6. Amplitude Modulation

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1 6. Amplitude Modulation Modulation is a proess by whih some parameter of a arrier signal is varied in aordane with a message signal. The message signal is alled a modulating signal. Definitions A bandpass signal is represented by s (t) = A(t) os θ(t) (6.1) where A(t) is the envelope and θ(t) = ω t + φ(t) = πf t + φ(t). φ(t) is alled the instantaneous phase deviation of s (t) and f is the arrier frequeny. For amplitude modulation, we an write s (t) = A(t) os πf t (6.) where A(t) is linearly related to the modulating signal m(t). A(t) is alled the instantaneous amplitude of s (t) and amplitude modulation is also referred to as linear modulation. Depending on the relationship between m(t) and A(t), we have the following types of amplitude modulation shemes: normal amplitude modulation (AM), double-sideband (DSB) modulation, single-sideband (SSB) modulation, and vestigial-sideband (VSB) modulation. Normal Amplitude Modulation (AM) A normal amplitude-modulated signal is given by s (t) = [A + m(t)] os πf t (6.3) s (t) = A os π f t +m (t )os π f t (6.4) arrier sidebands where A is a onstant and m(t) is the modulating signal. It is also very ommon to define a normal amplitude-modulated signal as s (t) = A[1 + m(t)] os πf t (6.5) The modulation index m is defined as [1] m = min m (t ) A (6.6) 6.1

2 Figure 6.1 shows normal AM signals for various values of modulation index. Clearly, the envelope of the modulated signals has the same shape as m(t) when m < 1. When m > 1, the arrier signal is said to be overmodulated and the envelope is distorted. Figure 6.1 Normal AM signals for various values of modulation index. The effiieny η of a normal AM signal is defined as [] η = P s P t x 1% (6.7) where P s is the power arried by the sidebands and P t is the total power of the normal AM signal. Spetrum of Normal AM Signals For normal amplitude modulation, s (t) = [A + m(t)] os πf t (6.8) s (t) = A os πf t + m(t) os πf t The Fourier transform of s (t) is S (f) = 1 A[δ(f-f ) +δ(f+f )] + 1 [M(f-f ) + M(f+f )] (6.9) Figure 6. shows the spetrum of a normal AM signal. Normal amplitude modulation simply shifts the spetrum of m(t) to the arrier frequeny f. The bandwidth of the modulated signal is f m Hz, where f m is the bandwidth of the modulating signal m(t). Generation of Normal AM Signals Figure 6. Spetrum of normal AM signal. A proess of generating a normal AM signal is shown in Figure 6.3. This type of modulation an be ahieved by using a non-linear devie, suh as a diode. This is shown in Figure 6.4 [3]. Figure 6.3 Generation of normal AM signal. Figure 6.4 Amplitude modulator using a diode. 6.

3 Let the input-output harateristi of a diode be approximated by a power series v o (t) = a v i (t) + b v i (t) (6.1) where a, b are onstants and v i (t) = os πf t + m(t) (6.11) Substituting equation (6.11) into (6.1), we have v o (t) = a[os πf t + m(t)] + b[os πf t + m(t)] = a m(t) + bos πf t + b m(t) + aos πf t + b m(t) os πf t (6.1) If we pass the signal v o (t) through a bandpass filter entred at + f, we obtain v' o (t) = [a + b m(t)] os πf t (6.13) = b[a + m(t)] os πf t (6.14) where A = a b. We generate a normal AM signal. A normal amplitude-modulated signal an also be obtained by multiplying m(t) by a periodi digital signal s(t). The modulator is alled a swithing modulator []. If we take a periodi retangular waveform s(t) of period T = 1/f, amplitude A m, and pulse width τ, the trigonometri Fourier series of s(t) is s(t) = A m τ T + (A T m τ sin πnf τ / n =1 πnf τ / ) os πnf t (6.15) s(t) = T + T n os πnf t (6.16) n =1 where = A m τ and n = A m τ sin πnf τ / πnf τ / series is. The orresponding omplex Fourier s(t) = 1 (A T m τ sin πnf τ / n = πnf τ / )e jπnf t (6.17) Figure 6.5 shows the periodi retangular waveform and its line spetrum. 6.3

4 Figure 6.5 (a) A periodi retangular waveform, and (b) its line spetrum. If the input signal is v i (t) = os πf t + m(t), the output of a swithing modulator is v o (t) = v i (t) s(t) = [os πf t + m(t)] s(t) = [os πf t + m(t)] ( T + T n os πnf t ) n =1 = [ os πf T t n os πnf t] + n =1 T m(t) + T os πf t + m(t) T n os πnf t n =1 v o (t) onsists of a d term, the omponent m(t), and an infinite number of normal AM signals at arrier frequenies f, f, 3f,... If we pass the signal v o (t) through a bandpass filter entred at + f, the filtered signal is v' o (t) = T os πf t + m(t) T 1 os πf t + T os πf t = 1 [A + m(t)] os πf T t where A = + 1. That is, we obtain a normal AM signal. Demodulation of Normal AM Signals [3] The proess of reovering the message signal from the modulated signal is alled demodulation or detetion. Two basi methods are available. Envelope Detetion. In this method, an envelope detetor is used to reover the message signal. An envelope detetor onsists of a diode and a resistor-apaitor ombination. This is shown in Figure 6.6. Figure 6.6 Envelope detetor. During the positive half-yle peaks of the modulated signal, the diode is forward biased, and the apaitor harges up to the peak value of the modulated signal. As the modulated 6.4

5 signal falls from its maximum, the diode turns off and the apaitor disharges through the resistor. The proess repeats in this way. For proper operation, the disharge time onstant RC must be hosen properly. Synhronous (Coherent) Detetion. Here, a produt detetor is used to onvert the bandpass signal to baseband. This is shown in Figure 6.7. Figure 6.7 Synhronous detetor. At the reeiving end, the bandpass signal is multiplied by a loally generated arrier signal os(πf t + φ ), where φ is an initial phase. The output of the multiplier is x(t) = [A + m(t)] os πf t os (πf t +φ ) =.5[A + m(t)] [os φ + os (4πf t +φ )] =.5[A + m(t)] os(4πf t +φ ) +.5Aos φ +.5m(t)os φ (6.18) If we suppress the first term by a low-pass filter, we get y(t) =.5Aos φ +.5m(t)os φ (6.19) It an be seen that we an reover the omponent m(t) if the initial phase φ is onstant and small. Suppose that the loal arrier signal is os[π(f + f)t], when the multiplier output beomes x(t) = [A + m(t)] os πf t os[π(f + f)t] =.5[A + m(t)] [os π f t + os π(f + f)t] =.5[A + m(t)] os π(f + f)t +.5Aos π f t +.5m(t)os π f t (6.) If we suppress the first term by a low-pass filter, we get y(t) =.5Aos π f t +.5m(t)os π f t (6.1) We annot reover the omponent m(t) unless the frequeny drift f is zero. Therefore, the loal arrier must not only be of the same frequeny but must be synhronised in phase with the arrier signal. If the arrier shifts in frequeny or phase, the resultant signal is distorted or attenuated. Synhronous detetion is sometimes alled oherent detetion. 6.5

6 Referenes [1] H. P. Hsu, Analog and Digital Communiations, MGraw-Hill, [] L. W. Couh II, Digital and Analog Communiation Systems, 5/e, Prentie Hall, [3] M. Shwartz, Information Transmission, Modulation, and Noise, 4/e, MGraw-Hill,

7 m (t ) max min m (t ) m (t ) s (t ) A (a) Envelope m < 1 Time Time -A (b) s (t ) A -A s (t ) A -A m (t ) -m (t ) () (d) m = 1 Time m > 1 Time Figure 6.1 Normal AM signals for various values of modulation index. 6.7

8 M (f ) -f m f m Frequeny Upper sideband Lower sideband S (f ) Lower sideband Upper sideband -f f - f m f f +f m Frequeny Figure 6. Spetrum of normal AM signal. m ( t ) Σ Normal AM 1 A A os π f t Figure 6.3 Generation of normal AM signal. m ( t ) os π f t D v i ( t ) R v o ( t ) ~ ~ ~ Bandpass filter +f v' o ( t ) Figure 6.4 Amplitude modulator using a diode. 6.8

9 s( t )... A m -T T τ τ T - - T (a)... Time Envelope n A m τ 1 T - 1 τ First zero rossing f 1 3 τ τ τ f (b) f Figure 6.5 (a) A periodi retangular waveform, and (b) its line spetrum. 6.9

10 s (t D ) R C y ( t ) Amplitude y ( t ) Envelope s ( t ) (a) (b) t Figure 6.6 Envelope detetor. s ( t ) x ( t ) LPF ~ y ( t ) os ( π f t + φ ) Loal arrier Figure 6.7 Synhronous detetor. 6.1

ENSC327 Communications Systems 4. Double Sideband Modulation. Jie Liang School of Engineering Science Simon Fraser University

ENSC327 Communications Systems 4. Double Sideband Modulation. Jie Liang School of Engineering Science Simon Fraser University ENSC327 Communiations Systems 4. Double Sideband Modulation Jie Liang Shool of Engineering Siene Simon Fraser University 1 Outline DSB: Modulator Spetrum Coherent Demodulator: Three methods Quadrature-arrier