Carrier frequency estimation ELEC-E54 Signal processing for communications
Contents. Basic system assumptions. Data-aided DA: Maximum-lielihood ML estimation of carrier frequency 3. Data-aided: Practical algorithms 4. on-data-aided DA estimation algorithms Based on U Mengali, A.. D Andrea, Synchronization Techniques for Digital Receivers, Plenum Press,997 Page
Example Consider a system operating at GHz. Assume 5 ppm instability in transmit and receive oscillators. Q: What is the maximum frequency difference Δf we can expect in the two oscillators? Q: After how long time can we expect the phase error to be 36 o? Timo I. Laaso and Stefan Werner Page 3
Basic system assumptions
Basic system assumptions Frequency recovery consists of two tass:. Deriving an estimate of the frequency error offset f In practice, a rough estimate for the carrier frequency is nown beforehand almost always. Compensating for this offset by counter rotating the received signal Assumption: Freq. offset f is much smaller than the symbol rate /T - Timing information can be recovered first and is nown for the frequency estimation algorithm - Phase rotation due to the frequency offset can be assumed constant during a symbol Page 5
Basic system assumptions... Typical passband pulse amplitude modulation PAM receiver structure Carrier demod. Free osc. Interpolator control Timing est. Decimator control x Analog filter A/D x Int. filt. MF Dec. x Symbol det&dec Freq. synth. Freq. integr. Freq. err.det. Phase est. Rough corr. Fine corr. Timo I. Laaso and Stefan Werner Page 6
Basic system assumptions... Frequency estimation approaches Consider first data-aided DA frequency recovery, assuming that a training signal is available Last, non-data-aided DA techniques are briefly discussed Timo I. Laaso and Stefan Werner Page 7
Basic concepts revisited
Pulse shaping with root raised cosine waveform bits QAM modulation Transmitter Root ^cos Root ^cos Demodulation Pulse shaping is needed to band limit the transmitted signal waveform h T t in the formulas that follow In simulations of communication systems it is often enough to use directly the raised cosine pulse.
Tx and Rx signals with root raised cosine pulse shaping.5 Message data Tx data Rx data.5 Amplitude -.5 - -.5 5 5 5 3 Time
Root raised cosine filter H jω src $ & & % & & ' cos[ π 4r ω ωc for ω ωc r, ω --r] for r + r, ω c for ω ωc + r. ormalized bandwidth is +r, where r is the roll-off factor Root raised cosine filters form a yquist pair such that there is no inter-symbol interference For example, 3G WCDMA specifies roll-off.3 root raised cosine pulse shaping in the transmitter. Max. adjacent channel leaage ratio is -45 db for base stations and -33 db for mobile stations 3//6
Raised cosine Convolution of two root raised cosine pulses o inter-symbol interference when sampled at the right time Raised cosine pulse train in time domain 3//6
Matched Filter Design of matched filter Maximize signal power i.e. power of g t g t* h t at t T Minimize noise i.e. power of n t w t* h t Given transmitter pulse shape gt of duration T, matched filter is given by h opt t g*t-t for all scalar gains - h opt t is scaled, time-reversed, and shifted version of gt Sampling time nown gt xt ht yt yt Pulse signal wt Matched filter t T
DA: ML estimation of carrier frequency
DA: ML Estimation of carrier frequency Transmit signal model: xt h T t a δt T a h T t T a data symbols to be transmitted h T t transmitted continuous-time waveform δt Dirac delta function Timo I. Laaso and Stefan Werner Page 5
DA: ML Estimation... Received continuous-time signal additive white Gaussian noise AWG channel: r t x t + w t rt received signal waveform wt additive noise The desired estimate for the received signal is of the form ~ j ft+ x π θ t e a h t T τ where f is to be estimated, θ is unnown and τ is assumed nown T Page 6
DA: ML Estimation... Consider first the PDF of the discrete-time signal r assuming AWG: Page 7 Continuous-time probability density function PDF: Replace variance by noise power spectral density PSD and sum by integral: / ~ exp, ; x r f p σ πσ θ r ~ exp x r C σ ~ exp, ; T dt t x t r C f p θ r
DA: ML Estimation... The continuous-time log-lielihood function LLF is: Λ r; f, θ C T r t ~ x t T T T C r t dt + ~ x t dt Re The first and the second integrals are independent of f and θ > the relevant part of LLF to be optimized is dt r t ~ x * t dt Λ r; f, θ Re T { r t ~ x * t } dt Timo I. Laaso and Stefan Werner Page 8
DA: ML Estimation... The integral can be expressed as Timo I. Laaso and Stefan Werner Page 9 where z is the integral sampled at t T + τ and L T T is the observation interval Λ ~ * Re, ; T dt t x t r f θ r Re L j z a e θ Re T ft j j dt T t h e t r a e τ π θ
DA: ML Estimation... By denoting L a z Z e jφ the LLF can be expressed as Λ r; f, θ Re Z { j φ θ Z e } cos φ θ and the corresponding LF PDF is p r; f, θ Z C exp cos φ θ Timo I. Laaso and Stefan Werner Page
DA: ML Estimation... The phase θ is a nuisance parameter and should be eliminated. That can be accomplished by averaging the PDF over [,π]: Timo I. Laaso and Stefan Werner Page cos exp, ; ; Z CI d Z C d f p f p θ θ φ π θ θ π π π r r where I x is the modified Bessel function of order zero: θ π π θ d e x I x cos
DA: ML Estimation Bessel function of first ind, order zero besseli,x in Matlab The function is monotonic for positive values, and it is only necessary to maximize the argument of I x
DA: ML Estimation... I x is an even, parabola-lie function and monotonic for positive values. As the argument is positive in our case, we can conclude that f max argmax r f The function to be maximized is thus with see p. 3 z t G f Z { p ; f } argmax{ Z } L a z jπfξ r ξ e ht ξ t dξ f Timo I. Laaso and Stefan Werner Page 3
Example of Gf ; fo /; rx expj**pi*fo*[:-]; xx -.5:.:.5; for ii :lengthxx yii absrx * exp- j**pi*xxii*[:-]'; end.5 -.4 -.3 -. -....3.4 Frequency offset 4
Example of Gf Location of the maximum depends on the sampling grid -.49:.:.49 R.W. 5
Example of Gf Estimation from a noisy signal. Shifted max. value R.W. 6
DA: ML Estimation... Maximization of Gf: Data symbols must be nown as originally assumed o closed-form solution available in general Use search algorithms instead Problem: several local maxima > Good initial estimate needed Timo I. Laaso and Stefan Werner Page 7
DA: Practical algorithms
DA: Practical algorithms Let us mae simplifying assumptions:. The Tx and Rx filters e.g. root raised cosine filters -> raised cosine filter form a yquist pulse. Data symbols are from phase shift-eying PSK constellation: a jα π e, α m, m,,,..., M M 3. Frequency offset f is small compared to the symbol rate - Together with. this allows the assumption of zero inter-symbol interference ISI - Typically, uncompensated CFOs of the order of -% of the symbol rate already cause ISI Page 9
DA: Practical algorithms... Received signal: r T i Matched filtering: jπft t e + θ aih t it τ + w t y r t ht t T τ dt jθ e a i e where the matched-filtered noise is n w t ht t T τ dt i jπft h T t it τ h T t T τ dt + n Timo I. Laaso and Stefan Werner Page 3
DA: Practical algorithms... Because f << /T, we can approximate : Timo I. Laaso and Stefan Werner Page 3 so that the MF output becomes: Data modulation is removed by multiplying y with a * T T τ τ τ π π + it t h e it t h e it f j ft j [ ]. n e a y T f j + + + θ τ π [ ] [ ] ' n e n a e a a y a z T f j T f j + + + + + + θ τ π θ τ π This signal is used for the design of the frequency estimation algorithms that follow
DA: Practical algorithms ote: Here T refers to symbol rate and {a } is a nown pilot/ training sequence In case unmodulated pilot tone is available a and T refers to sampling rate 3
DA: Algorithm Kay 989 Proposed by S. Kay, A fast and accurate single frequency estimator, IEEE Trans. on ASSP, Dec. 989. Rearrange Timo I. Laaso and Stefan Werner Page 33 where For high E s /, the random variables φ are approximately independent, zero-mean and Gaussian [ ] [ ] ' T f j T f j e n e z φ θ τ π θ τ π ρ + + + + + + [ ] θ τ π φ ρ + + + ' T f j j e n e
DA: Algorithm Kay 989 Approximation at high SR, i.e. low noise, needed to reach the Gaussian assumption in the previous page R.W. 34
DA: Algorithm Kay 989... Consider the product of consecutive samples the other conjugated: z z * ρ ρ e j [ πf T + τ + θ + φ ] j[ πf T + τ + θ + φ ] e ρ ρ e j [ πft+ φ φ ] The argument can be considered as a noisy measurement for frequency: { z z * } πft + φ φ v arg Page 35
DA: Algorithm Kay 989... Under the Gaussian assumption, the PDF pv;f can be expressed in closed form and solved for the MLE. The result is L fˆ γ arg{ z z * } πt where γ is a smoothing function: 3 L L γ,,,..., L L L Page 36
DA: Algorithm Covariance matrix of vector v is tri-diagonal Probability density function is given by Solving R - gives the weights γ 3//6 37
Kay s algorithm applied to Mies signal Uniform weighting simply calculates the mean of the angles Kay s weighting is more stable w.r.t. the number of samples than the uniform weighting although the differences are small CFO 394 393.5 393 39.5 39 39.5 39 Kay's weights Uniform weights 39.5 39 389.5 3 4 5 6 7 8 #samples 3//6 38
DA: Algorithm Fitz 99 Proposed by M. Fitz, Planar filtered techniques for burst mode carrier synchronization, GLOBECOM 9 conference, Phoenix, Arizona, Dec. 99. Define Using L R m z z * m, m L L m z m [ πf T + τ + θ ] j a y e + n' yields R m e j πmft + n'' m where n m is a zero-mean noise component. Timo I. Laaso and Stefan Werner Page 39
DA: Algorithm Fitz 99... The frequency estimate could be solved from the above expression as fˆ arg{ R m } πmt However, due to the mod π nature of the arg operation, the result may be erroneous. A better estimate is obtained by averaging. Define { R m } πmft e m arg Timo I. Laaso and Stefan Werner Page 4
DA: Algorithm Fitz 99... Timo I. Laaso and Stefan Werner Page 4 Under certain assumptions < / f MAX T where f MAX is the max frequency error, it holds that which yields { } m m m mft m R m e arg π { } arg + ft m R m π { } + m m R T f arg ˆ π
DA Algorithm comparison Fundamental difference between Kay and Fitz: Kay operates by first applying the hard arg arctan nonlinearity and then smooths/weights the result Fitz applies smoothing/filtering also before nonlinearity Fitz is less prone to errors and provides more accurate results in low SR conditions c.f. ML estimation of unnown phase of a sinusoid with nown frequency: angle of sample means instead of sample mean of angles Page 4
DA Algorithm comparison... Performance comparison QPSK system, α.5 excess bandwidth roll-off factor of root raised cosine pulse, L 3 ormalized Freq. Error Variance log scale Fitz Kay CRB 8 E S / db Figure shows that both algorithms attain the Cramer-Rao bound CRB at high SR For low SR, the performance of Kay s alg. breas down 8 s Timo I. Laaso and Stefan Werner Page 43
DA: Algorithm 3 Luise & Regiannini Timo I. Laaso and Stefan Werner Page 44 + m fmt j m m fmt j m e m n e m R " π π Recall ow average Using '' n e m R mft j + π ft j m fmt j e ft ft e sin sin + π π π π
DA: Algorithm 3 Luise & Regiannini and the fact that for f /T yields sin πft sin πft > f arg + R m π T m Accuracy degrades slowly as decreases while estimation range gets wider f /T Page 45
Fitz and L&R algorithms applied to Mies signal -38-38 -38-38 -384-384 CFO -386 CFO -386-388 -388-39 -39-39 5 5 5 3 35 4 45 #lags Fitz -39 3 4 5 6 7 8 9 #lags Luise&Regiannini 46
DA Algorithms
DA algorithm Proposed by Chuang & Sollenberger 99 When the data modulation cannot be eliminated by using training signals or decision feedbac, other techniques must be used Assume a small enough frequency error f << /T so that the matched filter MF output can again be approximated as y [ πf T + τ + θ ] j a e + n. jα π MPSK signal a e, α m, m,,,..., M M Use unit-magnitude property of MPSK signals to remove data modulation Page 48
DA algorithms... For every MPSK symbol it holds that so that y M M [ ] e a j [ πmf T + τ + Mθ ] + n'. ow we can use the same idea as in the DA algorithms: M jπmft [ y y * ] e + n'' where the noise component n is again assumed to be zero-mean Timo I. Laaso and Stefan Werner Page 49
DA algorithms... Better estimates are obtained by smoothing: L L M jπmft [ y y* ] e + Setting the averaged noise term to zero and solving for the frequency gives the estimate ice and simple! L fˆ L arg πmt Limitation: f << /MT [ y y* ] L M n'' Timo I. Laaso and Stefan Werner Page 5
DA algorithm performance Performance analysis QPSK system, excess bandwidth α.5, L ormalized Freq. Error Variance log scale CRB Chuang et al. 5 E S / db 3 The Cramer-Rao Bound CRB is not close even for high SR Useful for coarse estimation in bad conditions Timo I. Laaso and Stefan Werner Page 5
Chuang&Sollenberg with Mies signal Known data uses angle of the autocorrelation with lag QPSK applies yy*- 4 The difference in the estimated value is significant 3//6 5
Summary We discussed. Basic system assumptions. DA: ML estimation of carrier frequency 3. DA: Practical algorithms 4. DA estimation algorithms Timo I. Laaso and Stefan Werner Page 53