Fundamentals of Communication Systems SECOND EDITION

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1 GLOBAL EDITIO Fundamentals of Communication Systems SECOD EDITIO John G. Proakis Masoud Salehi

3 78 Effect of oise on Analog Communication Systems Chapter 6 The noise power is P n = ow we can find the output SR as = P o o P no S n ( f ) df = 0 4W = W 0. (6..) = A c 4 P M 4 W 0 = A c P M W 0. (6..) In this case, the received signal power, as given y Equation (3..), is P R = A c P M. Therefore, the output SR in Equation (6..) for DSB-SC AM may e expressed as ( S ) o DSB = P R 0 W, (6..3) which is identical to (S/), which is given y Equation (6..). Therefore, in DSB-SC AM, the output SR is the same as the SR for a aseand system. In other words, DSB- SC AM does not provide any SR improvement over a simple aseand communication system. 6..3EffectofoiseonSSBAM In this case, the modulated signal, as given in Equation (3..8), is u(t) = A c m(t) cos(π f c t) A c ˆm(t) sin(π f c t). (6..4) Therefore, the input to the demodulator is r(t) = A c m(t) cos(π f c t) A c ˆm(t) sin(π f c t) + n(t) = (A c m(t) + n c (t)) cos(π f c t) + ( A c ˆm(t) n s (t)) sin(π f c t). (6..5) Here we assume that demodulation occurs with an ideal phase reference. Hence, the output of the lowpass filter is the in-phase component (with a coefficient of )ofthe preceding signal. That is, y(t) = A c m(t) + n c(t). (6..6)

4 Section 6. Effect of oise on Amplitude Modulation Systems 79 We oserve that, in this case again, the signal and the noise components are additive, and a meaningful SR at the receiver output can e defined. Parallel to our discussion of DSB, we have and P o = A c 4 P M (6..7) P no = 4 P n c = 4 P n, (6..8) where Therefore, But in this case, P n = S n ( f ) df = 0 W = W 0. (6..9) = P o o P no = A c P M W 0. (6..0) P R = P U = A c P M; (6..) thus, = P ( ) R S =. (6..) ossb W 0 Therefore, the signal-to-noise ratio in a single-sideand system is equivalent to that of a DSB system. 6..4EffectofoiseonConventionalAM In conventional DSB AM, the modulated signal was given in Equation (3..6) as u(t) = A c [ + am(t)] cos π f c t. (6..3) Therefore, the received signal at the input to the demodulator is r(t) = [A c [ + am n (t)] + n c (t)] cos π f c t n s (t) sin π f c t, (6..4) where a is the modulation index and m n (t) is normalized so that its minimum value is. If a synchronous demodulator is employed, the situation is asically similar to the DSB case, except that we have + am n (t) instead of m(t). Therefore, after mixing and lowpass filtering, we have y (t) = [A c [ + am n (t)] + n c (t)]. (6..5) However, in this case, the desired signal is m(t), not + am n (t). The DC component in the demodulated waveform is removed y a DC lock and, hence, the lowpass filter output is y(t) = A cam n (t) + n c(t). (6..6)

5 80 Effect of oise on Analog Communication Systems Chapter 6 In this case, the received signal power is given y P R = A c [ + a P Mn ], (6..7) where we have assumed that the message process is zero mean. ow we can derive the output SR as = o AM 4 A c a P Mn 4 P n c = A c a P Mn 0 W = a P Mn + a P Mn = a P Mn P R + a P Mn 0 W = a P Mn + a P Mn = η A [ ] c + a P Mn 0 W ( S ), (6..8) where we have used Equation (6..) and η denotes the modulation efficiency. We can see that, since a P Mn < + a P Mn, the SR in conventional AM is always smaller than the SR in a aseand system. In practical applications, the modulation index a is in the range of The power content of the normalized message process depends on the message source. For speech signals that usually have a large dynamic range, P M is in the neighorhood of 0.. This means that the overall loss in SR, when compared to a aseand system, is a factor of or equivalent to a loss of db. The reason for this loss is that a large part of the transmitter power is used to send the carrier component of the modulated signal and not the desired signal. To analyze the envelope-detector performance in the presence of noise, we must use certain approximations. This is a result of the nonlinear structure of an envelope detector, which makes an exact analysis difficult. In this case, the demodulator detects the envelope of the received signal and the noise process. The input to the envelope detector is r(t) = [A c [ + am n (t)] + n c (t)] cos π f c t n s (t) sin π f c t; (6..9) therefore, the envelope of r(t) is given y V r (t) = [A c [ + am n (t)] + n c (t)] + n s (t). (6..30)

6 Section 6. Effect of oise on Amplitude Modulation Systems 8 ow we assume that the signal component in r(t) is much stronger than the noise component. With this assumption, we have therefore, we have a high proaility that After removing the DC component, we otain P (n c (t) A c [ + am n (t)]) ; (6..3) V r (t) A c [ + am n (t)] + n c (t). (6..3) y(t) = A c am n (t) + n c (t), (6..33) which is asically the same as y(t) for the synchronous demodulation without the coefficient. This coefficient, of course, has no effect on the final SR; therefore we conclude that, under the assumption of high SR at the receiver input, the performance of synchronous and envelope demodulators is the same. However, if the preceding assumption is not true, we still have an additive signal and noise at the receiver output with synchronous demodulation, ut the signal and noise ecome intermingled with envelope demodulation. To see this, let us assume that at the receiver input, the noise power is much stronger than the signal power. This means that V r (t) = [A c [ + am n (t)] + n c (t)] + n s (t) = A c ( + am n(t)) + n c (t) + n s (t) + A cn c (t)( + am n (t)) a (n c (t) + n s (t))[ + [ V n (t) + A cn c (t) V n (t) ( + am n(t)) A ] cn c (t) n c (t) + n s (t)( + am n(t)) ] = V n (t) + A cn c (t) V n (t) ( + am n(t)), (6..34) where (a) uses the fact that A c (+am n(t)) is small compared with the other components and () denotes n c (t) + n s (t) y V n(t), the envelope of the noise process; we have also used the approximation + ǫ + ǫ, for small ǫ, where ǫ = A cn c (t) n c (t) + n s (t)( + am n(t)). (6..35) We oserve that, at the demodulator output, the signal and the noise components are no longer additive. In fact, the signal component is multiplied y noise and is no longer distinguishale. In this case, no meaningful SR can e defined. We say that this system is operating elow the threshold. The suject of threshold and its effect on the performance By noise power at the receiver input, we mean the power of the noise within the andwidth of the modulated signal or, equivalently, the noise power at the output of the noise-limiting filter.

7 8 Effect of oise on Analog Communication Systems Chapter 6 of a communication system will e covered in more detail when we discuss the noise performance in angle modulation. Example 6.. We assume that the message is a wide-sense stationary random process M(t) with the autocorrelation function R M (τ) = 6 sinc (0,000τ). We also know that all the realizations of the message process satisfy the condition max m(t) = 6. We want to transmit this message to a destination via a channel with a 50-dB attenuation and additive white noise with the power spectral densitys n ( f ) = 0 = 0 W/Hz. We also want to achieve an SR at the modulator output of at least 50 db. What is the required transmitter power and channel andwidth if we employ the following modulation schemes?. DSB AM.. SSB AM. 3. Conventional AM with a modulation index equal to 0.8. Solution First, we determine the andwidth of the message process. To do this, we otain the power spectral density of the message process, namely, S M ( f ) =F[R M (τ)] = 6 ( ) 0,000 f, 0,000 which is nonzero for 0,000 < f < 0,000; therefore, W = 0,000 Hz. ow we can determine ( ) S as a asis of comparison: = P R 0 W = P R 0 0 = 08 P R. 4 Since the channel attenuation is 50 db, it follows that therefore, Hence, 0 log P T P R = 50; P R = 0 5 P T. = P T = 03 P T.. For DSB-SC AM, we have Therefore, and o = = 03 P T 0 3 P T 50 db = 0 5. = 0 5 P T = 00 Watts BW = W = 0,000 = 0,000 Hz 0 khz.

Problems from the 3 rd edition

(2.1-1) Find the energies of the signals: a) sin t, 0 t π b) sin t, 0 t π c) 2 sin t, 0 t π d) sin (t-2π), 2π t 4π Problems from the 3 rd edition Comment on the effect on energy of sign change, time shifting