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ECE 4371, Fall, 2014 Introduction to Telecommunication Engineering/Telecommunication Laboratory Zhu Han Department of Electrical and Computer Engineering Class 4 Sep. 8th, 2014 Overview Homework – 4.2.1, 4.2.3, 4.2.5, 4.2.7, 4.2.9 – 4.3.3, 4.3.8 – 4.4.2, 4.4.4 – 4.5.2 – 4.8.1 – Due 9/22/14 Phase-locked loop FM basics Carrier Recover Error DSB: e(t)=2m(t)cos(wct)cos((wc+ w)t+) e(t)=m(t) cos((w)t+) – Phase error: if fixed, attenuation. If not, shortwave radio – Frequency error: catastrophic beating effect SSB, only frequency changes, f<30Hz. – Donald Duck Effect Crystal oscillator, atoms oscillator, GPS, … Pilot: a signal, usually a single frequency, transmitted over a communications system for supervisory, control, equalization, continuity, synchronization, or reference purposes. Phase-Locked Loop Can be a whole course. The most important part of receiver. Definition: a closed-loop feedback control system that generates and outputs a signal in relation to the frequency and phase of an input ("reference") signal A phase-locked loop circuit responds both to the frequency and phase of the input signals, automatically raising or lowering the frequency of a controlled oscillator until it is matched to the reference in both frequency and phase. Voltage Controlled Oscillator (VCO) W(t)=wc+ce0(t), where wc is the free-running frequency Example Ideal Model Model LPF VCO – Si=Acos(wct+1(t)), Sv=Avcos(wct+c(t)) – Sp=0.5AAv[sin(2wct+1+c)+sin(1-c)] – So=0.5AAvsin(1-c)=AAv(1-c) Capture Range and Lock Range Carrier Acquisition in DSB-SC Signal Squaring method Costas Loop v1 ( t ) 2 1 2 Ac Al m ( t ) cos , 2 v 2 (t ) 1 1 1 v 3 ( t ) Ac Al m ( t ) cos sin Ac Al m ( t ) sin 2 2 2 2 SSB-SC not working 1 2 Ac Al m ( t ) sin v 4 ( t ) K sin 2 Costas receiver PLL Applications Clock recovery: no pilot Deskewing: circuit design Clock generation: Direct Digital Synthesis Spread spectrum: Jitter Noise Reduction Clock distribution FM Basics VHF (30M-300M) high-fidelity broadcast Wideband FM, (FM TV), narrow band FM (two-way radio) 1933 FM and angle modulation proposed by Armstrong, but success by 1949. Digital: Frequency Shift Key (FSK), Phase Shift Key (BPSK, QPSK, 8PSK,…) AM/FM: Transverse wave/Longitudinal wave Angle Modulation vs. AM Summarize: properties of amplitude modulation – Amplitude modulation is linear just move to new frequency band, spectrum shape does not change. No new frequencies generated. – Spectrum: S(f) is a translated version of M(f) – Bandwidth ≤ 2W Properties of angle modulation – They are nonlinear spectrum shape does change, new frequencies generated. – S(f) is not just a translated version of M(f) – Bandwidth is usually much larger than 2W Angle Modulation Pro/Con Application Why need angle modulation? – Better noise reduction – Improved system fidelity Disadvantages – Low bandwidth efficiency – Complex implementations Applications – FM radio broadcast – TV sound signal – Two-way mobile radio – Cellular radio – Microwave and satellite communications Instantaneous Frequency •Angle modulation has two forms - Frequency modulation (FM): message is represented as the variation of the instantaneous frequency of a carrier - Phase modulation (PM): message is represented as the variation of the instantaneous phase of a carrier s ( t ) Ac cos i ( t ) , w here Ac : carrier am plitude, i ( t ) : angle ( phase) f i (t ) 1 d i (t ) 2 s ( t ) Ac cos 2 f c t ( t ) w here ( t ) is a function of m essage signa l m ( t ). dt Phase Modulation PM (phase modulation) signal s ( t ) Ac cos 2 f c t k p m ( t ) ( t ) k p m ( t ), k p : phase sensitivity instantanous frequency f i ( t ) f c k p dm ( t ) 2 dt Frequency Modulation FM (frequency modulation) signal s ( t ) Ac cos 2 f c t 2 k f m ( ) d 0 t k f : freq u en cy sen sitivity in stan tan o u s freq u en cy f i ( t ) f c k f m ( t ) an g le i ( t ) 2 t 0 f i ( ) d 2 f c t 2 k f (Assume zero initial phase) t 0 m ( ) d FM Characteristics Characteristics of FM signals – Zero-crossings are not regular – Envelope is constant – FM and PM signals are similar Relations between FM and PM FM o f m ( t ) P M of m ( t ) PM of FM of t m ( ) d 0 dm ( t ) dt FM/PM Example (Time) FM/PM Example (Frequency) Matlab fc=1000; Ac=1; % carrier frequency (Hz) and magnitude fm=250; Am=0.1; % message frequency (Hz) and magnitude k=4; % modulation parameter % generage single tone message signal t=0:1/10000:0.02; % time with sampling at 10KHz mt=Am*cos(2*pi*fm*t); % message signal % Phase modulation sp=Ac*cos(2*pi*fc*t+2*pi*k*mt); % Frequency modulation dmt=Am*sin(2*pi*fm*t); % integration sf=Ac*cos(2*pi*fc*t+2*pi*k*dmt); % PM % Plot the signal subplot(311), plot(t,mt,'b'), grid, title('message m(t)') subplot(312), plot(t,sf,'r'), grid, ylabel('FM s(t)') subplot(313), plot(t,sp,'m'), grid, ylabel('PM s(t)') Matlab % spectrum w=((0:length(t)-1)/length(t)-0.5)*10000; Pm=abs(fftshift(fft(mt))); % spectrum of message Pp=abs(fftshift(fft(sp))); % spectrum of PM signal Pf=abs(fftshift(fft(sf))); % spectrum of FM signal % plot the spectrums figure(2) subplot(311), plot(w,Pm,'b'), axis([-3000 3000 min(Pm) max(Pm)]), grid, title('message spectrum M(f)'), subplot(312), plot(w,Pf,'r'), axis([-3000 3000 min(Pf) max(Pf)]), grid, ylabel('FM S(f)') subplot(313), plot(w,Pp,'m'), axis([-3000 3000 min(Pp) max(Pp)]), grid, ylabel('PM S(f)') Frequency Modulation FM (frequency modulation) signal s ( t ) Ac cos 2 f c t 2 k f m ( ) d 0 t k f : freq u en cy sen sitivity in stan tan o u s freq u en cy f i ( t ) f c k f m ( t ) an g le i ( t ) 2 t 0 f i ( ) d 2 f c t 2 k f m ( t ) Am cos(2 f m t ) (Assume zero initial phase) t m ( ) d 0 f i f c k f Am cos(2 f m t ) d 2 k f 1 d 1 d 2 f c t 1 fi 2 dt 2 dt 2 fc 1 2 2 k f Am cos(2 f m ) Let t t 0 Am cos(2 f m ) d dt Example Consider m(t)- a square wave- as shown. The FM wave for this m(t) is t shown below. FM ( t ) A cos( c t k f m( )d ). - t Assume m(t) starts at t 0. For 0 t T m(t) 1 , 2 m( )d t and 0 t for T 2 t T m(t) - 1 , ous frequency t 2 m( )d m( )d m( )d 0 The instantane T T 2 - (t - T ) T - t. 2 T 0 2 is i ( t ) c k f m ( t ) c k f for and i ( t ) c k f for T t T . 2 i max c k f and i min c k f m(t) 0 T 2T t FM ( t ) t 0 t T 2 Frequency Deviation Frequency deviation Δf – difference between the maximum instantaneous and carrier frequency f k f Am k f m ax | m ( t ) | – Definition: – Relationship with instantaneous frequency single-tone m ( t ) case: f i f c f cos(2 f m t ) general case: fc f fi fc f – Question: Is bandwidth of s(t) just 2Δf? No, instantaneous frequency is not equivalent to spectrum frequency (with non-zero power)! S(t) has ∞ spectrum frequency (with non-zero power). Modulation Index Indicate by how much the modulated variable (instantaneous frequency) varies around its unmodulated level (message frequency) A M (envelope): m ax | k a m ( t ) | , 1 A FM (frequency): m ax | k f m ( t ) | fm Bandwidth a (t ) t m ( ) d 2 2 kf 2 kf 3 ( t ) Re( ( t )) A cos w c t k f a ( t ) sin w c t a ( t ) cos w c t a ( t ) sin w c t ... 2! 3! Narrow Band Angle Modulation Definition k f a ( t ) 1 Equation ( t ) A cos w c t k f a ( t ) sin w c t Comparison with AM Only phase difference of Pi/2 Frequency: similar Time: AM: frequency constant FM: amplitude constant Conclusion: NBFM signal is similar to AM signal NBFM has also bandwidth 2W. (twice message signal bandwidth) Example Block diagram of a method for generating a narrowband FM signal. A phasor comparison of narrowband FM and AM waves for sinusoidal modulation. (a) Narrowband FM wave. (b) AM wave. Wide Band FM Wideband FM signal m ( t ) Am cos(2 f m t ) s ( t ) Ac cos 2 f c t sin(2 f m t ) Fourier series representation s ( t ) Ac J n ( ) cos 2 ( f c nf m ) t n S( f ) Ac 2 J n ( ) ( f f c nf m ) ( f f c nf m ) n J n ( ) : n -th order Bessel function of the firs t kind Example Bessel Function of First Kind 1 . J n ( ) ( 1) J n ( ) n 2 . If is sm all, th en J 0 ( ) 1, J1( ) Jn( ) 0 3. n J n ( ) 1 2 , 2 fo r all n 2 Spectrum of WBFM (Chapter 5.2) Spectrum when m(t) is single-tone s ( t ) Ac cos 2 f c t sin(2 f m t ) Ac J n ( ) cos 2 ( f c nf m ) t n S( f ) Ac 2 n Example 2.2 J n ( ) ( f f c nf m ) ( f f c nf m ) Spectrum Properties 1. frequencies: f c , f c f m , f c 2 f m , (for all n ). , f c nf m , T heoretically infinite bandw idth. 2. For << 1 (N BFM ), frequency: f c , f c f m J 0 ( ) 1, J 1 ( ) J 1 ( ), J n ( ) 0 for all n 2 3. M agnitude of f c nf m : Ac 2 J n ( ), depend on 4. C arrier ( f c ) magnitude J 0 ( ) can be 0 for some 5. A verage pow er: P n A 2 c 1 2 J ( ) 2 n 1 2 2 Ac Bandwidth of FM Facts – FM has side frequencies extending to infinite frequency theoretically infinite bandwidth – But side frequencies become negligibly small beyond a point practically finite bandwidth – FM signal bandwidth equals the required transmission (channel) bandwidth Bandwidth of FM signal is approximately by – Carson’s Rule (which gives lower-bound) Carson’s Rule Nearly all power lies within a bandwidth of – For single-tone message signal with frequency fm BT 2 f 2 f m 2( 1) f m – For general message signal m(t) with bandwidth (or highest frequency) W BT 2 f 2W 2( D 1)W w h ere D f is d eviatio n ratio (eq u ivalen t to ), W f m ax k f m ( t )