University of Manchester CS3282: Digital Communications Section 8: Carrier Modulated Transmission

Transcription

1 University of Manchester CS3282: Digital Communications Section 8: Carrier Modulated Transmission Convert binary data into form suited to channel characteristics; i.e. usable frequency band, gain & phase distortion within usable band anticipated noise characteristics frequency (e.g. Doppler) shifts Trans Channel Rec 24 Apr'06 CS3282 Sectn 8 1

2 Band-pass modulation Up to now, we have assumed a base-band channel. Frequency range from zero to B Hz. Suitably shaped pulses are symbols. Need transmission over channels which are not base-band: e.g. channel of bandwidth 200 khz centred on 900 MHz. Requires carrier modulated digital modulation. Approaches for base-band may be adapted to carrier modulated. Based on modulation techniques as used in radio. 24 Apr'06 CS3282 Sectn 8 2

3 8.1.1 Modulation of sine-wave carriers Pure sine-wave exists at just 1 frequency. Infinitessimally narrow bandwidth Some aspect varied in sympathy with baseband e.g. amplitude or frequency Detectable at receiver Spreads energy about the nominal frequency. No longer infinitessimally narrow bandwidth 24 Apr'06 CS3282 Sectn 8 3

4 8.1.2 Spread-spectrum modulation Use pseudo-random signal as carrier Wide bandwidth. Intended receiver knows the pseudo-random sequence. Has matched filter tuned to it. To other receivers the pseudo-random carrier is just noise. Increases their bit-error rate a little. More users allowed until accumulated noise gets too much. Known as DS-SSMA & CDMA. 24 Apr'06 CS3282 Sectn 8 4

5 8.1.3 Multi-carrier modulation Use set of sub-carriers instead of 1 carrier Currently sinusoidal Good for frequency selective fading in radio OFDM Used for DTV, DAB, WLAN, ADSL 64, 1024 or more sub-carriers OFDM based on FFT 24 Apr'06 CS3282 Sectn 8 5

6 8.2. Modulation Introduction to am and fm Most well known modulation techniques are am and fm as used for radio & TV. For am, multiply sine-wave by baseband signal.. For fm cause frequency to be modified by baseband. Baseband may be speech, music, or just a sine wave. With digital, baseband will be pulse sequence. 24 Apr'06 CS3282 Sectn 8 6

7 Multiplication of carrier by sine-wave volts volts t Multiply t 24 Apr'06 CS3282 Sectn 8 7

8 Frequency modulation (fm) by sine-wave volts t Modulate frequency t 24 Apr'06 CS3282 Sectn 8 8

9 Effect of modulation on frequency spectrum Power spectral density carrier frequency 24 Apr'06 CS3282 Sectn 8 9

10 Where do we get side-bands from? carrier * message A cos(ω C t) * cos(ω M t) = 0.5A cos(ω C t + ω M t) + 0.5A cos(ω C t - ω M t) = 0.5A cos( (ω C + ω M ) t ) A cos( (ω C - ω M ) t ) upper sideband lower sideband ω C = 2πf C, etc 24 Apr'06 CS3282 Sectn 8 10

11 Amplitude modulation Amplitude of sinewave can t be ve. Make bb purely +ve by adding constant. Always done with broadcast am radio stations Instead of cos(ω M t) use [1 + cos(ω M t)] A cos(ω C t ) 0.5A [cos( (ω C + ω M ) t ) + cos((ω C - s( (ω C + ω M ) t ) + cos((ω C - ω M )t) ] ier DSP am. er DSP am. 24 Apr'06 CS3282 Sectn 8 11

12 Large carrier DSP am modulator V V 1+cos(ω M t) t t Multiply V t 24 Apr'06 CS3282 Sectn 8 12

13 V Envelope detector for LC-DSB am t V t Rectify Low-pass filter V t 24 Apr'06 CS3282 Sectn 8 13

14 Coherent demodulation Envelope detection is non-coherent. Coherent demod needs local carrier at receiver. Exact in freq & phase. Derived from received signal. 24 Apr'06 CS3282 Sectn 8 14

15 V Coherent demodulation of am Received signal t V 1+cos(ω M t) t Mult Lowpass filter Derive local carrier V t Local carrier 24 Apr'06 CS3282 Sectn 8 15

16 Proof that coherent demodulation works Let received signal be A cos(ω C t).(1+cos(ω M t) ) Multiplying by local carrier gives A cos 2 (ω C t). ( 1+cos(ω M t) ) = 0.5A(1 + cos(2ω C t)).(1 + cos(ω M t) ) = 0.5A(1+cos(ω M t)) + 0.5A cos(2ω C t)(1+cos(ω M t) ) Low-pass filter removes this 24 Apr'06 CS3282 Sectn 8 16

17 Coherent demodulation again No longer requires modulating signal to be purely +ve Works with cos(ω M t) just as well as with 1+cos(ω M t) No longer large carrier & envelope detectn no good. When cos(ω M t) becomes ve, carrier amplitude remains +ve, but phase changes by 180 o With digital, modulating signal no longer sinewaves or music 24 Apr'06 CS3282 Sectn 8 17

18 8.2.2 Vector modulator & complex baseband Independently modulate cos(2πf C t) & sin(2πf C t) and sum. Coherent demodulatr for cos transmission blind to sin trans. And vice-versa. b I (t) Sin(2πf C t) Mult 2 channels for price of 1 Still single carrier ADD b R (t) Mult Cos(2πf C t) Complex baseband: b(t) = b I (t) + jb R (t) More about this later 24 Apr'06 CS3282 Sectn 8 18

19 Vector demodulator Sin(2πf C t) b R (t)cos(2πf C t) + b I (t)sin(2πf C t) Mult Lowpass filter b I (t) Derive local carrier (cos & sin) Mult Cos(2πf C t) Lowpass filter b R (t) 24 Apr'06 CS3282 Sectn 8 19

20 8.2.3 Modulation for digital transmission Generate base-band symbols from bit-stream (map to b_b) Use these symbols to modulate carrier. Modulation shifts b_b symbols up in frequency to transmission band of channel. Various forms of modulation may be used, e.g. amplitude modulation ( am ) frequency modulation ( fm ). Doubles bandwidth of base-band signal. 24 Apr'06 CS3282 Sectn 8 20

21 Mapping bit stream to base-band Generate impulses V t V Pulse-shaping filter t Map to base-band Stream of impulses produced according to bits & approach e.g. for unipolar: unit impulse for 1 & zero for 0. Pass impulse stream through pulse shaping filter. Impulses & filter may be analogue or digital (generally digital) 24 Apr'06 CS3282 Sectn 8 21

22 Techniques for digital transmission Can modulate amplitude, frequency &/or phase of cos(2πƒ C t). These 3 forms of modulation when used independently give us (a) (b) (c) amplitude shift keying (ASK) frequency shift keying (FSK) phase shift keying (PSK). There are many versions of each of these. Possible to use a combination of more than one form. Consider simplest binary forms first. 24 Apr'06 CS3282 Sectn 8 22

23 Binary frequency shift keying (B-FSK) volts t Map to base-band Modulate carrier t 24 Apr'06 CS3282 Sectn 8 23

24 Binary amplitude shift keying (B-ASK) volts t Map to base-band Multiply 24 Apr'06 CS3282 Sectn 8 24

25 Binary phase shift keying (B-PSK) volts t Map to base-band Multiply t 24 Apr'06 CS3282 Sectn 8 25

26 4-ary amplitude shift keying (ASK) volts volts t Map to base-band Multiply volts t 24 Apr'06 CS3282 Sectn 8 26

27 Combined multi-level ASK & PSK volts t Map to base-band Multiply 24 Apr'06 CS3282 Sectn 8 27

28 8.3. Amplitude shift keying b(t) r(t) b(t) cos(2πƒ c t) r(t) t t 24 Apr'06 CS3282 Sectn 8 28

29 ASK spectrum from BB spectrum Power spectral density PSD 50% RRC f 1 T 1 2T 1 T f -f C f C 1 f C T 24 Apr'06 CS3282 Sectn f C + T

30 ASK with effect of pulse shaping shown Shaping Map t cos(2πƒ c t) r(t) 24 Apr'06 CS3282 Sectn 8 30

31 Non-coherent detection of ASK Detection carried out without local carrier locked in frequency & phase with received carrier. A possible method is 'envelope detector. Diode & resistor produce 'half-wave rectified' voltage waveform when input voltage is ASK waveform. Smoothed by low-pass filter (or simple capacitor). Produces voltage waveform shown on next slide. Sampled at appropriate points in time to recover the bit-stream. 24 Apr'06 CS3282 Sectn 8 31

32 Coherent demodulation of ASK volts t Multiply Low pass Threshold detector Apr'06 CS3282 Sectn 8 32

33 Non-coherent detection of ASK Rectify & smooth Threshold detector t Apr'06 CS3282 Sectn 8 33

34 Envelope detector for ASK V V t t t Diode Resistor Low-pass filter (smoother) Sample 24 Apr'06 CS3282 Sectn 8 34

35 Constellation diagrams Show in phase and quadrature components as a graph as illustrated below for two examples: Quadrature to carrier Q In phase with carrier I 0 A 2A 3A Binary ASK with symbols 0 & Acos(..) 4-ary ASK with symbols 0, Acos(..), 2Acos(..), 3Acos(..) 24 Apr'06 CS3282 Sectn 8 35

36 Coherent demodulation of ASK Multiply by local carrier locked in frequency & phase with carrier received. s(t)cos(2πƒ c t) Lowpass filter Generate local carrier cos(2πƒ c t) Threshold detector s( t) cos 2 ( π f t) = 0.5s( t) + s( t) cos(4π c f c ) cos2θ= 2cos 2 θ -1 Removed by lowpass filter 24 Apr'06 CS3282 Sectn 8 36

37 Coherent versus non-coherent detection Let the signal be: b(t)cos(2πƒ c t). Noise is: N(t)cos(2πƒ c t + θ(t)) where N(t) is random envelope & θ(t) is random phase. This equals: N( t)cosθ ( t)cos(2πf t) + N( t)sinθ ( t)sin(2πf t) Half noise power in phase with cos(2πƒ c t ) & half with sin(2πƒ c t ). Non-coherent detection measures envelope of signal plus noise & is affected by full power of noise. Coherent detection multiplies by cos(2πƒ c t ) low-pass filters & thus eliminates half the noise power 3dB reduction in effective noise power as seen by detector. coherent detection tolerates 3dB more noise than non-coherent to achieve same BER. c c 24 Apr'06 CS3282 Sectn 8 37

38 8.4 Complx baseband & vector-modulator/demodulatr Vector modulator: sin(2πf C t) Map Map b I (t) b R (t) b R (t)cos(2πf C t) + b I (t)sin(2πf C t) cos(2πf C t) 24 Apr'06 CS3282 Sectn 8 38

39 Complex notation for vector-modulator b R (t) is in-phase component & b I (t) is quadrature component. Complex base-band signal is b R (t) + jb I (t) where j = (-1). Output is real part of: [ b R (t) + jb I (t)]. exp(-2πjf C t) since [ b R (t) + jb I (t)]. [cos(2πf C t) jsin(2πf C t) ] = [ b R (t) cos(2πf C t) + b I (t)sin(2πf C t) ] + j(..) Map b(t) Complx base-band Mult Complex signal. Take real part. exp(-2π j f C t) 24 Apr'06 CS3282 Sectn 8 39

40 Vector-demodulator Receives b R (t)cos(2πf C t) + b I (t)sin(2πf C t) Recovers b R (t) & b I (t) separately. b R (t) & b I (t) may be considered independent channels. If each transmits at 1 b/s/hz, we get 2 b/s per Hz. Two channels for price of one. Constellation diagrams becomes more interesting: 24 Apr'06 CS3282 Sectn 8 40

41 Vector demodulator (cont) Sin(2πf C t) Received signal r(t) Mult Low pass b I (t) Threshold Detector Derive local carrier (cos & sin) Mult Cos(2πf C t) Low pass b R (t) Threshold Detector Apr'06 CS3282 Sectn 8 41

42 Show why this works for cosine modulation Let r(t) = b R (t) cos(2π f C t) + b I (t) sin(2πf C t) ) Then r(t) cos(2π f C t) = b R (t)cos 2 (2π f C t) + b I (t) sin(2πf C t) )cos(2π f C t) = 0.5 b R (t)[1 + cos(4π f C t)] b I (t) sin(4πf C t) ) = 0.5b R (t) + 0.5b R (t) cos(4π f C t) b I (t) sin(4πf C t) ) Removed by lowpass filter Hence cosine demodulator recovers b R (t) & is blind to b I (t) 24 Apr'06 CS3282 Sectn 8 42

43 r(t)sin(2π f C t) Similarly for sine modulation = b R (t) cos(2π f C t)sin(2πf C t) + b I (t) sin 2 (2πf C t) ) = 0.5 b R (t) sin(4π f C t) b I (t) [1 - cos(4πf C t) ] = 0.5 b R (t) sin(4π f C t) b I (t) - 0.5b I (t)cos(4πf C t) Removed by lowpass filter Sine demodulator recovers b I (t) & is blind to b R (t) 24 Apr'06 CS3282 Sectn 8 43

44 Trig formulae This works because cos 2 (θ) & sin 2 (θ) have a constant (or DC) component 0.5 whereas sin(θ)cos(θ) does not. Relevant formulae are: cos 2 (θ) = cos(2θ) sin 2 (θ) = cos(2θ) sin(θ) cos (θ) = 0.5sin(2θ) 24 Apr'06 CS3282 Sectn 8 44

45 Constellation diags for ASK with complx baseband Quadrature to carrier In quadrature A 3A A In phase with carrier A 0 0 A 2A 3A In phas Binary ASK for b R (t) & b I (t) 4-ary ASK for b R (t) & b I (t) 24 Apr'06 CS3282 Sectn 8 45

46 Symbol allocation tables for binary & 4-ary ASK Bits b R b I A 1 0 A A A Bits b R b I A A A A A A A 2A A 3A A A A A 3A 24 Apr'06 CS3282 Sectn 8 46

47 8.5 Frequency Shift Keying (FSK) Can be straightforward form of digital modulation. Simple to generate and detect, Constant amplitude, insensitive to fluctuations of channel attenuation. Based on frequency modulation (fm) Uses set of distinct frequencies to represent symbols. Transmit constant amplitude sine-wave whose frequency varies between the frequencies assigned to each symbol. For binary signalling there are 2 frequencies, ƒ 0 & ƒ 1 say. Consider 3 generation methods. 24 Apr'06 CS3282 Sectn 8 47

48 8.5.1 Methods for generating FSK 1. Voltage controlled oscillator(vco) method. FM 1 Modulator (VCO) Better to have smoothly changing pulse for gradual transition. This is continuous phase form of FSK i.e. CPFSK. 2. Switched oscillator method of generating FSK. 1 0 FSK Clearly this may not produce a continuous phase output. 24 Apr'06 CS3282 Sectn 8 48

49 3. Vector-modulator method: For binary FSK with ƒ c +ƒ 1 & ƒ c -ƒ 1, apply cos (2πƒ 1 t) to Q and ±sin(2πƒ 1 t) to I. Sign determines the symbol. cos (2πƒ 1 t) Q input Sin(2πƒ c t) ±sin(2πƒ 1 t) I input Cos(2πƒ c t) 24 Apr'06 CS3282 Sectn 8 49

50 Exercise 8.1: Check that this works. Solution: When I=+sin(2πƒ 1 t), output is: sin(2πƒ 1 t)cos(2πƒ c t)+cos(2πƒ 1 t)sin(2πƒ c t) =sin(2π(ƒ c +ƒ 1 )t) When I=-sin(2πf 1 t) the output is: -sin(2πƒ 1 t)cos(2πƒ c t)+cos(2πƒ 1 t)sin(2πƒ c t) =sin(2π(ƒ c ƒ 1 )t) 24 Apr'06 CS3282 Sectn 8 50

51 Non-coherent detection of FSK at receiver (low bit-rates) Consider 3 methods 1. Set of band-pass filters with envelope-detectors; BPF (f0) Decide BPF (f1) 24 Apr'06 CS3282 Sectn 8 51

52 2. Discriminator followed by envelope-detector. Turns FSK into ASK for easier detection V t t t t f 1 f 0 Discriminator Low-pass filter (smoother) Gain f Resistor f 1 f 0 24 Apr'06 CS3282 Sectn 8 52

53 3. Phase Locked Loop detector for FSK. PLL is 'black box' with one input & 2 useful outputs: V Frequency modulated input t PLL VCO output VCO input (Voltage input frequency) t t 24 Apr'06 CS3282 Sectn 8 53

54 Phase-locked loop (PLL) PLL has VCO with frequency adapted to match that of FSK signal. VCO controlled by voltage generated by measuring phase difference between VCO output & incoming FSK signal. Voltage input frequency & can be used for detecting data bits t V t Low-pass filter VCO input voltage V VCO VCO output voltage 24 Apr'06 CS3282 Sectn 8 54

55 8.5.4 Non-coherent FSK detector for higher data rates: Zero crossing counter type of detector FSK Limiting Amplifier and Counter Decide Data Clock Reset 24 Apr'06 CS3282 Sectn 8 55

56 8.5.5 Coherent FSK detection: Similar to coherent ASK detection. Must have local carrier sine-waves at receiver. Must match exactly in frequency & phase the FSK symbols being received. For binary transmission there would be two locally generated sinewaves of frequency ƒ 0 and ƒ 1 respectively. The incoming signal is multiplied by both sine waves and the two signals which result are low-pass filtered. A comparator then has to decide which frequency ƒ 0 or ƒ 1 produced the larger output, and that determines the symbol. 24 Apr'06 CS3282 Sectn 8 56

57 8.5.6 Spectrum of FSK: At 1/T symbols/s, base-band signal has spectrum which is non-zero for 1/T<ƒ<1/T if 100% RC spectral shaping is applied Non-zero for 1/(2T)<ƒ<1/(2T) with 0% RC spectral shaping. When base-band signal is modulated to form FSK with signalling frequencies ƒ 1 & ƒ 0, one s form a DSB spectrum centred on ƒ 1 zero s form a DSB spectrum centred on ƒ 0. Resulting spectrum is sum of these two spectra. PSD PSD ƒ ƒ ƒ 0-1/T ƒ 0 ƒ 0 +1/T ƒ 1-1/T ƒ 1 ƒ 1 +1/T 24 Apr'06 CS3282 Sectn 8 57

58 PSD PSD + = ƒ ƒ ƒ 0-1/T ƒ 0 ƒ 0 +1/T ƒ 1-1/T ƒ 1 ƒ 1 +1/T PSD ƒ ƒ 0-1/T ƒ 0 ƒ 1 ƒ 1 +1/T 24 Apr'06 CS3282 Sectn 8 58

59 Sunde s FSK method Place ƒ 0 at ƒ 1 ±1/T & ƒ 1 at ƒ o ±1/T. 24 Apr'06 CS3282 Sectn 8 59

60 Minimum shift keying (MSK) Form of FSK where difference between ƒ 0 & ƒ 1 is 1/(2T) Hz. Makes MSK very efficient in its spectral utilisation. Price is increased complexity in generation & detection process. Non-coherent detection is difficult for MSK. The detection is recommended to be coherent (Sklar p152). Pulse-shaping filter: e.g. 100r % RRC, controls FSK spectrum. Placed just before the FSK modulator. Controls how frequency changes from ƒ 0 to ƒ 1 and vice-versa. In GSM phone systems the shaping is root-gaussian filter. This form of binary FSK is known as Gaussian MSK. 24 Apr'06 CS3282 Sectn 8 60

61 GMSK transmitter Map to impulse s FIR Gaussian shaping filter VCO GMSK 24 Apr'06 CS3282 Sectn 8 61

62 Gaussian minimum shift keying (GMSK) Spectrally efficient form of binary FSK with Gaussian pulse shaping. 2 bits/s /Hz Spectrum similar to ASK Used for GSM 24 Apr'06 CS3282 Sectn 8 62

63 Advantages & disadvantages of FSK Advantages: 1. Constant envelope hence not too sensitive to varying attenuation on the channel. 2. Detection based on frequency changes, so not very sensitive to frequency shifts of channel, (Doppler shifts etc). 3. Simple implementations possible for low bit-rates. Disadvantages of FSK: 1. Less bandwidth efficient than ASK or PSK (except MSK) 2. Bit-error rate performance in AWGN worse than PSK. 24 Apr'06 CS3282 Sectn 8 63

64 8.6. Phase shift keying (PSK) Send sinusoidal carrier with phase changes determined by bits Consider binary PSK with 1 bit/cycle, 0 0 & & rectangular pulse shaping phase shifts b(t) t Map ±cos(2πƒ c t) cos(2πƒ c t) 24 Apr'06 CS3282 Sectn 8 64

65 V A binary PSK waveform t Assuming 1 bit per cycle. 24 Apr'06 CS3282 Sectn 8 65

66 8.6.2 Coherent Detector for binary PSK ±cos(2πƒ C t) ±cos 2 (2πƒ c t) = ±0.5(1+cos4πƒ c t) Lowpass filter ±1/2 Threshold Detector Data +1/2: 1-1/2: 0 Generate local carrier cos(2πƒ C t) 24 Apr'06 CS3282 Sectn 8 66

67 Details of coherent PSK demodulator/detector Low-pass filter eliminates ±cos(4πƒ C t). Matched filter will achieve this because of orthogonality of ±cos(4πƒ c t) to sin(2πƒ c t). Local carrier must be generated from received signal. (Square incoming signal & divide frequency of result by 2). Spectrum of PSK similar to that of ASK. PSK multiplies carrier by bipolar base-band: ASK by unipolar. Shifts up base-band spectrum producing DSB spectrum centred on carrier frequency. 24 Apr'06 CS3282 Sectn 8 67

68 8.6.3 Differential PSK Phase shift of carrier with respect to previous symbol indicates current bit. Illustrate for 1 bit/cycle with 0 0 shift for 1 & for 0: V t (0 o ) (180 0 ) 24 Apr'06 CS3282 Sectn 8 68

69 90 0 & phase shifts often preferred with binary DPSK: V t 1 bit/cycle Discontinuities tell receiver when next symbol starts. Makes bit-synchronisation easier when symbol rate not fully synchronised with carrier (not exact no. of cycles/bit).. 24 Apr'06 CS3282 Sectn 8 69

70 8.6.4 Differential detection of binary DPSK Consider case where phase shifts are 0 0 & & there is an integer number (e.g. 1) of cycles per bit. Instead of generating local carrier, take previous symbol delayed as required carrier segment. Small penalty compared with a fully coherent technique. ±cos(2πƒ C t) ±cos 2 (2πƒ c t) = ±0.5(1+cos4πƒ c t) ±0.5 Lowpass filter Threshold detector Delay by T (Delay for 1 bit) 24 Apr'06 CS3282 Sectn 8 70

71 Lowpass filter output is +0.5 if carrier has been subject to 0 0 phase shift (logic 1 say) and 1/2 for (logic 0 ). Channel noise affects both data & delayed data used as carrier. Was used for modem data over telephone lines, 1200 b/s being possible over worst case lines. Increased to 2400bits/s using quaternary PSK (QPSK). 24 Apr'06 CS3282 Sectn 8 71

72 8.6.5 Detector for binary DSPK with 90 O & 270 O phase shifts rather than 0 and 180 O. LPF Detect Delay by T (Delay for 1 bit) 90 0 phase shift 24 Apr'06 CS3282 Sectn 8 72

73 8.6.6 Quaternary PSK (QPSK) Consider a vector modulator where b R (t) & b I (t) are bipolar Then b R (t)cos(2πf C t) & b I (t) sin(2πf C t) are both binary PSK. 2-channel modulation process is QPSK or 4-PSK. Sin(2πf C t) Map b I (t) Mult ADD Map b R (t) Mult Cos(2πf C t) 24 Apr'06 CS3282 Sectn 8 73

74 QPSK de-modulator Sin(2πf C t) Mult Low pass b I (t) Detect Mult Low pass Detect Detect carrier Cos(2πf C t) b R (t) 24 Apr'06 CS3282 Sectn 8 74

75 Two ways of looking at QPSK One way is vector modulation approach where cos(2πf C t) & sin(2πf C t) are binary PSK modulated independently. At receiver, coherent PSK detector for cos(2πf C t) channel is blind to transmission on sin(2πf C t) & vice-versa. Refer to b R (t) + j b I (t) as 'complex base-band' signal b(t). Transmitted QPSK signal is Re{ [b R (t) +j b I (t)] exp(-j2πf C t) } Map b(t) Complx base-band Mult exp(-2πjf C t) Transmit real part 24 Apr'06 CS3282 Sectn 8 75

76 Another way to look at QPSK QPSK sends 2 bits at once, using bipolar b R (t) & b I (t) Let b R (t) & b I (t) be rect pulses of amplitude -A or +A. Mapping to base-band may then be as follows (ω C =2πf C ) Bit1 bit2 b R (t) b I (t) QPSK symbol transmitted 0 0 A A Acos(ω C t) A sin(ω C t) = Acos(ω C t ) 0 1 A +A Acos(ω C t) + A sin(ω C t) = Acos(ω C t ) 1 0 +A A Acos(ω C t) A sin(ω C t) = Acos(ω C t 45 0 ) 1 1 +A +A Acos(ω C t) + A sin(ω C t) = Acos(ω C t ) Looking at a constellation diag for this mapping makes it clear why Acos(ω C t) + A sin(ω C t) = Acos(ω C t ) etc. 24 Apr'06 CS3282 Sectn 8 76

77 Constellation diagram for ±45 o, ±135 o QPSK Symbol allocation table: In quadrature with cos Bit1 bit2 b R (t) b I (t) 0 0 A A 0 1 A +A 1 0 +A A 1 1 +A +A 0,1 -V V 45 o V 1,1 In phase with cos (real pt) 0,0 1,0 24 Apr'06 CS3282 Sectn 8 77

78 Alternative constellation diag ( 0 o,90,180,270 o QPSK) Symbol allocation table: Bit1 bit2 b R (t) b I (t) 0 0 A A 1 0 -A A -A 1,0 Imag pt 0,1 1,1 0,0 Real 24 Apr'06 CS3282 Sectn 8 78

79 QPSK is 4-PSK. What about 8-PSK & 16-PSK? Can have 8-PSK (3 bits/symbol) & 16-PSK (4 bits/symbol). Constellation diagrams for shown below. Imag pt 8-PSK Real pt 16- PSK Re Differential forms of QPSK & M-PSK often used where changes in phase signify the data. Principle similar to DPSK. 24 Apr'06 CS3282 Sectn 8 79

80 Exercise 8.6: Consider how symbols for 8-PSK & 16-PSK may be associated with sequences of 3 or 4 bits, i.e. label the constellation diagrams. Use a form of 'Gray coding' With Gray coding, a symbol error generally causes just one bit-error 24 Apr'06 CS3282 Sectn 8 80

81 Exercise 8.6 (cont): What happens if we don t use Gray coding? If symbol 111 mistaken for 000 get 3 bit-errors 24 Apr'06 CS3282 Sectn 8 81

82 Advantage of Gray coding With Gray coding of multi-level symbols, bit-error rate may be assumed to be: symbol-error rate no. of bits/symbol except when the noise is exceptionally high. (We assume a symbol error just takes us to a nearby symbol which differs in just one bit with Gray coding) Repeat the labeling now for 16-PSK. 24 Apr'06 CS3282 Sectn 8 82

83 Exercise 8.7: Show how a vector-modulator may be used to generate the 8 or 16 symbols of 8-PSK & 16-PSK V Symbol b R (t) b I (t) 000 V V/1.4 V/ V/1.4 V/ V 100 V/1.4 -V/ V 110 -V V/1.4 -V/ Apr'06 CS3282 Sectn 8 83

84 Example 8.7 (cont) How would you detect 8-PSK with a vector demodulator & threshold detectors? Exercise 8.8: If radius of constellation diagram circle is V volts for QPSK, 8- PSK & 16-PSK calculate energy per bit for each of these schemes assuming rectangular pulses. Take 'noise immunity' as distance between each symbol on constellation diagram & nearest one to it, Estimate noise immunity for QPSK, 8-PSK & 16-PSK when radius is V in each case. 24 Apr'06 CS3282 Sectn 8 84

85 Exercise 8.9: How will pulse-shaping be applied to QPSK, 8-PSK and 16-PSK? With 100% RRC pulse shaping & symbol duration T, what is band-with efficiency (in b/s / Hz) for each of these techniques. What is theoretical maximum bandwidth efficiency in each case? 24 Apr'06 CS3282 Sectn 8 85

86 Single carrier digital modulation schemes ASK, FSK, PSK, DPSK, QPSK Differential QPSK Gaussian FSK & MSK Combined ASK & PSK (QAM, APK) etc. 24 Apr'06 CS3282 Sectn 8 86

87 Other modulation techniques Direct sequence spread spectrum techniques (DSSS) Frequency hopping (FHSS) Complementary code keying (CCK) 24 Apr'06 CS3282 Sectn 8 87

88 Pause End of slides on single carrier modulation 24 Apr'06 CS3282 Sectn 8 88

89 Multi-carrier modulation & OFDM Orthog frequency division multiplexing (OFDM) is relatively new multi-carrier modulation scheme. Used for DAB, ADSL & wireless LANs (IEEE a). Many, say 64 or 1024, carrier frequencies evenly spaced out over a range of frequencies. Used in IEEE802.11g/a/e with 64 carrier frequencies. High bandwidth efficiency & robust to freq. selective fading. First a few preliminaries & reminders 24 Apr'06 CS3282 Sectn 8 89

90 Quadrature amplitude modulation_qam QPSK combined with multi-level ASK With QPSK, ±A applied to cos & sin carriers With QAM, 0, ±A, ±2A applied Nicely represented by constellations 24 Apr'06 CS3282 Sectn 8 90

91 Constellation for QPSK Modulating sin 1,0 1,1 0,0 0,1 modulating cos Bit1 Bit2 b R b I 0 0 A A 0 1 A -A 1 0 -A A 1 1 -A -A 24 Apr'06 CS3282 Sectn 8 91

92 16-QAM _ symbol allocation table Bit1 bit2 bit3 bit4 VI VQ Bit1 bit2 bit3 bit4 VI VQ A A A A A -A A -A A 3A A 3A A -3A A -3A A A A A A -A A -A A 3A A 3A A -3A A -3A 24 Apr'06 CS3282 Sectn 8 92

93 16_QAM constellation 3A Imag (0010) (0100) A (0000) -A A 3A Real -A (0001) -3A (0011) 24 Apr'06 CS3282 Sectn 8 93

94 Vector-modulator Sin(2πf C t) Map b I (t) Mult ADD Map b R (t) Mult Cos(2πf C t) 24 Apr'06 CS3282 Sectn 8 94

95 Vector modulator in complex notation Take b(t) + jq(t) as a complex b-b signal. cos(2πf C t).b(t) + sin(2πf C t). q(t) = real { ( b(t) + jq(t) ) exp(-2πjf C t) } Map b(t) Complx base-band Mult exp(-2πjf C t) 24 Apr'06 CS3282 Sectn 8 95

96 A slight variation cos(2πf C t).b(t) sin(2πf C t). q(t) = real { ( b(t) + jq(t) ) exp(2πjf C t) } Instead of sin we modulate -sin: no real difference Map b(t) Complx base-band Mult exp(2πjf C t) 24 Apr'06 CS3282 Sectn 8 96

97 A slight variation (continued) Sin(2πf C t) Map b I (t) Mult ADD Map b R (t) Mult Cos(2πf C t) 24 Apr'06 CS3282 Sectn 8 97

98 Fast Fourier Transform FFT : {x[n]} 0,N-1 {X[k]} 0,N-1 X N 1 n= 0 [] [] j2πkn/ N k = x n e for k = 0, 1,..., N-1 x[n] X[k] 0 N Time-domain n N/2 f S /2 24 Apr'06 CS3282 Sectn N f S Frequency domain k

99 Implementation of FFT The FFT is fast in that it can be programmed or implemented in hardware very efficiently especially when N is a power of 2, e.g. 64, 256, 512, 1024, 24 Apr'06 CS3282 Sectn 8 99

100 Inverse Fast Fourier Transform IFFT: {X[k]} 0,N-1 {x[n]} 0,N-1 x N 1 1 n= 0 [] j2πkn/ N n = X[k] e for k = 0, 1,...,N X[k] N N/2 f S /2 N f S k x[n] 0 N 2N Time-domain 24 Apr'06 CS3282 Sectn n

101 End of preliminaries for multi-carrier modulation 24 Apr'06 CS3282 Sectn 8 101

102 Multi-carrier modulation Take 64 carrier frequencies over range f C to f C + 20 MHz: f C + 0, f C + f D, f C + 2f D,, f C +63f D with f D = 20MHz / 64 = khz. 24 Apr'06 CS3282 Sectn 8 102

103 10110 Map X 0 (t) Mult Multi-carrier modulation exp(2πjf C t) Map X 1 (t) exp(2πj(f C +f D )t) Mult Map X N-1 (t) exp(2πj(f C +(N-1)f D )t) Mult 24 Apr'06 CS3282 Sectn 8 103

104 Do multi-carrier modulation in two stages: Stage 1: Apply PSK, QPSK, QAM (or other) to obtain X 0 (t), X 1 (t),..., X N-1 (t) which remain constant for a symbol period T. (With QPSK we could represent 2N bits per symbol period). Then vector-modulate complex 'sub-carriers' of frequencies: 0, f D, 2f D,, (N-1)f D Stage 2: Vector-modulate exp(2πjf C ) with sum of modulated sub-carriers. 24 Apr'06 CS3282 Sectn 8 104

105 10110 Map X 0 (t) Stage Map X 1 (t) exp(2πjf D t) Mult Map X N-1 (t) exp(2πj((n-1)f D )t) Mult 24 Apr'06 CS3282 Sectn 8 105

106 Stage 2 N 1 m= 0 X m ( t) e (complex) 2πjmf D t N 1 m= 0 X m ( t) e 2πjm( f C + f D t) (complex but need only real part) exp(2πjf C ) Note that this is complex multiplication. 24 Apr'06 CS3282 Sectn 8 106

107 The 64 modulating signals: X 0 (t) = B 0 (t) + jq 0 (t) : modulating 0 Hz X 1 (t) = B 1 (t) + jq 1 (t) : modulating f D X 2 (t) = B 2 (t) + jq 2 (t) : modulating 2f D. X 63 (t) = B 63 (t) + jq 63 (t). : modulating 63f D With QPSK, each X i represents 2 bits. (IEE802.11a makes X 0 -X 5 & X 58 -X 63 all zero & uses 4 others for "pilot tones", leaving 48 to use.). 24 Apr'06 CS3282 Sectn 8 107

108 Adding these together we obtain: 63 x(t) = Σ X k (t) exp (2πjkf D t ) : - <t<. k=0 With symbol period T, assume sample x(t) at τ = T/64 and denote the samples by x[n]: 63 x(nτ) = x[n] = Σ X k (nτ) exp (2πjk f D nτ ) k=0 Make X k [nτ] =X k : constant for 0<n<63, i.e. for 1 symbol period 63 x[n] = Σ X k exp(jk(2π/n)n) : 0<n<63 k=0 Generates a set {x[0], x[1],, x[63]} of complex numbers. 24 Apr'06 CS3282 Sectn 8 108

109 This formula: 63 x[n] = Σ X k exp(jk(2π/n)n) : 0<n<63 k=0 takes 64 complex numbers {X 0, X 1,, X 63 } representing one symbol and generates a set {x[0], x[1],, x[63]} of complex numbers. It is inverse FFT formula (apart from a factor 1/64). Generates 64 complx samples of a time-domain waveform: a pulse. Repeat for next set of {X 0, X 1,..., X 63 }to get another pulse & so on. 24 Apr'06 CS3282 Sectn 8 109

110 With IEEE802.11, symbol period T = 4 µs, i.e. 250 k symbols/s. For each symbol we get 64 complex samples hence 16 M sample/s These 64 samples could form a single symbol capable of representing up to 128 bits with QPSK. The inverse FFT takes N frequency-domain samples & produces N time-domain samples. Here N=64. But what if we evaluated x[n] for n outside range 0 to 63? Samples 0 to 63 are repeated as 64 to 127 etc as shown next. 24 Apr'06 CS3282 Sectn 8 110

111 Real(x[n]} n Similarly for imaginary part. 24 Apr'06 CS3282 Sectn 8 111

112 Instead of taking n from 0 to 63, we take n from 0 to 79 or sometimes from -16 to 63. Taking n from -16 to 63 generates a cyclic prefix before n=0. From n= -16 to -1 we 16 extra samples which are a copy of x[48] to x[63]. Generates a set 80 time-domain complex numbers for each set {X 0, X 1,..., X 63 } rather than 64. If T remains at 4 us, we get 250 x 80 = 20 Mb/s for the timedomain waveform. Samples from -16 to -1 form the cyclic prefix. 24 Apr'06 CS3282 Sectn 8 112

113 Real(x[n]} n Similarly for imaginary part. 24 Apr'06 CS3282 Sectn 8 113

114 Orthogonal Freq Div Multiplexing (OFDM) Scheme described on previous slides is OFDM as used by IEEE OFDM also used for digital radio/tv but with different symbol timing and number of sub-carriers. With IEEE802.11a, time-domain OFDM "symbol" lasts 4us. Shape of symbol tell us the information. With QPSK on 48 carriers, x different symbol shapes. 250,000 such symbols strung together per second 24 Apr'06 CS3282 Sectn 8 114

115 Up-conversion Applying these complex samples to a vector-modulator with carrier exp(2πjf C ), taking real part we obtain required multi-carrier signal. To do this digitally (or in simulation) must up-sample to a sampling rate suitable for f C carrier. Could up-sample by a factor of 10 say by repeating each sample 10 times & digitally low-pass filtering result to one tenth of the new sampling frequency. If modulation in analogue form, real & imag parts of symbol stream must first be D to A converted. Again this may be best done by up-sampling first to simplify analogue filtering. 24 Apr'06 CS3282 Sectn 8 115

116 {x[n]} is inverse FFT of {X 0, X 1,., X 63 }. Normally generates complex sequence {x[0], x[1],., x[63]} With 63 samples, {x[0], x[1],., x[63]}, no information lost. {x[n]} 0,63 contains all the information in {X k }. 0,63 DFT of {x[n]} 0,63 gets back exactly to {X 0, X 1,., X 63 }. OFDM demodulator is FFT followed by detectors 24 Apr'06 CS3282 Sectn 8 116

117 But we calculate {x[0], x[1],, x[63], x[64],, x[79]} x[64] to x[79] is the "cyclic extension". Or we could calculate {x[-16],, x[63]}& have cyclic prefix Not much difference in reality. The extra samples may be thought of as a guard interval between one symbol & the next. But they are much more useful than just that. Useful for carrier and symbol synchronisation at receiver. Due to cyclic extension & cyclic nature of DFT & its inverse, even if exact synchronisation is not achieved at receiver, exact data can still be recovered with a phase shift. 24 Apr'06 CS3282 Sectn 8 117

118 Simulation of simplified OFDM trasmitter & receiver: Generate 8 random bits & use these to generate 1 OFDM symbol. Each symbol requires 4 complex numbers X 0 to X 3 which are transformed to time-domain {x[n]} 0,3 by 4-point inverse FFT. Take X 0 to X 3 to be complex numbers representing the I and Q channels of 4 QPSK modulators. Extend complex time-domain symbol to 6 samples {x[n]} 0,6 by cyclic extension & vector-modulate a 28 MHz carrier by the samples of x. Produce and plot the transmitted waveform. Show how original data can be recovered by 4-point FFT. 24 Apr'06 CS3282 Sectn 8 118

119 Example: Assume data is: Then X 0 =1+j, X 1 = 1- j, X 2 = -1+j, X 3 = -1-j X = [ 1+j 1-j -1+j -1-j ]; x=ifft(x) This generates the required symbol: j j j Including the cyclic extension, this becomes: j j j j j 24 Apr'06 CS3282 Sectn 8 119

120 With 64 = 2 6 sub-carrier frequencies, inverse DFT can be carried out very efficiently by FFT. OFDM works because of orthogonality of the 64 carriers. 24 Apr'06 CS3282 Sectn 8 120

121 Advantages of OFDM Good for channels affected by frequency selective fading because: (1) info can be spread out across sub-carriers in intelligent ways so that when some are lost, others will compensate. (2) guard-band allows for ISI; if 4us OFDM symbol rings on, it only affects beginning of next symbol, repeated at end. (Can have cyclic "prefix"). So no pulse-shaping necessary! (3) equalisation much easier than with single carrier systems. OFDM equalisation done by multiplication in frequency-domain after FFT. Easier than adaptive filtering used for single carrier. Works because of cyclic extension. 24 Apr'06 CS3282 Sectn 8 121

122 Disadvantages of OFDM Peak to mean" ratio of symbols can be very large by nature of FFT & its inverse. (Amplitudes can become very large in comparison to the mean) Shape of each OFDM symbol ( there are of them) is very complex & must be sent & received accurately. Transmitter & receiver must be linear to preserve shape..definitely not "constant envelope". Need class A amplifiers: less power efficient than those for constant envelope transmissions. Lot of power lost in the amplifiers. Not ideal for mobile phones, but fine for mobile computers with bigger batteries that are not sending data continuously. 24 Apr'06 CS3282 Sectn 8 122

123 Some details about IEEE a/g OFDM With IEEE802.11a and g, OFDM symbols transmitted in 4 µs giving maximum of 250 k symbols/second. Each symbol can carry up to 6 bits per carrier (using 64-QAM). Normally reduced to 4.5 bits per carrier by ¾ rate FEC. As there are 48 carriers carrying data, maximum bit-rate is 48 x 4.5 x 250 kb/s = 54 Mb/s. Distances over which this bit-rate achievable will be restricted. Lower bit-rates (48, 36, 24, 18, 12, 9 and 6Mb/s) available. Two lowest bit-rates (9 & 6Mb/s) use binary PSK & 3/4 or 1/2 rate FEC to achieve 48 x (3/4) x 250kb/s = 9 kb/s or 48 x (1/2) x 250 kb/s = 6 Mb/s. For 18 Mb/s & 12 Mb/s, QPSK is used on each of 48 data carriers. 24 Apr'06 CS3282 Sectn 8 123

124 Bandwidth efficiency of OFDM Consider an idealised OFDM system with 64 active carriers and QPSK used to achieve 128 bits per 4 µs (4 x 10-6 ) symbol. No cyclic extension. Each symbol generates 64 time-domain samples. So 128 x 250 kbits/second in 64 x 250 k = 16 Msymbols/s. 2 bits/s per sample/s 2 b/s per Hz when 16 Mbaud converted to 16 MHz signal. In practice 128 x 250 kb/s in 20 MHz. What about IEEE802.11a? 24 Apr'06 CS3282 Sectn 8 124

125 Forward error control (FEC) Extra bits inserted to allow bit-errors to be detected and corrected at the receiver. Block codes Convolutional codes Half rate FEC coder has M message bits in each 2M bits. 24 Apr'06 CS3282 Sectn 8 125

126 Soft-decision Viterbi FEC decoder. Soft decision means that instead of making a definite decision as to whether a bit is 0 or 1, threshold detector at receiver delivers a number between 0 & 1 indicating how certain it is about the decision. This may be illustrated for unipolar Voltage x Hard Soft decision x certain < x probably < x likely < x maybe < x maybe < x likely < x probably < x certain 1 24 Apr'06 CS3282 Sectn 8 126

127 The confidence of the decision is taken into account by the Viterbi decoder when it attempts to correct it-errors. Soft decision gains us about 2 db is SNR over hard. If there are too many bit-errors in the received coded bitstream, to be corrected by the Viterbi decoder will fail to correct these bit-errors.. 24 Apr'06 CS3282 Sectn 8 127

128 End of Section 8 24 Apr'06 CS3282 Sectn 8 128