This publication was conceived as a textbook for a first-year graduate course in the Signals and Systems Area of the Electrical Engineering Department at UCLA to introduce basic statistical concepts of detection and estimation and their applications to engineering problems to students in communication, telecommunication, control, and signal processing. Students majoring in electromagnetics and antenna design often take this course as well. It is not the intention of this book to cover as many topics as possible, but to treat each topic with enough detail so a motivated student can duplicate independently some of the thinking processes of the originators of these concepts. Whenever possible, examples with some numerical values are provided to help the reader understand the theories and concepts. For most engineering students, overly formal and rigorous mathematical methods are probably neither appreciated nor desirable. However, in recent years, more advanced analytical tools have proved useful even in practical applications. For example, tools involving eigenvalue–eigenvector expansions for colored noise communication and radar detection; non-convex optimization methods for signal classification; non-quadratic estimation criteria for robust estimation; non-Gaussian statistics for fading channel modeling; and compressive sensing methodology for signal representation, are all introduced in the book.

In general, the use of multiple antennas provides three types of gains: diversity gain, power/SINR gain (including interference suppression), and multiplexing (bandwidth or degree of freedom) gain. In Chapter 4, we studied the first two and learned that

1. • Antenna diversity changes the channel statistics and reduces the probability of the channel being in a deep fade, thereby improving the link reliability. Receive diversity is relatively easy to obtain. With space-time coding, transmit diversity can be achieved without the transmitter knowing the channel.

2. • Beamforming achieves power/SINR gain by increasing signal power and/or reducing interference. Maximal ratio combining is commonly used for receive beamforming and is applicable for transmit beamforming if the channel is known at the transmitter. Without precise channel knowledge, transmit beamforming is not possible in a single user link, but can be opportunistically achieved in a multiuser system with multiple receivers and limited SINR feedback. Power/SINR gain is mostly beneficial in the low SINR regime.

In general, the three types of gains can be simultaneously obtained, but there is a tradeoff of how much of each type a given communication scheme can achieve. A well-known example is the fundamental tradeoff of diversity and multiplexing studied in [186]. For the sake of simplicity, we focus on multiplexing gain in this chapter, and do not address the tradeoff.

Hypothesis testing is a concept originated in statistics by Fisher [1] and Neyman–Pearson [2] and forms the basis of detection of signals in noises in communication and radar systems.

Simple hypothesis testing

Suppose we measure the outcome of a real-valued r.v. X. This r.v. can come from two pdf's associated with the hypotheses, H0 or H1. Under H0, the conditional probability of X is denoted by p0(x) = p(x∣H0), −∞ < x < ∞, and under H1, the conditional probability of X is denoted by p1(x) = p(x|H1), − ∞ < x < ∞. This hypothesis is called “simple” if the two conditional pdf's are fully known (i.e., there are no unknown parameters in these two functions). From the observed x value (which is a realization of the r.v. X), we want to find a strategy to decide on H0 or H1 in some optimum statistical manner.

Example 3.1 The binary hypothesis problem in deciding between H0 or H1 is ideally suited to model the radar problem in which the hypothesis H0 is associated with the absence of a target and the hypothesis H1 is associated with the presence of a target. In a binary hypothesis problem, there are four possible states, whether H0 or H1 is true and whether the decision is to declare H0 or to declare H1. Table 3.1 summarizes these four states and the associated names and probabilities.

In this chapter we consider various analytical and simulation tools for system performance analysis of communication and radar receiver problems. In Section 8.1, we treat the analysis of receiver performance with Gaussian noise, first using the closure property of Gaussian vectors under linear operations. We then address this issue without using this closure property. Section 8.2 deals with the analysis of receiver performance with Gaussian noise and other random interferences caused by intersymbol interferences (ISI) due to bandlimitation of the transmission channel. Section 8.2.1 introduces the evaluation of the average probability of error based on the moment bounding method. Section 8.3 considers the analysis of receiver performance with non-Gaussian noises including the spherically invariant random processes (SIRP). By exploiting some basic properties of SIRP, Section 8.3.1 obtains a closed form expression for the receiver. We determine the average probability of error for the binary detection problem with additive multivariate t-distributed noise (which is a member of SIRP). Section 8.3.2 again uses some properties of SIRP to model wireless fading channels with various fading envelope statistics. By using Fox H-function representations of these pdfs, novel average probability of error expressions under fading conditions can be obtained. Section 8.3.3 treats the probabilities of a false alarm and detection of a radar problem with a robustness constraint. Section 8.4 first shows a generic practical communication/radar system, which may have various complex operations, making analytical evaluation of system performance in many cases difficult.

The central design idea of mobile broadband is to adapt wireless to the Internet, not vice-versa. Compared with its wireline counterpart, mobile broadband faces two major technical challenges: fading and interference, which make the wireless link less reliable, and mobility, which requires handoff from one cell to another as a user moves. The previous chapters describe the physical and MAC layer approaches of dealing with fading and interference and improving link reliability and system capacity. In this chapter, we will expand our scope to view the airlink as part of an end-to-end network system and address the handoff issue.

Network architecture describes the necessary functions of the network system, partitions them to a set of logical nodes, and defines the interfaces between the nodes. An end-to-end network system is usually quite complex. To handle the complexity in a scalable manner, a good design practice is to adopt a layered structure. For example, the famous open system interconnection (OSI) model defines a networking framework of implementing protocols in seven layers, namely the application, presentation, session, transport, networking, data link, and physical layers. When two nodes communicate with each other, control is passed down from a higher layer to a lower one in one node, all the way to the bottom physical layer, then over the physical channel to the other node, and finally moving up the hierarchy in that node. The TCP/IP model of the Internet simplifies the layering model to four layers, namely the application, transport, Internet, and network access layers.

Strictly speaking, OFDMA is merely a multiple access principle of sharing spectrum. To build a full-fledged mobile broadband communication system, one needs to design a whole set of system operations. We highlight a few important elements of the required system operations in Figure A.1 and provide a high level overview in this section.

Cell search, synchronization, and identification

After power-up, the user first detects the existence of a base station in the area by searching for some signal signature in the downlink. Below are some examples of commonly used signal signature:

1. • Correlation between cyclic prefix and the last portion of the OFDM symbol (see Section B.2 for cyclic prefix correlation).

2. • Synchronization channel, which consists of pre-defined signal waveforms with special properties, such as autocorrelation function close to a delta function (see Section 9.5 for an example of synchronization channel).

3. • Common pilot channel, which consists of a subset of high power tone-symbols distributed according to pre-defined patterns (see Section B.3 for an example of pilot channel).

4. • Beacon channel, which consists of a special OFDM symbol in which most of signal power is concentrated on one tone-symbol (see Section 9.5 for the use of beacon channel).

Once the user has discovered a base station, it synchronizes its carrier frequency, clock, and symbol time with the received downlink signal and then retrieves from the broadcast channel certain system information, such as system time, base station identifier, operator identifier.

In general scheduling addresses two problems:

1. • User selection: which users to transmit in the uplink or receive in the downlink.

2. • Resource allocation: what time-frequency bandwidth to be allocated to the selected users and what transmit power to be used.

Good scheduling strives to achieve two goals, namely quality of service (QoS) on the user level, measured by data rate, delay, loss, and fairness among users, and efficiency on the system level, measured by the total amount of traffic supported by the system.

A key feature in an OFDMA mobile broadband cellular system is scheduling by which a base station dynamically selects users and allocates time-frequency-power resource to them. In contrast, when a user is admitted to the system in a circuit-switched voice system, it is statically assigned a piece of bandwidth resource (time, frequency, or code) over which the voice traffic is transported without explicit dynamic scheduling. In a CDMA voice system, for example, the only dynamic resource allocation job is power allocation or power control [168]. The situation is quite different in mobile broadband, because data traffic is bursty and has different QoS requirements. Static resource allocation cannot simultaneously meet the QoS requirements and achieve high system efficiency. Scheduling becomes a necessity to dynamically match user selection and resource allocation with traffic needs and wireless channel conditions. On the other hand, scheduling has been well studied in wireline broadband networks (see [13, 137]), and many of those design and analysis ideas are applicable to the wireless counterpart.

In Chapter 4, we considered the detection of known binary deterministic signals in Gaussian noises. In this chapter, we consider the detection and classification of M-ary deterministic signals. In Section 5.1, we introduce the problem of detecting M given signal waveforms in AWGN. Section 5.2 introduces the Gram–Schmidt orthonormalization method to obtain a set of N orthonormal signal vectors or waveforms from a set of N linearly independent signal vectors or waveforms. These orthonormal vectors or signal waveforms are used as a basis for representing M-ary signal vectors or waveforms in their detection. Section 5.3 treats the detection of M-ary given signals in AWGN. Optimum decisions under the Bayes criterion, the minimum probability of error criterion, the maximum a posteriori criterion, and the minimum distance decision rule are considered. Simple minimum distance signal vector geometry concepts are used to evaluate symbol error probabilities of various commonly encountered M-ary modulations including binary frequency-shifted-keying (BFSK), binary phase-shifted-keying (BPSK), quadra phase-shifted-keying (QPSK), and quadra-amplitude-modulation (QAM) communication systems. Section 5.4 considers optimum signal design for M-ary systems. Section 5.5 introduces linearly and non-linearly separable and support vector machine (SVM) concepts used in classification of M deterministic pattern vectors. A brief conclusion is given in Section 5.6. Some general comments are given in Section 5.7. References and homework problems are given at the end of this chapter.

So far we have studied the system design principles of OFDMA-based mobile broadband under a conventional cellular network framework. The basic premises of the framework are:

1. • The base stations use high transmit power, and are placed at carefully chosen locations, ideally at the vertices of regular hexagons.

2. • A user is connected to the “best” base station. The best base station is usually the closest one that has the greatest downlink signal strength received at the user.

3. • A base station is open to all the users within a cell by providing “unrestricted” access service.

4. • Both the downlink and uplink communications are one-hop between the base station and the users. The users do not communicate directly even if they are nearby to each other.

5. • The time-frequency resource is reused spatially. Among cells reusing the same resource, a signal transmitted in one cell is treated as interference/noise in another cell.

6. • The spectrum to be used in a cell is fixed and known to both the base station and the users.

In this chapter, we explore several ideas that go beyond the conventional cellular framework in pursuit of the next performance leap.

The first such idea is heterogenous network topology. Inwireless, moving the transmitter and receiver close to each other increases signal strength, reduces required transmit power and thus interference to other transmissions, and allows dense spectrum reuse.

Communications systems are designed to send information from one point to another in the face of corrupting noise, signal loss and other degrading effects. Because these effects are statistical in nature, the field of signal detection and estimation was created to provide an analytical means to quantify link performance, establishing the quality of the information transfer. Although communications moved from point-to-point data links to networking among many users in the last decade, link analysis is still important to setting link performance even though networking can help overcome link performance limitations.

The most commonly used parameter for link analysis is the signal-to-noise ratio (SNR), which is normally defined as the ratio of the average signal power to the average noise power at some point in the receiver chain. Although to first order, optical and radio frequency (RF) communications systems operate essentially in the same way, their detection processes are different [1–3].

In an RF receiver, the first detector senses the signal and noise field strengths. The noise input to this detector is usually caused by thermal noise from the antenna and the associated field preamplifier.