Section 8: Carrier Modulated Transmission Convert binary data into - - PowerPoint PPT Presentation

24 Apr'06 CS3282 Sectn 8 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 Channel Trans Rec 10110 10111

24 Apr'06 CS3282 Sectn 8 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 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 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 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 6

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.

24 Apr'06 CS3282 Sectn 8 7

Multiplication of carrier by sine-wave

volts volts t Multiply t

24 Apr'06 CS3282 Sectn 8 8

Frequency modulation (fm) by sine-wave

volts Modulate frequency t t

24 Apr'06 CS3282 Sectn 8 9

Effect of modulation on frequency spectrum

carrier frequency Power spectral density

24 Apr'06 CS3282 Sectn 8 10

Where do we get ‘side-bands’ from?

carrier * message A cos(ωCt) * cos(ωMt) = 0.5A cos(ωCt + ωMt) + 0.5A cos(ωCt - ωMt) = 0.5A cos( (ωC + ωM) t ) + 0.5 A cos( (ωC - ωM) t ) upper sideband lower sideband ωC = 2πfC , etc

24 Apr'06 CS3282 Sectn 8 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(ωMt) use [1 + cos(ωMt)]

A cos(ωC t ) 0.5A [cos( (ωC + ωM) t ) + cos((ωC -

s( (ωC + ωM) t ) + cos((ωC - ωM)t) ]

ier DSP ‘am’. er DSP ‘am’.

24 Apr'06 CS3282 Sectn 8 12

Large carrier DSP ‘am’ modulator

• V

t V Multiply V t t 1+cos(ωMt)

24 Apr'06 CS3282 Sectn 8 13

‘Envelope detector’ for LC-DSB ‘am’

Low-pass filter

t V t V V t

Rectify

24 Apr'06 CS3282 Sectn 8 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 15

Coherent demodulation of ‘am’

V Received signal t V 1+cos(ωMt) t

Lowpass filter Mult

V t Local carrier

Derive local carrier

24 Apr'06 CS3282 Sectn 8 16

Proof that coherent demodulation works

• Let received signal be A cos(ωCt) .(1+cos(ωMt) )
• Multiplying by local carrier gives

A cos2 (ωCt) . ( 1+cos(ωMt) ) = 0.5A(1 + cos(2ωCt)) .(1 + cos(ωMt) ) = 0.5A(1+cos(ωMt)) + 0.5A cos(2ωCt)(1+cos(ωMt) ) Low-pass filter removes this

24 Apr'06 CS3282 Sectn 8 17

Coherent demodulation again

• No longer requires modulating signal to be purely +ve
• Works with cos(ωMt) just as well as with 1+cos(ωMt)
• No longer ‘large carrier & envelope detectn no good.
• When cos(ωMt) becomes −ve, carrier amplitude remains +ve,

but phase changes by 180o

• With digital, modulating signal no longer sinewaves or music

24 Apr'06 CS3282 Sectn 8 18

8.2.2 Vector modulator & complex baseband

• Independently modulate cos(2πfCt) & sin(2πfCt) and sum.
• Coherent demodulatr for ‘cos’ transmission blind to ‘sin’ trans.
• And vice-versa.

Mult Mult ADD Cos(2πfCt) Sin(2πfCt) bR(t) bI(t)

• “2 channels for price of 1”
• Still single carrier
• Complex baseband:

b(t) = bI(t) + jbR(t)

• More about this later

24 Apr'06 CS3282 Sectn 8 19

Vector demodulator

Mult Mult Cos(2πfCt) Sin(2πfCt) bR(t) bI(t) Derive local carrier (cos & sin) Lowpass filter Lowpass filter bR(t)cos(2πfCt) + bI(t)sin(2πfCt)

24 Apr'06 CS3282 Sectn 8 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 21

Mapping bit stream to base-band

Pulse-shaping filter ..1 1 0 1 0 ... Generate impulses t V V 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 22

Techniques for digital transmission

• Can modulate amplitude, frequency &/or phase of cos(2πƒCt).
• 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.

24 Apr'06 CS3282 Sectn 8 23

Binary frequency shift keying (B-FSK)

Modulate carrier Map to base-band 10110 t volts t

24 Apr'06 CS3282 Sectn 8 24

Binary amplitude shift keying (B-ASK)

Map to base-band 10110 t volts Multiply

24 Apr'06 CS3282 Sectn 8 25

Binary phase shift keying (B-PSK)

Map to base-band 10110 t volts Multiply t

24 Apr'06 CS3282 Sectn 8 26

4-ary amplitude shift keying (ASK)

Map to base-band 10110 t volts Multiply t volts volts

24 Apr'06 CS3282 Sectn 8 27

Combined multi-level ASK & PSK

Map to base-band 10110 t volts Multiply

24 Apr'06 CS3282 Sectn 8 28

8.3. Amplitude shift keying

r(t) cos(2πƒct) t b(t) r(t) t b(t)

24 Apr'06 CS3282 Sectn 8 29

ASK spectrum from BB spectrum

f

• fC

fC Power spectral density PSD

T 1 T 1 − T 2 1

f

T fC 1 + T fC 1 −

≈50% RRC

24 Apr'06 CS3282 Sectn 8 30

ASK with effect of pulse shaping shown

t Shaping

Map ..1 0 1.. r(t) cos(2πƒct)

24 Apr'06 CS3282 Sectn 8 31

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.

24 Apr'06 CS3282 Sectn 8 32

Coherent demodulation of ASK

10110 t volts Multiply Threshold detector Low pass

24 Apr'06 CS3282 Sectn 8 33

Non-coherent detection of ASK

t Rectify & smooth Threshold detector 10110

24 Apr'06 CS3282 Sectn 8 34

Envelope detector for ASK

Low-pass filter (smoother) t t t V V Sample Diode Resistor

24 Apr'06 CS3282 Sectn 8 35

8.3.3. Constellation diagrams

Show “in phase” and “quadrature” components as a graph as illustrated below for two examples:

Binary ASK with symbols 0 & Acos(..) In phase with carrier Quadrature to carrier Q I 4-ary ASK with symbols 0, Acos(..), 2Acos(..), 3Acos(..) 0 A 2A 3A

24 Apr'06 CS3282 Sectn 8 36

8.3.4. Coherent demodulation of ASK

Multiply by local carrier locked in frequency & phase with carrier received. Lowpass filter cos(2πƒct) s(t)cos(2πƒct) Generate local carrier Threshold detector

) 4 cos( ) ( ) ( 5 . ) ( cos ) (

2 c c

f t s t s t f t s π π + =

Removed by lowpass filter cos2θ= 2cos2θ - 1

24 Apr'06 CS3282 Sectn 8 37

8.3.5. Coherent versus non-coherent detection

Let the signal be: b(t)cos(2πƒct). Noise is: N(t)cos(2πƒct + θ(t)) where N(t) is random envelope & θ(t) is random phase. This equals:

) 2 sin( ) ( sin ) ( ) 2 cos( ) ( cos ) ( t f t t N t f t t N

c c

π θ π θ +

Half noise power in phase with cos(2πƒct ) & half with sin(2πƒct ). Non-coherent detection measures envelope of signal plus noise & is affected by full power of noise. Coherent detection multiplies by cos(2πƒct ) 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.

24 Apr'06 CS3282 Sectn 8 38

8.4 Complx baseband & vector-modulator/demodulatr

8.4.1 Vector modulator: ..11010.. Map sin(2πfCt) cos(2πfCt) bI(t) bR(t)

bR(t)cos(2πfCt) + bI(t)sin(2πfCt)

Map ..10010..

24 Apr'06 CS3282 Sectn 8 39

Complex notation for vector-modulator

• bR(t) is ‘in-phase’ component & bI(t) is ‘quadrature’ component.
• Complex base-band signal is bR(t) + jbI(t) where j = √(-1).
• Output is real part of:

[ bR(t) + jbI(t)] . exp(-2πjfC t) since [ bR(t) + jbI(t)] . [cos(2πfC t) − jsin(2πfC t) ] = [ bR(t) cos(2πfC t) + bI(t)sin(2πfC t) ] + j(..)

Mult b(t) 10110 Complex

• signal. Take

real part. Map Complx base-band 11011 exp(-2π j fCt)

24 Apr'06 CS3282 Sectn 8 40

8.4.2. Vector-demodulator

• Receives bR(t)cos(2πfC t) + bI(t)sin(2πfC t)
• Recovers bR(t) & bI(t) separately.
• bR(t) & bI(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 41

Vector demodulator (cont)

Mult Mult Threshold Detector Threshold Detector Cos(2πfCt) Sin(2πfCt) bR(t) bI(t) Low pass Low pass ..11010.. Derive local carrier (cos & sin) Received signal r(t) ..10010..

24 Apr'06 CS3282 Sectn 8 42

Show why this works for cosine modulation

Let r(t) = bR(t) cos(2π fC t) + bI(t) sin(2πfC t) ) Then r(t) cos(2π fC t) = bR(t)cos2(2π fC t) + bI(t) sin(2πfC t) )cos(2π fC t) = 0.5 bR(t)[1 + cos(4π fC t)] + 0.5 bI(t) sin(4πfC t) ) = 0.5bR(t) + 0.5bR(t) cos(4π fC t) + 0.5 bI(t) sin(4πfC t) ) Hence cosine demodulator recovers bR(t) & is blind to bI(t) Removed by lowpass filter

24 Apr'06 CS3282 Sectn 8 43

Similarly for sine modulation

r(t)sin(2π fC t) = bR(t) cos(2π fC t)sin(2πfC t) + bI(t) sin2(2πfC t) ) = 0.5 bR(t) sin(4π fC t) + 0.5 bI(t) [1 - cos(4πfC t) ] = 0.5 bR(t) sin(4π fC t) + 0.5 bI(t) - 0.5bI(t)cos(4πfC t) Removed by lowpass filter Sine demodulator recovers bI(t) & is blind to bR(t)

24 Apr'06 CS3282 Sectn 8 44

Trig formulae

This works because cos2(θ) & sin2(θ) have a constant (or DC) component 0.5 whereas sin(θ)cos(θ) does not. Relevant formulae are:

• cos 2 (θ) = 0.5 + 0.5 cos(2θ)
• sin 2 (θ) = 0.5 - 0.5 cos(2θ)
• sin(θ) cos (θ) = 0.5sin(2θ)

24 Apr'06 CS3282 Sectn 8 45

8.4.3. Constellation diags for ASK with complx baseband

Quadrature to In phase with carrier 0 A 2A 3A

Binary ASK for bR(t) & bI(t) 4-ary ASK for bR(t) & bI(t)

In phas

A A

3A A

In quadrature

carrier

24 Apr'06 CS3282 Sectn 8 46

Symbol allocation tables for binary & 4-ary ASK

Bits bR bI 0 0 0 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 Bits bR bI 0 0 0 0 0 1 0 A 1 0 A 0 1 1 A A

24 Apr'06 CS3282 Sectn 8 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, ƒ0 & ƒ1 say.
• Consider 3 generation methods.

24 Apr'06 CS3282 Sectn 8 48

8.5.1 Methods for generating FSK

• 1. “Voltage controlled oscillator(VCO)”method.

FM Modulator (VCO) 1 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.

1 FSK

Clearly this may not produce a continuous phase

• utput.

24 Apr'06 CS3282 Sectn 8 49

• 3. “Vector-modulator” method:

For binary FSK with ƒc+ƒ1 & ƒc-ƒ1, apply cos (2πƒ1t) to ‘Q’ and ±sin(2πƒ1t) to ‘I’ . Sign determines the symbol.

“Q” input “I” input Sin(2πƒct) Cos(2πƒct)

cos (2πƒ1t) ±sin(2πƒ1t)

24 Apr'06 CS3282 Sectn 8 50

Exercise 8.1: Check that this works. Solution: When I=+sin(2πƒ1t), output is: sin(2πƒ1t)cos(2πƒct)+cos(2πƒ1t)sin(2πƒct) =sin(2π(ƒc+ƒ1)t) When I=-sin(2πf1t) the output is:

• sin(2πƒ1t)cos(2πƒct)+cos(2πƒ1t)sin(2πƒct)

=sin(2π(ƒc–ƒ1)t)

24 Apr'06 CS3282 Sectn 8 51

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

24 Apr'06 CS3282 Sectn 8 52

• 2. Discriminator followed by envelope-detector.

Turns FSK into ASK for easier detection

Discriminator Low-pass filter (smoother) t t t V t f Gain Resistor f1 f0 f1 f0

24 Apr'06 CS3282 Sectn 8 53

• 3. Phase Locked Loop detector for FSK.

PLL is 'black box' with one input & 2 useful outputs:

PLL t V t t VCO input (Voltage ∝ input frequency) VCO

• utput

Frequency modulated input

24 Apr'06 CS3282 Sectn 8 54

8.5.3. 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

Low-pass filter VCO VCO input voltage VCO output t t V V voltage

24 Apr'06 CS3282 Sectn 8 55

8.5.4 Non-coherent FSK detector for higher data rates:

“Zero crossing counter” type of detector

Limiting Amplifier Clock Decide Counter Reset Data FSK

and

24 Apr'06 CS3282 Sectn 8 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 sine- waves of frequency ƒ0 and ƒ1 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 ƒ0 or ƒ1 produced the larger output, and that determines the symbol.

24 Apr'06 CS3282 Sectn 8 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 ƒ1 & ƒ0, ‘one’s form a DSB spectrum centred on ƒ1 ‘zero’s form a DSB spectrum centred on ƒ0. Resulting spectrum is sum of these two spectra.

PSD ƒ ƒ0-1/T ƒ0 ƒ0+1/T PSD ƒ ƒ1-1/T ƒ1 ƒ1+1/T

24 Apr'06 CS3282 Sectn 8 58

PSD ƒ

ƒ0-1/T ƒ0 ƒ0+1/T

PSD ƒ

ƒ1-1/T ƒ1 ƒ1+1/T

PSD

ƒ0-1/T ƒ0 ƒ1

ƒ

ƒ1+1/T

+ =

24 Apr'06 CS3282 Sectn 8 59

Sunde’s FSK method

Place ƒ0 at ƒ1±1/T & ƒ1 at ƒo±1/T.

24 Apr'06 CS3282 Sectn 8 60

8.5.7. Minimum shift keying (MSK)

• Form of FSK where difference between ƒ0 & ƒ1 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 ƒ0 to ƒ1 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 61

GMSK transmitter

FIR Gaussian shaping filter VCO Map to impulse s ..10110 .. GMSK

24 Apr'06 CS3282 Sectn 8 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 63

8.5.8. Advantages & disadvantages of FSK

Advantages:

• 1. Constant envelope hence not too sensitive to varying attenuation
• n 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 64

8.6. Phase shift keying (PSK)

• Send sinusoidal carrier with phase changes determined by bits
• Consider binary PSK with 1 bit/cycle, 00 & 1800 phase shifts

& rectangular pulse shaping cos(2πƒct) t ±cos(2πƒct) b(t) Map ..1010010..

24 Apr'06 CS3282 Sectn 8 65

A binary PSK waveform

t V 1 1 0 0 1 1 Assuming 1 bit per cycle.

24 Apr'06 CS3282 Sectn 8 66

8.6.2 Coherent Detector for binary PSK

±cos2(2πƒct) = ±0.5(1+cos4πƒct)

Data +1/2:”1”

• 1/2:”0”

Lowpass filter

Generate local carrier ±cos(2πƒCt)

Threshold Detector ±1/2

cos(2πƒC t)

24 Apr'06 CS3282 Sectn 8 67

Details of coherent PSK demodulator/detector

• Low-pass filter eliminates ±cos(4πƒC t).
• Matched filter will achieve this because of orthogonality of

±cos(4πƒct) to sin(2πƒct).

• 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
• n carrier frequency.

24 Apr'06 CS3282 Sectn 8 68

8.6.3 Differential PSK

Phase shift of carrier with respect to previous symbol indicates current bit. Illustrate for 1 bit/cycle with 00 shift for 1 & 1800 for 0:

V

1 1 0 0 0 (0o)

(1800) t

24 Apr'06 CS3282 Sectn 8 69

900 & 2700 phase shifts often preferred with binary DPSK:

t V

1 bit/cycle 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)..

24 Apr'06 CS3282 Sectn 8 70

8.6.4 Differential detection of binary DPSK

• Consider case where phase shifts are 00 & 1800 & 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.

±cos2(2πƒct) = ±0.5(1+cos4πƒct) ±0.5 ±cos(2πƒCt)

Lowpass filter Threshold detector Delay by T (Delay for 1 bit)

24 Apr'06 CS3282 Sectn 8 71

Lowpass filter output is +0.5 if carrier has been subject to 00 phase shift (logic 1 say) and –1/2 for 1800 (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 72

8.6.5 Detector for binary DSPK with 90O & 270O phase shifts rather than 0 and 180O.

LPF Detect Delay by T (Delay for 1 bit) 900 phase shift

24 Apr'06 CS3282 Sectn 8 73

8.6.6 Quaternary PSK (QPSK)

• Consider a vector modulator where bR(t) & bI(t) are bipolar
• Then bR(t)cos(2πfCt) & bI(t) sin(2πfCt) are both binary PSK.
• ‘2-channel’ modulation process is QPSK or 4-PSK.

Mult Mult ADD Map Map Cos(2πfCt) Sin(2πfCt) bR(t)

bI(t)

10110 11011

24 Apr'06 CS3282 Sectn 8 74

QPSK de-modulator

Sin(2πfCt) Mult Mult Cos(2πfCt) Detect Detect bR(t) bI(t) 10110 11011 Low pass Low pass Detect carrier

24 Apr'06 CS3282 Sectn 8 75

Two ways of looking at QPSK

• One way is ‘vector modulation’ approach where cos(2πfCt) &

sin(2πfCt) are binary PSK modulated independently.

• At receiver, coherent PSK detector for cos(2πfCt) channel is blind

to transmission on sin(2πfCt) & vice-versa.

• Refer to bR(t) + j bI(t) as 'complex base-band' signal b(t).
• Transmitted QPSK signal is Re{ [bR(t) +j bI(t)] exp(-j2πfCt) }.

10110 Mult exp(-2πj fCt) b(t) Map Transmit real part Complx base-band 11011

24 Apr'06 CS3282 Sectn 8 76

Another way to look at QPSK

• QPSK sends 2 bits at once , using bipolar bR(t) & bI(t)
• Let bR(t) & bI(t) be rect pulses of amplitude -A or +A.
• Mapping to base-band may then be as follows (ωC=2πfC)

Bit1 bit2 bR(t) bI(t) QPSK symbol transmitted 0 0 −A −A −Acos(ωCt) − A sin(ωCt) = Acos(ωCt−1350) 0 1 −A +A −Acos(ωCt) + A sin(ωCt) = Acos(ωCt+1350) 1 0 +A −A Acos(ωCt) − A sin(ωCt) = Acos(ωCt −450) 1 1 +A +A Acos(ωCt) + A sin(ωCt) = Acos(ωCt +450) Looking at a constellation diag for this mapping makes it clear why Acos(ωCt) + A sin(ωCt) = Acos(ωCt +450) etc.

24 Apr'06 CS3282 Sectn 8 77

Constellation diagram for ±45o, ±135o QPSK

In phase with cos (real pt) 1,1 1,0 0,1 0,0 45o V V

• V

In quadrature with cos Symbol allocation table: Bit1 bit2 bR(t) bI(t) 0 0 −A −A 0 1 −A +A 1 0 +A −A 1 1 +A +A

24 Apr'06 CS3282 Sectn 8 78

Alternative constellation diag ( 0o,90,180,270o QPSK)

Real

Symbol allocation table: Bit1 bit2 bR(t) bI(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

24 Apr'06 CS3282 Sectn 8 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.

Re Real pt 8-PSK 16- PSK Imag pt

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 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'.

000 001 011 010 110 111 101 100 With Gray coding, a symbol error generally causes just one bit-error

24 Apr'06 CS3282 Sectn 8 81

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

24 Apr'06 CS3282 Sectn 8 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 83

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

V Symbol bR(t) bI(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

24 Apr'06 CS3282 Sectn 8 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 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 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 87

Other modulation techniques

• Direct sequence spread spectrum techniques (DSSS)
• Frequency hopping (FHSS)
• Complementary code keying (CCK)

24 Apr'06 CS3282 Sectn 8 88

Pause

• End of slides on single carrier modulation

24 Apr'06 CS3282 Sectn 8 89

Multi-carrier modulation & OFDM

• Orthog frequency division multiplexing (OFDM) is relatively new

‘multi-carrier’ modulation scheme.

• Used for DAB, ADSL & wireless LANs (IEEE 802.11a).
• 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 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 91

Constellation for QPSK

modulating cos 0,0 0,1 1,0 1,1 Bit1 Bit2 bR bI 0 0 A A 0 1 A -A 1 0

• A

A 1 1

• A -A

Modulating sin

24 Apr'06 CS3282 Sectn 8 92

16-QAM _ symbol allocation table

Bit1 bit2 bit3 bit4 VI VQ Bit1 bit2 bit3 bit4 VI VQ 0 0 0 0 A A 1 0 0 0 3A A 0 0 0 1 A -A 1 0 0 1 3A -A 0 0 1 0 A 3A 1 0 1 0 3A 3A 0 0 1 1 A -3A 1 0 1 1 3A -3A 0 1 0 0 -A A 1 1 0 0

• 3A A

0 1 0 1

• A -A 1 1

0 1 -3A -A 0 1 1 0

• A 3A 1 1 1 0
• 3A 3A

0 1 1 1

• A -3A 1 1 1 1
• 3A -3A

24 Apr'06 CS3282 Sectn 8 93

‘16_QAM’ constellation

A 3A

• A
• 3A

A 3A Real Imag (0000)

• A

(0001) (0010) (0011) (0100)

24 Apr'06 CS3282 Sectn 8 94

Vector-modulator

Mult Mult ADD Map Map Cos(2πfCt) Sin(2πfCt) bR(t) bI(t) 10110 11011

24 Apr'06 CS3282 Sectn 8 95

Vector modulator in complex notation

Take b(t) + jq(t) as a complex b-b signal. cos(2πfCt).b(t) + sin(2πfCt). q(t) = real { ( b(t) + jq(t) ) exp(-2πjfCt) } Mult Map exp(-2πjfCt) b(t) 10110 11011 Complx base-band

24 Apr'06 CS3282 Sectn 8 96

A slight variation

cos(2πfCt).b(t) − sin(2πfCt). q(t) = real { ( b(t) + jq(t) ) exp(2πjfCt) } Instead of sin we modulate -sin: no real difference Mult Map exp(2πjfCt) b(t) 10110 11011 Complx base-band

24 Apr'06 CS3282 Sectn 8 97

A slight variation (continued)

Mult Mult ADD Map Map Cos(2πfCt) −Sin(2πfCt) bR(t) bI(t) 10110 11011

24 Apr'06 CS3282 Sectn 8 98

Fast Fourier Transform

FFT : {x[n]}0,N-1 → {X[k]}0,N-1

[ ] [ ]

1 1 for

1 / 2

N- , ..., , k = e n x k X

N n N kn j

∑

− = −

=

π

k |X[k]| N N/2 fS/2 fS x[n] n N Time-domain

Frequency domain

24 Apr'06 CS3282 Sectn 8 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 8 100

Inverse Fast Fourier Transform

IFFT: {X[k]}0,N-1

→ {x[n]}0,N-1

[ ]

1 1 for 1

1 / 2

, ...,N- , k = e X[k] N n x

N n N kn j

∑

− =

=

π

|X[k]| N N/2 fS/2 fS k 2N x[n] n N Time-domain

24 Apr'06 CS3282 Sectn 8 101

End of preliminaries for multi-carrier modulation

24 Apr'06 CS3282 Sectn 8 102

Multi-carrier modulation

• Take 64 carrier frequencies over range fC to fC + 20 MHz:

fC + 0, fC + fD, fC + 2fD, … , fC +63fD

with fD = 20MHz / 64 = 312.5 kHz.

24 Apr'06 CS3282 Sectn 8 103

Mult Map exp(2πjfCt) X0(t) 10110 11011 01001 Mult Map exp(2πj(fC+fD)t) X1(t) 11001 Multi-carrier modulation XN-1(t) Map 11001 Mult 11110 exp(2πj(fC+(N-1)fD)t)

24 Apr'06 CS3282 Sectn 8 104

Do multi-carrier modulation in two stages: Stage 1:

Apply PSK, QPSK, QAM (or other) to obtain

X0(t), X1(t), ..., XN-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 , fD, 2fD , …, (N-1)fD Stage 2:

Vector-modulate exp(2πjfC) with sum of modulated sub-carriers.

24 Apr'06 CS3282 Sectn 8 105

Stage 1

10110 01001 Mult Map exp(2πjfDt) X1(t) 11001 X0(t) Map 11011 XN-1(t) Map 11001 Mult 11110 exp(2πj((N-1)fD)t)

24 Apr'06 CS3282 Sectn 8 106

Stage 2

∑

− = + 1 ) ( 2

) (

N m t f f jm m

D C

e t X

π

∑

− = 1 2

) (

N m t jmf m

D

e t X

π

(complex but need only real part) (complex)

exp(2πjfC) Note that this is complex multiplication.

24 Apr'06 CS3282 Sectn 8 107

The 64 modulating signals:

X0(t) = B0(t) + jQ0(t) : modulating 0 Hz X1(t) = B1(t) + jQ1(t) : modulating fD X2(t) = B2(t) + jQ2(t) : modulating 2fD …. X63(t) = B63(t) + jQ63(t). : modulating 63fD With QPSK, each Xi represents 2 bits. (IEE802.11a makes X0-X5 & X58-X63 all zero & uses 4 others for "pilot tones", leaving 48 to use.).

24 Apr'06 CS3282 Sectn 8 108

• Adding these together we obtain:

63 x(t) = Σ Xk(t) exp (2πjkfD 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 Xk(nτ) exp (2πjk fD nτ ) k=0

• Make Xk[nτ] =Xk : constant for 0<n<63, i.e. for 1 symbol period

63 x[n] = Σ Xk 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 8 109

This formula:

63 x[n] = Σ Xk exp(jk(2π/N)n) : 0<n<63 k=0

takes 64 complex numbers {X0, X1, …, X63 } 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 {X0, X1, ..., X63}to get another pulse & so on.

24 Apr'06 CS3282 Sectn 8 110

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 8 111

63 127

• 64
• 128

Similarly for imaginary part. Real(x[n]} n

24 Apr'06 CS3282 Sectn 8 112

• 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

{X0, X1, ..., X63} rather than 64.

• If T remains at 4 us, we get 250 x 80 = 20 Mb/s for the time-

domain waveform.

• Samples from -16 to -1 form the ‘cyclic prefix’.

24 Apr'06 CS3282 Sectn 8 113

63 127

• 64
• 128

Similarly for imaginary part.

• 16

Real(x[n]} n

24 Apr'06 CS3282 Sectn 8 114

Orthogonal Freq Div Multiplexing (OFDM)

• Scheme described on previous slides is OFDM as used by

IEEE802.11.

• 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, 296 ≈ 8 x 10 28 different symbol shapes.
• 250,000 such symbols strung together per second

24 Apr'06 CS3282 Sectn 8 115

Up-conversion

• Applying these complex samples to a vector-modulator with

carrier exp(2πjfC), 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 fC 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 8 116

• {x[n]} is inverse FFT of {X0, X1, …., X63} .
• 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 {Xk}0,63 .
• DFT of {x[n]}0,63 gets back exactly to {X0, X1, …., X63}.
• OFDM demodulator is FFT followed by detectors

24 Apr'06 CS3282 Sectn 8 117

• 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 8 118

Simulation of simplified OFDM trasmitter & receiver:

• Generate 8 random bits & use these to generate 1 OFDM symbol
• . Each symbol requires 4 complex numbers X0 to X3 which are

transformed to time-domain {x[n]}0,3 by 4-point inverse FFT.

• Take X0 to X3 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 8 119

Example: Assume data is: 00 01 10 11 Then X0 =1+j, X1 = 1- j, X2 = -1+j, X3 = -1-j X = [ 1+j 1-j -1+j -1-j ]; x=ifft(X) This generates 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

24 Apr'06 CS3282 Sectn 8 120

• With 64 = 26 sub-carrier frequencies, inverse DFT can be carried
• ut very efficiently by FFT.
• OFDM works because of orthogonality of the 64 carriers.

24 Apr'06 CS3282 Sectn 8 121

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 8 122

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 1014 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 8 123

Some details about IEEE 802.11a/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

• r 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 8 124

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 8 125

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 8 126

‘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 ≤ 0.125 000 certain 0 0.125 < x ≤ 0.25 001 probably 0 0.25 < x ≤ 0.375 010 likely 0 0.375 < x ≤ 0.5 011 maybe 0 0 .5 < x ≤ 0.625 1 100 maybe 1 . 0.625 < x ≤ 0 .75 1 101 likely 1 0.75 < x ≤ 0.875 1 110 probably 1 0.875 < x 1 111 certain 1

24 Apr'06 CS3282 Sectn 8 127

• 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 bit-

stream, to be corrected by the Viterbi decoder will fail to correct these bit-errors. .

24 Apr'06 CS3282 Sectn 8 128

End of Section 8