Modulation Techniques Modulation: translates an information-bearing signal (message signal) to a new spectral location (frequency domain) fc: carrier frequency ↔ a(t) A(f) A( f-fc ) Frequency (f) Frequency (f) -fw 0 +fw 0 (fc –fw)fc (fc+fw) Baseband signal Bandpass signal • Communication channels Æ bandpass transfer (frequency) response ⇒ translates the message signal to be within the channel transfer response Selected frequency bands H(f) Frequency Band Carrier frequency Bandwidth Longwave Radio 100kHz 2kHz BW Shortwave Radio 5MHz 100kHz VHF 100MHz 2MHz f Microwave 5GHz 100MHz f0 Millimeterwave 100GHz 2GHz • Facilitates antenna reception Optical 5x1014 Hz 1013Hz ERG2310A-II p. II-1 Modulation Techniques If more than one message signal utilizes a channel Æmodulation allows translation of different signals to different spectral locations Æmultiplexing allows two or more message signals to be transmitted by a single transmitter (frequency division multiplexing) A3c(f) Ædesired modulated signal can be selected by a receiver f A (f) fc1 1c A1(f) Low-pass filter modulator f f fc1 f c3 fc2 A2(f) A (f) demodulator modulator 2c f channel f fc2 f demodulator A (f) fc3 f 3 A3c(f) fc2 modulator f Low-pass filter f fc1 fc2 fc3 fc3 f A2c(f) ERG2310A-II f p. II-2 Frequency Translation π ↔ j2 fct ↔ − Recall: If a(t) A( f ) , then a(t)e A( f fc ) (Fourier transform pairs) Consider a message signal x(t) , which is bandlimited to the frequency range 0 to W π and has its Fourier transform is X(f) , is multiplied by cos (2 fc t) . 1 ℑ{}x(t)cos(2π f t) = []X ( f + f ) + X ( f − f ) c 2 c c Baseband signal Frequency translated signal ERG2310A-II p. II-3 Recovery of Baseband Signal To recovery the baseband signal, we can simply multiply the translated signal with π cos (2 fc t). x(t) x(t) []x(t)cos(2πf t) cos(2πf t) = x(t)cos 2 (2πf t) = + cos(4πf t) c c c 2 2 c Frequency translated signal 1 1 X ( f ) + []X ( f − 2 f ) + X ( f + 2 f ) 2 2 c c Baseband signal We obtain the baseband signal x(t) and a signal whose spectral range extends from (2fc-W) to (2fc+W). As fc >> W, the extra signal is removed by a low-pass filter. ERG2310A-II p. II-4 Analog (Continuous-Wave) Modulation A parameter of a high-frequency sinusoidal carrier is varied proportionally to the message signal x(t) . General modulated signal: = []ω + φ s(t) A(t)cos ct (t) ω c : carrier frequency A(t) : instantaneous amplitude φ(t) : instantaneous phase deviation When A(t) is linearly related to the modulating (message) signal = []ω + φ ∝ Æ Amplitude modulation (AM) s(t) A(t)cos ct o where A(t) x(t) When φ(t) is linearly related to the modulating signal = []ω + φ φ ∝ Æ Phase modulation (PM) s(t) Ac cos ct (t) where (t) x(t) When time derivative of φ(t) is linearly related to the modulating signal dφ(t) Æ Frequency modulation (FM) s(t) = A cos[]ω t + φ(t) where ∝ x(t) c c dt *FM & PM are commonly called Angle Modulation ERG2310A-II p. II-5 Analog (Continuous-Wave) Modulation Unmodulated carrier frequency Message signal Amplitude-modulated signal Angle-modulated signal (frequency- modulated) ERG2310A-II p. II-6 Analog (Continuous-Wave) Modulation Message signal Unmodulated carrier Phase-modulated signal Frequency-modulated signal ERG2310A-II p. II-7 Amplitude Modulation The envelope of the modulated carrier has the same shape as the message signal. x(t) s(t) envelope Ac π The amplitude of the carrier wave [ Ac cos(2 fct) ] varies linearly with the baseband message signal x(t). The standard form of an amplitude-modulated (AM) signal is given by: = []+ ~ π s(t) Ac 1 ma x(t) cos(2 f ct) ~ Where x(t) is the normalized message signal and ma is called the modulation index. ERG2310A-II p. II-8 Amplitude Modulation: DSB-LC = []+ ~ π AM signal: s(t) Ac 1 ma x(t) cos(2 f ct) = π + ~ π Ac cos(2 f ct) Ac ma x(t)cos(2 f ct) = ω + ~ ω ω = π Ac cos( ct) Ac ma x(t)cos( ct) where c 2 f c A A m S( f ) = c []δ ( f + f ) + δ ( f − f ) + c a []X ( f + f ) + X ( f − f ) 2 c c 2 c c Double-sideband – large carrier (DSB-LC) ERG2310A-II p. II-9 Amplitude Modulation: DSB-LC = []+ ~ π s(t) Ac 1 ma x(t) cos(2 f ct) Ac Distorted signal ! So, Ac has to be large enough or we have to control the modulation index ma. ERG2310A-II p. II-10 Amplitude Modulation: DSB-LC = []+ ~ π s(t) Ac 1 ma x(t) cos(2 f ct) DSB-LC + x(t) Effect of modulation index ma : ma < 1 m = 1 m > 1 Ac(1+ma) a a Ac Ac(1-ma) A − A Define modulation depth = max c Ac For a sinusoidal message signal, Amax=Ac(1+ma), thus the modulation depth is ma . ERG2310A-II p. II-11 Amplitude Modulation: DSB-LC = []+ ~ π Generation of DSB-LC signal: s(t) Ac 1 ma x(t) cos(2 f ct) Product modulator: Nonlinear Square-law modulator: device + v = a v + a v 2 x(t) filter s(t) out 1 in 2 in v v = + ω in out vin x(t) cos ct ω cos ct 2a = + 2 + 2 ω + + 2 ω where A =a and m =2a /a vout a1 x(t) a2 x (t) a2 cos ct a1 1 x(t) cos ct c 1 a 2 1 a1 142444 43444 s(t) Chopper/rectifier modulator: ω c + + x(t) Band x(t) Band - pass - pass vo(t) vo(t) + filter + filter ω ω A cosω t at c A cosω t at c c c - c c - ERG2310A-II p. II-12 Amplitude Modulation: DSB-LC Chopper/rectifier modulator: The chopper or rectifier can generate a periodic waveform whose fundamental ω frequency is c rad/sec. ω Let f(t) = Accos ct + x(t) The periodic signal p (t) can be represented as ∞ ω = jn ct p(t) ∑ Pn e n=−∞ Consider p(t)f(t): ∞ ω = jn ct f (t) p(t) ∑ Pn f (t)e . n=−∞ Applying the frequency translation property of the Fourier transform, we get ∞ ℑ = ω − ω { f (t) p(t)} ∑ Pn F( n c ). n=−∞ ERG2310A-II p. II-13 Amplitude Modulation: DSB-LC Demodulation of DSB-LC signal: By envelope detector: the diode cuts off the negative part of the DSB-LC signal while RC acts as a lowpass filter to retrieve the envelope. >> >> 1/W RC 1/ f c where W is the message signal bandwidth ERG2310A-II p. II-14 Amplitude Modulation: DSB-LC = []+ ~ π DSB-LC signal: s(t) Ac 1 ma x(t) cos(2 f ct) Consider the average power of s(t) : = 2 Ps s (t) = 2 []+ ~ 2 2 ω Ac 1 ma x(t) cos ( ct) = 2 []+ 2 ~ 2 + ~ 1 + ω Ac 1 ma x (t) 2ma x(t) (1 cos(2 ct) 2 A2 = c []1+ m 2 ~x 2 (t) + 2m ~x(t) + []1+ m 2 ~x 2 (t) + 2m ~x(t) cos(2ω t) 2 a a a a c A2 = c []1+ m 2 ~x 2 (t) if ~x(t) = 0 and as cos(2ω t) = 0 2 a c A2 = c []1+ m 2 P where P = ~x 2 (t) 2 a x x = + where P : average power per sideband Pc 2Psb sb 1 1 1 Thus P = A2 ; P = A2 m 2 P = m 2 P P c 2 c sb 4 c a x 2 a x c 1 1 For m x(t) ≤ 1 ⇒ m 2 P ≤ 1 ⇒ P ≤ P ⇒ P = P − 2P ≥ P a a x sb 2 c c s sb 2 s At least 50% of total transmitted power resides in the carrier term which conveys no information Æ wasteful of power ERG2310A-II p. II-15 Amplitude Modulation: DSB-LC Fraction of total transmitted power contained in the sidebands is: 2P m 2 P P m 2 P µ = sb = a x c = a x + + 2 + 2 Pc 2Psb Pc ma Px Pc 1 ma Px 1 If x(t) is a single sinusoid, i.e. cosω t, then P = x 2 (t) = m x 2 m 2 Thus, µ = a and is known as the transmission efficiency of DSB-LC AM system. + 2 2 ma Example: A given AM (DSB-LC) broadcast station transmits an average carrier power output of 40kW and uses a modulation index of 0.707 for sine-wave modulation. Calculate (a) the total average power output; (b) the transmission efficiency; and (c) the peak amplitude of the output if the antenna is represented by a 50-ohm resistive load. = + = + 2 Solution: (a) The total average power output is Ps Pc 2Psb Pc (1 ma / 2). = + = For ma = 0.707,Ps 40(1 1/ 4) 50 kW. (0.707)2 0.5 (b) The transmission efficiency is µ = = = 20%. 2 + (0.707)2 2.5 A2 (c) Consider P = ,⇒ A2 = 2RP = 4×106. c 2R c + = The peak amplitude of the output is (1 ma )A 3414 V. ERG2310A-II p. II-16 Amplitude Modulation: DSB-SC The “wasted” carrier power in DSB-LC can be eliminated by setting ma=1 and suppressing the carrier. = ~ π Thus the modulated signals becomes s(t) Ac x(t)cos(2 f ct) In this case, the carrier frequency component is suppressed, thus it is called double-sideband suppressed-carrier modulation (DSB-SC) .