DIGITAL COMMUNICATIONS SYSTEMS. MSc in Electronic Technologies and Communications

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1 DIGITAL COMMUNICATIONS SYSTEMS MSc in Electronic Technologies and Communications

2 Bandpass binary signalling The common techniques of bandpass binary signalling are: - On-off keying (OOK), also known as amplitude shift keying (ASK), which consists in activating and deactivating a sinusoidal carrier wave by using a unipolar binary signal. It is equivalent to a DSB-SC signal, where the modulating signal is a unipolar binary signal. - Binary phase shift keying (BPSK), which consists in shifting the phase of a sinusoidal carrier wave 0º or 80º by using a unipolar binary signal. It is equivalent to PM with a unipolar digital signal or modulating a DSB-SC signal by means of a polar digital waveform. - Frequency shift keying (FSK), which consists in shifting the frequency of a sinusoidal wave from a mark frequency (corresponding to sending a binary digit ) until a space frequency (corresponding to sending a binary digit 0 ), according to a baseband digital signal. It is equivalent to modulating an FM carrier by using a binary digital signal.

3 Bandpass binary signalling 3

4 Amplitude shift keying (ASK) - An OOK signal is given by: s( t) A m( t)cos t c c - The complex envelope of the OOK signal is: g( t) Ac m( t) for OOK - The power spectral density of this complex envelope will be given by: P ( f g ) Ac ( f ) T b sin ft ftb b for OOK when m(t) is a unipolar signal with peak amplitude of, in such a way that s(t) has an average normalized power equal to A c /. The power spectral density of the bandpass signal is obtained by means of a simple shift of the spectrum of the complex envelope towards frequencies f c and f c, additionally to multiplying them by a scale factor of ¼. 4

5 Amplitude shift keying (ASK) - When R = /T b is the data rate, the null-to-null bandwidth for an OOK signal is R, that is, exactly the double of that of the baseband signal (B T = B). - If we use a raised-cosine filter, for the case of binary signalling (D = R) we will have B r) D ( r) R BT ( r) R ( 5

6 Amplitude shift keying (ASK) - An OOK signal can be demodulated by using an envelope detector (non-coherent detection) or by using a product detector (coherent detection), since it is basically an AM signal. However, for optimal detection of an OOK signal corrupted by additive white Gaussian noise (AWGN), it is required from product detection and processing with a correlation filter. 6

7 Binary phase shift keying (BPSK) - A BPSK signal is given by s( t) A cos t D m( t) c c p where m(t) is a baseband polar data signal. If we expand the previous expression, we will be able to see that it consists in an AM signal s( t) Ac cos Dpm( t) cos ct Acsin Dpm( t) sin c - If we suppose that m(t) only takes values of, and that cos(x) and sin(x) are even and odd functions of x, the representation of the BPSK signal is reduced to - The peak phase deviation D p = Dq sets the value for the pilot carrier term. The digital modulation index, h, is defined as where Dq is the maximum peak-to-peak phase deviation (radians) along a symbol period (in the case of binary signals, the bit period). t A cos D cos t A m( t)sin D s( t) c p c c p sin ct pilot carrier term h Dq data term 7

8 Binary phase shift keying (BPSK) - It is clear that the smaller is D p the larger the power is wasted in the pilot carrier. The efficiency of the modulation can be increased to its maximum value by making D p = Dq = / (h = ). In that case, the BPSK signal is transformed in s( t) A m( t)sin t c c - Hence, BPSK in this optimal case is equivalent to a DSB-SC signal with a baseband polar waveform. The complex envelope of this signal is g( t) ja m( t) c for BPSK and its power spectral density P ( f ) g sin ft A ctb ftb b for BPSK where m(t) takes values of, in such a way that s(t) has an average normalized power of A c /. The spectrum of the BPSK signal can be easily obtained from the complex envelope in the same way as it was previously described. The BPSK 8 signal has a null-to-null bandwidth of R, just like the OOK signals.

9 Differential phase shift keying (DPSK) - The demodulation of BPSK signals requires from synchronous detection. However, despite a BPSK signal cannot be incoherently demodulated, if this one is differentially encoded previous to transmission, it will be possible to recover the data signal by using the differential decoder shown in the figure. The differentially encoded BPSK signal is known as DPSK. For optimal detection, we have to substitute the low-pass filter by a correlation filter with integrating and dumping, and the DPSK input signal has to be pre-filtered with a bandpass filter with impulse response function h(t)=p[(t-0.5t b )/T b ]cos( c t). In practice, it is usual to work with DPSK instead of BPSK, because it does not require from a carrier recovering circuit. 9

10 Frequency shift keying (FSK) - We can distinguish between two kinds of FSK signals, discontinuous-phase FSK and continuous-phase FSK. In the first case, we only need a switching device, which commutes, according to the value of the baseband binary signal, between two oscillators working at different frequencies. For this reason, the phase of the signal is usually discontinuous. The discontinuous-phase FSK is given by Ac cos t q s( t) Ac cos t q,, when a binary digit '' when a binary digit '0' is sent is sent where f is the mark frequency and f the space frequency. The continuous-phase FSK signals are generated by feeding a frequency modulator with the data signal. This continuous-phase FSK signal is given by s( t) A c s( t) Re cos ct D f t m( ) d for FSK jct jq ( t) g ( t) e, g( t) Ace, q ( t) D f where m(t) is the baseband signal. Although m(t) is discontinuous in the commutation instant, q(t) is continuous, because it is proportional to the integral of m(t). If the modulating signal is binary, it is called binary FSK. t m( ) d 0

11 Frequency shift keying (FSK) - The spectra of FSK signals, in the same way as those of FM, are difficult to evaluate, since g(t) is a non-linear function of m(t). However, we are going to suppose that we use as binary modulating signal a square waveform whose period is T 0 = T b, being the data rate R = /T b. The peak frequency deviation is DF = max[(/)dq(t)/dt] = D f / when m(t) takes values of. From the input square waveform, we will obtain a triangular phase function, and the digital modulation index is Dq h DFT where the equality with the modulation index for FM is only fulfilled if we take as bandwidth B = /T 0. The Fourier series of the complex envelope is n jn 0 g( t) c n e DF / T DF B / h n t A c sin / h n cn where f 0 = /T 0 = R/ and D = DF = h/t 0. 0 DF R 0 n sin f Frequency modulation index (FM) / h n / h n

12 Frequency shift keying (FSK) - The spectrum of the complex envelope is G( f ) cn ( f nf 0 ) cn n n f nr - From the previous expression is easy to deduce the spectrum of the FSK signal. Notice that it will be constituted by a series of delta functions which are R/ apart and grouped in the vicinity of the mark and space frequencies. The bandwidth of a FSK signal is given by the Carson s rule: B T = ( + )B, where = DF/B. Therefore, if we consider B = R (first null bandwidth) B T DF B - FSK signals can be demodulated with a frequency detector (non coherent) or by using a product detector (coherent detection). In this second case, we need two product detectors tuned to the mark and space frequencies, each followed by a low-pass filter. The output signals form the filters are subtracted by using a linear adder so as to obtain the demodulated binary signal. For optimal detection in front of AWGN, we need coherent detection and processing with a correlation filter in conjunction with a threshold device (comparator). DF R

13 Frequency shift keying (FSK) 3

14 Estimated spectrum for an ASK signal (R =, f c = 0) 4

15 Estimated spectrum for a BPSK signal (R =, f c = 0) 5

16 Estimated spectrum for an FSK signal (R =, f = 0, f = 0) 6

17 Estimated spectrum for an FSK signal (R =, f = 9, f = ) 7

18 Estimated spectrum for an FSK signal (R =, f = 9.5, f = 0.5) 8

19 Estimated spectrum for an FSK signal (R =, f = 9.9, f = 0.) 9

20 Quadrature phase shift keying (QPSK) and M-ary phase shift keying (M-PSK) - If by means of a binary signal, making use of a DAC, we generate a multilevel signal of M levels and apply this one to a PM transmitter, we will have the so-called M-ary phase shift keying (M-PSK). M-PSK for M = 4 is a special case known as quadrature phase shift keying (QPSK). In the figure are shown the two possible symbol constellations for a QPSK signal, that is, the allowed values for the complex envelope. 0

21 Quadrature phase shift keying (QPSK) and M-ary phase shift keying (M-PSK) - M-PSK transmission can also be generated by means of two carrier waves in quadrature, which modulate the x and y components of the complex envelope (instead of using a phase modulator) where: g( t) A e x y i i A A c c c jq ( t) cosq sen q i i x( t) jy( t) being i =,,..., M, and q i are the phase angles which are allowed for the M-PSK signal.

22 Quadrature amplitude modulation (QAM) - The signal generated using two quadrature carriers (see previous figure) is called quadrature amplitude modulation (QAM). The generalized QAM signal is s( t) x( t)cos t c y( t) sen t c - In the figure is shown a rectangular QAM constellation of 6 symbols (M = 6 levels), where the relationship between (R i, q i ) and (x i, y i ) can easily be evaluated from this figure. Components x i and y i are allowed to have four levels by dimension.

23 3 DIGITAL SYSTEMS FOR Quadrature amplitude modulation (QAM) - The x and y waveforms are represented by where D = R/l and (x n, y n ) denote one of the allowed values (x i, y i ) during the time which is necessary to transmit a symbol centred in t = nt s = n/d (we need T s s for sending each symbol), h (t) is the pulse waveform used for each symbol. Sometimes the synchronization between components x(t) and y(t) is compensated by T s / = /(D) s, then y(t) would be given by n n n n D n t h y t y D n t h x t x ) ( ) ( n n D D n t h y t y ) (

24 Quadrature amplitude modulation (QAM) - A popular type of staggered modulation for the case of QPSK (QAM where M = 4) is the so-called offset QPSK (OQPSK), where the data stream is divided in even and odd-numbered bits, each being transmitted via the cosine or the sine of the carrier, respectively. Moreover, an offset of T s / s (T b s) is introduced between both components. With this technique the maximum phase jumps are equal to ±90º, instead of the phase jumps of ±80º which can be obtained with conventional QPSK schemes. The phase jumps of 80º during symbol transitions can yield problems during the signal reception under highly dispersive channels or due to non-linear amplification, etc. A special case of OQPSK when h (t) is a sine pulse is minimum shift keying (MSK). 4

25 QPSK OQPSK y x m DIGITAL SYSTEMS FOR Quadrature amplitude modulation (QAM) - Example. Comparison between QPSK and OQPSK signals x y x y x y x y x y

26 Power spectral density of MPSK and QAM - The complex envelope of an MPSK or QAM is given by g ( t) n c f t n nt s - where c n is a complex random variable that represents the multilevel value of the pulse during the nth symbol, f(t) = P(t/Ts) is equivalent to a rectangular pulse with duration T s, D = /T s is the symbol rate (or baud rate). The Fourier transform of the square pulse is F( f ) T s sin ft s sin lft lt b fts lftb where T s = lt b (there exist l bits corresponding to each allowed multilevel value). In the case of symmetrical modulation, the power spectral density of the complex envelope for MPSK or QAM with data modulation with rectangular bit pattern is: P ( f ) g sin lft K lftb b b for MPSK and QAM 6

27 Power spectral density of MPSK and QAM - This power spectral density coincides with the power spectral density of QPSK when l =. The null-to-null bandwidth for MPSK or QAM is: B T R / l D - The spectral efficiency is: h R BT l bits/s Hz 7

28 Minimum shift keying (MSK) - Minimum shift keying (MSK) is another bandwidth-conservative technique, which is equivalent to OQPSK using sine pulses h (t). The MSK signal is a continuousphase FSK (CPM, continuous phase modulation) with the minimum modulation index (h = 0.5) which ensures orthogonality of the modulated signals. - The peak frequency deviation is DF h T - The complex envelope of the MSK signal is - where m(t) =, 0 < t < T b. Then b 4T b R 4 jq ( t) jdf g t A e A e 0 ( ) c c for MSK t m( ) d jq ( t) jt /(Tb ) g( t) Ac e Ac e x( t) jy( t), 0 t where the signs denote the possible data values during the time interval (0, T b ). T b 8

29 Minimum shift keying (MSK) - Therefore: x( t) A y( t) Acsin and the MSK signal is c cos t T t T b b,, 0 t T b 0 t T b s( t) x( t)cos t c y( t) sin t c - It can be observe as the change in the sign of m(t) during the interval (0, T b ) only affects to y(t), but not to x(t), in the signalling interval (0, T b ). Moreover, the pulse sin[t/(t b )] of y(t) is T b -second width. In addition, we can see as the sign of m(t) during the interval (T b, T b ) only affects to x(t) but not to y(t) in the signalling interval (T b, 3T b ). That is, the data modulate alternatively the components x(t) and y(t), hence MSK is equivalent to OQPSK in which the pulse shape is one-half cycle of a sinusoid. 9

30 Minimum shift keying (MSK) - Example. MSK signalling y x y x y x y x y x 30

31 Minimum shift keying (MSK) - Example. Comparison of the spectra of the MSK, QPSK and OQPSK signals 3

32 Error performance in bandpass communication systems - In the same way as in baseband communication systems, the optimal detector for bandpass communication systems is the correlation filter. In the figure is shown the receiver structure based on the correlation filtering for the binary case, where two parallel correlation branches are required. - The output signals from both correlators can be subtracted, such as is shown in the figure: z( T) z( T) z( T) 3

33 Error performance in bandpass communication systems - The output signal z(t) is given by a signal component a i (T) which is related with the transmitted symbol in that moment (for the binary case, a or a ) plus a noise component n 0 (T): z ( T ) a ( T ) n 0( T i ) - The detector has to decide if s or s has been sent according to if z(t) is bigger or smaller than g 0 : ( ) s a a z T g 0 s 33

34 Error performance in bandpass communication systems - Any detector which uses the previous structure to determine which symbol has been sent will present an optimal error performance under AWGN. Suppose, for example, the case of M-ary PSK (MPSK). In this case, we have that s i (t) can be expressed as: E i 0 t T s i ( t) cos0t T M i,, M - The factor E /T is added to normalize the expected output of the detector, then making the energy per symbol of the MPSK signal equal to E. On the other hand, we have that any of the previous signals s i (t) can be written in terms of the next set of orthonormal basis functions: ( t) cos( 0 T t ( t) sin( 0t) T ) 34

35 35 DIGITAL SYSTEMS FOR Error performance in bandpass communication systems - We will have that: being f i = i/m the phase shift of the transmitted symbol, with respect to zero degrees. M i T t t E t E t M i E t M i E t a t a t s i i i i i,, 0 ) ( sen ) ( cos ) ( sen ) ( cos ) ( ) ( ) ( f f

36 Error performance in bandpass communication systems - Observe as, after the coherent demodulation by using the two correlation branches, we obtain the in-phase and quadrature baseband components X and Y of the complex envelope for the received symbol. Therefore, it is logical to think of obtaining an error performance of this bandpass system equivalent to that of its baseband counterpart. Thus, for binary schemes we did have that: P B Q E b ( ) N 0 36

37 Error performance in bandpass communication systems - In the previous expression, is the cross-correlation coefficient between the transmitted symbols, which for antipodal-signalling such as BPSK is equal to -. Thus, the bit error probability of a BPSK communication system is: P B Q E N 0 b 37

38 Error performance in bandpass communication systems - For binary ASK or OOK, we have that = 0, hence: P B Q E b N 0 - For binary FSK, due to modulate the possible transmitting symbols by using orthogonal basis functions, we will have that = 0 again and, therefore, the bit error rate coincides with that obtained for OOK when coherent detection is used. Both ASK and FSK signals can be non-coherently demodulated by using envelope detectors and oscillators tuned to the frequencies corresponding to that of the transmitted symbols. These detectors, since they do not require from phase-synchronization with the received carrier waves, are less complex but, on the other hand, present a worse error performance than their optimal coherent counterparts. At best, the bit error rate of a non-coherent detector of FSK signals (and for OOK signals too) will be given by: P B exp E b N0 38

39 Error performance in bandpass communication systems 39

40 Error performance in bandpass communication systems - When we are working with M-ary schemes, the symbol error performance P E (M) for coherently detected MPSK signals, for large energy-to-noise ratios, can be expressed as P E ( M ) Q E N 0 s sin M where E s = E b log M is the energy per symbol and M = k is the size of the symbol set, being k the number of bits per symbol. For the case of MPSK signalling, the relationship between the probability of symbol error P E and the probability of bit error P B, when Gray code assignment is used, is approximately, M P B PE log M P k E (for P E ) - Hence, for QPSK (OQPSK and MSK), considering the previously said, we have that: P P P E, QPSK E, QPSK B, BPSK P B, QPSK PB, BPSK log M 40

41 Error performance in bandpass communication systems 4

42 Error performance in bandpass communication systems - Taking into account that the symbol rate R s is decreased by increasing the number of bits k represented per symbol, for a given data rate R: R s R / k and, in the case of using Nyquist filtering, the minimum-required bandwidth for transmitting at a symbol rate R s is: B T T s R s - The spectral efficiency for MPSK systems, and MQAM in general, taking into account the previous relationships, is: h R B T R R s R R / k k log M bit/s/hz 4

43 Error performance in bandpass communication systems - For rectangular constellation, Gaussian channel and matched filter reception, the bit-error probability of M-QAM, where M = k and k is even, is: P B ( / L) Q log L 3log L L E N 0 b 43

44 44 DIGITAL SYSTEMS FOR Error performance in bandpass communication systems - For coherently detected MFSK signals, the symbol-error probability satisfies that: - For non-coherently detected MFSK, an upper bound for the error probability is the following (this upper bound of P E is, logically, also valid for coherent detection): - For MFSK signalling, the relationship between the probability of symbol error P E and the bit-error probability P B is: 0 ) ( N E Q M M P s E 0 exp ) ( N E M M P s E / bits number of bits erroneous number of k k k E B k k E k k E E B M M P P P k k P P P

45 Error performance in bandpass communication systems 45

46 Error performance in bandpass communication systems - The MFSK schemes have the inconvenient, in contrast to MPSK and MQAM schemes, that by increasing M the required bandwidth is larger and larger. For non-coherently detected MFSK, the minimum tone spacing (separation between the frequencies of the different tones or carriers) is R s = /T s, being R s the symbol rate and T s the symbol duration. Hence, if we use M carriers, the minimum required bandwidth will be: B T M T s MR M - Therefore, the spectral efficiency for MFSK schemes is: h R BT s log M M R k MR log M bit/s/hz 46

47 Error performance in bandpass communication systems - Comparison between the spectral efficiency for the different schemes of bandpass digital signalling 47

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