5 Modulation and demodulation 5.1 MAXIMUM LIKELIHOOD ESTIMATION We start again with the ML principle defined in Section 3.1 of Chapter 3. After the signal despreading, vector of parameters θ to be estimated includes timing of the received symbols τ 0 , phase of the received carrier θ 0 , frequency offset of the received signal ν 0 , amplitude of the signal A 0 and data symbols a n θ(τ 0 ,θ 0 ,ν 0 ,A 0 ,a n )(5.1) After despreading, the narrowband signal can be represented as r(t) = s(t, θ) + w(t) (5.2) The likelihood becomes L( ˜ θ) = C 1 exp  − C 2 N 0  T 0 |(r(t) − s(t, ˜ θ)| 2 dt  (5.3) In the sequel, we will use a linear-modulated complex-signal format given by s(t, ˜ θ) = A 0 exp(j ˜ θ 0 )  ˜a 0 h(t − nT −˜τ 0 ) + j ˜ b n h(t − nT − εT −˜τ 0 ) (5.4) where h( ) is the pulse shape and for ε = 0 or 1/2 we have quadrature phase shift keying (QPSK) or offset QPSK (OQPSK) signals, respectively. The likelihood function defined by equation (3.5) now becomes λ(θ) = R(T 0 , ˜ θ) = Re   T 0 0 r(t)s ∗ (t, ˜ θ) dt  (5.5) If we define the filters matched to the pulse shape in I and Q channel as p(n, ˜τ)=  ∞ −∞ r(t)h(t − nT −˜τ)dt q(n, ˜τ,ε)=  ∞ −∞ r(t)h(t − nT − εT −˜τ)dt (5.6) Adaptive WCDMA: Theory And Practice. Savo G. Glisic Copyright ¶ 2003 John Wiley & Sons, Ltd. ISBN: 0-470-84825-1 124 MODULATION AND DEMODULATION then equation (5.5) becomes R(N, ˜ θ) = Re  exp(−j ˜ θ) N  n=1 ˜a n p(n, ˜τ)  + Im  exp(−j ˜ θ) N  n=1 ˜ b n q(n, ˜τ,ε)  (5.7) In the special, important case of nonstaggered signals (ε = 0), we find q = p.Ifwedefine c n = a n + jb n , the correlation integral becomes R(N, ˜ θ) = Re  exp(−j ˜ θ) N  n=1 ˜c ∗ n p(n, ˜τ)  (5.8) 5.1.1 Phase and frequency correction: phase rotations and NCOs For a given phase error θ (n), the complex signal sample (sampling index n) z in (n)iscor- rected by multiplying the sample by a complex correlation factor exp(jθ(n)) as follows: z in (n) = x in (n) + jy in (n) z 0 (n) = z in (n) × exp(j θ (n)) (5.9) By using exp(j θ ) = cos θ + j sin θ ,weget z 0 = x in cos θ − y in sin θ + j(x in sin θ + y in sin θ) (5.10) The operation is known as phase rotation and the block diagram for the realization of equation (5.10) is shown in Figure 5.1. Frequency corrections (translations) can be performed by the same circuitry but now the phase correction will change in time. For the frequency error ν, the correction becomes z 0 = z in exp(j2πnvT s )(5.11) + _ + + x 0 ( n ) x in ( n ) q( n ) y 0 ( n ) sin q cos q y in ( n ) Sine/ Cosine ROM Figure 5.1 Phase rotation. FREQUENCY-ERROR DETECTION 125 A simultaneous phase rotation and frequency translation is performed as exp[j(2πnvT s + θ)] (5.12) In the next section we will focus on the problem of detecting phase and frequency error. The circuit from Figure 5.1 will be used for error corrections, given an error value of phase or frequency. 5.2 FREQUENCY-ERROR DETECTION We start again with the likelihood function in the following form: L( ˜ θ) = exp  2C 2 N 0  T 0 Re[r(t)s ∗ (t, ˜ θ)]dt  (5.13) To emphasize the existence of frequency error, the signal defined by equation (5.4) is rewritten as s ∗ (t, ˜ θ) = ∞  n=−∞ [˜a n h(t − nT −˜τ)− j ˜ b n h(t − nT − εT −˜τ)] exp[−j( ˜ θ + 2π ˜vt)] (5.14) In this case equation (5.13) becomes L( ˜ θ) ∼ = N−1  m=0 exp  2C 2 N 0 ˜a m Re   ∞ −∞ r(t)h(t − mT −˜τ)e −j( ˜ θ+2π ˜vt) dt  × exp  2C 2 N 0 (− ˜ b m ) Im   ∞ −∞ r(t)h(t − mT − εT −˜τ)e −j( ˜ θ+2π ˜vt) dt  (5.15) L( ˜ θ) ∼ = N−1  m=0 exp  2C 2 N 0 ˜a m Re[p(m)]  × exp  2C 2 N 0 (− ˜ b m ) Im[q(m)]  (5.16) Joint maximization of equation (5.16) with respect to all the parameters would be rather complex for practical implementation. To remove data from equation (5.16), we use aver- aging of the function. For M-ary modulation this can be represented as L a,b = N−1  m=0  m  i=1 1 M exp  2C 2 N 0 a i Re[p(m)]   ×  m  i=1 1 M exp  − 2C 2 N 0 b i Im[q(m)]   (5.17) 126 MODULATION AND DEMODULATION In the simple case of binary modulation we have L a,b = N−1  m=0 cosh  2C 2 N 0 Re[p(m)]  · cosh  2C 2 N 0 Im[q(m)]  (5.18) For nonoffset QPSK modulation, q(m) = p(m) and equation (5.18) becomes L a,b = N−1  m=0 cosh  2C 2 N 0 Re[p(m)]  · cosh  2C 2 N 0 Im[p(m)]  (5.19) By taking the logarithm of equation (5.19), we have  a,b  ln[L a,b ] = N−1  m=0  N−1  m=0 ln cosh  2C 2 N 0 Re[p(m)]  + ln cosh  2C 2 N 0 Im[p(m)]   (5.20) The following approximations are used at this point: ln cosh(x) ∼ = x 2 2 ,|x|1 ∼ = |x|,|x|1 (5.21) For the small value of the argument we have  a,b ∼ = C 3 N−1  m=0 ({Re[p(m)]} 2 +{Im[p(m)]} 2 ) = C 3 N−1  m=0 |p(m)| 2 (5.22) Equation (5.22) can be maximized by changing ˜ν in p(m) in the open loop search. By taking the derivative of equation (5.20), we get the tracker for the QPSK signal. ∂ a,b ∂˜v = N−1  m=0 2C 2 N 0 Re[p v (m)]tanh  2C 2 N 0 Re[p(m)]  + N−1  m=0 2C 2 N 0 Im[p v (m)]tanh  2C 2 N 0 Im[p(m)]  (5.23) where p v (m)∂p(m)/∂ ˜v. A sample of frequency-error detector control signal is u v (n) = Re[p v (n)]tanh  2C 2 N 0 Re[p(n)]  + Im[p v (n)]tanh  2C 2 N 0 Im[p(n)]  (5.24) FREQUENCY-ERROR DETECTION 127 Re Im Re Im v ~ Signal matched filter r ( t ) Frequency rotator Filter strobes p and q are complex Non offset signals: p ( n ) = q ( n ) Offset signal: q ( n ) is time-offset from p ( n ) by e T h (− t ) − j 2p th (− t ) Frequency loop filter Frequency error detector { p v ( n )}, { q v ( n )} u v ( n ) { p ( n )}, { q ( n )} Figure 5.2 ML-derived frequency detector. and its time average U v = E n [u v (n)] is called the detector characteristic or S-curve. The block diagram realizing equation (5.24) is shown in Figure 5.2. 5.2.1 QPSK tracking algorithm: practical version Further simplification is obtained if we use tanh(x) ∼ = x,|x|1 ∼ = sgn(x),|x|1 (5.25) which for nonoffset QPSK results in u v (n) = Re[p(n)]Re[p v (n)] + Im[p(n)]Im[p v (n)] = Re[p(n)p ∗ v (n)] (5.26) 5.2.2 Time-domain example: rectangular pulse After considerable labor, the S-curve defined as u v (n) = E n [U v (n)] for a unit-amplitude rectangular pulse in time domain and random data is found to be [1] U v (v, τ ) = C c A 0  τ sin πvτ πv 2  cos πvτ − sin πvτ πvτ  + (T − τ ) sin πv(T − τ ) πv 2 ×  cos πv(T − τ ) − sin πv(T − τ ) πv(T − τ )  (5.27) For random data the S-curve is shown in Figure 5.3. 128 MODULATION AND DEMODULATION If the data pattern is assumed to rotate by 90 ◦ from one symbol to the next, that is, c m+1 = jc m , the S-curve is as given in Figure 5.4. For binary phase shift keying (BPSK) dotting signal where c m+1 =−c m ,theS-curve is as given in Figure 5.5. One should notice that for random data, S-curve demonstrates regular shape with the slope decreasing with the timing error. Even for nonsynchronized systems with τ /T = 0.5, the system would operate. For rotating data the impact of timing is larger. For dotting signal, if the timing error becomes large, the S-curve not only has a reduced slope, which is equivalent to reducing the signal-to-noise ratio in the loop, but also changes the sign resulting in the devastating effects of generating the control signal that would cause loss of synchronization. 0 1 −1 −2 0.2 0.3 0.4 0.5 Frequency error, ∆ vT U v ∆t / T = 0 0.1 2 Figure 5.3 Frequency detector S-curves for rectangular pulses (nonoffset signal). 0 1 −1 0.125 0.25 Frequency error, ∆ vT U v ∆τ/ T = 0 Figure 5.4 S-curve for rotating pattern: square pulses. CARRIER PHASE MEASUREMENT: NONOFFSET SIGNALS 129 0.125 0.25 0.5 0 −1 Frequency error, ∆ vT ∆τ / T = 0 U v 1 Figure 5.5 S-curve for BPSK dotting pattern; square pulses. 5.3 CARRIER PHASE MEASUREMENT: NONOFFSET SIGNALS In this case possible solutions will depend very much on a number of parameters. Regarding the signal format, there will be differences for single amplitude [M-ary phase- shift keying (MPSK)] versus multiamplitude [M-ary quadrature amplitude modulation (MQAM)] or offset versus nonoffset signal. Different representations such as rectangular versus polar representation of phase error or parallel versus serial representation of signal (offset only) will result in different solutions. Additional knowledge such as clock timing (clock-aided) or data or decisions [data-aided (DA) or decision-directed (DD)] will also be of great importance. Configurations such as feedforward (FF) versus feedback (FB) will also offer different advantages and drawbacks. 5.3.1 Data-aided (DA) operation In this case a preamble c n and timing τ 0 are available and equation (5.8) becomes R(N, ˜ θ)= Re  exp(−j ˜ θ) N  n=1 c ∗ n p(n)  (5.28) At the maximum point, ∂R/∂θ vanishes and we have Im  exp(−j ˆ θ)  n c ∗ n p(n)  = 0 ⇔  n Im[exp(−j ˆ θ)c ∗ n p(n)] = 0 (5.29) 130 MODULATION AND DEMODULATION and we define again the sample of the S-curve as u θ (n) = Im  exp(−j ˜ θ)  n c ∗ n p(n)  (5.30) Under the ideal conditions the matched filter output pulse becomes p(n) = A 0 c n exp(jθ 0 ) + v(n) (5.31) where v(n) is a sample of noise (assumed zero-mean Gaussian) and A 0 is signal amplitude. Averaging gives the S-curve as U 0 = EIm{exp(−j ˜ θ)c ∗ n [A 0 c n exp(jθ 0 ) + v(n)]} = A 0 E|c n | 2 Im{exp[j(θ 0 − ˜ θ)]}=A 0 E|c n | 2 sin θ (5.32) where θ = θ 0 − ˜ θ. 5.3.2 Decision-directed (DD) operation If preamble is not available, detected data can be used instead, resulting in a DD solution. Implementation of equation (5.30) for such a case is shown in Figure 5.6. Specific DD algorithm If we represent the output of the phase rotator p(n)e −j ˜ θ as the complex sequence {x(n), y(n)}, the output of the decision algorithm as ˆc n =ˆa n + j ˆ b n = sgn(x n ) + jsgn(y n ) * * ~ Timing Phase rotator Decisions Data out Integrator Complex signal Real signal Complex conjugate Matched filter r ( t ) p ( n ) Sample Loop filter Im( ) h (− t ) u q ( n ) c n e _ j q ~ ˆ c n * t ˆ q Figure 5.6 Decision-directed carrier tracking. CARRIER PHASE MEASUREMENT: NONOFFSET SIGNALS 131 where sgn(v)=+1(−1) if v is greater than 0 (v<0), then equation (5.30) is given by u n (n) = Im(x n + jy n )(ˆa n + j ˆ b n )=y n sgn(x n ) − x n sgn(y n )(5.33) This is known as four-phase hard-limiting Costas detector that is so widely used in QPSK systems. Rectangular representation If we use the following steps: • exp(−j ˜ θ)→ a rectangular representation in equation (5.28) λ( ˜ θ)= Re  (cos ˜ θ − j sin ˜ θ) N  n=1 ˆc ∗ n p(n)  (5.34) differentiate with respect to ˜ θ • bring all the expressions into the summation sign • take the real part of the derivative • the ML estimate ˆ θ occurs for the value of ˜ θ at which the derivative goes to zero N  n=1 sin ˆ θ Re[ˆc ∗ n p(n)] − N  n=1 cos ˆ θ Im[ˆc ∗ n p(n)] = 0 Solving for the angle gives ˆ θ(n) = arctan              n−1  i=n−M Im[ˆc ∗ i p(i)] n−1  i=n−M Re[ˆc ∗ i p(i)]              (5.35) Implementation of equation (5.35) is shown in Figure 5.7. 5.3.3 Nondecision-aided measurements Why might one choose to avoid DD measurements? First of all there are some circum- stances, such as acquisition intervals or low signal-to-noise ratios, for which data decisions are of poor quality and should not be used. One can show that in BPSK the equivalent signal-to-noise ratio in such systems will be reduced by factor (1–2P e ) 2 where P e is the bit error rate. Omitting the decision operation might reduce equipment complexity. (Not likely to be a good reason in a digital implementation. Indeed, we find digital DD meth- ods are often simpler than non-DD methods.) In this case the likelihood function will be averaged out with respect to data. 132 MODULATION AND DEMODULATION * Re Im Timing Sample Matched filter Phase rotator Decisions Storage arctan Table Data out ~ r ( t ) p ( n ) h (− t ) e _ j q ~ c n ˆ c n * t ˆ ÷ q Figure 5.7 DD arctan phase recovery. The starting point is equation (5.18) in the form L( ˜ θ)= C 3 exp  2C 2 N 0 Re  e −j ˜ θ N  n=1 c ∗ (n)p(n)  ⇔ L( ˜ θ)= C 3 N  n=1 exp  2C 2 N 0 Re  e −j ˜ θ N  n=1 c ∗ (n)p(n)  (5.36) Averaging with respect to data results in L c ( ˜ θ)= E c [L( ˜ θ)] L c ( ˜ θ)= C 3 N  n=1 ξ(n) (5.37) For BPSK, c(n) = a(n)+ j0, where a(n) =±1 and equation (5.37) gives ξ(n) = cosh  2C 2 N 0 Re[e −j ˜ θ p(n)]  (5.38) The log-averaged likelihood function is  c ( ˜ θ)= ln C 3 + N  n=1 ln cosh  2C 2 N 0 Re[e −j ˜ θ p(n)]  (5.39) [...]... 8022/88/NL/DG 10 Ascheid, G and Meyr, H (1982) Cycle slips in phase-locked loops: a tutorial survey IEEE Trans Commun., COM-30, 2228–2241 146 MODULATION AND DEMODULATION 11 Kobayashi, H (1971) Simultaneous adaptive estimation and decision algorithm for carrier modulated data transmission systems IEEE Trans Commun., C0M-19, 268–280 12 Lindsey, W C and Meyr, H (1977) Complete statistical description of the . −∞ r(t)h(t − nT −˜τ)dt q(n, ˜τ,ε)=  ∞ −∞ r(t)h(t − nT − εT −˜τ)dt (5.6) Adaptive WCDMA: Theory And Practice. Savo G. Glisic Copyright ¶ 2003 John Wiley