1. 24 Apr'06CS3282 Sectn 81 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 Channel Trans Rec 10110 10111

2. 24 Apr'06CS3282 Sectn 82 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.

3. 24 Apr'06CS3282 Sectn 83 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

4. 24 Apr'06CS3282 Sectn 84 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.

5. 24 Apr'06CS3282 Sectn 85 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

6. 24 Apr'06CS3282 Sectn 86 8.2. Modulation 8.2.1 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.

7. 24 Apr'06CS3282 Sectn 87

8. 24 Apr'06CS3282 Sectn 88 Frequency modulation (fm) by sine-wave Modulate frequency t volts t

9. 24 Apr'06CS3282 Sectn 89 Effect of modulation on frequency spectrum carrier frequency Power spectral density

10. 24 Apr'06CS3282 Sectn 810 carrier * message A cos(  C t) * cos(  M t) = 0.5A cos(  C t +  M t) + 0.5A cos(  C t -  M t) = 0.5A cos( (  C +  M ) t ) + 0.5 A cos( (  C -  M ) t ) upper sideband lower sideband  C = 2  f C, etc Where do we get ‘side-bands’ from?

11. 24 Apr'06CS3282 Sectn 811 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(  M t) use [1 + cos(  M t)] A cos(  C t ) 0.5A [cos( (  C +  M ) t ) + cos((  C -  M )t) ] Large carrier DSP ‘am’. More easily demodulated (by envelope detector)

12. 24 Apr'06CS3282 Sectn 812 Large carrier DSP ‘am’ modulator V t V Multiply V t t 1+cos(  M t)

13. 24 Apr'06CS3282 Sectn 813 ‘Envelope detector’ for LC-DSB ‘am’ Rectify Low-pass filter t V t V V t

14. 24 Apr'06CS3282 Sectn 814 Coherent demodulation Envelope detection is ‘non-coherent’. ‘Coherent’ demod needs local carrier at receiver. Exact in freq & phase. Derived from received signal.

15. 24 Apr'06CS3282 Sectn 815 Coherent demodulation of ‘am’ V t V Mult V t t 1+cos(  M t) Lowpass filter Local carrier Received signal Derive local carrier

16. 24 Apr'06CS3282 Sectn 816 Proof that coherent demodulation works Let received signal be A cos(  C t).(1+cos(  M t) ) Multiplying by local carrier gives A cos 2 (  C t). ( 1+cos(  M t) ) = 0.5A(1 + cos(2  C t)).(1 + cos(  M t) ) = 0.5A(1+cos(  M t)) + 0.5A cos(2  C t)(1+cos(  M t) ) Low-pass filter removes this

17. 24 Apr'06CS3282 Sectn 817 Coherent demodulation again No longer requires modulating signal to be purely +ve Works with cos(  M t) just as well as with 1+cos(  M t) No longer ‘large carrier & envelope detectn no good. When cos(  M t) becomes  ve, carrier amplitude remains +ve, but phase changes by 180 o With digital, modulating signal no longer sinewaves or music

18. 24 Apr'06CS3282 Sectn 818 8.2.2 Vector modulator & complex baseband Independently modulate cos(2  f C t) & sin(2  f C t) and sum. Coherent demodulatr for ‘cos’ transmission blind to ‘sin’ trans. And vice-versa. Mult ADD Cos(2  f C t) Sin(2  f C t) b R (t) b I (t) “2 channels for price of 1” Still single carrier Complex baseband: b(t) = b I (t) + jb R (t) More about this later

19. 24 Apr'06CS3282 Sectn 819 Vector demodulator Mult Cos(2  f C t) Sin(2  f C t) b R (t) b I (t) Derive local carrier (cos & sin) Lowpass filter Lowpass filter b R (t)cos(2  f C t) + b I (t)sin(2  f C t)

20. 24 Apr'06CS3282 Sectn 820 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.

21. 24 Apr'06CS3282 Sectn 821 Mapping bit stream to base-band Pulse-shaping filter..1 1 0 1 0... Generate impulses t VV 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)

22. 24 Apr'06CS3282 Sectn 822 Techniques for digital transmission Can modulate amplitude, frequency &/or phase of cos(2  C t). These 3 forms of modulation when used independently give us (a)amplitude shift keying (ASK) (b)frequency shift keying (FSK) (c)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.

23. 24 Apr'06CS3282 Sectn 823 Binary frequency shift keying (B-FSK) Modulate carrier Map to base-band 10110 t volts t

24. 24 Apr'06CS3282 Sectn 824 Binary amplitude shift keying (B-ASK) Map to base-band 10110 t volts Multiply

25. 24 Apr'06CS3282 Sectn 825 Binary phase shift keying (B-PSK) Map to base-band 10110 t volts Multiply t

26. 24 Apr'06CS3282 Sectn 826 4-ary amplitude shift keying (ASK) Map to base-band 10110 t volts Multiply t volts

27. 24 Apr'06CS3282 Sectn 827 Combined multi-level ASK & PSK Map to base-band 10110 t volts Multiply

28. 24 Apr'06CS3282 Sectn 828 8.3. Amplitude shift keying r(t) cos(2  c t) b(t) t r(t) t

29. 24 Apr'06CS3282 Sectn 829 8.3.2. 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.

30. 24 Apr'06CS3282 Sectn 830 Coherent demodulation of ASK 10110 t volts Multiply Threshold detector Low pass

31. 24 Apr'06CS3282 Sectn 831 Non-coherent detection of ASK Rectify & smooth Threshold detector t 10110

32. 24 Apr'06CS3282 Sectn 832 Low-pass filter (smoother) t t t V V Sample Diode Resistor Envelope detector for ASK

33. 24 Apr'06CS3282 Sectn 833 8.3.3. Constellation diagrams Show “in phase” and “quadrature” components as a graph as illustrated below for two examples:

34. 24 Apr'06CS3282 Sectn 834 8.3.4. Coherent demodulation of ASK Multiply by local carrier locked in frequency & phase with carrier received. Lowpass filter cos(2  c t) s(t)cos(2  c t) Generate local carrier Threshold detector Removed by lowpass filter cos2  = 2cos 2  - 1

35. 24 Apr'06CS3282 Sectn 835 8.3.5. Coherent versus non-coherent detection Let the signal be: b(t)cos(2  c t). Noise is: N(t)cos(2  c t +  (t)) where N(t) is random envelope &  (t) is random phase. This equals: Half noise power in phase with cos(2  c t ) & half with sin(2  c t ). Non-coherent detection measures envelope of signal plus noise & is affected by full power of noise. Coherent detection multiplies by cos(2  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.

36. 24 Apr'06CS3282 Sectn 836 8.4 Complx baseband & vector-modulator/demodulatr 8.4.1 Vector modulator:..11010.. Map sin(2  f C t) cos(2  f C t) b I (t) b R (t) b R (t)cos(2  f C t) + b I (t)sin(2  f C t) Map..10010..

37. 24 Apr'06CS3282 Sectn 837 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(-2  jf C t) since [ b R (t) + jb I (t)]. [cos(2  f C t)  jsin(2  f C t) ] = [ b R (t) cos(2  f C t) + b I (t)sin(2  f C t) ] + j(..) Mult Map exp(-2  j f C t) b(t) 10110 11011 Complex signal. Take real part. Complx base-band

38. 24 Apr'06CS3282 Sectn 838 8.4.2. Vector-demodulator Receives b R (t)cos(2  f C t) + b I (t)sin(2  f 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:

39. 24 Apr'06CS3282 Sectn 839 Vector demodulator (cont) Mult Threshold Detector Threshold Detector Cos(2  f C t) Sin(2  f C t) b R (t) b I (t) Low pass Low pass..11010.. Derive local carrier (cos & sin) Received signal r(t)..10010..

40. 24 Apr'06CS3282 Sectn 840 Show why this works for cosine modulation Let r(t) = b R (t) cos(2  f C t) + b I (t) sin(2  f C t) ) Then r(t) cos(2  f C t) = b R (t)cos 2 (2  f C t) + b I (t) sin(2  f C t) )cos(2  f C t) = 0.5 b R (t)[1 + cos(4  f C t)] + 0.5 b I (t) sin(4  f C t) ) = 0.5b R (t) + 0.5b R (t) cos(4  f C t) + 0.5 b I (t) sin(4  f C t) ) Hence cosine demodulator recovers b R (t) & is blind to b I (t) Removed by lowpass filter

41. 24 Apr'06CS3282 Sectn 841 Similarly for sine modulation r(t)sin(2  f C t) = b R (t) cos(2  f C t)sin(2  f C t) + b I (t) sin 2 (2  f C t) ) = 0.5 b R (t) sin(4  f C t) + 0.5 b I (t) [1 - cos(4  f C t) ] = 0.5 b R (t) sin(4  f C t) + 0.5 b I (t) - 0.5b I (t)cos(4  f C t) Removed by lowpass filter Sine demodulator recovers b I (t) & is blind to b R (t)

42. 24 Apr'06CS3282 Sectn 842 Trig formulae This works because cos 2 (  ) & sin 2 (  ) have a constant (or DC) component 0.5 whereas sin(  )cos(  ) does not. Relevant formulae are: cos 2 (  ) = 0.5 + 0.5 cos(2  ) sin 2 (  ) = 0.5 - 0.5 cos(2  ) sin(  ) cos (  ) = 0.5sin(2  )

43. 24 Apr'06CS3282 Sectn 843 8.4.3. Constellation diags for ASK with complx baseband In phase with carrier Quadrature to carrier 0 A 2A 3A Binary ASK for b R (t) & b I (t) 4-ary ASK for b R (t) & b I (t) In quadrature In phas A A 3A A 0

44. 24 Apr'06CS3282 Sectn 844 Symbol allocation tables for binary & 4-ary ASK Bits b R b I 0 0 0 1 0 A 1 0 A 0 1 1 A A Bits b R b I 0 0 0 0 0 0 1 0 A 0 0 1 0 0 2A 0 0 1 1 0 3A 0 1 0 0 A 0 0 1 0 1 A A 0 1 1 0 A 2A 0 1 1 1 A 3A 1 0 0 0 2A 0 1 0 0 1 2A A..... 1 1 1 1 3A 3A

45. 24 Apr'06CS3282 Sectn 845 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,  0 &  1 say. Consider 3 generation methods.

46. 24 Apr'06CS3282 Sectn 846 FM Modulator (VCO) 1010 0 1 0 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. 1010 FSK 1. “Voltage controlled oscillator(VCO)”method. Clearly this may not produce a continuous phase output. 8.5.1 Methods for generating FSK

47. 24 Apr'06CS3282 Sectn 847 3. “Vector-modulator” method: For binary FSK with  c +  1 &  c -  1, apply cos (2  1 t) to ‘Q’ and  sin(2  1 t) to ‘I’. Sign determines the symbol. “Q” input “I” input Sin(2  c t) Cos(2  c t) cos (2  1 t)  sin(2  1 t)

48. 24 Apr'06CS3282 Sectn 848 Exercise 8.1: Check that this works. Solution: When I=+sin(2  1 t), output is: sin(2  1 t)cos(2  c t)+cos(2  1 t)sin(2  c t) =sin(2  (  c +  1 )t) When I=-sin(2  f 1 t) the output is: -sin(2  1 t)cos(2  c t)+cos(2  1 t)sin(2  c t) =sin(2  (  c –  1 )t)

49. 24 Apr'06CS3282 Sectn 849 8.5.2. Non-coherent detection of FSK at receiver (low bit-rates) Consider 3 methods 1. Set of band-pass filters with envelope-detectors; BPF (f0) BPF (f1) Decide

50. 24 Apr'06CS3282 Sectn 850 Discriminator Low-pass filter (smoother) t t t V t f Gain Resistor f 1 f 0 2. Discriminator followed by envelope-detector. Turns FSK into ASK for easier detection

51. 24 Apr'06CS3282 Sectn 851 PLL t V t t VCO input (Voltage  input frequency) VCO output Frequency modulated input 3. Phase Locked Loop detector for FSK. PLL is 'black box' with one input & 2 useful outputs:

52. 24 Apr'06CS3282 Sectn 852 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 Low-pass filter VCO VCO input voltage VCO output voltage t t V V 8.5.3. Phase-locked loop (PLL)

53. 24 Apr'06CS3282 Sectn 853 8.5.4 Non-coherent FSK detector for higher data rates: “Zero crossing counter” type of detector Limiting Amplifier Clock DecideCounter Reset Data FSK and

54. 24 Apr'06CS3282 Sectn 854 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 sine- waves of frequency  0 and  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  0 or  1 produced the larger output, and that determines the symbol.

55. 24 Apr'06CS3282 Sectn 855 8.5.6 Spectrum of FSK: At 1/T symbols/s, base-band signal has spectrum which is non-zero for –1/T<  <1/T if 100% RC spectral shaping is applied Non-zero for –1/(2T)<  <1/(2T) with 0% RC spectral shaping. When base-band signal is modulated to form FSK with signalling frequencies  1 &  0, ‘one’s form a DSB spectrum centred on  1 ‘zero’s form a DSB spectrum centred on  0. Resulting spectrum is sum of these two spectra. PSD   0 -1/T  0  0 +1/T PSD   1 -1/T  1  1 +1/T

56. 24 Apr'06CS3282 Sectn 856 PSD   0 -1/T  0  0 +1/T PSD   1 -1/T  1  1 +1/T PSD  0 -1/T  0  1   1 +1/T + =

57. 24 Apr'06CS3282 Sectn 857 Sunde’s FSK method Place  0 at  1  1/T &  1 at  o  1/T.

58. 24 Apr'06CS3282 Sectn 858 8.5.7. Minimum shift keying (MSK) Form of FSK where difference between  0 &  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  0 to  1 and vice-versa. In GSM phone systems the shaping is root-Gaussian filter. This form of binary FSK is known as “Gaussian MSK”.

59. 24 Apr'06CS3282 Sectn 859 FIR Gaussian shaping filter VCO Map to impulse s..10110.. GMSK GMSK transmitter

60. 24 Apr'06CS3282 Sectn 860 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

61. 24 Apr'06CS3282 Sectn 861 8.5.8. 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.

62. 24 Apr'06CS3282 Sectn 862 8.6. Phase shift keying (PSK) cos(2  c t) t  cos(2  c t) b(t) Map..1010010.. Send sinusoidal carrier with phase changes determined by bits Consider binary PSK with 1 bit/cycle, 0 0 & 180 0 phase shifts & rectangular pulse shaping

63. 24 Apr'06CS3282 Sectn 863 A binary PSK waveform t V 1 1 0 0 1 1 0 Assuming 1 bit per cycle.

64. 24 Apr'06CS3282 Sectn 864 8.6.2 Coherent Detector for binary PSK Lowpass filter cos(2  C t)  cos 2 (2  c t) =  0.5(1+cos4  c t) Threshold Detector Data +1/2:”1” -1/2:”0”  1/2  cos(2  C t) Generate local carrier

65. 24 Apr'06CS3282 Sectn 865 Low-pass filter eliminates  cos(4  C t). Matched filter will achieve this because of orthogonality of  cos(4  c t) to sin(2  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. Details of coherent PSK demodulator/detector

66. 24 Apr'06CS3282 Sectn 866 90 0 & 270 0 phase shifts often preferred with binary DPSK: t V 1 1 0 1 1 0 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).. 1 bit/cycle

67. 24 Apr'06CS3282 Sectn 867 8.6.4 Differential detection of binary DPSK Consider case where phase shifts are 0 0 & 180 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. Lowpass filter  cos(2  C t)  cos 2 (2  c t) =  0.5(1+cos4  c t) Threshold detector Delay by T (Delay for 1 bit)  0.5

68. 24 Apr'06CS3282 Sectn 868 Lowpass filter output is +0.5 if carrier has been subject to 0 0 phase shift (logic 1 say) and –1/2 for 180 0 (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).

69. 24 Apr'06CS3282 Sectn 869 8.6.5 Detector for binary DSPK with 90 O & 270 O phase shifts rather than 0 and 180 O. LPFDetect Delay by T (Delay for 1 bit) 90 0 phase shift

70. 24 Apr'06CS3282 Sectn 870 8.6.6 Quaternary PSK (QPSK) Consider a vector modulator where b R (t) & b I (t) are bipolar Then b R (t)cos(2  f C t) & b I (t) sin(2  f C t) are both binary PSK. ‘2-channel’ modulation process is QPSK or 4-PSK. Mult ADD Map Cos(2  f C t) Sin(2  f C t) b R (t) b I (t) 10110 11011

71. 24 Apr'06CS3282 Sectn 871 QPSK de-modulator Mult Detect Cos(2  f C t) Sin(2  f C t) b R (t) b I (t)10110 11011 Low pass Low pass Detect carrier

72. 24 Apr'06CS3282 Sectn 872 Two ways of looking at QPSK One way is ‘vector modulation’ approach where cos(2  f C t) & sin(2  f C t) are binary PSK modulated independently. At receiver, coherent PSK detector for cos(2  f C t) channel is blind to transmission on sin(2  f 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(-j2  f C t) }. Mult Map exp(-2  j f C t) b(t) 10110 11011 Transmit real part Complx base-band

73. 24 Apr'06CS3282 Sectn 873 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 (  C =2  f C ) Bit1 bit2 b R (t) b I (t) QPSK symbol transmitted 0 0  A  A  Acos(  C t)  A sin(  C t) = Acos(  C t  135 0 ) 0 1  A +A  Acos(  C t) + A sin(  C t) = Acos(  C t+135 0 ) 1 0 +A  A Acos(  C t)  A sin(  C t) = Acos(  C t  45 0 ) 1 1 +A +A Acos(  C t) + A sin(  C t) = Acos(  C t +45 0 ) Looking at a constellation diag for this mapping makes it clear why Acos(  C t) + A sin(  C t) = Acos(  C t +45 0 ) etc.

74. 24 Apr'06CS3282 Sectn 874 Constellation diagram for  45 o,  135 o QPSK In phase with cos (real pt) 1,1 1,0 0,1 0,0 45 o V V -V In quadrature with cos Symbol allocation table: Bit1 bit2 b R (t) b I (t) 0 0  A  A 0 1  A +A 1 0 +A  A 1 1 +A +A

75. 24 Apr'06CS3282 Sectn 875 Real Alternative constellation diag ( 0 o,90,180,270 o QPSK) Symbol allocation table: Bit1 bit2 b R (t) b I (t) 0 0 A 0 0 1 0 +A 1 0 -A 0 1 1 -A -A Imag pt 0,1 1,1 0,0 1,0

76. 24 Apr'06CS3282 Sectn 876 Re Real pt 8-PSK 16- PSK Imag pt 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. Differential forms of QPSK & M-PSK often used where changes in phase signify the data. Principle similar to DPSK.

77. 24 Apr'06CS3282 Sectn 877 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'. 000 001 011 010 110 111 101 100 With Gray coding, a symbol error generally causes just one bit-error

78. 24 Apr'06CS3282 Sectn 878 Exercise 8.6 (cont): What happens if we don’t use Gray coding? 000 001 010 011 100 101 110 111 If symbol 111 mistaken for 000 get 3 bit-errors

79. 24 Apr'06CS3282 Sectn 879 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.

80. 24 Apr'06CS3282 Sectn 880 Exercise 8.7: Show how a vector-modulator may be used to generate the 8 or 16 symbols of 8-PSK & 16-PSK. 000 001 011 010 110 111 101 100 Symbol b R (t) b I (t) 000 V 0 001 V/1.4 V/1.4 010 -V/1.4 V/1.4 011 0 V 100 V/1.4 -V/1.4 101 0 -V 110 -V 0 111 -V/1.4 -V/1.4 V

81. 24 Apr'06CS3282 Sectn 881 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.

82. 24 Apr'06CS3282 Sectn 882 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?

83. 24 Apr'06CS3282 Sectn 883 Single carrier digital modulation schemes ASK, FSK, PSK, DPSK, QPSK Differential QPSK Gaussian FSK & MSK Combined ASK & PSK (QAM, APK) etc.

84. 24 Apr'06CS3282 Sectn 884 Other modulation techniques Direct sequence spread spectrum techniques (DSSS) Frequency hopping (FHSS) Complementary code keying (CCK)

85. 24 Apr'06CS3282 Sectn 885 8.7. Introduction to multi-carrier modulation & OFDM Introduces concept of multi-carrier modulation Compares with single carrier modulation to determine some advantages & disadvantages. Orthogonal frequency division multiplexing (OFDM) introduced as highly efficient form of multi-carrier modulation widely used in broadcasting, ADSL & wireless LANs. Implementation of OFDM using FFT & inverse FFT. Parameters of 802.11 OFDM implementation investigated. First, revise some important aspects of single carrier modulation.

86. 24 Apr'06CS3282 Sectn 886 8.8 Matched filtering & equalization for single carrier ‘Map to base-band’ at transmitter has ‘pulse shaping filter’. Generates sinc-like pulses of correct amplitude & polarity at right time. Pulse added to previous pulses & modulated onto carrier. Diagram below illustrates generation & modulation of a single pulse With ASK, ‘sinc like’ pulse shape becomes ‘envelope’. Pulse shaping filter Excite Pulse-s filter b(t)..11101.. Multiply t Volts t volts Map to base-band volts t

87. 24 Apr'06CS3282 Sectn 887 T -3T sinc T (t) t 1 -T 2T -2T 3T 4T-4T Formulae for sinc(x) & sinc T (x)

88. 24 Apr'06CS3282 Sectn 888 Single carrier PSK with pulse shaping Pulse shaping filter Excite Pulse-s filter b(t)..11101.. Multiply t Volts t volts Map to base-band Volts t envelope

89. 24 Apr'06CS3282 Sectn 889 Output of transmitter with two PSK pulses Volts t

90. 24 Apr'06CS3282 Sectn 890 ‘Single carrier’ receiver Receiver must demodulate to obtain base-band b(t). Pulse shapes distorted & affected by noise. Sample & detect for rectangular pulses discussed in last lecture. May work for low bit-rates over channels with little distortion or noise Performance can be improved by introduction of – a matched filter optimally tuned to shape of transmitted pulses to minimise effect of noise (AWGN). – a channel equaliser to cancel out distortion introduced by channel...1100.. Matched filter Demodulator Channel equaliser Sample & detect b(t) Channel signal + AWGN

91. 24 Apr'06CS3282 Sectn 891 Matched filter & RRC pulses Matched filter & channel equaliser may have complex input signals. Multi-level pulses may be used instead of binary. Pulse shapes seen at input to ‘sample & detect’ block be Nyquist; i.e. centre of each pulse must coincide with zero-crossings of all others. e.g. R% ‘raised cosine’. Matched filter multiplies received pulse shape by a copy of itself. So transmitter must now send root raised cosine (RRC) pulses. Look very similar & ‘sinc-like’. Transmitted pulse is ‘squared’ by matched filter in receiver. If transmitter sent RC pulses, detector would see squared RC pulses. These would not have zero-crossings in the right places.

92. 24 Apr'06CS3282 Sectn 892 Channel equaliser Channel equaliser’ is an ‘adaptive filter’ Programmed to correct any differences between pulses seen at output of matched filter & ideal RC pulses required by detector. Aims to cancel out effect of the channel, In particular the effects of frequency selective fading. Received amplitude reduced at some frequencies & reinforced at others. Equalizer must do opposite of this. Must adapt to changes in fading channel characteristics. A demanding filtering task, and it cannot always be successful. If there is a very deep fade, it will just not be possible to reverse it. Trying to do so will just emphasize noise at frequency of deep fade. Single carrier sine-wave modulation still widely used.

93. 24 Apr'06CS3282 Sectn 893 8.9. Spread spectrum modulation Use of a single sine-wave as a carrier is not the only possible choice. Could use a ‘pseudo-random’ carrier known at transmitter & receiver. Bandwidth much wider than that of a modulated sine-wave. This may appear very wasteful of bandwidth. It will appear as noise to receivers not tuned to its exact characteristics. Transmission is ‘coded’ by pseudo-random carrier & security is a bonus. Transmitter-receivers using different pseudo-random carriers can co-exist. This is direct sequence spread spectrum multiplexed access (DS-SSMA) Also referred to as ‘code division multiplexed access’ (CDMA). Basis of most 2G mobile phone systems in the USA. 3G mobile telephony will be based on enhanced form of CDMA.

94. 24 Apr'06CS3282 Sectn 894 8.10. Multi-carrier modulation Assume we have 20 MHz radio channel centred on 2.46 GHz. Could apply single carrier modulation to a sine-wave carrier at 2.46 GHz. With QPSK, max achievable bandwidth efficiency is 2 b/s per Hz Allows 40 Mb/s to be transmitted with 0% RRC pulses (pure sinc). 50% RRC pulses would reduce bandwidth efficiency to 1.33 bits/s per Hz. Only 26.7 Mbits/s now possible, but generating the pulses is much easier. In both cases, whole 20 MHz used by the single carrier modulated signal.

95. 24 Apr'06CS3282 Sectn 895 Alternative to single carrier modulation An alternative is to divide the 20 MHz band into sub-bands with a sinusoidal ‘sub-carrier’ in centre of each band. Instead of one carrier we now have many ‘sub-carriers’. IEEE802.11 divides 20 MHz into 64 sub-bands each of 312.5 kHz. Now 64 sub-carriers at frequencies F+f 0, F+f 1, …, F+f 63 Hz. F is lowest frequency of the 20MHz band f 0 = 156.25 Hz, f 1 = 468.75 Hz, …, f 63 = 19843.75 Hz. Modulating each sub-carrier with QPSK with 0% RRC pulse shaping would achieve 625 kb/s per sub-band. Total bit-rate = 625 x 64 = 40 Mb/s (same as with single carrier) But now the bits are divided into 64 parallel sub-streams. Bit-rate of each sub-stream is 1/64 of the total. This is multi-carrier modulation.

96. 24 Apr'06CS3282 Sectn 896 Pulse shaping again To see the main advantage of multi-carrier modulation, look again at the demands of pulse shaping For single carrier, it is necessary to have a band-limited spectrum. Use ‘sinc-like’ pulses with zero-crossings at t=  T,  2T, etc. Pure sinc pulse has rectangular & strictly band-limited spectrum. Rectangular pulse of duration T would have a ‘sinc-like’ frequency spectrum with zero-crossings at f =  1/T,  2/T,  3/T, etc. Unsuitable for single-carrier modulation. But (as we shall see) may be suitable for multi-carrier. Study the graphs on the next slide.

97. 24 Apr'06CS3282 Sectn 897 Spectra of rect & sinc pulses T.sinc 1/T (f) t T/2-T/2 1 rect T (t) f T 1/T -1/T 2/T -2/T 3/T -3/T 4/T -4/T Fourier transform Real part shown Imag part = 0 f 1/(2T)-1/(2T) T T.rect 1/T (f) sinc T (t) t 1 T -T 2T -2T 3T -3T 4T -4T Fourier transform Real pt shown Imag pt = 0

98. 24 Apr'06CS3282 Sectn 898 Spectra of 50% RC pulses & spectra RC(f) t T/2-T/2 1 rc(t) f T 1/T -1/T 2/T -2/T 3/T -3/T 4/T -4/T Fourier transform Real part shown Imag part = 0 -3T/4 3T/4 f 1/(2T) -1/(2T) T RC(f) rc(t) t 1 T -T 2T -2T 3T -3T 4T -4T Fourier transform Real pt shown Imag pt = 0 3/(4T) -3/(4T)

99. 24 Apr'06CS3282 Sectn 899 Sub-band spectral interference (ICI) With single carrier, R% RC (or RRC) pulses are used at expense of decreasing band-width efficiency. With multi-carrier, pulse shapes close to rectangular may be used. Their spectra are ‘sinc-like’ & of very wide bandwidth. With 64 adjacent sub-bands, there is clearly a danger of inter spectrum interference, or ‘inter-sub-carrier interference (ICI). Also a danger of spectrum leaking outside the 20 MHz band. Both these dangers may be avoided.

100. 24 Apr'06CS3282 Sectn 8100 Eliminating ICI by OFDM Rectangular pulses may be used if peak of spectrum for each sub-band corresponds to zero crossings for all other modulated sub-carriers. Interference avoided in frequency-domain rather than time-domain. Looking at previous graphs, ICI is avoided if adjacent sub-carriers are spaced exactly 1/T Hz apart when sub-band bit-rate is 1/T b/s. This is orthogonal frequency division multiplexing (OFDM) Highly efficient because sub-carriers are as close together as they can possibly be without introducing spectral interference. Each modulated sub-carrier is ‘orthogonal’ to all others which means that they do not interfere with each other.

101. 24 Apr'06CS3282 Sectn 8101 t T/2-T/2 1 rect T (t) Modulate F t T/2-T/2 1 rect T (t) Modulate F+2/T t T/2-T/2 1 rect T (t) Modulate F+1/T T.sinc 1/T (..) f T F+2/T F T.sinc 1/T (f-F) f T F+1/T F T.sinc 1/T (..) f F+2/T F+1/TF+3/T Assume purely real spectrum Combining OFDM sub-bands

102. 24 Apr'06CS3282 Sectn 8102 OFDM spectrum Fourier transform SUM f 1/T 3/T Combine real spectra Assume purely real spectra

103. 24 Apr'06CS3282 Sectn 8103 Use of sub-carriers Bit-rate (1/T) for each sub-channel is 1/64 times total bit-rate Zero-crossings of sinc spectra (at  1/T  2/T,..) much closer together. So the sinc spectra ‘die away’ must faster. Ones in centre of 20 MHz band die away almost completely at edges. Ones near edges not modulated. Out of 64 sub-carriers, do not modulate first six, last five & no. 32. Four other sub-carriers reserved as ‘pilots’, Leaves 48 sub-carriers that can be modulated with data. In IEEE802.11 standard, sub-carriers 0 & 27 to 37 not modulated & 4 others are designated as pilots. Again this leaves 48 sub-carriers for data. Depending on processing, the 2 approaches are probably the same.

104. 24 Apr'06CS3282 Sectn 8104 Modulation of sub-carriers With IEEE802.11, each OFDM sub-carrier modulated by choice of: –binary-PSK, (1 bit per pulse) –QPSK, (2 bits per pulse) –16-QAM (4 bits per pulse) –64-QAM (6 bits per pulse) 16-QAM & 64-QAM are multi-level schemes. Implement by vector-modulator according to ‘constellations’. Illustrate for QPSK & 16-QAM ‘Gray coding’ for 16-QAM makes nearest dots differ in just 1 bit. Differential PSK, QPSK & QAM used where the difference between the current & previous pulse specifies the bit pattern.

105. 24 Apr'06CS3282 Sectn 8105 Constellation for QPSK modulating cos 0,0 0,1 1,0 1,1 Bit1 Bit2 b R b I 0 0 A A 0 1 A -A 1 0 -A A 1 1 -A -A Modulating sin

106. 24 Apr'06CS3282 Sectn 8106 ‘16_QAM’ constellation A 3A -A -3A A3A Real Imag (0000) -A (0001) (0010) (0011) (0100) (1000) (1001) (1010) (1011) (1100) (1101) (1110) (1111) (0110) (0101) (0111) (modulates cos) (modulates sin)

107. 24 Apr'06CS3282 Sectn 8107 Vector-modulator as used for 16-QAM Mult ADD Map Cos(2  f C t) Sin(2  f C t) 3A,-3A,.. -3A,-A,.. 1011 1101.. t V t V Re{..}

108. 24 Apr'06CS3282 Sectn 8108 Vector modulator in complex notation Take b(t) + jq(t) as a complex b-b signal. cos(2  f C t).b R (t) + sin(2  f C t).b I (t) = real { ( b R (t) + jb I (t) ) exp(-2  jf C t) } Mult Map exp(-2  jf C t) b(t) 1011 1101.. Complx base-band Take real pt Sometimes people make this exp(2  jf C t). Makes little difference as long as they are consistent.

109. 24 Apr'06CS3282 Sectn 8109 Fast Fourier Transform & its inverse FFT : {x[n]} 0,N-1  {X[k]} 0,N-1 Inverse FFT: {X[k]} 0,N-1  {x[n]} 0,N-1 Both are ‘fast’ in that they can be programmed or implemented in hardware very efficiently especially when N is a power of 2, e.g. 64, 512, 1024

110. 24 Apr'06CS3282 Sectn 8110 Take 64 sub-carrier frequencies over range F to F + 20 MHz: f C + 0, f C + f D, f C + 2f D, …, f C +63f D with f D = 20MHz / 64 = 312.5 kHz f C = F + 176.25 kHz For orthogonality (correct freq-domain zero crossings) sub-carriers must be 1/T Hz apart. So f D = 1/T & pulse duration T = 3.2 x 10 -6 s = 3.2  s Could transmit 1/(3.2  s) = 312.5 k pulses per second, but we don’t. Extend each pulse to 4  s with a 0.8  s ‘guard-interval’. Transmit 250 k ‘extended pulses’ per second. Guard-interval’ extension could be 0.8  s of zero voltage. But it’s not. Its a ‘cyclic extension’ as we will see later. 8.11 OFDM implementation

111. 24 Apr'06CS3282 Sectn 8111 Bandwidth efficiency of IEEE802.11 OFDM Theoretical maximum is 1 pulse/s per Hz. Using only 48 out of 64 sub-channels loses 25% of total capacity. Lose another 20% (=0.8/4) because of guard-interval (cyclic extension) Max bandwidth efficiency is 60% (=3/4 x 4/5) of 1 pulse/s per Hz. = 0.6 x 2 =1.2 b/s per Hz, if QPSK used for all 48 sub-carriers. With QPSK, bit-rate in 20 MHz will be 24 Mb/s. With 64-QAM, bit-rate achieved is 72 Mb/s. Reduced to 36 Mb/s by half rate convolutional coder. IEEE specifies ¾ rate ‘punctured coder’ for 64-QAM. Gives bit-rate of 72 x3/4 = 54 Mb/s. A ¾ rate punctured conv coder is half rate coder with 2 out of every 6 bits erased to reduce bit-rate to 4/3 times the original.

112. 24 Apr'06CS3282 Sectn 8112 Mult Map exp(2  jf C t) X 0 (t) 10110.. Mult Map exp(2  j(f C +f D )t) X 1 (t) 11001.. Mult Map exp(2  j(f C +63f D )t) X N-1 (t) 11001.. Multi-carrier vector-modulation (in principle)

113. 24 Apr'06CS3282 Sectn 8113 Multi-carrier modulation in practice: Stage 1: Apply PSK, QPSK, QAM (or other) to obtain X 0 (t), X 1 (t),..., X 63 (t) which remain constant for a ‘pulse (symbol) period’ T. Then vector-modulate complex 'sub-carriers' of frequencies: 0, f D, 2f D, …, 63f D Stage 2: Vector-modulate exp(2  jf C t) with output from Stage 1

114. 24 Apr'06CS3282 Sectn 8114 Map X 0 (t) 10110.. Mult Map exp(2  jf D t) X 1 (t) 11001.. Mult Map exp(2  j63f D t) X 63 (t) 11001.. Stage 1 x(t) t X 0 (t )

115. 24 Apr'06CS3282 Sectn 8115 exp(2  jf C t) Stage 2 Complex multiplication. = x(t) (complex) (complex but need only real part) OFDM

116. 24 Apr'06CS3282 Sectn 8116 Stage 1: 63 x(t) =  X m (t) exp (2  jmf D t ) with f D = 1/T m=0 Take 64 samples of x(t) pulse of duration T Let  = T/64 & denote x(n  ) by x[n] for n = 0, 1,..., 63. Set X m (n  ) =X m : constant for 0<n<63 63 x(n  ) = x[n] =  X m exp (2  jm n  /T ) m=0 63 x[n] =  X m exp(jm(2  /64)n) : 0 < n < 63 m=0 Generates a set {x[0], x[1], …, x[63]} of complex numbers.

117. 24 Apr'06CS3282 Sectn 8117 Use of inverse FFT to generate x(t) Can now take 64 complex numbers {X 0, X 1, …, X 63 } representing one symbol & generate 64 complex samples {x[0], x[1], …, x[63]} of x(t). 63 x[n] =  X m exp(jm(2  /N)n) : 0<n<63 m=0 This is ‘inverse FFT’ formula (apart from a factor 1/64). Pulse is of duration T = 3.2  s. It is sampled at T/64 = (1/20)  s or 20 MHz (20 x 10 6 complex samples/second) Real & imag pts of {x[n]} 0,63 could be D to A converted & applied to analogue implementation of Stage 2. Call {x[n]} 0,63 ‘base-band OFDM pulse’ Repeat for next set of {X 0, X 1,..., X 63 } to get another pulse & so on.

118. 24 Apr'06CS3282 Sectn 8118 Stage 2 Real part of x(t) multiplies cos(2  f C t) & imag part multiplies sin(2  f C t). Real part of output is OFDM symbol starting at f C Hz rather than zero. More convenient to implement Stage 2 digitally exp(2  jf C t) must be sampled & x(t) ‘up-sampled’ to same sampling rate. Assume fc = 100 MHz & cos(2  f C t) & sin(2  f C t) are sampled at 400 MHz. Must increase sampling rate of x(t) by a factor of 20; i.e. 63 x[n] =  X m exp(jm(2  /1280)n) : 0 < n < 1279 m=0 which is more conveniently written as 1279 x[n] =  Y m exp(jm(2  /1280)n) : 0 < n < 1279 m=0

119. 24 Apr'06CS3282 Sectn 8119 Implementing Stage 2 digitally The ‘up-sampling’ is achieved by increasing I-FFT order by factor 20. Instead of 64 point I-FFT, we need a 1280 point I-FFT. 1280 is not a power of 2, but there are fast algorithms for such an I-FFT. Applying 1280 point I-FFT to {Y m } 0,1279 which is a ‘zero-padded’ version of {X m } 0,63 gives a version of x(t) sampled at 400 MHz. Since exp(2  jf C t) is also sampled at 400 MHz, we can now implement ‘Stage 2’ digitally by multiplying x(t) by exp(2  jf C t) sample by sample. Taking the real part of the result we obtain 100 MHz sinusoidal carrier modulated by a base-band OFDM signal. The result is sampled at 40 MHz. Converting to analogue & removing all frequencies above about 130 MHz leaves an analogue version of the required OFDM signal. Up-sampling x(t) is useful even in analog implentations to simplify DAC.

120. 24 Apr'06CS3282 Sectn 8120 Shape of OFDM symbol conveys the bit-sequence. With QPSK on 48 carriers, 2 96  10 29 different symbol shapes. With 16-QAM there are  10 60 different pulse shapes (with single carrier binary PSK there are just two!) OFDM pulses must be accurately represented & processed by linear circuits. (with a small number pulses, linearity is not so important) Highly linear amplifiers (Class A) are very power inefficient. Amplifiers used in 2G mobile phones (for GMSK - a form of binary FSK) are not very linear but extremely power efficient. GMSK is ‘constant envelope’ - OFDM is definitely not! OFDM symbols (pulse shapes)

121. 24 Apr'06CS3282 Sectn 8121 Each 3.2  s pulse is extended to 4  s by prefixing a 0.8  s ‘guard time’ The prefix is made to be a copy of the final 0.8 us (16 samples) of the pulse. It is called a ‘cyclic prefix’ or ‘cyclic extension’. Generate 80 time-domain complex numbers for each ‘extended pulse’ Each extended pulse takes 4 us, so we send 250 k extended pulses/second. Cyclic extension Real{x[n]} n 80 160 -80 Similarly for imaginary part. 16 Cyclic prefix 3.2  s pulse Cyclic prefix 3.2  s pulse

122. 24 Apr'06CS3282 Sectn 8122 Coherent demodulator with sampler, sync & symbol extraction. Apply FFT to recover {X 0, X 1, …, X 63 }. Channel distortion cancelled out by equaliser applied to FFT output. OFDM receiver exp(-2  jf C t) Complex multiplication. OFDM Derive local carrier Sample & extract 4  s ext- symbol FFT Detector 20 kHz lowpass filter Equa liser

123. 24 Apr'06CS3282 Sectn 8123 Detectors FFT of {x[n]} 0,63 gets back to {X 0, X 1, …., X 63 }. Detect sequence of 1,2, 4 or 6 bits by finding nearest dot on the appropriate constellation diagram. (B-PSK, QPSK, 60-QAM or 64-QAM), ‘Nearest dot’ detector for each complex number generated by FFT is required. 0,0 0,1 1,0 1,1 Re Illustrate for QPSK Im

124. 24 Apr'06CS3282 Sectn 8124 Cyclic extension as ‘guard interval’ Eliminates inter-symbol interference between 3.2  s OFDM symbols. 0.8  s is longer than any delay between a direct path & any reflected paths within a building. As speed of radio waves  300  10 6 m/s, allows for a path-length difference of 0.8  300 = 250 m. Any reflected path up to 250 m longer than direct path will not cause one 3.2  s OFDM symbol to interfere with the next. Multipath propagation may still distort structure of OFDM symbols. Equaliser required to reverse this distortion.

125. 24 Apr'06CS3282 Sectn 8125 Cyclic extension for equalisation A guard interval could be 0.8  s of zero voltage. Cyclic extension is more than just a guard interval. With the FFT, it greatly simplifies equalisation process. Multi-path propagation causes radio channel to act like a ‘filter’. Single carrier demodulator employs adaptive filter to cancel it. Filtering is computationally intensive. Filtering in time-domain becomes multiplication in frequency-domain. FFT is part of OFDM demodulator, so equalisation, using multiplication rather than filtering, can be applied to FFT output. Difference between ‘cyclic’ filtering with FFT & ‘linear’ filtering. Disappears when input to FFT is result of applying cyclically extended signal to channel. Cyclic extension to OFDM symbol allows equalisation by ‘cyclic’ filtering by FFT & complex multiplication.

126. 24 Apr'06CS3282 Sectn 8126 Cyclic extension for synchronisation The cyclic extension is also useful for carrier & symbol synchronisation at the receiver since, if the first 16 samples of an extended pulse are the same as last 16, we are synchronised.

127. 24 Apr'06CS3282 Sectn 8127 Exercise: generation of OFDM with 4 sub-carriers Given 8-bits, 00011011, show how one OFDM base-band symbol {x[n]} may be generated by a 4-point inverse FFT. Use QPSK to modulate the 4 sub-carriers. Extend to 6 samples {x[n]} 0,6 by cyclic extension & explain how a high frequency carrier would be modulated by the samples of x. Show how original data can be recovered by 4-point FFT.

128. 24 Apr'06CS3282 Sectn 8128 Solution: Data is: 00 01 10 11 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 ]; % array of 4 complex numbers Perform 4 point IFFT on X to obtain array x x=ifft(X) % This does it in MATLAB Array x now contains the 4 samples of the required symbol: [ 0 0.5 + 0.5j j 0.5 - 0.5j ] Including the cyclic extension, this becomes: [ j 0.5 - 0.5 j 0 0.5 + 0.5j j 0.5 - 0.5j ]

129. 24 Apr'06CS3282 Sectn 8129 8.12 Advantages of OFDM Spectrally efficient because of orthogonality of the 64 carriers. Good for channels affected by frequency selective fading because: (i) Effects of fading, affecting a small range of frequencies, can be spread out using ‘interleaving’ so that FEC can more easily correct any bit-errors. (ii) Cyclic extension as a guard-interval, eliminates ISI caused by multi-path propagation. Simpler way of eliminating ISI than pulse-shaping as used in single carrier systems. (iii) Equalisation is easier than with single carrier systems which use adaptive filtering. OFDM receiver can amplify real & imag parts of FFT outputs such that they have same amplitudes. Possible because of the cyclic extension as explained earlier.

130. 24 Apr'06CS3282 Sectn 8130 Disadvantages of OFDM ‘Peak to mean’ ratio of symbols can be very large by nature of FFT & Inv-FFT. (Amplitudes can become very large in comparison to the mean) Shapes OFDM symbols very complex & must be sent & received accurately. With QPSK on each sub-carrier,  10 29 shapes & even more with 64-QAM 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. Sensitive to ‘Doppler’ frequency shifts.

131. 24 Apr'06CS3282 Sectn 8131 8.13 Some more details about IEEE 802.11a/g OFDM With IEEE802.11a & g, OFDM symbols take 4  s;  250 k symbols/second. Each symbol can carry 1-6 bits per carrier (BPSK, QPSK, 16- & 64-QAM). Highest bit-rate with 64-QAM & 3/4 rate conv coder: 48 x 6 x (3/4) 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 6 Mb/s) available. Two lowest bit-rates (9 & 6 Mb/s) use binary PSK & 3/4 or 1/2 rate FEC : 48 x (3/4) x 250kb/s = 9 Mb/s 48 x (1/2) x 250 kb/s = 6 Mb/s. For 18 & 12 Mb/s, QPSK is used on each of 48 data carriers. For 36 & 24 Mb/s use 16-QAM. With 1/2 rate coder 64-QAM would give 36 Mb/s, so use 2/3 rate for 48 Mb/s.

132. 24 Apr'06CS3282 Sectn 8132 8.14. Conclusions and learning outcomes Matched filtering affects pulse-shaping in single carrier modulation. Channel equalisation, required to cancel effects of frequency selective fading, is a computationally expensive adaptive filtering task. OFDM is highly efficient form of multi-carrier modulation. Single carrier uses  sinc pulses & eliminates inter-symbol interference OFDM uses  rect pulses & eliminates inter-spectral interference. FFT & I- FFT implement OFDM directly. Channel equalisation much easier to implement - no adaptive filter needed. Need for highly linear amplification & wide range of peak-to-mean ratios cause practical problems especially for battery powered mobile equipment. Parameters of 802.11 OFDM implementation have been analysed.

133. 24 Apr'06CS3282 Sectn 8133 8.15 Problems & discussion points 1. What is the max bit-rate that can be transmitted without ISI on a 1 MHz channel using (i) B-PSK, (ii) QPSK, (iii) 16-QAM. 2. What is the max bit-rate that can be transmitted with arbitrarily low bit-errors over a noise-less channel of 1 MHz bandwidth [Ans:  ] 3. Repeat Q.2 for a noisy channel where the SNR is 30 dB. 4. How does spectrum of a 50% RC pulse differ from that of a pure sinc pulse. 5. Why are RRC rather than RC pulses used in single carrier transmissions. 6. How many different OFDM symbol shapes are there with 64-QAM? 7. Why are the first & last few sub-carriers left unmodulated? 8. With 16-QAM, why are the 4-bit numbers arranged in ‘Gray coder’ order? 9. Derive a constellation for 64-QAM. 10. Why are interleaving & FEC very important with OFDM?

134. 24 Apr'06CS3282 Sectn 8134 Problems & discussion points (cont) 11.Given that their bandwidth was 30 kHz & in cities B C  30 kHz, why was an equaliser not needed in a ‘1G’ mobile phone. Why is an equaliser definitely needed in a WLAN receiver when single carrier modulation is used? 12. Explain why bandwidth efficiency of 802.11 OFDM is 0.6 symbols/s per Hz without FEC. What is bandwidth efficiency when ¾ rate convolutional coder is used? 13. If a single carrier modulation scheme is used with R% RRC pulse shaping, what value of R would give a bandwidth efficiency of 0.6 pulses (symbols) per Hz ? 14. How are 24 & 36 Mb/s achieved over an IEEE802.11g WLAN? 15. Some non-standard versions of 802.11 claim to achieve 108 Mb/s. How is this done? 16. 802.11g claims max bit-rate of 54Mb/s. But cost of sending sync preambles & headers reduces this bit-rate even in ideal conditions. Assuming ideal conditions, estimate max average bit-rate (i) where close to max length packets (  2000 byte payload) always sent, (ii) where packets contain only 160 bytes of payload (20 ms of G711 speech).