Introduction to Analog And Digital Communications

Transcription

1 Introdution to Analog And Digital Communiations Seond Edition Simon Haykin, Mihael Moher

2 Chapter 9 Noise in Analog Communiations 9.1 Noise in Communiation Systems 9. Signal-to-Noise Ratios 9.3 Band-Pass Reeiver Strutures 9.4 Noise in Linear Reeivers Using Coherent Detetion 9.5 Noise in AM Reeivers Using Envelope Detetion 9.6 Noise in SSB Reeivers 9.7 Detetion of Frequeny Modulation (FM) 9.8 FM Pre-emphasis and De-emphasis 9.9 Summary and Disussion

3 Noise an broadly be defined as any unknown signal that affets the reovery of the desired signal. The reeived signal is modeled as r ( = s( + w( (9.1) s( is the transmitted signal w( is the additive noise 3

4 Lesson 1 : Minimizing the effets of noese is a prime onern in analog ommuniations, and onsequently the ratio of signal power is an important metri for assessing analog ommuniation quality. Lesson : Amplitude modulation may be deteted either oherently requiring the use of a synhronized osillator or non-oherently by means of a simple envelope detetor. However, there is a performane penalty to be paid for non-oherent detetion. Lesson 3 : Frequeny modulation is nonlinear and the output noise spetrum is paraboli when the input noise spetrum is flat. Frequeny modulation has the advantage that it allows us to trade bandwidth for improved performane. Lesson 4 : Pre-and de-emphasis filtering is a method of reduing the output noise of an FM demodulator without distorting the signal. This tehnique may be used to signifiantly improve the performane of frequeny modulation systims. 4

5 9.1 Noise in Communiation Systems The mean of the random proess Both noise and signal are generally assumed to have zero mean. The autoorrelation of the random proess. With white noise, samples at one instant in time are unorrelated with those at another instant in time regardless of the separation. The autoorrelation of white noise is desribed by N0 R w ( τ ) = δ ( τ ) (9.) The spetrum of the random proess. For additive white Gaussian noise the spetrum is flat and defined as N0 S w ( f ) = (9.3) To ompute noise power, we must measure the noise over a speified bandwidth. Equivalent-noise bandwidth is N = N B (9.4) Fig T B T 5

6 Fig. 9.1 Bak Next 6

7 9. Signal-to-Noise Ratios The desired signal, s (, a narrowband noise signal, n( x ( = s( + n( (9.5) For zero-mean proesses, a simple measure of the signal quality is the ratio of the varianes of the desired and undesired signals. Signal-to-noise ratio is defined by E[ s ( ] SNR = E[ n ( ] (9.6) The signal-to-noise ratio is often onsidered to be a ratio of the average signal power to the average noise power. 7

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9 Fig. 9. 9

10 Fig. 9. Bak Next 10

11 If the signal-to-noise ratio is measured at the front-end of the reeiver, then it is usually a measure of the quality of the transmission link and the reeiver front-end. If the signal-to-noise ratio is measured at the output of the reeiver, it is a measure of the quality of the reovered intormation-bearing signal whether it be audio, video, or otherwise. Referene transmission model This referene model is equivalent to transmitting the message at baseband. Fig

12 Fig. 9.3 Bak Next 1

13 1. The message power is the same as the modulated signal power of the modulation sheme under study.. The baseband low-pass filter passes the message signal and rejets outof-band noise. Aordingly, we may define the referene signal-tonoise ratio,, as SNR ref SNR ref = average power of the modulatedmessagesignal average power of noise mesured in the message bandwidth (9.11) A Figure of merit Figure of merit = post detetion SNR referenesnr Fig

14 Fig. 9.4 Bak Next 14

15 The higher the value that the figure of merit gas, the better the noise performane of the reeiver will be. To summarize our onsideration of signal-to-noise ratios: The pre-detetion SNR is measured before the signal is demodulated. The post-detetion SNR is measured after the signal is demodulated. The referene SNR is defined on the basis of a baseband transmission model. The figure of merit is a dimensionless metri for omparing sifferent analog modulation-demodulation shemes and is defined as the ratio of the post-detetion and referene SNRs. 15

16 9.3 Band-Pass Reeiver Strutures Fig. 9.5 shows an example of a superheterodyne reeiver AM radio transmissions Common examples are AM radio transmissions, where the RF hannels frequenies lie in the range between 510 and 1600 khz, and a ommon IF is 455 khz FM radio Another example is FM radio, where the RF hannels are in the range from 88 to 108 MHz and the IF is typially 10.7 MHz. s( = s ( os(πf s ( sin(πf I Q (9.1) The filter preeding the loal osillator is entered at a higher RF frequeny and is usually muh wider, wide enough to enompass all RF hannels that the reeiver is intended to handle. With the same FM reeiver, the band-pass filter after the loal osillator would be approximately 00kHz wide; it is the effets of this narrower filter that are of most interest to us. Fig

17 Fig. 9.5 Bak Next 17

18 9.4 Noise in Linear Reeivers Using Coherent Detetion Double-sideband suppressed-arrier (DSB-SC) modulation, the modulated signal is represented as s( = A m( os(π ft + θ ) f is the arrier frequeny m( is the message signal The arrier phase θ (9.13) In Fig. 9.6, the reeived RF signal is the sum of the modulated signal and white Gaussian noise w( After band-pass filtering, the resulting signal is x ( = s( + n( (9.14) Fig

19 Fig. 9.6 Bak Next 19

20 In Fig.9.7 The assumed power spetral density of the band-pass noise is illustrated For the signal s( of Eq. (9.13), the average power of the signal omponent is given by expeted value of the squared magnitude. The arrier and modulating signal are independent E[ s ( ] = E[( A os(π f t + θ )) ]E[ m ( ] (9.15) P = E[ m ( ] (9.16) A P E[ s ( ] = (9.17) Pre-detetion signal-to-noise ratio of the DSB-SC system A noise bandwidth B T The signal-to-noise ratio of the signal is DSB A P SNR = (9.18) pre N B 0 T Fig

21 Fig. 9.7 Bak Next 1

22 The signal at the input to the oherent detetor of Fig. 9.6 x( = s( + n ( os(πf n ( sin(πf I Q (9.19) v( = x( os(πf = + 1 ( A m( + n 1 ( A m( + n I ( ) I ( )os(4πf 1 n Q ( sin(4πf (9.0) 1+ osa os A os A = and sin Aos A = sin A These high-frequeny omponents are removed with a low-pass filter 1 y( = ( A m( + n I ( ) (9.1)

23 The message signal m( and the in-phase omponent of the filtered noise ( appear additively in the output. n I The quadrature omponent of the noise is ompletely rejeted by the demodulator. Post-detetion signal to noise ratio 1 The message omponent is A m(, so analogous to the omputation of the predetetion signal power, the post-detetion 1 signal power is A P where P is the average message power as 4 defined in Eq. (9.16). 1 The noise omponent is n I ( after low-pass filtering. As desribed in Setion 8.11, the in-phase omponent has a noise spetral density of N0 over the bandwidth from B. If the low-pass filter T / to BT / has a noise bandwidth W, orresponding to the message bandwidth, whih is less than or equal to /, then the output noise power is B T W E[ n I ( ] = N df W 0 = N W 0 (9.) 3

24 Post-detetion SNR of SNR DSB post = 1 4 ( A (N ) P W) 0 A P = N W (9.3) Post-detetion SNR is twie pre-detetion SNR. Figure of merit for this reeiver is DSB SNR post Figure of merit = = 1 SNR We lose nothing in performane by using a band-pass modulation sheme ompared to the baseband modulation sheme, even though the bandwidth of the former is twie as wide. ref 4

25 9.5 Noise in AM Reeivers Using Envelope Detetion The envelope-modulated signal s( = A (1 + k m( )os(πf a (9.4) The power in the modulated part of the signal is E[(1 + k a m( ) ] = E[1 + k = 1+ k = 1+ k a a P a m( + k E[ m( ] + k a a m E[ m ( ] ( ] (9.5) The pre-detetion signal-to-noise ratio is given by SNR AM pre = A (1 N + 0 k B a T P) (9.6) Fig

26 Fig. 9.8 Bak Next 6

27 Model the input to the envelope detetor as x( = s( + n( = [ A + A k a m( + n I ( ]os(πf n Q ( sin(πf (9.7) The output of the envelope detetor is the amplitude of the phasor representing x( and it is given by y( = envelope of x( = {[ A (1 + k m( ) + n ( ] a I + n Q ( } 1/ (9.8) Using the approximation A + B A when A >> B, y( A + A k m( n ( a + I (9.9) Fig

28 Fig. 9.9 Bak Next 8

29 The post-detetion SNR for the envelope detetion of AM, SNR AM post A k P = a N W 0 (9.30) This evaluation of the output SNR is only valid under two onditions: The SNR is high. is adjusted for 100% modulation or less, so there is no distortion of the signal envelope. The figure of merit for this AM modulation-demodulation sheme is AM SNR post ka P Figure of merit = = (9.31) SNR 1+ k P ref a Fig Fig

30 Fig Bak Next 30

31 Fig Bak Next 31

32 In the experiment, the message is a sinusoidal wave m( = Asin(πf m, We ompute the pre-detetion and post-detetion SNRs for samples of its signal. These two measures are plotted against one another in Fig. 9.1 for k a = 0.3. The post-detetion SNR is omputed as follows: The output signal power is determined by passing a noiseless signal through the envelope detetor and measuring the output power. The output noise is omputed by passing plus noise through the envelope detetor and subtrating the output obtained form the lean signal only. With this approah, any distortion due to the produt of noise and signal omponents is inluded as noise ontribution. From Fig.9.1, there is lose agreement between theory and experiment at high SNR values, whih is to be expeted. There are some minor disrepanies, but these an be attributed to the limitations of the disrete time simulation. At lower SNR there is some variation from theory as might also be expeted. Fig

33 Fig. 9.1 Bak Next 33

34 9.6 Noise in SSB Reeivers The modulated wave as A A s( = m( os(π f + mˆ ( sin(πf (9.3) We may make the following observations onerning the in-phase and quadrature omponents of s( in Eq. (9.3) : 1. The two omponents m ( and mˆ ( are unorrelated with eah other. Therefore, their power spetral densities are additive.. The Hilbert transform m ˆ ( is obtained by passing m( through a linear filter with transfer funtion jsgn( f ). The squared magnitude of this transfer funtion is equal to one for all f. Aordingly, m ( and mˆ ( have the same average power p. 34

35 35 The pre-detetion signal-to-noise ratio of a oherent reeiver with SSB modulation is The band-pass signal after multipliation with the synhronous osillator output is After low-pass filtering the, we are left with (9.33) W 4 SNR 0 SSB pre N P A = (9.34) ) sin(4 ) ( ) ˆ ( ) os(4 ) ( ) ( ) ( ) ( ) )os( ( ) ( t f t n m t A t f t n m t A t n m t A t f t x t v Q I I π π π = = (9.35) ) ( ) ( ) ( 1 + = t n m t A t y I v( ) os( t πf

36 The spetrum of the in-phase omponent of the noise n I ( is given by s N I ( f ) = S 0, N ( f f ) + S N ( f + f ), B f B (9.36) otherwise sn I ( f ) = 0 N 0,, W f otherwise W (9.37) The post-detetion signal-to-noise ratio SNR SSB post A = 4N P W The figure of merit for the SSB system Figure of merit = 0 SNR SNR SSB post ref (9.38) = 1 (9.39) 36

37 Comparing the results for the different amplitude modulation shemes There are a number of design tradeoffs. Single-sideband modulation ahieves the same SNR performane as the baseband referene model but only requires half the transmission bandwidth of the DSC-SC system. SSB requires more transmitter proessing. 37

38 9.7 Detetion of Frequeny Modulation (FM) The frequeny-modulated signal is given by s( = A os + ( ) (9.40) t πf t πk f m τ dτ 0 Pre-detetion SNR The pre-detetion SNR in this ase is simply the arrier power A C / divided by the noise passed by the bandpass filter, N 0 B T ; namily, SNR = AM pre A N B 0 T 1. A slope network or differentiator with a purely imaginary frequeny response that varies linearly with frequeny. It produes a hybridmodulated wave in whih both amplitude and frequeny vary in aordane with the message signal.. An envelope detetor that reovers the amplitude variation and reprodues the message signal. Fig

39 Fig Bak Next 39

40 Post-detetion SNR The noisy FM signal after band-pass filtering may be represented as x ( = s( + n( (9.41) n( = n ( os(πf n ( sin(πf I (9.4) We may equivalently express n( in terms of its envelope and phase as Q n( = r( os[π f t + φ ( ] n (9.43) Where the envelope is And the phase is r( = [ n φ n ( = ( I + tan 1 n n n Q I Q ( ] ( ( 1/ (9.44) (9.45) Fig

41 Fig Bak Next 41

42 We note that the phase of s( is φ( = πk f 0 t m( τ ) dτ (9.46) The noisy signal at the output of the band-pass filter may be expressed as x( = = s( + n( A os[π f t + φ( ] + r( os[πf t + φ ( ] n (9.47) The phase θ ( of the resultant is given by θ ( = φ( + tan 1 A r( sin( ψ ( ) + r( os( ψ ( ) (9.48) Fig

43 Fig Bak Next 43

44 Under this ondition, and noting that tan 1 ξ ξ sin eξ << 1, the expression for the phase simplifies to r( θ ( = φ( + sin[ ψ ( ] A Then noting that the quadrature omponent of the noise is n Q ( = r( sin[ φ ( ], n we may simplify Eq.(9.49) to nq ( θ ( t ) = φ( + A (9.50) (9.49) t nq ( θ ( t ) πk f m( τ ) dτ + 0 A (9.51) The ideal disriminator output 1 dθ ( v( = π dt = k m( n ( (9.5) f + d 44

45 The noise term ( is defined by n d n d ( = 1 πa dn Q dt ( (9.53) The additive noise at the disriminator output is determined essentially by the quadrature omponent n Q ( of the marrowband noise n(. jπf jf G ( f ) = = (9.54) π A A The power spetral density S ( f ) of the quadrature noise ompinent N Q n Q ( as follows; S ( f ) = G( f ) S ( f ) N d = f A S N Q N Q ( f ) (9.55) 45

46 Power spetral density of the noise ( is shown in Fig.9.16 S N d N0 f ( f ) = A 0, BT, f < (9.56) otherwise Therefore, the power spetral density S ( f ) of the noise n N ( t 0 ) 0 appearing at the reeiver output is defined by n d S N 0 N0 f ( f ) = A 0,, f < W (9.57) otherwise Average post - detetion noise power N = A 0 W W f N0W = 3A 3 df (9.58) Fig

47 Fig Bak Next 47

48 3A k P FM SNR post = f 3 N W 0 (9.59) Figure of merit Figure of merit = SNR SNR FM post ref k f P = 3 W = 3D = 3A k N0W A N W 0 f P 3 (9.60) The figure of merit for an FM system is approximately given by Figure of merit 3 BT 4 W (9.61) 48

49 Thus, when the arrier to noise level is high, unlike an amplitude modulation system an FM system allows us to trade bandwidth for improved performane in aordane with square law. 49

50 50

51 51

52 Threshold effet At first, individual liks are heard in the reeiver output, and as the pre-detetion SNR dereases further, the liks merge to a rakling or sputtering sound. At and below this breakdown point, Eq.(9.59) fails to aurately predit the post-detetion SNR. Computer experiment : Threshold effet with FM Complex phasor of the FM signal is given by { } jπk m( τ ) d s( = A exp τ Similar to the AM omputer experiment, we measure the predetetion and post-detetion SNRs of the signal and ompare the results to the theory developed in this setion. f t 0 Fig

53 Fig Bak Next 53

54 9.8 FM Pre-emphasis and De-emphasis To ompensate this distortion, we appropriately pre-distort or preemphasize the baseband signal at the transmitter, prior to FM modulation, using a filter with the frequeny response 1 H ( f ) pre = f < H ( f ) de W (9.6) The de-emphasis filter is often a simple resistane-apaitane (RC) iruit with 1 H ( f ) = de (9.63) f 1+ j f3db At the transmitting end, the pre-emphasis filter is H ( f ) = 1+ pre j f f 3dB (9.64) Fig

55 Fig Bak Next 55

56 56 The modulated signal is approximately Pre-emphasis an be used to advantage whenever portions of the message band are degraded relative to others. + + = + + = ) ( ) ( os ) ( ) ( os ) ( 0 0 m t k ds s m k t f A ds ds s dm s m k t f A t s f t f t f α π π π α π π

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58 9.9 Summary and Disussion We analyzed the noise performane of a number of different amplitude modulation shemes and found: 1. The detetion of DSB-SC with a linear oherent reeiver has the same SNR performane as the baseband referene model but requires synhronization iruitry to reover the oherent arrier for demodulation.. Non-suppressed arrier AM systems allow simple reeiver design inluding the use of envelope detetion, but they result in signifiant wastage of transmitter power ompared to oherent systems. 3. Analog SSB modulation provides the same SNR performane as DSB- SC while requiring only half the transmission bandwidth. In this hapter, we have shown the importane of noise analysis based on signal-to-noise ratio in the evaluation of the performane of analog ommuniation systems. This type system, be it analog or digital. 58