (ω c t + φ(t)) h BPF [ A 2 c = a a A 2 ] c 2 2 cos(2ω ct +2φ(t) h BPF = a 2 A 2 c 2 cos(2ω ct +2φ(t)) (11) carrier frequency and the modulation (dispersion) index have been doubled in the above process. The above example is for a square-law device. If we consider an n-th order 28

15 device where BPF is tuned to nf c. [a n i n (t)] h BPF (t) nf c,nβ Remark: if f m is fixed and β nβ, then f n f implies β = f/f m. Using a frequency multiplier increases the carrier frequency and β. For example, the frequency range of the message signal is [5Hz, 15 khz]. NBFM generates FM signal with f = 25Hz, then the corresponding β range is [25/15, 25/5] = [.167,.5]. Now requires f c 91.2Mhz, f 75kHz, then the required β ranges [75/15, 75/5] = [5, 15], which is WBFM. We can generate it using following Armstrong FM generator. 29 f m :[5Hz,15kHz] $S_{NBFM}(t)$ f c1 =2kHz f 1 =25Hz f c2 =12.8 MHz f 2 =1.6 khz f c3 =1.9 MHz f 3 =1.6 khz f c4 =91.2 MHz f 4 =76.8 khz m(t) k f a(t) Frequency multiplier x64 Frequency converter Frequency multiplier x48 A c sin(ω c t) π/2 ~ Crystal oscillator 1.9 MHz Figure 5: Armstrong indirect FM transmitter. 3

16 Demodulation of FM Ideal differentiator When apply ψ FM (t) to an ideal differentiator, the output is ψ FM (t) = d { ]} A c cos [ω c t + k f m(α)dα dt ] = A c [ω c + k f m(t)] sin [ω c t + k f m(α)dα (12) The signal ψ FM (t) is both amplitude and frequency modulated. Since ω = k f m p <ω c, ω c + k f m(t) > for all t. Therefore, m(t) can be obtained by envelope detection of ψ FM (t) The amplitude A c is assumed to be a constant. If A c is not a constant, the envelope of ψ FM (t) would be A c (t)[ω c + k f m(t)], and the ED output would be proportional to m(t)a c (t). This variation in A c should be removed before applying the signal to the FM detector. Bandpass limiter (Hard limiter+ bandpass filter) to eliminate the amplitude variations. 31 Let the input as ] v i (t) =A(t) cos [ω c t + k f m(α)dα = A(t) cos θ(t) The output of the hard limiter v o (t) can be expressed as 1 cos θ> v o (θ) = 1 cos θ< v o as a function of θ is a periodic square-wave function with period 2π, and can be expressed by a Fourier series as v o (θ) = (cos θ cos 3θ + cos 5θ + ) π Substituting θ = ω c t + k f m(α)dα, we have v o [(θ(t)] = 4 { ] cos [ω c t + k f m(α)dα 13 [ω π cos 3 c t + k f + 1 [ ] } 5 cos 5 ω c t + k f m(α)dα + ] m(α)dα (13) Let the bandpass filter with a center frequency ω c and a bandwidth B FM, the filter output 32

17 e o (t) is the desired angle-modulated carrier with a constant amplitude e o (t) = 4 [ ] π cos ω c t + k f m(α)dα Using PLL to demodulate FM signal When the incoming FM signal is A sin[ω c t + θ i (t)], where Then θ i (t) =k f θ o (t) =θ i (t) θ e = k f assuming a small error θ e, from (1)the LPF output is m(α)dα m(α)dα θ e e o (t) = 1 c θ o (t) k f c m(t) Thus, the PLL acts as an FM demodulator. If the incoming signal is a PM wave, θ o (t) =θ i (t) θ e = k p m(t) θ e and e o (t) =k p ṁ(t)/c. In this case we need to integrate e o (t) to obtain the desired signal. 33