MMSE-Based Algorithm for Joint Signal Detection, Channel and Noise Variance Estimation for OFDM Systems (eBook)

Vincent Savaux, Yves Lou?t (Autoren)

eBook Download: EPUB

2014 | 1. Auflage
100 Seiten
Wiley-Iste (Verlag)
978-1-119-00790-6 (ISBN)

Infos

This book presents an algorithm for the detection of an orthogonal frequency division multiplexing (OFDM) signal in a cognitive radio context by means of a joint and iterative channel and noise estimation technique. Based on the minimum mean square criterion, it performs an accurate detection of a user in a frequency band, by achieving a quasi-optimal channel and noise variance estimation if the signal is present, and by estimating the noise level in the band if the signal is absent.

Organized into three chapters, the first chapter provides the background against which the system model is presented, as well as some basics concerning the channel statistics and the transmission of an OFDM signal over a multipath channel. In Chapter 2, the proposed iterative algorithm for the noise variance and the channel estimation is detailed, and in Chapter 3, an application of the algorithm for the free-band detection is proposed. In both Chapters 2 and 3, the principle of the algorithm is presented in a simple way, and more elaborate developments are also provided. The different assumptions and assertions in the developments and the performance of the proposed method are validated through simulations, and compared to methods of the scientific literature.

Autor

Vincent Savaux is a post doctorate at Supélec in Rennes, France. He has worked in the SCEE (Signal, Communications and Embedded Electronics) team since the beginning of 2014. Yves Louët is Professor at Supélec (SCEE team) in Rennes, France. His research activities concern the physical layer of multicarrier communication systems applied to intelligent and green radio.

Inhalt

Introduction ix

Chapter 1. Background and System Model 1

1.1. Channel model 1

1.1.1. The multipath channel 1

1.1.2. Statistics of the channel 2

1.2. Transmission of an OFDM signal 7

1.2.1. Continuous representation 7

1.2.2. Discrete representation 9

1.2.3. Discrete representation under synchronization mismatch 12

1.3. Pilot symbol aided channel and noise estimation 12

1.3.1. The pilot arrangements 12

1.3.2. Channel estimation 15

1.3.3. Noise variance estimation 19

1.4. Work motivations 22

Chapter 2. Joint Channel and Noise Variance Estimation in the Presence of the OFDM Signal 25

2.1. Presentation of the algorithm in an ideal approach 25

2.1.1. Channel covariance matrix 25

2.1.2. MMSE noise variance estimation 27

2.1.3. Proposed algorithm: ideal approach 27

2.1.4. Simulation results: ideal approach 41

2.2. Algorithm in a practical approach 48

2.2.1. Proposed algorithm: realistic approach 48

2.2.2. Convergence of the algorithm 51

2.2.3. Simulations results: realistic approach 60

2.3. Summary 65

Chapter 3. Application of the Algorithm as a Detector For Cognitive Radio Systems 67

3.1. Spectrum sensing 67

3.1.1. Non-cooperative methods 69

3.1.2. Cooperative methods 71

3.2. Proposed detector 73

3.2.1. Detection hypothesis 73

3.2.2. Convergence of the MMSE-based algorithm under the hypothesis H0 74

3.2.3. Decision rule for the proposed detector 79

3.3. Analytical expressions of the detection and false alarm probabilities 82

3.3.1. Probability density function of M under H1 82

3.3.2. Probability density function of M under H0 85

3.3.3. Analytical expressions of Pd and Pfa 86

3.4. Simulations results 88

3.4.1. Choice of the threshold sigmaf 88

3.4.2. Effect of the choice of esigma on the detector performance 89

3.4.3. Detector performance under non-WSS channel model and synchronization mismatch 92

3.4.4. Receiver operating characteristic of the detector 94

3.5. Summary 98

Conclusion 99

Appendices 101

Bibliography 109

Index 119

Textauszug

1

Background and System Model

In this first chapter, some basics regarding the propagation channel and the wireless transmission of an orthogonal frequency division multiplexing (OFDM) signal are recalled. Moreover, a brief state of the art of the pilot aided channel estimation methods is provided. Although the latter cannot be exhaustive, it covers some relevant techniques, in particular in an OFDM context.

1.1. Channel model

1.1.1. The multipath channel

The transmission channel (or propagation channel) is the environment situated between the transmitting and the receiving antennas. Whether an indoor or outdoor environment is considered, the signal transmitted over the channel suffers from some perturbations of different kinds: reflection, diffraction or diffusion. These phenomena are due to obstacles in the propagation environment, like buildings or walls. Besides, the transmitter, the receiver or both of them may be in motion, which induces Doppler effect.

In certain contexts, the transmitter and the receiver are in line of sight (LOS), so the channel is not destructive for the signal. On the contrary, in non-line of sight (NLOS) transmissions, the signal goes through several paths before reaching the receiving antenna. In that case, the propagation environment is called a multipath channel, and is mathematically written as a sum of weighted delayed Dirac impulses δ(τ):

[1.1]

where the channel impulse response (CIR) h(t, τ) depends on the number of paths L, the complex gains hl and the delays τl. In this work, we will instead take an interest in NLOS transmissions. The channel frequency response (CFR) is obtained from [1.1] by means of the Fourier transform (FT) operation denoted by FT:

[1.2]

where the subscript in FT(.) denote the variable on which the Fourier transform is processed. Figure 1.1 illustrates this relationship ((a): h(t, τ), and (b): H(t, f)). We can observe that the FT is made on the delay τ, which makes the frequency response H(t, f) a time-varying function. When the channel does not vary, it is called static, and when the variations are very slow, the channel is called quasi-static. In this book, we will assume the latter scenario.

1.1.2. Statistics of the channel

1.1.2.1. Rayleigh channel

As numerous natural phenomena, the transmission channel is subject to random variations. Therefore, the instantaneous CIR [1.1] and CFR [1.2] are not sufficient to completely describe the channel. It becomes relevant to use the statistical characterization of the CIR and the CFR to study this random process.

Figure 1.1. Illustration of a time-varying impulse response h(t, τ) and a frequency response H(t, f) of a multipath channel. For a color version of the figure, see www.iste.co.uk/savaux/mmse.zip

In an NLOS transmission, due to the channel, the signal comes from all possible directions at the receiving antenna that is assumed to be isotropic. Thus, each delayed version of the received signal is considered as an infinite sum of random components. By applying the central limit theorem, h(t) is then a zero-mean Gaussian complex process whose gain |h(t)| follows a Rayleigh distribution [PAT 99] pr, Ray(r) of variance :

[1.3]

where r is a positive real value. The probability density function (PDF) of the phase of a Rayleigh process follows a uniform distribution, noted pϕ, Ray (θ):

[1.4]

The Rayleigh channel model is very frequently used, particularly in theoretical studies, since it is relatively close to reality, and the literature on Rayleigh distribution is very extensive. For these reasons, Rayleigh channels are considered all along this work. However, it does not cover all the possible scenarios: in a LOS context, the direct path adds a constant component to the previous model. In that case, |h(t)| follows a Rice distribution, which is described in [RIC 48]. More recently, the Weibull model [WEI 51] has been proposed in order to describe real channel measurements with more accuracy. Nakagami model [NAK 60], later generalized in [YAC 00] by the κ − μ distribution, is also a global model from which Rayleigh’s and Rice’s are particular cases.

1.1.2.2. WSSUS model

The channel being a time-frequency varying random process, it is relevant to characterize it through its first and second-order statistic moments. According to Bello’s work [BEL 63], let us assume a wide sense stationary uncorrelated scattering (WSSUS) model, defined as follows:

– WSS: each path hl(t) in [1.1] is a zero mean Gaussian complex process, i.e. E{hl(t)} = 0, ∀t, with E{.} the statistical expectation. Consequently, the mean of each path is independent from the time variations. Furthermore, the time correlation function can be only written with the difference Δt = t1 − t2, i.e.

[1.5]

Each path hl(t) of the channel is then wide sense stationary.

– US: the paths are uncorrelated, so for l1 ≠ l2, we have

[1.6]

This model is used in the following to apply the proposed detection and channel estimation algorithm. However, it does not necessarily match the reality, so we will also study the performance of the proposed method under channel model mismatch, particularly in Chapter 3.

Let us also define two very useful statistical functions that characterize the channel along the delay and the frequency axes:

– The intensity profile Γ(τ). A commonly used model is the decreasing exponential [EDF 98, STE 99, FOE 01].

– The frequency correlation function of the channel RH (Δf), whose expression will be detailed later.

These two functions are linked by Fourier transform:

[1.7]

Figure 1.2 depicts the decreasing intensity profile and the real and imaginary parts of the frequency correlation function.

Figure 1.2. Link between the channel intensity profile and the frequency correlation function

1.2. Transmission of an OFDM signal

When combined with a channel coding, the transmission of data using a frequency multiplexing is very robust against the frequency selective channels, in comparison with single-carrier modulations [SCO 99, DEB]. The use of orthogonal subcarriers has been proposed since the 1950s, in particular for military applications, but the acronym OFDM appeared in the 1980s, when the evolution of the technology of semiconductors enabled a great development of the implementation of complex algorithms, especially the algorithms based on large size FFT/IFFT. This kind of modulation is now used in a large number of wired and wireless transmission standards.

1.2.1. Continuous representation

In the continuous formalism, the baseband OFDM signal is written as:

[1.8]

where sn(t) is the nth OFDM symbol, Π(t) is the rectangular function of duration Ts as

[1.9]

where is the subcarrier spacing, M is the number of subcarriers such as, if we denote by B the bandwidth, we have Fs = B/M. The scalar Cm,n with m = 0, 1, …, M − 1 are the information symbols coming from a set Ω of a given constellation, such as the binary phase shift keying (BPSK) or the four-quadrature amplitude modulation (4-QAM). The different subcarriers of the OFDM symbols are orthogonally arranged, thus, no interference occurs in the frequency domain (see Figure 1.3). The received signal u(t) is the convolution of s(t) and h(t), plus the white Gaussian noise denoted by w(t). In the frequency domain, due to the Fourier transform property, the convolution becomes a simple product:

[1.10]

[1.11]

So as to cancel the intersymbol interferences (ISIs) due to the delayed paths of the channel, the solution consists of adding a guard interval (GI) at the head of each OFDM symbol. If the GI length is greater than the maximum delay of the channel, it contains all the interferences from the previous symbol, and the GI removal cancels the ISI. In the following, let us assume that the GI is a cyclic prefix, i.e. the end of each OFDM symbol is copied at its head. As noted later, in addition to the ISI cancellation, the use of a CP gives a cyclic structure to the OFDM symbols. Let us denote by TCP the duration of the CP.

Figure 1.3 shows the effects of the channel on the OFDM signal in the time and the frequency domains. Figure 1.3(a) illustrates, in the time domain, the ISI cancellation due to the CP removal. The frequency orthogonality is displayed in Figure 1.3(b). The...

Produkt-Details

• Erscheint lt. Verlag: 25.9.2014
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Informatik ► Programmiersprachen / -werkzeuge
• Themenwelt: Informatik ► Theorie / Studium ► Algorithmen
• Themenwelt: Mathematik / Informatik ► Mathematik
• Themenwelt: Technik ► Elektrotechnik / Energietechnik
• Themenwelt: Technik ► Nachrichtentechnik
• Schlagworte: Electrical & Electronics Engineering • Elektrotechnik u. Elektronik • Numerical Methods & Algorithms • Numerische Methoden u. Algorithmen
• ISBN-10: 1-119-00790-9 / 1119007909
• ISBN-13: 978-1-119-00790-6 / 9781119007906