Signal Transmission and Modulation

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1 Signal ransmission and Modulation Prof. Dr.-Ing. Ingolf Willms Partly based on the script of Prof. homas Kaiser Prof. Dr.-Ing. I. Willms ransmission and Modulation S. 1

2 Chapter 4 Digital M 4.1 Basics of detection theory Prof. Dr.-Ing. I. Willms ransmission and Modulation S.

3 4.1 Basics of Detection heory Digital M: - required is to transmit data streams with lowest bit error rate (BER) - thus optimal distinguishment of binäry data on the base of a sequence of predefined functions are needed - combining M binary data leads to distinguishing M symbols zu unterscheiden Prof. Dr.-Ing. I. Willms ransmission and Modulation S. 3

4 4.1 Basics of Detection heory M method applied now: Signal x(t) is transmitted in case (H 1 ) of logical 1, no signal in case (H 0 ) of logical 0. Signal x(t) has a duration of and an energy of E x. Additive Gausssian noise n(t) is assumed Für die beiden Fälle H 0 und H 1 gilt damit: H : yt () nt () H : yt () xt () nt () 0 1 Ex x () t dt 0 Prof. Dr.-Ing. I. Willms ransmission and Modulation S. 4

5 4.1 Basics of Detection heory he receiver has to decide on the base of y(t) for the case of H 0 or H 1. Instead of y(t) now coefficients of a set of orthonormal functions f n (t) describing it are evaluated for which hold: yt () Yf() t mit Y yt () f() tdtund E 1n n n n n f n0 t0 Due to linear relations of coefficients and the functions hold: xt () X f () t mit X x() t f () tdt n n n n n0 t0 nt () N f () t mit N n() t f () tdt n n n n n0 t0 n Y X N n n n Prof. Dr.-Ing. I. Willms ransmission and Modulation S. 5

6 4.1 Basics of Detection heory he best detection situation can be obtained by choosing a proper set of functions f n (t) with: his gives: f () ()/ 0 t x t Ex xt () X f() t X f() t E f() t n0 n n 0 0 x 0 xt () Ex 0 x t f0 t dt x t dt t0 t0 Ex Ex due to X ( ) ( ) ( ) which leads to X E. Important is: X 0 0 x 0 Prof. Dr.-Ing. I. Willms ransmission and Modulation S. 6

7 4.1 Basics of Detection heory For making the decision now the follwing data have to be compared: H : Y N Y N Y N H : Y X N Y X N N Y N hus only on the basis of 1 Y y() t f () t dt y() t x() t dt 0 0 t0 Ex t0 the decision can be made. In case of H, Y is always larger than in case of H! Prof. Dr.-Ing. I. Willms ransmission and Modulation S. 7

8 4.1 Basics of Detection heory Example with separation into sine functions and matching x(t) he X n here represent the Fourier coefficients b n xt () rectt ( / -0.5)sin( t) E x xt () X f() t mit X xt () f() tdt n n n n n0 t0 0 0 sin ( t) dt x f ( t) x( t) / E rect( t/ - 0.5)sin( t) fn() t rect( t/ 0-0.5)sin( n0t) 0 Prof. Dr.-Ing. I. Willms ransmission and Modulation S. 8

9 4.1 Basics of Detection heory For M orthonormal functions x m (t) and for possibly different signal energies the following block diagram for AWGN channels results. yt () x1( t) x () t dt... dt 1 E 1 x 1 E x Choose the Largest xm true () t xm () t 0... dt 1 E M x Prof. Dr.-Ing. I. Willms ransmission and Modulation S. 9

10 4.1 Basics of Detection heory Alternative method for realising the multiplication&integration: he operations are realised by a matched filter. 1 Y0 y() t x() t dt. E 0 x 0 1 he operation replaces Y0 y( ) x( ) d E x 0 1 by Y0 y( ) h ( ) d with x( ) h ( ) or h( ) x ( ) E x his is achieved by a filter and sampling the output signal at t : t 1 1 Y0 g() t t y( ) h( t ) d y( ) h( ) d E x t 0 Ex 0 Prof. Dr.-Ing. I. Willms ransmission and Modulation S. 10

11 4.1 Basics of Detection heory 1 E 1 x x ( t) 1 + y() t x ( t) + 1 E x yt () Choose the Largest xm true () t x ( t) M sample = t + 1 E M x Prof. Dr.-Ing. I. Willms ransmission and Modulation S. 11

12 Chapter 4 Digital M 4. Pulse Amplitude Modulation (PAM) Mit freundlicher Unterstützung von Pearson-Studium Die Abbildungen sind z.. entnommen aus dem Buch Grundlagen der Kommunikationstechnik von J.G. Proakis, M. Salehi Prof. Dr.-Ing. I. Willms ransmission and Modulation S. 1

13 4. PAM Criteria for good transmission Robustness concerning noise Low bandwith usage Low x power Prof. Dr.-Ing. I. Willms ransmission and Modulation S. 13

14 4. PAM ransmission of data stream by amplitude modulated signals PAM with 0 and A as signal amplitude Bipolar (binary antipodal) with amplitudes A and +A Non binary PAM methods use M amplitude values Binary and quaternary signals Prof. Dr.-Ing. I. Willms ransmission and Modulation S. 14

15 4. PAM Binary Modulation Each binary value corresponds to a signal with duration D hus no additional coding required Non binary Modulation L bits are combined to one symbol hus L symbols result ransmission time for one symbol is S = L * D L less changes of signal levels For equal maximum level L = L smaller amplitude values need to be detected Compromise of S/N ratio and bandwith usage is required Prof. Dr.-Ing. I. Willms ransmission and Modulation S. 15

16 0 d 4. PAM Binary Modulation x t x t ASK BPSK d D 3d d Quaternary Modulation x 4ASK t S D 7d 5d 3d d Octonary Modulation x 8ASK t S 3 D Prof. Dr.-Ing. I. Willms ransmission and Modulation S. 16

17 4. PAM Minimisation of bandwith usage Avoiding of rectangular signal form due to si-form of spektrum Alternatives of signal forms in time-domain: si-function, Gauss function Prof. Dr.-Ing. I. Willms ransmission and Modulation S. 17

18 4. PAM Bandpass signals Modulation is typically required (matching channel properties) Sequence of Gauss funktions form the signal in base band Modulation of this signal by AM, PM or FM Gauss signal and its Spectrum of baseband (a) and AM-Modulator AM signal (b) Prof. Dr.-Ing. I. Willms ransmission and Modulation S. 18

19 Chapter 4 Digital ransmission 4.3 ransmission using orthogonal signals Prof. Dr.-Ing. I. Willms ransmission and Modulation S. 19

20 4.3 ransmission using orthogonal signals Extended example of S.8 with separation into sine and cosine functions and corresponding x m (t) functions X im correspond to Fourier coefficients a n and b n x () t X f () t mit m1, m 0m 0m x () t rect( t/ -0.5)cos( t) E x1 cos ( 0t) dt 0 f () t x ()/ t E rect( t/ -0.5)cos( t) 01 1 x f () t rect( t/ -0.5)cos( n t) n x () t rect( t / -0.5)sin( t) E x sin ( 0t) dt 0 f () t x ()/ t E 0 x f () t rect( t / -0.5)sin( n t) n Prof. Dr.-Ing. I. Willms ransmission and Modulation S. 0 Ein

21 4.3 ransmission using orthogonal signals M orthogonal functions x m (t) are used in detection yt () x () t 1 x () t dt... dt 1 E 1 x 1 E x Choose the Largest xm true () t x () M t 0... dt 1 E M x Prof. Dr.-Ing. I. Willms ransmission and Modulation S. 1

22 4.3 ransmission using orthogonal signals Example for coding in the transmitter For binary value 0 x 1 (t) is transmitted For binary value 1 x (t) is transmitted he value of 1 Y yt () f () tdt ytx () () tdtmit m1, 0m 0m m t0 Ex t0 gives the criteria for choosing one hypothesis For noise free transmission it results: For binary value 0 : Y 01 = Y K > 0, Y 0 = 0 For binary value 1 : Y 01 = 0, Y 0 = Y K > 0 with 1 1 K () () = m m() x mit 1, m Ex t0 Ex t0 Y y t x t dt x t dt E m m Prof. Dr.-Ing. I. Willms ransmission and Modulation S. m m

23 4.3 ransmission using orthogonal signals Constellation diagram he values Werte für Y 01 und Y 0 can be represented in a graphic for good hus the degree of robustness can be figured out (0,Y K ) Y 0 (Y K,0) Y 01 Distance between marked points indicates degree of robustness against noise Prof. Dr.-Ing. I. Willms ransmission and Modulation S. 3

24 4.3 ransmission using orthogonal signals Other orthogonal base band signals Sine and cosine functions are orthogonal his also holds for sine/cosine signals of n times the fundamental frequency Walsh functions are orthogonal Other functions are known and can be specified examples of 4 orthogonal functions Prof. Dr.-Ing. I. Willms ransmission and Modulation S. 4

25 Chapter 4 Digital M 4.4 Quadrature AM Prof. Dr.-Ing. I. Willms ransmission and Modulation S. 5

26 4.4 QAM Principle of QAM QM uses different amplitude and phase levels of the carrier he number of M symbols depends on possibilities of combing amplitude and phase levels x () t a cos( t) b sin( t) QAM m c m c a b cos( tarctan( b / a ) mit m 1,,... M m m c m m Prof. Dr.-Ing. I. Willms ransmission and Modulation S. 6

27 4.4 QAM Example with amplitude and phase values 4 different base band signals result his QAM version can also be understood as phase shift keying (4-PSK) verstanden werden he constellation diagram describes here real and imaginary part of the equivalent LP signal of the carrier (0,Y K ) ( Y K,0) Y 0 (Y K,0) Y 01 (0, Y ) a) Constellationsdiagramm for M = 4 K b) he same degree of robustness results for a rotated diagram, see next slide Prof. Dr.-Ing. I. Willms ransmission and Modulation S. 7

28 4.4 QAM 4 Signals of the constellation diagram a) und b): Zu a): x () t cos( t) rect( t/ 0.5) 1 x () t sin( t) rect( t/ 0.5) x () t cos( t) rect( t/ 0.5) 3 x () t sin( t) rect( t/ 0.5) 4 c c c c Zu b): x () t 1/ (cos( t ) sin( t )) rect( t / 0.5) cos( t /4) rect( t / 0.5) 1 x ( t) 1 / ( cos( t ) sin( t )) rect( t / 0.5) cos( t 3 / 4) rect( t / 0.5) x () t 1/ ( cos( t ) sin( t )) rect( t / 0.5) cos( t 3 / 4) rect( t / 0.5) 3 4 c c c c c c c c x () t 1/ (cos( t ) sin( t )) rect( t / 0.5) cos( t /4) rect( t / 0.5) c c c c Prof. Dr.-Ing. I. Willms ransmission and Modulation S. 8

29 /4 1 x k 4PSK t t k x x i,4psk q,4psk t t Prof. Dr.-Ing. I. Willms ransmission and Modulation S. 9 t t t

30 4.4 QAM Additional extensions for M > 4 Additional amplitude and phase levels give more symbols Example for M = 16 x () t ( 5/3 i /3) rect( t / -0.5)cos( t) 1i 4 i x () t ( 5/3 i /3) rect( t / -0.5)sin( t) 5i 8 i 0 0 c c Constellation diagram for M = 16 Prof. Dr.-Ing. I. Willms ransmission and Modulation S. 30

31 M=4 M=8 4.4 QAM x1 x x3 x x x1 x x3 x 4 x5 x6 x he graphics show the constellation diagrams for comparison of 4- ASK and 8-ASK Prof. Dr.-Ing. I. Willms ransmission and Modulation S. 31

32 4.4 QAM Further variants In principle there is no limit for the number of symbols Different constellation diagrams can be selected Large M values lead to high sensitivity to noise Reason: Low distances between points in constellation diagram Prof. Dr.-Ing. I. Willms ransmission and Modulation S. 3

33 4.4 QAM Modulator for QAM he filters specify the behaviour of impulses in time domain Prof. Dr.-Ing. I. Willms ransmission and Modulation S. 33

34 4.4 QAM Receiver for QAM signals Prof. Dr.-Ing. I. Willms ransmission and Modulation S. 34

35 Chapter 4 Digital M 4.5 PSK and FSK Prof. Dr.-Ing. I. Willms ransmission and Modulation S. 35

36 4.5 PSK and FSK Principles for PSK/FSK Constant amplitude for all symbols bei suitable modulation of I/Q carrier Constellation diagram thus always includes a circle Again distance of points corresponds to robustness Constellation diagrams for different number of symbols for PSK Prof. Dr.-Ing. I. Willms ransmission and Modulation S. 36

37 4.5 PSK and FSK Properties Similar to PM/FM of analog signals (e,g. conc. power efficiency) Problem: Jumps due to chnges in the phase enlarge the bandwidth Modulated carrier signal for M = 4 (4-PSK or QPSK) Prof. Dr.-Ing. I. Willms ransmission and Modulation S. 37

38 4.5 PSK and FSK Gray-Coding In principle freedom is given conceerning relation of coed and symbols Gray coding reduces also here bit errors k for Dn 0 and Dn for Dn1 and Dn for Dn 1 and Dn for Dn 0 and Dn 11 4 Example for a relation of bit combinations and phase values of the modulated carrier Prof. Dr.-Ing. I. Willms ransmission and Modulation S. 38

39 4.5 PSK and FSK PSK-Demodulation For that I- and Q-signal have to be determined A PLL takes care of carrier phase estimation Demodulator for PSK signals Prof. Dr.-Ing. I. Willms ransmission and Modulation S. 39

40 Offset-PSK Phase changes lead to short-time drop of carrier amplitude An offset (delay ) between I- and Q-Signal prevents this d d d 4.5 PSK and FSK d d f 1 d d d f 0 Possible signal changes of 4-PSK Signal changes of offset-psk Prof. Dr.-Ing. I. Willms ransmission and Modulation S. 40

41 d x D k 4OPSK t t k d d d d d x x i,4opsk q,4opsk t t Prof. Dr.-Ing. I. Willms ransmission and Modulation S. 41 t t t

42 4.5 PSK and FSK Differential PSK (DPSK) In DPSK only phase chnages are evaluated In DQPSK theer are only phase jumps of ±45 and ±135 Phase jumps of 180 are avoided his is achieved by alternating angle values (different for even and odd clocks) For DQPSK holds: 3 k 0,,, k 1,,, For phase differences hold: k k 1 k Prof. Dr.-Ing. I. Willms ransmission and Modulation S. 4

43 Advantages of DPSK Almost constant amplitude DPSK needs no knowledge of exact carrier phase! x q d 4.5 PSK and FSK Even K Odd K d d d x i he constellation diagram with possible phase values of DQPSK Prof. Dr.-Ing. I. Willms ransmission and Modulation S. 43

44 4.5 PSK and FSK FSK signals Modulation comparable to FM In FSK also phase jumps can happen due to symbol changes Only discrete frequencies are transmitted Signals are almost orthogonal Für x signal (with non-symmetrical frequency bands with regard to carrier frequency holds: t S xm t cos C mdt mrect S with as minimum difference frequency and m = 0, 1,... M-1 d By changing of frequencies by (M-1)ω d / a symmetrical spektrum can be realised. Prof. Dr.-Ing. I. Willms ransmission and Modulation S. 44

45 4.5 PSK and FSK Orthogonality property of FSK signals Here signals of different symbols are tested for othogonalityt his test leads to: m k x t x t dt S 0 C d C d cos m t cos k t dt S S 1 1 cos C m k dt dt cos dm ktdt 0 0 sin C m kds S sin dm k S S. mk mk C d S d S he first expression drops close to zero for large argument, the second expression is zero for: d S Prof. Dr.-Ing. I. Willms ransmission and Modulation S. 45.

46 4.5 PSK and FSK Demodulation of M-FSK signals Prof. Dr.-Ing. I. Willms ransmission and Modulation S. 46

47 4.5 PSK and FSK Continuos PSK and FSK An FSK modulator can be realised by switching M different oszillators Avoiding phase jumps reduces bandwidth he same holds for jumps of instantanous frequency Realisation by memories and by Continuos phase jumps Prof. Dr.-Ing. I. Willms ransmission and Modulation S. 47

48 4.5 PSK and FSK Points in constellation diagram moves with time! FSK gives movement at constant radius In CPFSK phase does not jump Shown points represent state at end of a clock Carrier phase for binary CPFSK Prof. Dr.-Ing. I. Willms ransmission and Modulation S. 48

49 4.5 PSK and FSK Carrier phase for quaternary CPFSK Prof. Dr.-Ing. I. Willms ransmission and Modulation S. 49

50 4.5 PSK and FSK MSK signals (Special case of CPFSK) Orthogonality for smallest frequency change is looked for equivalent LP carrier signal s are consideerd Phase values at end of the clock are m I times of π I S x t e x t e S 1 m mi j t j t S Due to complex valued LP signals holds: S * jmi x t x t dt x t dt e si m 0 1 m I is minimum value for orthogonality 1 1 S I 0 0! Phasen rising about / in period S gives instantaneous frequency of d. Prof. Dr.-Ing. I. Willms ransmission and Modulation S. 50 S

51 4.5 PSK and FSK a) I-Signal (real part of equiv. LP signal of MSK-modulated carrier) b) Q-Signal (imaginary part of it) c) MSK-Signal Prof. Dr.-Ing. I. Willms ransmission and Modulation S. 51

52 4.5 PSK and FSK Power spectrum desnity of MSK compared to offset QPSK [Gronemeyer und McBride, 1976] Prof. Dr.-Ing. I. Willms ransmission and Modulation S. 5

53 4.5 PSK and FSK GMSK signals Also MSK still produces frequency jumps Further reduction of bandwidth results by LP filterung A Gaussian filter is typically used ransfer function of this filter: ln For its impulse response holds: g H e with as cut-off frequency GAUSS g hgauss t e ln g t ln g Prof. Dr.-Ing. I. Willms ransmission and Modulation S. 53

54 4.5 PSK and FSK Impulse response of Gaussian filters for different cut-off frequencies with ω 3dB = ω g Prof. Dr.-Ing. I. Willms ransmission and Modulation S. 54

55 4.5 PSK and FSK D t t GMSK MSK GMSK Comparison of phase of binary MSK and GMSK Prof. Dr.-Ing. I. Willms ransmission and Modulation S. 55 t

56 4.5 PSK and FSK Application of GMSK in mobile phones Prof. Dr.-Ing. I. Willms ransmission and Modulation S. 56

57 Chapter 4 Digital ransmission 4.6 Aspects of Symbol Interference Prof. Dr.-Ing. I. Willms ransmission and Modulation S. 57

58 4.6 Symbolinterferenz Relation bandwidth to data rate For the AWGN situation an optimal matched filter reception is assumed (see chap. 4.1) ASK is the chosen transmission method Without noise with yt ( ) xt ( ) for each symbol the value 1 Y0 y( ) h( ) d = with x( t) h( t) is determined which corresponds to filtering E x 0 (correlation) of the signal yt ( ) with yt ( ) x ( - t). For a symbol sequence Sk ( ) at the matched filter output the signal results: ( Skxt ( ) ( - k)) * x ( - t) Sk ( ) ( t- - k) with the ACF ( t) x( ) xt ( ) d xx xx k0 k0 Prof. Dr.-Ing. I. Willms ransmission and Modulation S. 58

59 4.6 Symbolinterferenz Due to sampling the following sequence of values results: xx S(k) ( k) k0 xx For preventing an overlapping of neighboured symbol values (i.e. no symbol interference) and because of the even ACF it must hold: ( k) (k) 0 k 0 (1st Nyquist criteria) xx Example: Binary sequence with only non-zero values S(k) (k) (k 1) k0 0 0 For that the following output signals result: S(k) (t - - k) S(0) (t - ) S(1) (t - ) xx xx xx For the sample values therefore hold: At t : S(k) xx( - ) xx( ) xx(0) xx(-) At t : S(k) ( -) ( ) () (0) xx xx xx xx Prof. Dr.-Ing. I. Willms ransmission and Modulation S. 59

60 4.6 Symbolinterferenz Possible ACF s: Signals limited to symbol duration such as the rect-funktion Rectangular signals give a ACF s in triangular form Disadvantage: For a perfect transmission in principle an infinitely large bandwidth is needed he ideal frequency limited signal is the si function Disadvantage of the si function: Symbol interference in case of unprecise sampling points xx t si( t/ ) S his signal demands a bandwidth of g (50% of the clock rate) Prof. Dr.-Ing. I. Willms ransmission and Modulation S. 60

61 4.6 Symbolinterferenz Problems in the application of the si-function High demands on exact sampling points he problem is eased by a modification of the si-function Disadvantge of this method: Not optimised use of the bandwidth Original and modified si-function and its spectra Prof. Dr.-Ing. I. Willms ransmission and Modulation S. 61

62 4.6 Symbolinterferenz Other effects In practice non-ideal channels are often given with strong changes in the frequency response even in the case of transmission at a small bandwidth his has similar effects as a non-si-form of the ACF Counter measure: Application of a bandwith efficient method called Orthogonal Frequency Division Multiplex (OFDM) Basic idea: Separation of the whole channel into many sub-channels with a constant frequency response and equal bandwidth Prof. Dr.-Ing. I. Willms ransmission and Modulation S. 6

63 4.6 Symbolinterferenz FSK signals in the subchannels All K subchannels will be used simultaneously Each subchannel has its own carrier he symbolrate and the corresponding period = K S of each subchannel will be adjusted to the K-times lower bandwidth (in comparison to the transmission without OFDM) Also the different S/N ratios can be taken into account Carriers in the k's subchannel: x t cos t with k 0,1,... K -1 k k m All carriers are orthogonal to each other due to: 0 cos t cos t dt 0 i k k k i i For K > 5 in the x and Rx devices FF algorithms are applied instead of a bank of band-pass filters.. Prof. Dr.-Ing. I. Willms ransmission and Modulation S. 63

64 4.6 Symbolinterferenz Block diagram of an OFDM transmission system Prof. Dr.-Ing. I. Willms ransmission and Modulation S. 64

Fund. of Digital Communications Ch. 3: Digital Modulation

Fund. of Digital Communications Ch. 3: Digital Modulation Klaus Witrisal witrisal@tugraz.at Signal Processing and Speech Communication Laboratory www.spsc.tugraz.at Graz University of Technology November