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Modulation Techniques

Modulation Techniques

Techniques

Modulation: translates an information-bearing signal (message signal) to a new spectral location ( domain)

fc: carrier frequency ↔ a(t) A(f) A( f-fc )

Frequency (f) Frequency (f) -fw 0 +fw 0 (fc –fw)fc (fc+fw) 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 100kHz 2kHz BW 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 (frequency division ) 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 Recoveryπ of Baseband Signal π

To recovery the baseband signal, we can simplyπ multiply the translated signal with π cos (2 fc t). x(t) x(t) []x(t)cos(2 f t) cos(2 f t) = x(t)cos 2 (2 f 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-) 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 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 (PM) s(t) Ac cos ct (t) where (t) x(t) When time derivative of (t) is linearly related to the modulating signal ω dφ(t) Æ (FM) s(t) = A cos[]t + φ(t) where ∝ x(t) φ c c dt *FM & PM are commonly called 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 [ 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 = []+ ~ π AM signal: s(t) Ac 1 mωa x(t) cos(2 f ct) = + ~ Ac cos(2 f ct) Ac ma x(t)cos(π2 f ct) = + ~ ω ω = π Ac cos( ct) Ac ma x(t)cos( ct) where c 2 f c

δ A A m S( f ) = c []( f + f ) + δ ( f − f ) + c a []X ( f + f ) + X ( f − f ) 2 c c 2 c c Double- – 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 ω v v = + ω in out vin x(t) cos ct ω cos ct  2a  = + 2 + 2 + + 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/ 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 ∞ ω ℑ = − 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 : the 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:ω 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 Ac 1 ma x(t) cos ( ct) ω

= 2 []+ 2 ~ 2 + ~ 1 +  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(2 t) 2 a a a a ω 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 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)Example: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

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) . A = c []+ + − Its spectral density is: S( f ) X ( f f c ) X ( f f c ) 2

Average Power of the modulated signal: = 2 Ps s (t) = 2 ~ 2 2 ω Ac x (t)cos ( ct) 1 = A2 P where P = ~x 2 (t) 2 c x x = 2Psb 1 ∴ P = A2 P sb 4 c x

ERG2310A-II p. II-17 Amplitude Modulation: DSB-SC

± ω A signal spectrum can be translated an amount c rad/sec in frequency by multiplying ω the signal with any periodic waveform whose fundamental frequency is c rad/sec.

The periodic signal p (t) can be represented as ∞ ω = jn ct p(t) ∑ Pn e n=−∞ Consider p(t)x(t): ∞ ω = jn ct x(t) p(t) ∑ Pn x(t)e . n=−∞

Applying the frequency translation property of the Fourier transform, we get ∞ ω ℑ = − ω {x(t) p(t)} ∑ Pn X ( n c ). n=−∞

ERG2310A-II p. II-18 Amplitude Modulation: DSB-SC

Example: A periodic signal consists of the exponentially decreasing waveform e-at, 0 ≤ t < T, repeated every T seconds. A given signal f(t) is multiplied by this periodic signal. Determine an expression describing the spectrum and the time waveform of the resulting amplitude-modulated signal if all components except those centered at ±ω ω π c, c =2 /T, are discarded. Solution: The Fourier series for the given periodic signal can be written as ∞ −aT jnω t 1 T − − ω 1 1− e p(t) = P e c , where = at jn ct = ∑ n Pn e e dt . =−∞ ∫0 + ω n T T a jn c The spectrum of the product p(t)f(t) is

1 −  1 1  (1− e aT ) F(ω− ω ) + F(ω+ ω ) .  + ω c − ω c  T a j c a j c  The corresponding terms in the Fourier series are

1 −  1 ω 1 − ω  (1− e aT ) f (t)e j ct + f (t)e j ct .  + ω − ω  T a j c a j c  Combining yields the time waveform

−aT 2 1− e −1 ω = π ω + θ θ = tan (−ω / a), c 2 /T. f (t)cos( ct 0 ), where 0 c T 2 + ω 2 a c ERG2310A-II p. II-19 Amplitude Modulation: DSB-SC

Generation of DSB-SC signal by balanced modulator:

 + 1  ω Ac 1 x(t) cos ct 1  2  x(t) AM 2 Modulator A cosω t + c c ω + x(t)Ac cos ct - 1 − x(t) AM 2 Modulator  − 1  ω Ac 1 x(t) cos ct  2 

ERG2310A-II p. II-20 Amplitude Modulation: DSB-SC

Demodulation of DSB-SC signal: Assuming that the transmitted signal is s(t) = x(t)cosω t ω c

To demodulate the signal,ω we have = 2 s(t)cos ct x(t)cos ct = 1 + 1 ω 2 x(t) 2 x(t)cos 2 ct

Taking theω Fourier transform of both sides, we get ω ℑ{s(t)cos t} = 1 X ( ) c 2 ω + 1 X ( +ω2 ) 4 ω c + 1 − ω 4 X ( 2 c )

ERG2310A-II p. II-21 Amplitude Modulation: DSB-SC

∆ω θ Consider a small frequency error, , and a phase error, 0, are introduced in the locally generated carrier signal at the receiver. The signal at the receiver becomes ω + ∆ω +θ = ω ω ω + ∆ω +θ s(t)cos[( c )t 0 ] x(t)cos ct cos[(θ c )t 0 ] = 1 x(t)cos[(∆ω)t + ] 2 ω0 + 1 + ∆ +θ 2 x(t)cos[(2 c )t 0 ]. After passing via the low-pass filter, the output is

= 1 ∆ω +θ eo (t) 2 x(t)cos[( )t 0 ].

Phase error and frequency error results in undesirable . In some cases, they vary randomly, resulting in unacceptable performance.

Remedy: Using a synchronized oscillator to recover the original signal f(t) from the modulated signal φ(t). (Synchronous detection, or coherent detection)

ERG2310A-II p. II-22 Amplitude Modulation: DSB-SC

The original signal x(t) can be recovered from the modulated signal s(t) by multiplying ω s(t) by cos ct (i.e. synchronous detection). The same circuits as those used for modulation can be used for demodulation with the following minor differences.

1. Since the desired output spectrum is centered about ω=0 and therefore a low-pass filter is needed at the output.

2. The oscillator in the demodulator must be synchronized to the oscillator in the demodulator to achieve proper demodulation.

This is usually accomplished by either a direct connection if the modulator and demodulator are in close proximity or by supplying a sinusoid displaced in frequency but related to the modulator- oscillator frequency. The sinusoid is called a “pilot carrier”.

ERG2310A-II p. II-23 Amplitude Modulation: DSB-SC

Pilot Carrier Systems It is a common method used in DSB-SC modulation to maintain synchronization between modulator and demodulator. In this case, a sinusoidal tone whose frequency and phase are related to the carrier frequency is generated and is sent outside the pass-band of the modulated signal so it will not alter the frequency response capability of the system. A tuned circuit in the receiver detects the tone, translate it to the proper frequency, and uses it to correctly demodulate the DSB-SC signal. e.g. Stereo-multiplex system + L-R L ∑ x - To 38kHz 19kHz + ÷2 frequency + transmitter cosω t Atten. ∑ R c divider + + L+R ∑ + Spectrum used for stereo multiplexing before transmission

Audio (mono) er

i DSB-SC

r

r

a

c

t

o l

L+R i L-R (lower L-R (upper P sideband) sideband) in kHz ERG2310A-II 0 15 19 23 38 53 f p. II-24 Amplitude Modulation: DSB-SC

Phase-Locked Loop (PLL) In pilot tone system, phase-locked loop is used to synchronize one sinusoidal to another.

A simplified phase-locked loop stereo demodulator.

ERG2310A-II p. II-25 Amplitude Modulation: QAM

Quadrature Multiplexing ω Using the orthogonality of sines and cosines , it is possible to transmit and receive two different signals simultaneously on the same frequency.ω =ω ω + ω s(t) f1 (t)cos ct f 2 (t)sin cωt ω ω ω ω ω cos ct cos ct = 2 + s(t)cos ct f1 (t)cos ct ωf 2 (t)sin c cos ct = 1 + 1 ω + 1 ω 2 f1 (t) 2 f1 (t)cos 2 ct 2 f 2 (t)sin 2 ct f1(t) x x LPF ½ f1(t) ω = + 2 + s(t)sin ct f1 (t)cos ct sin ct f 2 (t)sin ct ∑ s(t) = 1 + 1 − 1 ω 2 f1 (t)sin 2 ct 2 f 2 (t) 2 f 2 (t)cos 2 ct + f (t) x x LPF ½ f (t) 2 2 In the low-pass filter, all terms at 2ωc are attenuated, yielding ω ω = 1 sin ct sin ct e1(t) 2 f1(t), = 1 e2 (t) 2 f2 (t).

Thus, each signal can be recovered by synchronous detection of the received signal using carriers of the same frequency but in phase quadrature.

ERG2310A-II p. II-26 ERG2310A-II p. II-27 Frequency Division Multiplexing (FDM)

Frequency-division multiplexing is the positioning of signal spectra in frequency such that each signal spectrum can be separated out from all the others by filtering.

ERG2310A-II p. II-28 Frequency Division Multiplexing (FDM)

Example: commercial radio and television receiver

ERG2310A-II p. II-29 (IF)

Heterodyning means the translating or shifting in frequency. In the receiver the incoming modulated signal is translated in frequency, thus occupying an equal bandwidth centered about a new frequency, known as an intermediate frequency (IF), which is fixed and is not dependent on the received signal center frequency.

The signal is amplified at the IF before demodulation.

If this intermediate frequency is lower than the received carrier frequency but above the final output signal frequency, it is called a .

ERG2310A-II p. II-30 Intermediate Frequency (IF)

Advantage: The amplification and filtering is performed at a fixed frequency regardless of station selection. Disadvantage: Image-frequency problem Two ways to solve this problem i. Choose the intermediate frequency as high as possible and practical. ii. Attenuate the image frequency before heterodyning.

The intermediate frequency chosen must be free from other strong transmissions or otherwise the receivers will amplify these spurious signals as they leak into the high-gain IF stages.

ERG2310A-II p. II-31 Intermediate Frequency (IF)

Example: A given radar receiver operating at a frequency of 2.80 GHz and using the super- heterodyne principle has a local oscillator frequency of 2.86 GHz . A second radar receiver operates at the image frequency of the first and interference results.

(a) Determine the intermediate frequency of the first radar receiver. (b) What is the carrier frequency of the second receiver? (c) If you were to redesign the radar receiver, what is the minimum intermediate frequency you would choose to prevent image-frequency problems in the 2.80-3.00 GHz radar band?

Solution: = − = − = (a) f IF f LO fc 2.86GHz 2.80GHz 60MHz.

= + = + = (b) f IMAGE fc 2 f IF 2.80GHz 0.12GHz 2.92GHz.

≥ − = − = ≥ (c) 2 f IF fMAX fMIN 3.00GHz 2.80GHz 0.20GHz; f IF 100MHz.

ERG2310A-II p. II-32

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