Mobile and Satellite Communications Lecture 7 Modulation Modulation he process of inserting our information signal onto a carrier wave he carrier wave is better suited to propagation over the channel Systematically vary a parameter of the carrier wave: s( t) Ac cos( π f ct + θc) Frequency, Amplitude, or Phase
Basic ASK, PSK, & FSK Good Modulation Scheme ASK, PSK, & FSK are basic modulation techniques We wish to perform modulation such that robust to receiver noise bandwidth efficient robust to fading easy to detect and recover
M-arymodulation ransmit one of M possible symbols per symbol period Each symbol carries Symbol rate relates to bit rate: bits Signal bandwidth is fundamentally proportional to the symbol rate Spectral Efficiency η k log R b BW M R k s R b BWαRs s Example - QASK ransmitted signal is s( t) [ a gt n cos πf t a gt n sin πf t ] n [ n] ( ) ( c ) Q[ n] ( ) ( c ) I I and Q components each have one of four possible levels (transmitting bits each)
Example 6-QASK ransmit 4-bits per symbol Increased bit rate for the same signal bandwidth his comes at a cost of signalling power (higher voltage amplitudes ±3V) Recall, our communication resources: S C B log + N Optimal Receivers Consider simple binary ASK over AWGN channel A single symbol, the received signal will be r ( t) a g( t) n( t) m m + where g(t)is the shape of the transmitted waveform at the receiver (after the channel) Our aim is to detect the transmitted amplitude ± a m
Optimal Receivers A linear receiver could always be modelled by an impulse response, h(t): Output of the receiver is: Aim to maximise the signal part over the noise part of the output Maximise: y ( τ t) g( τ) dτ h ( t) am h( τ t) g( τ) dτ + h( τ t) n( τ) dτ Optimal Receivers Maximise this with the constraint of fixed receiver power: h ( t) dt Simple mathematical problem: what function, when multiplied by our given function g(t), when integrated, will give the largest result? max f ( x) f α ( x) g( x) dx g( x)
Optimal Receivers Optimal solution is always a scaled version of the function itself (it will agree in all places) For our receiver, the impulse response should be a scaled version of the received pulse shape - > Matched filter h( t) g( t) α Equivalent correlator form of the receiver (more suited to digital systems) Received signal r( t) am g( t) + n( t) ψ ( t) g( t) Eg dt Discriminator input, r Matched filter h( t) g( t) Eg Discriminator input, r Generalised Optimal Receiver ransmitted symbol must be one of M possible signals { s ( t )} M m m Can construct a basis for the space spanned by these signals (the signal basis): { ψ ( t )} N k k Signal vector: ( t) smkψk( t) k Symbol Energy must satisfy: s m s m N ( s, s,, s ) m m K mn E N m sm( t) dt k s mk
Generalised Optimal Receiver Optimal receiver should correlate received signal against the signal basis: his determines the received signal vector: k rm( t) ψk( t) dt smk nk r + ψ ( t) ( r K, r ) ( s, K, s ) + ( n,, n ) s n r +, N m mn K N m Received signal r ( t) s( t) + n( t) ψ ( t ) dt dt r r Discriminator inputs, r k ψ K (t) dt r K Symbol Discrimination he received signal vector is then the sent symbol vector plus a N-dimensional AWGN noise vector Detection method choose symbol closest to received signal vector in the Euclidean sense: arg min{ d( r, s )} arg min{ r s } m m m m m his is called Maximum Likelihood detection (ML) If we know the probability distribution of sent symbols we could do a bit better -> Maximum A-priori detection (MAP)
Example - QASK Consider 4-QASK (same as 4-QPSK) Signal constellation (set of possible sent signals) { s ( ) ( ) cos( ) m t aig t πfct aqg ( t) sin( πfct) ai ± ; aq ± } An appropriate basis: g ( ) ( t) g ( ) ( ) ( t) ψ t cos πfct, ψ t sin( πfct) Eg Eg Every symbol can be expressed as a linear combination of basis signals { s ( t) a E ψ ( t) + a E ψ ( t) a ± ; a } m I g i Q g I Q ± Bandwidth Efficiency Recall basic Fourier theory, fort x ( t) X( f), his basic example gives our fundamental result in communications theory Bandwidth otherwise sin πf ( πf) usedα α Symbol rate Symbol duration
Bandwidth Efficiency We wish to minimise signal bandwidth: So more users can share the channel (say FDMA) o allow greater data rates to be communicated o reduce out of band power (and so minimise adjacent channel interference) Reduce distortion due to ISI (Inter-symbol Interference) Nyquist Criterion For a communication channel channel bandwidth will limit data rate that can be achieved r( t) akg t k + Received signal must be: k Use a matched correlator receiver, to recover symbol when k, he receiver output is: r [ ] r( t), ψ ( t) a[ ] g ( t), ψ ( t) + a[ k] g ( t k ), ψ ( t) + n( t), ψ ( t) k [ ] ( ) n( t) ψ ( t ) g ( t) Eg
Nyquist- ISI For no inter-symbol interference, we require the second term to be zero: g ( t k), ψ ( t) g ( t) g ( t k) dt for allk An obvious choice would be to have the symbol waveforms in different symbol periods non-overlapping -> but after passage through a band-limited channel, this is not trivial to obtain Eg Nyquist- ISI Re-arrange the condition for no-isi on the channel to give: g tg ~ τ t dt g g ~ t κ t k where g ~ ( t) g( t ) ( ) ( ) ( )( ) δ( k) τ k his looks like a sampled version of the convolution of the shaping pulse with its time-reversed copy Fourier transform to express in the frequency domain: G ( f + kr) κ k
Nyquist Criterion Using the sampling theorem (that the Fourier of a sampled signal consist of spectral copies of the spectrum of the original signal, spaced at multiples of the sampling frequency G(f)is the spectrum of the received pulse his is the condition for there to be no-isi in the communication system, when expressed in the frequency domain k G ( f + kr) κ his is an important result -> allows us to construct systems that satisfy this condition Nyquist Criterion Demonstrates the importance of channel on achievable data rates Case I: (R > B) ISI is inevitable G(f) B X(f) f -3R/ -R/ R R/ 3R/ f
Nyquist Criterion Case II: (R B) theoretically possible by G(f) taking f G( f) κπ R B f Not physically realisable -3R/ -R/ R R/ 3R/ (infinite support) Case III: (R < B) can satisfy criteria. Only if Ris less than Bcan we communicate G(f) without ISI B f X(f) X(f) f -3R/ -R/ R R/ 3R/ f G ( f) Nyquist Pulses A common, practical pulse shape that satisfies the Nyquist criterion Known as root-raised cosine (RRC) pulses Frequency response is: R if f ( β) f π β β R R ( β) ( + + cos β) + if f R R if f ( + β) heir bandwidth is thus: ( βis the roll-off factor) R B ( +β)
Signal Bandwidth Consider QASK/QPSK for illustration A transmitted signal can be expressed as: s( t) ai[ n] g( t n) cos( πfct) aq[ n] g( t n) sin( πfct) n Under the assumptions that symbol sequence { a I[ n], aq[ n] } is uncorrelated and uniformly distributed Can easily show, by Fourier transforming the time-averaged autocorrelation function, that the power spectral density of the signal is: S (( f) R ( E E ) G( f f ) + G( f + f ) S [ ] s g { c c } Pulse shaping he important point is that the bandwidth of the shaping pulse determines the signal bandwidth rue for all modulation schemes (FSK), but analysis and expression is more complicated Shaping pulse determines spectral characteristics
Basic Fourier theory results Signal with discontinuities PSD decays as (db per decade) -Example: square pulse: Continuous signal with discontinuous first 4 derivative PSD decays as f (4dB/decade) -Example: triangular pulse: t, for t Continuous signal with continuous first 6 derivative decays at least as fast as f (6dB/decade) ( t) x, for t, otherwise X ( f) x ( t) X( f) sinc ( πf), otherwise f sin πf ( πf) Pulse Shaping Illustrations of the results for square and triangular pulse below o minimise bandwidth we want to make the signal as smooth as possible Digital modulation the more marked the discontinuities, the easier it is for the receiver to recover the data
ime domain: Square Pulse g( t) u ( t) ift (, ) otherwise PSD: G( f ) sin c 9% Bandwidth: ( πf) Roll-off: db/decade B 9.65 99 Half-sinusoid Pulse ime domain: g( t) u ( t) πt sin PSD: G( f ) 9% bandwidth: 4 π cos ( πf) ( f) [ ] B99.8.5.5 Half-sinusoid pulse 4dB/decade roll-off...3.4.5.6.7.8.9
Raised Cosine pulse ime domain: g( t) u ( t) πt cos 3 PSD: G( f ) 6 sin c ( πf) ( f) ( ).8.6.4 Raised-cosine pulse 9% bandwidth: B.49 99..8.6.4 4dB/decade roll-off....3.4.5.6.7.8.9 Gaussian Pulse Shapes ime domain: Pulse spectrum: Spectrally efficient but violates the Nyquist criterion t g( t) erfc π B3 erfc π B3 log log H ( f) e log f B 3 Degree of ISI introduced is quantified by the 3-dB bandwidth-symbol period product B 3 Effectively this is the ratio between the symbol period and the Gaussian pulse width GSM uses this with B 3.3 ( t )
Pulse Shaping - example Pulse shaping can smooth out signal discontinuities, reducing bandwidth FSK - comments Can be generated in a way to have continuous phase (but discontinuous frequency) waveform: E b ( t) u ( t) cos[ π f t+ θ( t) ] s he instantaneous frequency carries the information symbols: fi( t) fc + h ang( t n) Instantaneous carrier phase is the integral of the instantaneous frequency: t Note that carrier phase is continuous, which means the signal is continuous too c n ( t) π fi( τ) dτ πfct + πh anq( t n) θ i n
he signal derivative is: FSK - comments If can smooth the frequency waveform, the signal derivative will be continuous too Gradual frequency transitions (freq. shaping pulse) ( t) ds dt dθi Ac π fc+ c + dt sin Produces a signal with a roll-off of at least 4dB/decade FSK with the continuous phase property have inherently good spectral performance ( πft θ ( t) ) i Carrier Synchronisation he receiver will always need to recover the carrier s phase and frequency Inaccuracies result in a reduction of receiver performance -> increased BER Carrier phase must be tracked in real-time: Unknown/changing x-rx distance Oscillator drift wo options: Pilot assisted transmit a reference carrier (pilot tone) Non-pilot assisted use the information signal itself to lock
Carrier Synchronisation Pilot tone transmitted stable carrier for the receiver to synchronise to. Uses available signalling power Can be efficient if multiple receivers can share the same pilot tone (on the cellular Down-link) Use a simple PLL to lock receiver Local Oscillator onto this pilot tone Non-coherent Receiver For the up-link, it is inefficient to have each MS transmitting its own pilot and the BS running a separate PLL for each MS Use a non-coherent detection scheme
Non-coherent Detection - QPSK Non-coherent detection removes the need for a pilot tone, at the cost of reduced SNR performance he above structure works for FSK (or really any orthogonal signalling scheme) QPSK requires a different methodology Differential Encoding can be used for QPSK Differential Encoding map the transmitted symbol to the change in the carrier phase, and not the actual carrier phase Differential Encoding ransmitted carrier phase is determined by: where symbol m θ + mn m[ n] m[ n ] [ ] M [ n] {,, KM } is the transmitted Receiver then only needs to track carrier phase changes (and not the precise carrier phase) Can easily be implemented with a detector with memory θ π
r(t) DPSK Receiver phase Shift, 9 º g( t) ( ) dt E g cos( π f c t+φ) local oscillator sin( π t+φ) f c g( t) ( ) dt E g r I [n] [ n ] he decisions are made according to: ri ri[ n] ri[ n ] + rq[ n] rq[ n ] rq rq[ n] ri[ n ] ri[ n] rq[ n ] he matched filter outputs are: ri [ n] Es cos( θm[ n] φ) + ni[ n] rq [ n] Es sinθ ( m[ n] φ) + nq[ n] So, detector outputs: r I Es cos( θm[ n] θm[ n ]) + n I r Q Es sin( θm n θmn ) + n Q [ ] [ ] delay mixer delay r I r Q [n] [ n ] r Q r I r Q phase detector DQPSK - Example Consider a QPSK system with differential encoding, and signal mapping as shown Input data stream: Carrier phase change(δθ): -π/ π π/ π ransmitted phase: -π/ π/ π Non-coherent detector outputs: Decision points: θ(n) (x(n),y(n)) Output bits (+, ) π/ (, +) π (-, ) -π/ (, -) x ( n) E s cos( θm[ n] θm[ n ] ) [ n] E s sin( θm[ n] θm[ n ] ) y
Bit Clock Synchronisation If there is a long sequence of bits that don t change, it is difficult to detect when bits begin and end A clever solution in DQPSK is π/4-dqpsk Rotate the constellation every symbol period by π/4 Ensures the carrier phase changes every symbol Input Bit Pair Phase change of carrier π/4 3π/4-3π/4 -π/4 Bit Clock Synchronisation Can be determined similar to carrier phase synchronisation (effectively using a PLL) Performed at the baseband Often can be implemented digitally (on a sampled waveform) Other techniques exist such as Early-Late Gate Estimation
Error Performance Measured as the Bit Error Rate (BER) probability of a bit error Noise robustness of the signalling scheme Depends on the model of noise adopted he simplest is AWGN: Noise is additive Gaussian distribution White spectrum Auto-correlation S N ( f ) r(t) s(t) + n(t) P( n) N e πσ N N R N ( τ ) F n σ δ( τ) Binary Signalling Ultimately, noise performance is determined by how far apart points are in the signal constellation For example, binary PSK with coherent detection Per symbol is: Pr Error probability: { error } Pr{ n> d / } Pr { error} Pr{ n> d / } Bit Error Rate: since, E b g d / Q Q σ n P e Q E N b d N
BER Example -QPSK QPSK with coherent detection Can assume that noise effects each component independently. Probability a symbol is detected in error: - a Q + - + a I Pr { symbol} Q d N nn Nearest neighbour distance: here are two bits per symbol: d nn g E Es b Map symbols such that neighbours differ by a single bit: P eb P es -> BER for QPSK with coherent detection g P e Q E N b BER General Constellation A general constellation, expressed in terms of its signal basis Correlator output: Received signal r ( t) s( t) + n( t) rm( t) ψk( t) dt smk nk Received signal vector: r ( r K, r ) ( s, K, s ) + ( n,, n ) s + n k r +, N m mn K N m ψ ( t ) ψ ( t ) ψ K (t) dt dt dt r r r K Discriminator inputs, r k Noise vector is an iid Gaussian random vector. Each component is independent
BER General Constellation ML decision thresholds regions closest to each symbol point Error probability prob. that the m-dimensional noise vector causes received signal vector to be closer to M another symbol. Pes Pr does not lie in Zi mi sent M ( ) ( r m ) dr he exact BER can be a quite complex integral over an m-dimensional region M - i M M M P( r lies in Zi mi sent) i i Zi P i BER General Constellation Nearest-neighbour approximation accurate for large SNR values Assume if symbol is detected in error it will be detected as one of its nearest neighbours only: Forms an upper bound: P e ( m ) i P UAij P( Aij) j i j i Pair-wise error probabilities: P ( ) ij A ij Q N d Assuming symbols are all equally likely: Pe M M Q d ij N i j n.n i
BER General Constellation Can relate symbol error rate to BER in several ways:.hierarchical constellation some points are closer to others Use a Gray code (eg. QASK) hen, for M-ary scheme log M eb P e.non-hierarchical distances between points are uniform Example an orthogonal constellation (FSK) Every other symbol is equally likely Note: as M gets large -> Peb P e P P eb M Pe K M K Pe BER Results for Common Signalling Schemes Some common signalling schemes (all with coherent detection) BPSK binary FSK QPSK Eb P e Q N E b P e Q N Eb P e Q N M-ary PSK M-ary FSK P eb Q log M P eb log M E sin N b π ( ) M E log M ( ) b M Q N Note: all involve Q-function - asymptotic Peb Q E N κ b κeb e N
Non-coherent Detection Much more complex Usually involve (noise)^ terms no longer Gaussian Often analytic expressions can t be obtained Some useful results: Binary FSK P eb e Eb N Binary PSK P eb e Eb N BER Key Results he exact expression for the BER of a signalling scheme is determined by: Modulation scheme Detection method/receiver structure General trends Will always be a function of energy to noise PSD per bit, E b N As E b N increases, BER decreases Asymptotic behaviour -> exponential (large SNR) Peb Q κeb κeb N e N
Rayleigh Channel Model For channels other than the AWGN the analysis is more involved he next simplest is the slow, flat-fading Rayleigh channel Slow, flat Rayleigh Model At any instant the channel is AWGN he channel SNR varies (slowly changes much slower than a symbol period), and follows an exponential distribution (this models fading), Pγ e Γ γ Γ ( ) BER for Rayleigh Model the average BER over this channel can be calculated as: Often this integral must be evaluated numerically Analytical results for some common modulation schemes Coherent BPSK: e Γ γ Γ Coherent binary-fsk: Non-coherent BPSK: P ray eb P P eb AWGN eb P eb ( γ) dγ Γ +Γ P eb ( +Γ ) Γ +Γ Asymptotic behaviour -> P eb E N b
Non-linear Effects in Modulation Variations in x-rx distance, and fading events, imply variable amplification is needed Difficult to build amplifiers that are linear over a wide voltage range (not to mention tuneable too) Recall WA CONSAN AMPLIUDE MODULAION Seek modulation schemes that feature a constant amplitude envelope Signal information is not associated with amplitude of waveform envelope Constant Amplitude Modulation FSK inherently has the constant amplitude property BPSK with a smooth amplitude pulse (to improve bandwidth performance) does not have this g(t) property P(t) t t
Offset - QPSK A variation of QPSK to achieve a constant amplitude waveform o combat the effects of fading and non-linear distortion in amplifiers Offset the I and Q phases by half a symbol period Offset QPSK When the half-sinusoid shaping pulse is used, the resulting waveform will have constant amplitude O-QPSK signal: ( ) st ai g( t n ) ( ) ( ( + ) ) ( )] [ ] b cos πfct aq gt n [ ] b sin πfct m n m n [ n Power profile: ( t) g ( t nb) + g ( t ( n+ ) b) P n n If use half-sinusoid pulse: Resulting profile: g πt ( t) u ( t) sin ( ) ( ) ( ( ) ) π t n b + + π t n b Pt sin sin n b b ( ) ( ) π t nb π t nb sin + cos n b b [ ] n -
Sophisticated Modulation Schemes wo good examples of modulation schemes that combine the ideas we ve discussed:.π/4 ODQPSK π/4 -Offset Differential Quadrature Phase Shift Keying Used in the USDC (G Cellular standard in the US).GMSK Gaussian Minimum Shift Keying Used in GSM Features: π/4-odqpsk π/4 offset in the symbol constellation, to improve symbol synchronisation performance Offset I and Q phases, to smooth power profile -> robustness to fading and non-linear amplification Differential encoding, to allow non-coherent detection (say for the up-link). Also could be employed with a pilot tone and coherent detection (would improve BER performance)
π/4-odqpsk ransmitter structure Receiver Structure (Coherent) θ Gaussian Minimum Shift Keying Form of FSK with continuous phase he continuous phase property is an example of a modulation scheme with memory Phase waveform πh ( t) θ( ) ± t, for t Additional Gaussian filter employed to smooth out the phase transitions
GMSK Gaussian filter response log f B 3 ( ) H f e h( t) π 3 π B log B t 3e log 3dB bandwidth is B 3 t g( t) erfc π B3 erfc π B 3 log log ( t ) Reduces bandwidth but introduces ISI he smaller the bandwidth, the wider the pulses Symbol period 3dB bandwidth product quantifies the level of ISI introduced into the modulation scheme GMSK Symbol period-3db bandwidth product B 3 he smaller the value the more significant the ISI GSM uses GMSK with B 3.5 Minimum Shift Keying binary FSK where the frequency separation is chosen so as to be minimum that makes the symbol waveforms orthogonal
MSK Consider two frequency signals hey will be orthogonal if: s o minimise the frequency separation: Recall modulation index of FM: -> h cos s E b ( t) u ( t) cos( πft) for i, i ( πft) cos( πf t) dt sinc( π ( f f )) k f s, k,,3,k f f ( ) h f f f i s s MSK Modulation with memory Can show that it is equivalent to Offset-QPSK with the half-sinusoid shaping pulse: E his implies an easy structure to generate GMSK E 3π / π π / t - 3π / π π / - - 3π / - - π π / - 3π / π π / t t t 3 b b ( t) cos( θ( t) ) cos( πf t) sin( θ( t) ) sin( πf t) s c c
GMSK GMSK can be recovered using either Coherent or Non-coherent techniques: GMSK Noise performance is slightly worse than BPSK/QPSK his is due to the ISI introduced by the Gaussian filter For Coherent detection: for B 3.5 ρ.68 Q ρe N ρ.85 for pure MSK, Spectral performance is superior to QPSK (6dB roll-off performance) P eb b B 3
3G Standards Employ basic BPSK or QPSK oo difficult to employ more sophisticated modulation scheme at that chip rate and over that bandwidth Signal processing: CDMA BER Performance Recall SNR expression for CDMA SINR N B Implies BER, if assume coherent BPSK/QPSK: CDMA is interference limited P G SNR SNR G s ( K ) + ( K ) -> even as improve SNR, the BER will never improve beyond P eb > Q P s data+ G K P Q E b K + N G eb