Digital modulations (part 1)

Digital modulations (part 1) Outline : 1. Digital modulations definition. Classic linear modulations.1 Power spectral density. Amplitude digital modulation (ASK).3 Phase digital modulation (PSK).4 Quadrature phase digital modulation (QPSK) 3. Exercices Exercise session 4 : Digital modulations (part 1) 1

1. Digital modulations definition Let m(t) a baseband signal (NRZ typically). Modulated digital signal : } s(t) = Re ψ [m(t)] e j(π f ct+ϕ c ) where ψ(.) = ψ I (.) + j ψ Q (.) defines the modulation type. Other form for the modulated signal : s(t) = ψ I [m(t)] cos(π f c t + ϕ c ) ψ Q [m(t)] sin(π f c t + ϕ c ) or s(t) = ψ [m(t)] cos(π f c t + ϕ c + argψ [m(t)]) Exercise session 4 : Digital modulations (part 1)

Generally, we may distinguish two modulation types : Linear modulations : Angular modulations : ψ [m(t)] = linear fonction of m(t) ψ [m(t)] = e jϕ[m(t)] where ϕ [m(t)] = linear fonction of m(t). Linear digital modulations : s(t) = Re e j(π f ct+ϕ c ) d k (t) e j(θ k π f c kt ) where the d k (t) signals contain the information to be transmitted and θ k is a constant phase. } Two types of linear modulations : Classic modulations : θ k = π f c kt Offset modulations : θ k = π f c kt + k π Exercise session 4 : Digital modulations (part 1) 3

. Classic linear modulations Modulated signal : } s(t) = Re e s (t) e j(π f ct+ϕ c ) Complex envelope of the modulated signal : e s (t) = = d k (t) D k g k (t kt ) where g k (t) = real modulating waveform signal. For the sake of simplicity, we will choose g k (t) = g(t), k. D k = complex random variable which contains the digital information to be transmitted : D k = A k + j B k Exercise session 4 : Digital modulations (part 1) 4

Where So, e s (t) = s I (t) + j s Q (t) s I (t) = s Q (t) = A k g(t kt ) B k g(t kt ) Other form for the modulated signal : s(t) = s I (t) cos(π f c t + ϕ c ) s Q (t) sin(π f c t + ϕ c ) Or s(t) = [ ] A k g(t kt ) cos(π f c t + ϕ c ) [ ] B k g(t kt ) sin(π f c t + ϕ c ) Quadrature modulation for two digital baseband signals (NRZ type). Exercise session 4 : Digital modulations (part 1) 5

.1 Power spectral density (PSD) Reminder : X (t) is wide sense stationnary (WSS) if : µ X independant of t, and Γ XX (t,t τ) = EX(t)X (t τ)} depends only on τ Γ XX (τ) PSD of the modulated signal S(t) Such that, s(t) non-stationnary : We have to stationnarize : S(t) = Re M (t) e jπ f ct } S(t) = Re } M (t) e j(π f ct+θ) where Θ = Uniform random variable on [0,π[. Exercise session 4 : Digital modulations (part 1) 6

We showed that µ S = 0 Γ SS (t,t τ) = 1 ReΓ MM(t,t τ)e jπ f cτ } If M (t) is WSS, then S(t) is WSS. We can write Γ SS (τ) = 1 4 [ ΓMM (τ)e jπ f cτ + Γ MM(τ)e jπ f cτ ] we deduce finally γ S ( f ) = γ M ( f f c ) + γ M ( f f c ) 4 Exercise session 4 : Digital modulations (part 1) 7

PSD of the complex envelope M(t) M (t) = D k g(t kt ) D k is caracterized by its mean : µ D = ED k } its variance : σd = E(D k µ D )(D k µ D ) } = E D k } If the random variables D k are not correlated, then [ G ( f ) γ M ( f ) = σd + µ D 1 ( T T δ f m ) ] T which is real and symetric. m= We can write γ S ( f ) = γ M ( f f c ) + γ M ( f + f c ) 4 Exercise session 4 : Digital modulations (part 1) 8

. Amplitude digital modulation (ASK : Amplitude Shift Keying) Caracteristics D k purely real (B k 0). So, e s (t) = s I (t) = purely real (s Q (t) = 0). A k g(t kt ) Rectangular modulating waveform impluse : g(t) = rect (0,T ) (t) Exercise session 4 : Digital modulations (part 1) 9

Envelope and phase of the modulated signal We remind ourselves that, e s (t) = a(t) e jϕ(t) We can then write A k = A k e j π (1 sign(a k)) So, a(t) = ϕ (t) = A k rect (0,T ) (t kt ) π (1 sign(a k)) rect (0,T ) (t kt ) Observations : The signal envelope is not constant. Phase jumps of π discontinuous phase. Exercise session 4 : Digital modulations (part 1) 10

Exemple : ASK- Modulation Ak +A, A} constant envelope. T = T b Constellation diagram Complex plan of e s (t) s Q ( A, 0) (A,0) s I Exercise session 4 : Digital modulations (part 1) 11

Signals T b I(t) 1 1 0 1 1 0 0 1 0 1 1 0 0 1 +1 1 t T = T b s I (t) 1 1 1 1 1 1 1 1 0 0 0 0 0 0 +A A t s Q (t) t a(t) +A t ϕ(t) 0 0 π 0 0 π π 0 π 0 0 π π 0 t Exercise session 4 : Digital modulations (part 1) 1

Power spectral density Hypothesis : both signals ±A have an equal probability. Mean Variance µ D = ED k } = 0 σ D = E D k } = EA k} = A Modulating waveform signal g(t) = rect (0,Tb )(t) G ( f ) = e jπ f T b Tb sinc( f T b ) PSD of the complex envelope γ es ( f ) = A T b sinc ( f T b ) PSD of the modulated signal γ s ( f ) = A T b 4 sinc [( f f c )T b ] + sinc [( f + f c )T b ] } Exercise session 4 : Digital modulations (part 1) 13

.3 Digital phase modulation (PSK : Phase Shift Keying) Caracteristics General shape for the modulated signal s(t) = A rect (0,T ) (t kt ) cos(π f c t + ϕ c + ψ k ) where ψ k = constant random variable on[kt,(k + 1)T [ : ψ k ψ ψ = ϕ 0 + i πm }, i = 0,...,M 1 Exercise session 4 : Digital modulations (part 1) 14

Shaping of the modulated signal So, s(t) = A = rect (0,T ) (t kt ) [cos(π f c t + ϕ c )cosψ k sin(π f c t + ϕ c )sinψ k ] [ ] Acosψ k rect (0,T ) (t kt ) cos(π f c t + ϕ c ) [ ] Asinψ k rect (0,T ) (t kt ) sin(π f c t + ϕ c ) e s (t) = s I (t) + j s Q (t) = A Classic linear digital modulation with rect (0,T ) (t kt ) (cosψ k + j sinψ k ) D k = Ae jψ k g(t) = rect (0,T ) (t) Exercise session 4 : Digital modulations (part 1) 15

Envelope and phase of the modulated signal a(t) = A ϕ (t) = rect (0,T ) (t kt ) ψ k rect (0,T ) (t kt ) Observations : Constant signal envelope. Phase jump discontinuous phase. Exercise session 4 : Digital modulations (part 1) 16

Exemple : PSK- or BPSK modulations ψ k 0,π} D k Ae j0,ae jπ} BPSK ASK- Identical constellation diagram Power spectral density Identical to the ASK- modulation : γ s ( f ) = A T b 4 sinc [( f f c )T b ] + sinc [( f + f c )T b ] } Exercise session 4 : Digital modulations (part 1) 17

.4 Quadrature phase digital modulation (QPSK : Quadrature Phase Shift Keying) Caracteristics Phase modulation with 4 states (PSK-4) : ψ k 3π/4, π/4,+π/4,+3π/4} D k Ae j 3π 4,Ae j π 4,Ae j π 4,Ae j 3π 4 } Rectangular modulating waveform impulse : g(t) = rect (0,T ) (t) = rect (0,Tb ) (t) Exercise session 4 : Digital modulations (part 1) 18

Constellation diagram s Q (t) ( A,+ A ) 01 11 (+ A,+ A ) s I (t) ( A, A ) 00 10 (+ A, A ) Exercise session 4 : Digital modulations (part 1) 19

Inphase and quadrature components Let where I (t) = I k = We build the two sequences I k δ (t kt b ) +1 for the bit 1 1 for the bit 0 s I (t) = s Q (t) = A I k g(t kt ) = A k g(t kt ) A I k+1 g(t kt ) = B k g(t kt ) where T = T b, and A k = I k A even bits for the sequence I k B k = I k+1 A odd bits of the sequence I k Exercise session 4 : Digital modulations (part 1) 0

Modulated signal envelope and phase e s (t) = s I (t) + j s Q (t) = = We can then write a(t) = = (A k + j B k ) rect (0,T ) (t kt ) A (I k + j I k+1 ) rect (0,T ) (t kt ) s I (t) + s Q (t) A Ik + I k+1 rect (0,T ) (t kt ) = A rect (0,T ) (t kt ) and ϕ (t) = rect (0,T ) (t kt ) arctan ( Ik+1 I k ) Exercise session 4 : Digital modulations (part 1) 1

Observations : Constant. signal envelope Phase jumps of π or π/ discontinuous phase. Signals T b I(t) 1 1 0 1 1 0 0 1 0 1 1 0 0 1 +1 1 t T = T b s I (t) 1 1 1 0 0 0 0 + A A t s Q (t) 1 1 1 1 1 0 0 + A A t a(t) +A t ϕ(t) π 4 3π 4 π 4 3π 4 3π 4 π 4 3π 4 t Exercise session 4 : Digital modulations (part 1)

1 Modulation QPSK (a) 0 1 0 1 4 6 8 10 1 (b) 0 1 0 1 4 6 8 10 1 (c) 0 1 0 1 4 6 8 10 1 (d) 0 1 0 1 4 6 8 10 1 (e) 0 (f) 1 0 4 6 8 10 1 1 0 1 0 4 6 8 10 1 (a) Binary sequence I(t) (b) s I (t) (c) s Q (t) (d) s I (t) cos(π f c t) (e) s Q (t) sin(π f c t) (f) Modulated signal s(t) Exercise session 4 : Digital modulations (part 1) 3

QPSK Modulator s I (t) s I (t)cos(π f c t) cos(π f c )t I(t) série + s(t) parallèle π sin(π f c )t s Q (t) s Q (t)sin(π f c t) QPSK Demodulator cos(π f c )t 1 s I(t) Filtre adapté et décision s(t) parallèle 1 I(t) π série sin(π f c )t 1 s Q(t) Filtre adapté et décision Exercise session 4 : Digital modulations (part 1) 4

Power spectral density Hypothesis : The four states have an equal probability. D k = (± A,± A ) Mean Variance µ D = ED k } = 0 σ D = E D k } = A Modulating waveform signal g(t) = rect (0,Tb )(t) G ( f ) = T b e jπ f T b b sinc( f T b ) Complex envelope signal γ es ( f ) = A T b sinc ( f T b ) Modulated signal PSD γ s ( f ) = A T b sinc [( f f c )T b ] + sinc [( f + f c )T b ] } Exercise session 4 : Digital modulations (part 1) 5

3. Exercices 1. Let the classic linear digital modulation 16-QAM (or 16-QASK) whose constellation diagram is given by 1000 1100 0100 0000 1001 1101 0101 0001 1011 1111 0111 0011 1010 1110 0110 0010 The modulating waveform impulse is a 4T b duration rectangular signal. (a) For the binary sequence 1011001011011001, determine the inphase and quadrature components, the envelope and the phase of the modulated signal. (b) (c) (d) Determine the power spectral density of the modulated signal (hypothesis : all symbols have an equal probability). Determine the bandwith of s(t) function of R b and the spectral efficiency. Determine the type of modulation. (e) What do you think of the repartition of the symbol in the constellation diagram? Is it wise in terms of bandwidth, power consumption and/or error probability? (f) If the modulated signal is expressed by s(t) = I(t) Q(t) with I(t) = s I (t)cos(π f c t + ϕ) Q(t) = s Q (t)sin(π f c t + ϕ) What is the relation between γ I ( f ), γ Q ( f ) and γ s ( f ). Are they correlated? Exercise session 4 : Digital modulations (part 1) 6

. Consider a classic linear modulation caracterized by the following constellation diagram A/4 3A/4 111 101 011 001 010 000 A/ A 110 100 The modulating waveform impulse is rectangular and extend from 0 to T. The carrier frequency is given by f c. (a) (b) (c) Represent graphically the temporal evolution of the inphase component, the quadrature component, the amplitude and the phase of the modulated signal for the binary sequence : 101111010011000. If the symbols beginning by 0 have a probability two times higher than those beginning by 1, compute the power spectral density of the modulated signal. If a bit has a 10 µs duration, determine the bit rate R b and the bandwith of the modulated signal. Exercise session 4 : Digital modulations (part 1) 7

3. We achieve a classic linear digital modulation with a circuit comprising an 8-ways switch activated every 3T b seconds depending on the binary sequence to be transmitted (T b is the inverse of the bit rate R b ). The 8 inputs of the switch receive signals s 000 (t), s 001 (t),... derived from the carrier cos(π f c t) ; ( s 000 (t) = cos π f c t + 5π 6 s 001 (t) = 3cos(π fc t π) s 010 (t) = ( sin π f c t 4π ) 3 s 011 (t) = sin(π π f c t) s 100 (t) = ( ) π sin 3 π f ct s 101 (t) = 3sin ) ( π f c t π ( s 110 (t) = cos π f c t 5π 6 ( s 111 (t) = 3cos π f c t + π ) The switch delivers an output signal s(t) which is the modulated digital signal. ) ) (a) (b) (c) (d) Determine and draw the constellation diagram for this modulation. What is the number of states in this constellation? Draw the inphase component, the quadrature component, the envelope and the phase of the modulated signal for the following binary sequence : 011001101011000010111. Expressed, in terms of R b, the bandwidth of the modulated signal. Determine the power spectral density of the modulated signal if the symbols have the same probability and are not correlated. Exercise session 4 : Digital modulations (part 1) 8

4. Let the classic linear modulation with the following constellation diagram 1[V] 11 1[V] 10 01 00 The emission probability for the symbols are p(00) = p(11) = 1/6, p(10) = p(01) = 1/3 and the symbols are not correlated. The modulating waveform is a rectangular impulse with an unit amplitude and a duration of T b where T b is the bit duration. The resulting modulated signal can be written s(t) = I(t) Q(t) with I(t) = s I (t)cos(π f c t + ϕ) Q(t) = s Q (t)sin(π f c t + ϕ) where s I (t) et s Q (t) are respectively the inphase and quadrature components of the modulated signal and ϕ is a random variable with an uniform probability density function on the interval [0, π]. (a) (b) (c) (d) (e) Compute the value of the inphase and quadrature components for the modulated signal for the following binary sequence : 01011110000110. Compute, in terms of the bit rate R b (=1/T b ), the bandwith of the modulated signal. Compute the power spectral density of the modulated signal s(t). Compute the power spectral density of the signals I(t) and Q(t). With the help of the two previous points, determine the relation between the spectral density γ I ( f ), γ Q ( f ) and γ s ( f ). What s your conclusion concerning the correlation between the signals I(t) and Q(t)? Exercise session 4 : Digital modulations (part 1) 9

Answer 1. (a) (b) (c) B = R b 4, η = 4[bit/s/Hz] (d) Hybrid modulation (f) γ s ( f ) = 10T b sinc [4T b ( f f c )] + sinc [4T b ( f + f c )] } γ I ( f ) = γ Q ( f ) = 5T b sinc [4T b ( f f c )] + sinc [4T b ( f + f c )] } = 1 γ s( f ) So I(t) and Q(t) are not correlated. (a) (b) γ s ( f ) = 11A 3 T b sinc [3T b ( f f c )] + sinc [3T b ( f + f c )] } (c) R b = 100[kbit/s], B = 33.3[kHz] 3. (a) 8 states (b) (c) B = R b 3 (d) and γ M ( f ) = 3T b sinc [3T b f ] [ 89 16 1 16 m= ( 1 δ f m ) ] 3T b 3T b 4. (a) (b) B = R b (c) (d) γ s ( f ) = 3T b γ I ( f ) = 4T b 3 γ Q ( f ) = T b 6 γ S ( f ) = γ M ( f f c ) + γ M ( f + f c ) 4 sinc [T b ( f f c )] + sinc [T b ( f + f c )] } sinc [T b ( f f c )] + sinc [T b ( f + f c )] } sinc [T b ( f f c )] + sinc [T b ( f + f c )] } (e) γ s ( f ) = γ I ( f ) + γ Q ( f ) So γ I ( f ) and γ Q ( f ) are not correlated. Exercise session 4 : Digital modulations (part 1) 30