Residual carrier frequency offset and symbol timing offset in narrow-band OFDM modulation

Residual carrier frequency offset and symbol timing offset in narrow-band OFDM modulation Master s thesis iko Ohukainen 2264750 Department of Mathematical Sciences University of Oulu Spring 2017

Contents Introduction 2 1 Preliminaries 3 1.1 Fourier transform......................... 3 1.2 yquist-shannon sampling theorem............... 5 1.3 Cramer-Rao lower bound..................... 6 1.4 Kalman filtering.......................... 9 2 Orthogonal frequency-division multiplexing system 14 2.1 Transmitter............................ 15 2.2 Channel.............................. 18 2.3 Receiver.............................. 20 2.4 Symbol timing and frequency offsets.............. 23 2.4.1 Symbol timing offset................... 24 2.4.2 Frequency offset...................... 26 2.5 Received signal model...................... 28 3 Timing and frequency offset estimation 34 3.1 Cramer-Rao lower bound..................... 35 3.2 Maximum likelihood estimate.................. 38 3.3 Frequency domain correlation based estimate.......... 41 3.4 Kalman filter based estimate................... 43 4 umerical simulations 45 4.1 Simulations............................ 46 4.2 Conclusion............................. 51 References 57 1

Introduction The technological progress in wireless communication has enabled us to be reachable anywhere without the help of electrical conductors, such as wires. The modern cellular network offers almost ubiquitous connectivity and wireless local area networks are found in homes, offices and commercial spaces. The development has led to the point where each thing is equipped with wireless connectivity. This expanding of Internet and networks to cover thing like meters, buildings and health care appliances is called the Internet of Things (IoT). The requirements for the connected things are different from the requirements for mobile phones and computers that have fixed power source or are easily recharged. Mobile phones and computers are characterized by the need for high data rates and very small end-to-end transmission delays. These devices have enough computing power to handle complex transmission schemes and can use resources to transmit with high power. For things like temperature meters and location beacons, this is not possible. The devices might be required to run from a small battery for years. In addition, these devices might need to transmit or receive only small amounts of data with long breaks in between. From the device point of view, fast synchronization is vital to be able to save power when transmission is not needed. The possibly low signal strength together with the low computation power lay a challenge for the synchronization process because the parameters need to be inferred from the received signal. Orthogonal frequency-division multiplexing (OFDM) is a digital modulation method used in existing wireless standards such as Long-Term Evolution (LTE) [1], IEEE 802.11 [2] and Digital Video Broadcasting (DVB) [3]. The 3rd Generation Partnership Project (3GPP) is defining a radio access technology based on OFDM that is targeted to enable IoT devices ubiquitous networking [1, chapter 10]. To reach the requirement of low cost and low power consumption, the transmission bandwidth is narrowed and repetitions are introduced to enable communications in situations where receiver suffers from very low signal strength. This specification is used as a guideline for the numerical simulations in section 4. In the first section, a few concepts are introduced that give the foundation to the reviews in following sections. The Fourier transform is defined together with its discrete version, which gives the basis for OFDM modulation. The yquist-shannon sampling theorem proves that a continuous signal can be constructed from its discrete samples without losing information of the original signal under certain requirements. This thesis focuses on the estimation of synchronization parameters in OFDM system and the Cramer-Rao 2

lower bound is used to give a benchmark for the derived estimator schemes. Lastly the Kalman filtering concept is presented which is used in one of the estimators derived in section 3. The second section defines the transmission system using OFDM modulation to transmit data. The transmitted signal is derived together with the perturbations caused by the multi-path channel. An OFDM receiver is responsible for demodulating the signal and de-multiplexing the data streams carried on different sub-carriers. In this section the effect of non-idealities in the OFDM system are introduced and their effect is incorporated to the signal model derived in the last part of this section. The estimation of the synchronization parameters is discussed in the third section. Three estimators are derived according to the signal model presented in the second section and the Cramer-Rao lower bound is derived to give a benchmark for the estimators. The estimators are developed for two purposes, to estimate the parameters from a short signal sample and to track the timevariant change in the estimation parameters during continuous reception. In the last section the derived estimators performance is numerically evaluated in a simulation environment. The estimator mean, mean squared error performance and continuous tracking capabilities are studied. 1 Preliminaries 1.1 Fourier transform The Fourier transform is used to represent a function of time with the frequencies it is composed of. Taking the Fourier transform of a function gives a complex-valued function; where the magnitude represents the amount of that frequency in the original function and the phase represent the phase offset from the basic sinusoid of that frequency. Definition 1.1. If function s : R C is such that s(t) dt <, then its Fourier transform S : R C is the bounded function on R defined by S(f) = F(s)(f) = e j2πtf s(t)dt. (1) For a function S, S(f) df <, the inverse Fourier transform F 1 (S) is defined as s(t) = F 1 (S)(t) = e j2πtf S(f)df. (2) The Fourier transform and the inverse Fourier transform are, as the naming suggests, inverse operations. 3

Theorem 1.2. For a function s : R C, that has a Fourier transform F(s) = S : R C, it holds that Proof. See Theorem 5.3 in [4]. F 1 (F(s)) (t) = s(t). (3) The corresponding operation in discrete domain is the discrete Fourier transform. It transforms a finite, equally spaced sequence of time domain samples to an equally spaced frequency domain sequence. The interval at which the discrete Fourier transform is sampled is the reciprocal of the duration of the time domain sequence. Definition 1.3. Let s[n] C be a finite sequence of equally spaced values of a function s. The discrete Fourier transform S[k] C is defined as S[k] = F(s)[k] = 1 1 e j2πkn/ s[n]. (4) n=0 The inverse discrete Fourier transform is defined as s[n] = F 1 (S)[n] = 1 1 e j2πkn/ S[k]. (5) The discrete Fourier transform can be presented as a linear mapping by defining the discrete Fourier transform matrix F M, where the elements are given by F n,k = 1 e j2πkn/. (6) The Fourier transform is then given as k=0 S = Fs, (7) where s and S are the discrete sequences in vector form. In addition, the discrete Fourier transform and the inverse discrete Fourier transform are each other s inverse operations. Theorem 1.4. Let s[n] C n be a finite sequence. Then F 1 (F(s)) [n] = s[n]. (8) 4

Proof. By definition F 1 (F(s)) [n] = 1 = 1 1 k=0 1 m=0 1 e j2πkn/ 1 s[m] k=0 m=0 e j2πkm/n s[m] e j2πk(n m)/. The sum 1 k=0 ej2πk(n m)/ evaluates to when n m = 0 and zero otherwise. Then F 1 (F(s)) [n] = s[n]. 1.2 yquist-shannon sampling theorem The following theorem is by Shannon [5]. It states that a continuous signal can be completely recovered from its samples, when they are taken with a rate of twice the signal bandwidth. Theorem 1.5 (yquist-shannon sampling theorem). Let s(t) be a bandlimited signal such that it s spectrum S(f) = F(s)(f) is limited to an interval [ B, B]. Then the signal can be recovered from it s samples if it is sampled at a frequency of 2B. Proof. Let S(f) be the frequency spectrum of s(t). Then S(f) = 0, if f > B. We know that a signal is completely defined by it s spectrum by s(t) = S(f)e j2πft df = B B S(f)e j2πft df. Then, if we sample the signal at time instances t = n, we get 2B ( ) n B B s = S(f)e j2πf n nf 2B df = jπ S(f)e B df. 2B B B On the other hand, the spectrum S(f) can be represented by it s Fourier series as nf jπ S(f) = c n e B, n= where the coefficients c n are given by c n = 1 B nf jπ S(f)e B df. 2B B 5

We see that the integral is exactly a sample of the signal at time index t = n 2B and therefore S(f) = 1 2B n= ( ) n nf jπ s e B. 2B ow, we get a representation of the signal s(t) from it s samples as s(t) = 1 B ( ) n nf jπ s e B e j2πft df 2B B n= 2B = 1 ( ) n B s e j2πf(t 2B) n df 2B n= 2B B = 1 ( ) n e j2πf(t 2B) n s 2B n= 2B j2π ( B ) t n f= B 2B ( ) n e jπ(2bt n) e jπ(2bt n) = s n= 2B j2π(2bt n) ( ) n = s sinc(π(2bt n)). n= 2B So, the signal s(t) is completely determined by the samples taken at time indices t = n 2B. 1.3 Cramer-Rao lower bound The Cramer-Rao lower bound gives the lower bound for the variance of an unbiased estimate. An estimator that reaches the Cramer-Rao lower bound has the lowest possible mean squared error amongst unbiased estimators. This fact is given by the following theorem provided by Kay [6]. Theorem 1.6. Let f(x; Θ) be a probability density function with real valued parameters Θ R p. Let X = [X 1,..., X n ] T C n, be a vector of independent random variables with density function f(x; Θ). Assume that the set S = {x C n f(x; Θ) > 0} doesn t depend on Θ. Let ˆΘ = ˆΘ(x) be an unbiased estimator of Θ. Assume that following regularity conditions are satisfied 1. Derivatives f(x; Θ)/ θ i are integrable and f(x; Θ) f(x; Θ)dx = dx for all i = 1, 2,..., p. (9) θ i S S θ i 2. Second derivatives 2 f(x; Θ)/( θ i θ j ) are integrable and 2 θ i θ j S f(x; Θ)dx = 6 S 2 f(x; Θ) θ i θ j dx (10)

for all i = 1, 2,..., p and j = 1, 2,..., p. 3. We can change the order of integration and differentiation as θ i for all i = 1, 2,..., p. S ˆΘ(x)f(x; Θ)dx = S Then, the covariance matrix of ˆΘ satisfies f(x; Θ) ˆΘ(x) dx (11) θ i C ˆΘ F 1 (Θ) 0, (12) where 0 means that the matrix is positive semi-definite and where the Fisher information matrix F(Θ) is given by { 2 } ln f(x; Θ) [F(Θ)] ij = E. (13) θ i θ j Further, the variance of estimator ˆθ i is lower bounded by Proof. Because ˆΘ is unbiased we get E{ ˆθ i θ i 2 } [F 1 (Θ)] ii. (14) Θ = E{ ˆΘ} = S ˆΘ(x)f(x; Θ)dx. Differentiating both sides with respect to Θ and using the regularity condition (11) yields Θ Θ = I = S ˆΘ(x) where the superscript ( ) T denotes transpose. Define a random vector Z = T f(x; Θ) dx, Θ 1 f(x; Θ) f(x; Θ) + 1 S c(x) Θ, where 1, x S c 1 S c = 0, x S. 7

Then, the expectation of Z is given by E{Z} = = S S 1 f(x; Θ) f(x; Θ)dx f(x; Θ) Θ f(x; Θ) Θ dx = f(x; Θ)dx = 0, Θ S where the regularity condition (9) was used. Evaluate the expression E{( ˆΘ Θ)Z T } to get E{( ˆΘ Θ)Z T } =E{ ˆΘZ T } ΘE{Z T } T 1 f(x; Θ) = ˆΘ(x) f(x; Θ)dx S f(x; Θ) Θ T f(x; Θ) = ˆΘ(x) dx = I. S Θ Let a R p and b R p be arbitrary vectors. ow a T b is a scalar and (a T b) 2 =(a T Ib) 2 = (a T E{( ˆΘ Θ)Z T }b) 2 = (E{a T ( ˆΘ Θ)Z T b}) 2. Applying Cauchy-Schwartz inequality gives (E{a T ( ˆΘ Θ)Z T b}) 2 E{(a T ( ˆΘ Θ)) 2 }E{(Z T b) 2 } =E{a T ( ˆΘ Θ)( ˆΘ Θ) T a}e{b T ZZ T b} =a T C ˆΘ abt C Z b, where C ˆΘ and C Z are the covariance matrices of ˆΘ and Z. Since b was arbitrary we can choose b = C 1 Z a to have (a T b) 2 = (a T C 1 Z a) 2 a T C ˆΘ aat C 1 Z C Z C 1 Z a = a T C ˆΘ aat C 1 or a T C 1 Z aa T C 1 Z a a T C ˆΘ aat C 1 Z a. Because covariance matrices are positive semi-definite and symmetric, the term a T C 1 Z a is a non-negative real number and we have a T C 1 Z a a T C ˆΘ a. Then it follows that C ˆΘ C 1 Z is positive semi-definite. To prove that C Z = F(Θ) we use the regularity conditions (9) and (10) to find that 0 = 1 = f(x; Θ)dx θ i θ j θ i θ j S 8 Z a

So = ln f(x; Θ) f(x; Θ)dx θ i S θ j ( 2 ) ln f(x; Θ) ln f(x; Θ) ln f(x; Θ) = f(x; Θ) + f(x; Θ) dx. S θ i θ j θ i θ j E{Z i Z j } = E { 2 } ln f(x; Θ) θ i θ j or [C Z ] ij = [F(Θ)] ij. The last part of this theorem states that the variance of the estimate ˆθ i is lower bounded by [F 1 (Θ)] ii. This is obtained by the fact that diagonal elements of a positive semi-definite matrix are non-negative and [C ˆΘ F 1 (Θ)] ii 0 or E{ ˆθ i θ i 2 } [F 1 (Θ)] ii. 1.4 Kalman filtering Kalman filter is an algorithm to find the solution for a linear quadratic estimation problem. The Kalman filter recursively calculates new estimates of a systems state by using knowledge of previous state and observations. The system model and noise covariances are assumed to be known. In a situation where the model is linear and accurate the Kalman filter is shown to be the optimal estimator in the sense that it minimizes the mean-square error [7]. Consider a stochastic process X k that evolves according to the following state transition equation x k = f(x k 1, u k, v k ), (15) where the current state x k depends only on the state x k 1 of the previous time step, known control input u k and zero mean white Gaussian noise v k with covariance Q k. Observations of the system are provided by y k = h(x k, n k ), (16) where n k is zero mean white Gaussian noise with covariance matrix R k. The Kalman filter is used to estimate the state x. First, the knowledge from previous time step is used to predict the state vector and then new observation is used to make the estimate more accurate. We denote the estimate of state x at time index n with observations up to time index m as ˆx n m. The prediction of the new state is given by ˆx k k 1 = E{f(x k 1, u k, v k )}, (17) 9

with a prediction error covariance P k k 1 = E{(x k ˆx k k 1 )(x k ˆx k k 1 ) T }. (18) When an observation of the system is made available, it is used to improve the estimate. The update is given by ˆx k k = ˆx k k 1 + K k (y k ŷ k ), (19) where K k is the Kalman gain and the observation is predicted by ŷ k = E{h(ˆx k k 1, n k )}. (20) The Kalman gain is computed such that the mean squared error E{ x k ˆx k k 2 } is minimized. This corresponds to minimizing the trace of the covariance matrix P k k = E{(x k ˆx k k )(x k ˆx k k ) T }. Using equation (19), the covariance matrix can be written as P k k = E{(x k ˆx k k 1 K k (y k ŷ k ))(x k ˆx k k 1 K k (y k ŷ k )) T } = P k k 1 P xy K T k K k P T xy + K k P y K T k, (21) where P xy = E{(x k ˆx k k 1 )(y k ŷ k ) T } is the cross-correlation matrix and P y = E{(y k ŷ k )(y k ŷ k ) T } is the observation covariance. The trace is given by E{ x k ˆx k k 2 } = tr(p k k ) = tr(p k k 1 ) 2tr(P T xyk k ) + tr(k k P y K k ). (22) Taking the derivative of the trace with respect to the Kalman gain K k and setting it to zero gives tr(p k k ) = 2P xy + 2K k P y = 0 K k or K k = P xy P 1 y. (23) Using Kalman gain given in equation (23), we can write the update for the error covariance in equation (21) as P k k = P k k 1 K k P T xy. (24) For other Kalman gains the form of equation (21) has to be used. 10

Original Kalman filter The original Kalman filter published by Kálman [8] assumes that the state transition and observation can be evaluated with linear functions. The state transition function is now given by the following difference equation x k = A k x k 1 + B k u k + v k (25) and the observation is derived from the current state as y k = H k x k + n k, (26) where A k is the state transition matrix, B k is the control matrix and H k is the observation matrix. When the dynamic model is given by equations (25) and (26), the state prediction is given by ˆx k k 1 = E{A k x k 1 + B k u k + v k } = A k E{x k 1 } + B k u k = A kˆx k 1 k 1 + B k u k. (27) The covariance matrix of prediction error can be written as P k k 1 = E{(x k ˆx k k 1 )(x k ˆx k k 1 ) T } = E{(A k (x k 1 ˆx k 1 k 1 ) + v k )(A k (x k 1 ˆx k 1 k 1 ) + v k ) T } = A k P k 1 k 1 A T k + Q k. (28) is After the observation is made available, the update of the state estimate ˆx k k = ˆx k k 1 + K k (y k ŷ k ) and the estimation error covariance is = ˆx k k 1 + K k (y k H kˆx k k 1 ) (29) P k k = P k k 1 KP T xy = (I K k H k )P k k 1, (30) where the cross-correlation matrix is given by P xy = E{(x k ˆx k k 1 )(y k ŷ k ) T } = E{(x k ˆx k k 1 )(x k ˆx k k 1 ) T H T k } = P k k 1 H T k. (31) The optimal Kalman gain is given by K k = P xy P 1 y = P k k 1 H T k (E{(y k ŷ k )(y k ŷ k ) T }) 1 11

= P k k 1 H T k (E{(H k (x k ˆx k k 1 ) + n k )(H k (x k ˆx k k 1 ) + n k ) T }) 1 = P k k 1 H T k (H k P k k 1 H T k + R k ) 1. (32) The algorithm is initialized by assigning ˆx 0 0 = E{x 0 } and P 0 0 = E{(x 0 ˆx 0 0 )(x 0 ˆx 0 0 ) T }. If the linear state transition and observation models are accurate and the noise statistics are known, the estimate ˆx k k of the state x k minimizes the mean squared error [7]. Unscented Kalman filter The original Kalman filter assumes that the system model can be accurately modeled with linear functions. The unscented Kalman filter has been developed to apply the linear estimation scheme on systems that exhibit non-linear state transition or observation models. The unscented Kalman filter uses a sampling technique called scaled unscented transform [9] to estimate the true mean and covariance. A minimum number of points, called sigma points, around the estimated mean are selected. These sigma points are fed through the non-linear system model and the weighted ensemble mean and covariance are used to estimate the true mean and covariance matrix. In this thesis the unscented Kalman filter notation follows the scheme introduced by Van Der Merwe in his dissertation [7]. The state of the stochastic process now evolves according to a non-linear function and the observations are given by x k = f(x k 1, u k, v k ) (33) y k = h(x k, n k ). (34) The filters state at each time step k is given by augmented versions of the state vector and covariance matrix, where the process and observation noises are incorporated. The augmented state estimate vector is given by ˆx k k and the augmented covariance matrix is given by ˆx k k a = E{v k } (35) E{n k } P k k 0 0 P a k k = 0 Q k 0. (36) 0 0 R k 12

Let the dimension of augmented state vector be L. The sigma points around the estimated mean of state vector are selected according to Xk 1 k 1 a = [ˆx k 1 k 1, a ˆx k 1 k 1 a + γ P a k 1 k 1, ˆxa k 1 k 1 γ ] P a k 1 k 1 = Xk 1 k 1 x Xk 1 k 1 v Xk 1 k 1 n, (37) where the square root denotes the Cholesky decomposition and γ = L + λ. (38) Here λ is a scaling parameter given by λ = α 2 (L + κ) L, where α is a parameter for how far from the mean the sigma points are selected and κ is selected to ensure positive definiteness of covariance matrix. The weights for computing means and covariances are given by W m i W0 m = λ L + λ, (39) W0 c = λ L + λ + 1 α2 + β and (40) = Wi c 1 = 2(L + λ) i = 1,..., 2L. (41) The superscript denotes whether the weight term is used for calculating the mean or covariance. The extra term in covariance weight (40) is used to help minimize higher order errors in the estimated covariance by selecting a suitable value for parameter β. For Gaussian distributed state vector the optimal value is known to be β = 2 [9]. In addition, the unscented Kalman filter works in two phases. First, a prediction phase is carried out to give the prediction of the state. The sigma points are passed through the state transition function X x k k 1 = f(x x k 1 k 1, u k, X v k 1 k 1) (42) and the predicted state and prediction error covariance are estimated with weighted ensemble mean and covariance as and P k k 1 = 2L i=0 ˆx k k 1 = 2L i=0 Wi m Xi,k k 1 x (43) ( ) ( ) Wi c X x i,k k 1 ˆx k k 1 X x T k k 1 ˆx k k 1. (44) 13

In the update phase, the transitioned sigma points are propagated through the observation model and again weighted ensemble mean and covariance are used to estimate the cross-correlation and observation covariance. The observation sigma points are given by from which the observation estimate is given by Y k = h(x x k k 1, X n k 1 k 1), (45) the observation covariance ŷ k = 2L i=0 W m i Y i,k, (46) P y = 2L i=0 and the cross-correlation matrix is P xy = 2L i=0 W c i (Y i,k ŷ k ) (Y i,k ŷ k ) T (47) ( ) Wi c Xi,k k 1 ˆx k k 1 (Yi,k ŷ k ) T. (48) The update to the state vector and error covariance are given by the equations (19), (24) and (23) as derived earlier. Here they are given again K k = P xy P 1 y, (49) ˆx k k = ˆx k k 1 + K k (y k ŷ k ) and (50) P k k = P k k 1 K k P T xy. (51) 2 Orthogonal frequency-division multiplexing system Orthogonal frequency-division multiplexing (OFDM) is a digital modulation method where the transmitted symbol carries multiple data symbols simultaneously. The low rate data is used to modulate sub-carriers that are orthogonal in a sense that transmission over one sub-carrier does not affect the transmission over adjacent sub-carriers. Modulation and demodulation is accomplished by taking a discrete Fourier transform of symbol samples. The advantages of using OFDM modulation is that it is robust against multipath fading and demodulation can be done without complex equalization. The spectral efficiency is good because no guard interval is used between sub-carriers in frequency and it is computationally efficient because of 14

discrete Fourier transform can be computed by a fast Fourier transform algorithm. The use of cyclic prefix makes the modulation scheme robust against timing synchronization errors. The challenges of OFDM are its sensitivity to frequency synchronization errors and Doppler shift. Imperfect synchronization results in loss of orthogonality which causes phase rotation and attenuation of the signal, as well as introduces additional noise due to inter-carrier interference (ICI). 2.1 Transmitter The transmitter is responsible for multiplexing different logical channels, modulation and radiating the signal from antennas. The OFDM symbol is a collection of orthogonal information bearing sub-carriers summed together. Definition 2.1. Orthogonal frequency-division multiplexing symbol carrying a block of symbols c = [c, c 2 +1,..., c 2 2 1]T is given by s(t) = 1 c k e j2πf kt, 0 t < T u, (52) k= 2 where e j2πfkt = w fk (t) is the sub-carrier waveform, f k = k T u is the sub-carrier frequency, T u is the symbol duration and is the number of sub-carriers. We assume that is a power of two and sub-carrier index zero denotes the center sub-carrier. The orthogonality means that even though the frequency spectrum of the sub-carriers overlap they do not contribute energy to the adjacent subcarriers in demodulation. The orthogonality is proved by following theorem. Theorem 2.2. Let the effective OFDM symbol duration be T u. Then the subcarrier waveforms w f (t) = e j2πft are orthogonal if the spacing is a multiple of 1 T u. Proof. Two signals, f, g : R C, f g, are orthogonal on the interval [t 0, t 0 + T u ] if f, g [t0,t 0 +T u] = 1 t0 +T u T u t 0 f(t)g (t)dt = 0. Let f 1 and f 2 denote the two sub-carrier frequencies such that f 1 f 2 = k T u, k Z. Then w f1 (t), w f2 (t) [t0,t 0 +T = 1 t0 +T u u] e j2πf1t e j2πf2t dt T u = 1 t0 +T u e j2π(f 1 f 2 )t dt = 1 T u t 0 T u t 0 t0 +T u t 0 15 e j2πkt/tu dt = ej2πkt/tu j2πk t 0 +T u t=t 0

= ej2πkt 0/T u e j2πk e j2πkt 0/T u j2πk = e jπk(1+2t 0/T u) sin(πk) πk = e j2πkt 0/T u e jπk ejπk e jπk j2πk = e jπk(1+2t 1, k = 0, 0/T u) sinc(πk) = 0, k 0. With this property the demodulation of sub-carrier k can be carried out by correlating the OFDM symbol s(t) with the sub-carrier waveform conjugate as T u Tu 0 s(t)wf k (t)dt = = T u Tu 0 1 k = 2 c k w fk (t)w f k (t)dt c k wfk (t), w fk (t) = c k. (53) k = 2 Generating OFDM symbol using DFT The OFDM symbols are generated by multiplexing data symbols d l,k and reference symbols x l,k in time and frequency to OFDM symbol l and sub-carrier k and forwarded to an inverse discrete Fourier transform. The resources to be mapped to sub-carrier k in OFDM symbol l are given by d l,k, if (l, k) D, c l,k = x l,k, if (l, k) P, (54) 0, else, where the sets D and P are the index sets containing data symbol and reference symbol locations respectively. The input block to the size inverse discrete Fourier transforms is put into vector form as c l = [ c l,, c 2 l, +1,..., 2 ] T c l,. The output of inverse Fourier transform is the baseband OFDM 2 1 symbol block where F = 1 s l = [s l,0, s l,1,..., s l, 1 ] T = F H c l, (55) 1 1 1 e j2π( 2 ) 1 e j2π( +1) 1 2 e j2π( 1) 1 2...... e j2π( 2 ) 1 e j2π( 1 +1) 2 e j2π( 2 16 1) 1 (56)

is the discrete Fourier transform matrix of size. The elements of s l are then given by s l,n = 1 kn j2π c l,k e. (57) k= 2 In the beginning of OFDM symbol block a cyclic prefix is added which is a copy of the last g samples. The operation can be carried out by multiplying OFDM symbol block with matrix [ ] C = 0g ( g) I g g (58) and we get I s cp,l = Cs l = [s l, g,..., s l, 1, s l,0,..., s l, 1 ] T. (59) Cyclic prefix is used to eliminate inter-symbol interference caused by multipath propagation, where a channel consists of multiple propagation paths that arrive at different times to the receiver. Last delayed samples of a symbol arrive at the same time as the first samples of the next symbol causing interference but the prefix acts as a buffer zone between two symbols. Additionally the cyclic prefix helps to reduce the problems due to time synchronisation errors because a cyclic shift in discrete Fourier transform input corresponds to linear phase shift in output. After applying the prefix, the OFDM symbol samples are put to serial form and converted from digital to analogue domain, see Theorem 1.5. Let T be the sample time interval and let T u = T, T g = g T and T s = ( + g )T be the effective symbol duration, prefix duration and the duration of the complete OFDM symbol. The baseband OFDM symbol stream is given by [10] where s(t) = 1 l= k= 2 c l,k e j2π k Tu (t Tg lts) u(t lt s ), (60) 1, 0 < t < T s u(t) = 0, else is a rectangular windowing function. Then, the signal is divided into real and imaginary part and quadrature mixed to passband frequency and radiated through antennas. The quadrature-mixed signal is given by s c (t) = Re(s(t)) cos(2πf c t) Im(s(t)) sin(2πf c t), (61) 17

where f c is the carrier frequency. It is worth mentioning that the signal s c (t) is real valued. 2.2 Channel The signal has to pass through a radio channel. In this work, the channel is assumed a Rayleigh fading multipath channel. The channel consists of multiple physical propagation paths, which cause the receiver to see the superposition of multiple instances of the sent waveform that arrive at different times and are attenuated and phase shifted differently. The movement of user equipment, base station or reflecting surfaces lead to Doppler effect that causes frequency shifts to the different received instances of the signal. The transmission conditions might vary with time and frequency. The time dependent change in channel is called time selectivity or fading and frequency dependent change is called frequency selectivity. In addition, additional noise from surrounding environment is introduced to the signal. The channel can be represented in time domain by its impulse response h(τ, t) which is a complex valued function of path delay τ and time t. A completely analogous representation is the response in frequency domain H(f, t), which relates to the impulse response such that they are Fourier transform pairs, H(f, t) = h(τ, t)e j2πτf dτ. The channel response h(τ, t) is applied to a signal s(t) by convolution as r(t) = (h(τ, t) s(t))(t) = h(τ, t)s(t τ)dτ. Because Fourier transform of a convolution of two signals is equal to the product of Fourier transforms of the signals, we get that R(f) = F(r(t)) = H(f, t) S(f), where S(f) = F(s(t)) and R(f) = F(r(t)). Therefore, the channel effect can be applied by convolution in time domain or by multiplication in frequency domain. Useful correlation functions The channel impulse response h(τ, t) is assumed to be zero mean wide-sensestationary stochastic process in the variable t. The behavior of the channel can be characterized by the following covariance functions [11]. Definition 2.3. The channel s time domain covariance function is defined as c(τ 1, τ 2, t 1, t 2 ) =E {(h(τ 1, t 1 ) E{h(τ 1, t 1 )}) (h(τ 2, t 2 ) E{h(τ 2, t 2 )})} =E{h (τ 1, t 1 )h(τ 2, t 2 )} = E{h (τ 1, 0)h(τ 2, t 2 t 1 )} c(τ 1, τ 2, t), (62) 18

where t = t 2 t 1. We make the assumption that the propagation path with delay τ 1 is uncorrelated with the propagation path with delay τ 2 τ 1 so c(τ 1, τ 2, t) = c(τ 1, τ 2, t)δ(τ 2 τ 1 ) c(τ, t), (63) where δ(t) is the Dirac delta function. This assumption is valid because signals arriving at different times might have travelled a completely different path to the receiver. If we set t = 0, the covariance function gives the average power output of the channel as a function of delay τ. That is why the function c(τ, 0) is called the delay power spectrum of the channel. Definition 2.4. The frequency domain covariance function is defined as C(f 1, f 2, t 1, t 2 ) =E {(H(f 1, t 1 ) E{H(f 1, t 1 )}) (H(f 2, t 2 ) E{H(f 2, t 2 )})} =E {H (f 1, t 1 )H(f 2, t 2 )} { } =E h (τ 1, t 1 )e j2πτ 1f 1 dτ 1 h(τ 2, t 2 )e j2πτ 2f 2 dτ 2 = = = e j2π(τ 2f 2 τ 1 f 1 ) E{h (τ 1, t 1 )h(τ 2, t 2 )}dτ 1 dτ 2 e j2π(τ 2f 2 τ 1 f 1 ) c(τ 1, τ 2, 0, t 2 t 1 )δ(τ 2 τ 1 )dτ 1 dτ 2 e j2π(f 2 f 1 )τ c(τ, t)dτ C( f, t). The covariance function in frequency domain is seen to be the Fourier transform of the time domain covariance function. The function C( f, t) is called spaced-time, spaced-frequency correlation function and it gives a measure for coherence time and coherence bandwidth of the channel. Setting t = 0 in C( f, t) gives the spaced frequency correlation function C( f, 0) which relates to the delay power spectrum c(τ, 0) such that they are Fourier transform pairs. Therefore the reciprocal of the width of the delay power spectrum T τ gives a measure of the coherence bandwidth of the channel ( f) c. A coarse measure is given by ( f) c 1 T τ. The meaning of the coherence bandwidth is that two signals that are spaced further than the coherence bandwidth ( f) c are affected differently by the channel, when signals spaced less than coherence bandwidth are affected similarly by the channel. Taking the Fourier transform of spaced-time, spaced-frequency correlation function with respect to t gives the function S( f, λ). If we set f = 0 we get the function S(0, λ) which gives the signal s intensity as a function of the Doppler frequency λ. Therefore the function S(0, λ) is called the Doppler power spectrum of the channel. The Doppler power spectrum relates to the 19

spaced-time correlation function such that they are Fourier transform pairs. The Doppler spread B d is the range over which the Doppler power spectrum is non-zero and its reciprocal gives a measure to the coherence time of the channel ( t) c. A coarse measure is ( t) c 1 B d. Two more Fourier transform pairs can be found by defining a function s(τ, λ) called the scattering function of the channel. It is the Fourier transform pair of the covariance function c(τ, t) with respect to t. It is also the Fourier transform pair of the function S( f, λ) with respect to the variable f. It gives the average power output of the channel as a function of the delay τ and Doppler frequency λ. Rayleigh fading In this work, the channel is assumed to exhibit Rayleigh fading. The impulse response h(τ, t) is assumed to have zero mean and phase evenly distributed between 0 and 2π. The envelope of the impulse response is then Rayleigh distributed. The probability density function of the Rayleigh distribution is f(x) = x x2 e 2σ σ2 2. (64) The scale parameter σ is given by the average power of the impulse response, σ 2 = E t { h(τ, t) 2 }. Because the arrival of different propagation paths are grouped into clusters, the channel impulse usually is essentially non-zero at the delays τ i, where τ i is the delay of the cluster i. Then the channel impulse response can be approximated by h(τ, t) = L i=1 h i (t)δ(τ i τ), where L is the number of significant clusters. This simplification introduces modelling error, which is neglected in this work. Example 2.5. Here we illustrate the effect of channel on the signal, when channel impulse response is given by h(τ, t) = L i=1 h i (t)δ(τ i τ). The channel impulse is applied to a signal by convolution as h(τ, t) s(t) = h(τ, t)s(t τ)dτ = L h i (t)δ(τ τ i )s(t τ)dτ i=1 L = h i (t) L δ(τ τ i )s(t τ)dτ = h i (t)s(t τ i ). i=1 i=1 2.3 Receiver The receiver is responsible for capturing the signal, compensating for distortions, demodulation, de-multiplexing and passing received samples to de- 20

coder. After capturing the signal with antennas, the radio frequency signal is band pass filtered to reject unwanted frequencies and given by L r c (t) = h i(t)s c (t τ i ) + w c (t), (65) i=0 where h i and w c are real valued channel response and noise. We assume that the channel impulse response is constant during one OFDM symbol and the duration τ L τ 1 is shorter than the cyclic prefix T g. Quadrature-mixing down to baseband frequency is accomplished by first multiplying the received signal with carrier frequency sinusoids that are separated by a π phase shift. In this work, any imbalance between in-phase and 2 quadrature components is assumed to be ideally removed. Using equation (61), the in-phase component is given by L I(t) =r c (t) cos(2πf c t) = h i(t)s c (t τ i ) cos(2πf c t) + w c (t) cos(2πf c t) i=1 L = h i(t)[re(s(t τ i )) cos(2πf c (t τ i )) cos(2πf c t) i=1 Im(s(t τ i )) sin(2πf c (t τ i )) cos(2πf c t)] + w c (t) cos(2πf c t) L h = i(t) i=1 2 [Re(s(t τ i))(cos( 2πf c τ i ) + cos(2πf c (2t τ i ))) Im(s(t τ i ))(sin( 2πf c τ i ) + sin(2πf c (2t τ i )))] + w c (t) cos(2πf c t). (66) The signal is low-pass filtered to reject the double frequency and suitable amplification is used to get L I(t) = h i(t) [Re(s(t τ i )) cos( 2πf c τ i ) i=1 Im(s(t τ i )) sin( 2πf c τ i )] + w (t), (67) where w (t) is filtered noise. Similarly, for quadrature part, we get after filtering and amplification L Q(t) = h i(t) [Re(s(t τ i )) sin( 2πf c τ i ) i=1 + Im(s(t τ i )) cos( 2πf c τ i )] + w (t). (68) 21

The quadrature-mixer outputs, in-phase and quadrature component, are next combined to get the received baseband OFDM symbol stream. The signal is given by L r(t) =I(t) + jq(t) = h i(t)[re(s(t τ i ))(cos( 2πf c τ i ) + j sin( 2πf c τ i )) i=1 + j Im(s(t τ i ))(cos( 2πf c τ i ) + j sin( 2πf c τ i ))] + w (t) + jw (t) L = h i(t)e j2πfcτ i s(t τ i ) + w(t) i=1 L = h i (t)s(t τ i ) + w(t), (69) i=1 where h i (t) = h i(t)e j2πfcτ i is the complex valued channel impulse response and w(t) = w (t) + jw (t) complex valued noise. The delays of the different propagation paths are unknown and the carrier frequency is usually very big, so the phase of the channel impulse response can be regarded uniformly distributed. If the real valued channel response has a Rayleigh distribution, then h i (t) exhibits Rayleigh fading. The noise w(t) is white Gaussian noise because the real and imaginary components are uncorrelated with equal power. This results from the fact that the in-phase and quadrature waveforms are orthogonal. The samples of OFDM symbol l are recovered from the received symbol stream by sampling at time indices t n = nt + T g + lt s, n {0, 1,..., 1}. Here, the time variable t is advanced so that t = 0 denotes the time instant when the first sample of OFDM symbol 0 has reached the receiver. Then τ i denotes the relative delay of path i compared to the first path and τ 1 = 0. Because the channel impulse is assumed constant during one OFDM symbol, the index n can be omitted and h i (nt +T g +lt s ) = h i (l). The sampling stage incorporates the removal of cyclic prefix. The received time domain OFDM symbol block l is given by where r l = [r l,0, r l,1,..., r l, 1 ] T, (70) L r l,n = h i (l)s(nt + T g + lt s τ i ) + w(nt + T g + lt s ) i=1 = L i=1 h i (l) 1 l = k = 2 k c l,k ej2π Tu (nt τ i+(l l )T s) 22

u(nt + T g τ i + (l l )T s ) + w(t n ). (71) The windowing function evaluates to unit only for l = l and the delayed paths of the last samples of the OFDM symbol that would fall outside sampling window are recovered from the delayed parts of the cyclic prefix. Then r l,n = 1 = 1 k = 2 c l,k e j2πk n/ L i=1 h i (l)e j2π τ i T k / + w(t n ) c l,k H l,k e j2πk n/ + w(t n ). (72) k = 2 Then the received OFDM symbol block can be written as r l = F H H l c l + w l, (73) where H is the diagonal channel frequency response matrix and w l is a vector of noise samples. The demodulation of received samples is done by calculating the discrete Fourier transform. For the received symbol samples y l we get y l = Fr l = H l c l + Fw l = H l c l + W l. (74) After demodulation, each sub-carrier experiences frequency-flat channel because the channel matrix is diagonal. For example, zero-forcing or minimum mean squared error equalizer can be used to retrieve the transmitted data symbols [12]. Equalized data samples are then detected and forwarded to encoding to find the transmitted information. 2.4 Symbol timing and frequency offsets The transmitter and receiver do not share a common frequency and timing reference, so a synchronization task has to be performed before demodulation can be carried out. The synchronization process is divided into two parts. First, an acquisition phase is carried out, where the frequency and timing are acquired without any prior knowledge with sufficient accuracy. Then, during the data reception, the frequency and timing are constantly being monitored and corrected. The estimation of the residual carrier frequency offset and symbol timing offset are addressed in this work. After initial synchronization, relatively small errors are present in the data transmission phase which need to be compensated for. 23

2.4.1 Symbol timing offset Symbol timing errors are introduced in the sampling phase of receiver processing. Incorrect information of the time instant a OFDM symbol starts causes first samples of the symbol to be lost and last samples are populated with samples from the next symbol or vice versa. The samples from adjacent OFDM symbol are disturbance to the current symbol and is called intersymbol interference (ISI). Losing samples causes attenuation of the signal as not all of the transmitted energy is recovered and the orthogonality between sub-carriers is lost causing inter-carrier interference (ICI). Cyclic prefix is used to give a buffer to mitigate the effect of too early sampling as received OFDM symbols are only circularly shifted versions of the correct symbol. This applies only in the case when the significant part of multipath channels impulse response is shorter than prefix duration. The next examples from [10] show the effect of symbol timing offset to the received demodulated samples. We denote the symbol timing offset by ε R. Example 2.6. Assume an OFDM transmission where there is no cyclic prefix, g = 0, and perfect frequency synchronisation. Let channel introduce only additive white Gaussian noise. Assume also that the symbol timing offset is not greater than the symbol duration, i.e. ε < s. Because of the timing offset, the sampling of OFDM symbol l is shifted from the optimal position and done at time indices t n = nt + lt u + εt, n {0, 1,..., 1}. Then, the received samples of transmitted signal are r l,n =s(nt + lt u + εt ) + w(t n ) = 1 + w(t n ) = 1 l= k = 2 l= k = 2 k c l,k ej2π Tu (nt +εt +(l l )T u) u(nt + εt + (l l )T u ) c l,k ej2π k n e j2π εk u(nt + εt + (l l )T u ) + w(t n ). Demodulation of the samples with discrete Fourier transform gives y l,k = 1 1 r l,n e j2πkn/ = 1 1 n=0 n=0 l = k = 2 c l,k ej2π k n e j2π εk e j2π kn u(nt + εt + (l l )T u ) 24

+ 1 1 w(t n )e j2πkn/ n=0 = 1 1 ej2πεk/ c l,k + 1 + 1 + w l,k k = 2 k k l = l 0 n=0 u(nt + εt ) e j2πεk / c l,k k = 2 = ε e j2πεk/ c l,k + + k = 2 k k k = 2 + w l,k e jπ(k k) ε 1 e jπ(k k) ε 1 1 n=0 1 e j2πεk / c l,k e j2π(k k)n/ u(nt + εt ) n=0 e j2π(k k)n/ u(nt + εt + (l l )T s ) sin ( ) π(k k) ε / sin(π(k k)/) ej2πεk c l,k sin ( ) π(k k) ε / sin(π(k k)/) ej2πεk c l+sgn(ε),k = ε e j2πεk/ c l,k + wl,k ICI + wl,k ISI + w l,k. So, the received demodulated block can be written as where Φ = diag{e j2πεk/ } 2 1 k= 2 and W ISI l y l = ε Φc l + Wl ICI + Wl ISI + W l, is the diagonal phase shift matrix and Wl ICI are the additional noise terms due to inter-carrier and inter-symbol interference. We see that the timing offset causes attenuation due to loss of signal energy and linear phase rotation. Inter-symbol and inter-carrier interference can be seen as additional noise. Example 2.7. Consider the OFDM system from previous example. If we assign a cyclic prefix of g samples and we assume that the symbol timing offset is g ε 0. Then, there are no samples from adjacent symbols causing inter-symbol interference and due to cyclic prefix the orthogonality 25

of sub-carriers is preserved. The received signal is therefore only affected by the sub-carrier specific rotation. The demodulated samples are given by 2.4.2 Frequency offset y l =Φc l + W l. The frequency synchronization errors are due to carrier frequency offset (CFO) and sampling frequency offset (SFO). Carrier frequency offset is caused by the difference between transmitter and receiver oscillator frequencies and channel induced Doppler effect. When the signal is down-mixed with erroneous carrier frequency, the sub-carrier frequencies and the reference frequencies in discrete Fourier transform do not differ by a multiple of 1 T u. This means that the orthogonality between sub-carriers is lost in the receiver and intercarrier interference is introduced. Error in sampling clock frequency causes the sampling time instant to drift gradually too early or late. If the sampling frequency offset is severe or it is not corrected, the discrete Fourier transform input window will drift away from optimal position and samples from other OFDM symbols will result in inter-symbol interference. The carrier frequency offset in hertz is denoted by f. We normalise it with sub-carrier spacing in frequency and denote ν = ft u. The sampling clock frequency offset is given by f s. Then the receiver sampling time interval is T = 1 f s+ f s = (1 + ξ)t, where T is the transmitter sampling interval and ξ = T T = fs T f s+ f s. Example 2.8 (See [10]). Here we illustrate the effect of carrier and sampling clock frequency offset to the received samples y l. We assume perfect symbol timing, ε = 0. Let the transmitted signal experience additive white Gaussian noise channel. The received passband signal is given by r c (t) = s c (t) + w c (t). The down-mixing is done with erroneous carrier frequency f c = f c + f, where f is the frequency offset in hertz. Then, the filtered and amplified in-phase and quadrature outputs are given by I(t) = Re(s(t)) cos( 2π ft) Im(s(t)) sin( 2π ft) + w (t) Q(t) = Re(s(t)) sin( 2π ft) + Im(s(t)) cos( 2π ft) + w (t). and Combining the in-phase and quadrature components gives the received baseband signal as r(t) = I(t) + jq(t) = e j2π ft s(t) + w(t). 26

Sampling is done at time indices t n = (n + l s )(1 + ξ)t + T g, n {0, 1,..., 1}, so that r((n + l s )(1 + ξ)t + T g ) = r l,n. The demodulated samples are y l,k = 1 1 r l,n e j2πnk/ n=0 = 1 1 e j2π ftn s(t n )e j2πnk/ + 1 1 w(t n )e j2πnk/ = 1 n=0 1 l = k = n=0 2 n=0 e j2π f[(n+ls)(1+ξ)t +Tg] k j2π e Tu (n+ls)(1+ξ)t l T s) c l,k u((n + l s)(1 + ξ)t + T g l T s )e j2πnk/ + w l,k. Here we concentrate on the exponents. If we denote φ k,k = (1+ξ)( ν+k ) k and φ k = φ k,k, we can write the above equation in the following form y l,k = 1 1 l = k = n=0 2 e j2πφ k,k n/ e j2πνg/ e j2πlφ k s/ e j2π(l l )k s/ c l,k u((n + l s )(1 + ξ)t + T g l T s ) + w l,k. The gradual drift of the sampling window causes inter-symbol interference. We assume that the sampling frequency offset ξ is so small that during one OFDM symbol the sampling window moves at most one sample forwards or backwards. This requires ξ s < 1 or ξ < 1 s. If the sampling instances t n [ T g +lt s, (1+l)T s [ there are no samples from the previous or following OFDM symbols and inter-symbol interference doesn t occur. We concentrate only on the received OFDM symbols l for which the sampling window is overlapping with corresponding sent symbol, i.e. the timing was correct for OFDM symbol l = 0 and we assume sufficiently small l in the following inspections. 1. The sampling window is inside optimal interval [ T g + lt s, (1 + l)t s [: y l,k = 1 = k = 2 k = 2 e j2πνg/ e j2πlφ k s/ c l,k 1 n=0 e j2πνg/ e j2πlφ k s/ c l,k e jπφ k,k 1 e j2πφ k,k n/ + w l,k sin(πφ k,k) sin(πφ k,k/) + w l,k = sin(πφ k) sin(πφ k /) e j2πνg/ e jπφ 1 k e j2πlφ k s/ c l,k + wl,k ICI + w l,k. 27

We see that the received symbols c l,k have been phase rotated and attenuated. Additionally, the orthogonality between sub-carriers has been lost and additional noise term wl,k ICI due to inter-carrier interference is introduced. For small φ k the attenuation can be approximated by sinc sin(πφ function as k ) sinc(πφ sin(πφ k /) k) and is close to unit. The phase shift depends on the OFDM symbol index l and sub-carrier index k. 2. Samples from adjacent OFDM symbol fall into the sampling window. Denote the offset of the sampling window outside the optimal interval by n. Demodulated samples are given by y l,k = 1 = k = 2 e j2πνg/ e j2πlφ k s/ c l,k u((n + l s )(1 + ξ)t + T g lt s ) + 1 1 n=0 + w l,k k = 2 + k= 2 k = 2 1 e j2πφ k,k n/ n=0 e j2πνg/ e j2πlφ k s/ e sgn( n)j2πk s/ c l+sgn( n),k e j2πφ k,k n/ u((n + l s )(1 + ξ)t + T g (l + sgn( n))t s ) e jπφ k,k n 1 e j2πνg/ e j2πlφ k s/ c l,k e jπφ k,k n 1 sin ( ) πφ k,k n sin(πφ k,k/) e j2πνg/ e j2πlφ k s/ e sgn( n)j2πk s/ c l+sgn( n),k sin ( ) πφ k,k n sin(πφ k,k/) + w l,k ) = sin ( n πφ k sin(πφ k /) e j2πνg/ e jπφ n 1 k e j2πlφ k s/ c l,k + w ICI l,k + w ISI l,k + w l,k. Compared to the previous case, the attenuation is more severe and additional noise term w ISI l,k due to inter-symbol interference is introduced. 2.5 Received signal model Gathering the effects of symbol timing and frequency offsets, we arrive to the signal model that contains the effects of multipath channel, symbol timing 28

offset, carrier frequency offset and sampling frequency offset [10]. Theorem 2.9. Let symbol timing offset ε be so small that no inter-symbol interference is introduced. Assume that carrier frequency offset ν < 1 and 2 sampling clock frequency offset ξ 1. Let the cyclic prefix duration be longer than the channel impulse response and assume that the channel impulse response is constant during one OFDM symbol. Then, the demodulated received samples are given by y l =Φ(ε, ν, ξ)θ l (ν, ξ)h l c l + W ICI l + W l. (75) The elements of diagonal matrix Φ(ε, ν, ξ) are given by Φ k = sin(πφ k) [ ] sin(πφ k /) e jπ 1 φ k +2φ k,0 ε g 2ν, where φ k = (1 + ξ)( ν + k) k and k =, + 2 2 1,..., 1. This causes attenuation and phase shift depending on the timing 2 ε and frequency offsets ν and ξ. The elements of the diagonal matrix Θ l (ν, ξ) are given by Θ l,k = e j2πlφ k s, k {, +1,..., 1}, causing frequency 2 2 2 offset ν and ξ and OFDM symbol index l dependent phase shift. The channel matrix H l is diagonal and contains the frequency response samples of the channel. The term Wl ICI is noise due to inter-carrier interference, given by w ICI l,k = k = 2 ;k k [ sin(πφ k,k) sin(πφ k,k/) ejπ φ k,k 1 +2φ k,0 ε ] g 2ν e j2πlφ k s cl,k H l,k. (76) Term W l is additive zero mean white Gaussian noise modelled as w l,k = 1 1 w((n + ε + l s )(1 + ξ)t + T g )e j2πkn/, (77) n=0 with E{ w l,k 2 } = E{ w(t) 2 } = σ 2 w. Proof. Let s(t) denote the continuous modulated signal that is transmitted. The signal is modelled as s(t) = 1 l= k= 2 where u(t) is a windowing function given by 1, 0 < t < T s, u(t) = 0, else. c l,k e j2π k Tu (t Tg lts) u(t lt s ), 29

After passing through the channel, the signal is affected by channel impulse response h(τ, t) = L i=1 h i (t)δ(τ i τ) and additional noise w(t) is introduced. Mismatch between transmitter and receiver carrier frequencies and channel induced Doppler effect cause carrier frequency offset f, which is seen as a phase rotation of the signal. The received signal is given as r(t) = e j2π ft L i=1 h i (t)s(t τ i ) + w(t). The received signal is sampled at time indices t n = (n + ε + l s )(1 + ξ)t + T g, n {0, 1,..., 1}. The received samples are given by r l,n =r(t n ) = e j2π ftn =e j2π ftn L i=1 L i=1 h i (l) 1 h i (t n )s(t n τ i ) + w(t n ) u(t n τ i l T s ) + w(t n ) = 1 L i=1 l = k = 2 l = k = 2 k c l,k ej2π Tu (tn τ i T g l T s) [ ] c l,k ej2π ft n+ k Tu (tn Tg l T s) h i (l)e j2π τ i T k / u(t n τ i l T s ) + w(t n ). The term in the exponent can be written in equivalent form as ( ) ft n + k tn T g l s = 1 [ ν(n + ε + l s)(1 + ξ) ν g + k ((n + ε + l s )(1 + ξ) l s )] = 1 [(1 + ξ)( ν + k )n + l(1 + ξ)( ν + k ) s l k s + (1 + ξ)( ν + k )ε ν g ] n + lφ s k + k (l l ) s + φ ε k,0 ν g, =φ k,0 where we denote ν = ft, φ k,k = (1 + ξ)( ν + k ) k and φ k = φ k,k. The received samples are demodulated by passing through a discrete Fourier transform and we get y l,k = 1 1 r l,n e j2πkn/ n=0 30

= 1 2 1 1 c l,k ej2π[φ k,0 n/+lφ k s/+k (l l ) s/+φ k,0 ε/ ν g/] l = k = n=0 2 L i=1 h i (l)e 2π τ i T k / u(t n τ i l T s )e j2πkn/ + 1 1 w(t n )e j2πkn/ = 1 = 1 n=0 2 1 1 c l,k ej2π[φ k,k n/+lφ k s/+k (l l ) s/+φ k,0 ε/ ν g/] l = k = n=0 2 L i=1 + w l,k h i (l)e j2π τ i T k / u((n + ε)(1 + ξ)t + T g + (l l )T s + lξt s τ i ) 2 1 c l,k ej2π[lφ k Ss/+k (l l ) s/+φ k,0 ε/ ν g/] l = k = 2 1 e j2πφ k,k n/ L n=0 i=1 h i (l)e j2π τ i T k / u((n + ε)(1 + ξ)t + T g + (l l )T s + lξt s τ i ) + w l,k. With the assumption that symbol timing and sampling clock frequency offsets are adequately small and that the cyclic prefix is longer than the channel impulse response, the windowing function u(t) will evaluate unit only when l = l. The equation simplifies to y l,k = 1 c l,k e j2π[lφ k s/+φ k,0 ε/ νg/] k = 2 1 e j2πφ k,k n/ L n=0 i=1 + w l,k. h i (l)e j2π τ i T k / u((n + ε)(1 + ξ)t + T g + lξt s τ i ) Because the cyclic prefix is an exact copy of the last part of the OFDM symbol the delayed samples of the cyclic prefix compensate for the loss of delayed samples of the last part. Then y l,k = 1 k = 2 c l,k e j2π[lφ k s/+φ k,0 ε/ νg/] 31