What is a mixer?

A mixer is a three-port circuit that employs a non-linear or time-varying device in order to perform the critical frequency translation function in wireless communication systems. The non-linear or time-varying parameter can be either a conductance/resistance or a transconductance. If the time-varying element is a resistance or conductance, the mixer is called resistive. Mixers that rely on a time-varying transconductance are known as active mixers.

When used in a transmitter, the mixer acts as an upconverter by shifting the data signal from a low frequency to the carrier frequency, making it suitable for transmission by the antenna. In the receiver, it serves as a downconverter by separating the data signal from the carrier and shifting it to a low frequency, where it can be demodulated and processed in a cost-effective manner. Ideally, in both cases, the signal at the output is a replica of the signal at one of the mixer inputs, translated to a lower or higher frequency, with no loss of information and no added distortion.

Most IC mixers are implemented with switches. In addition, image-reject mixers also require 90 degree phase shifters and in-phase power combiners or splitters. Finally, mixers can be employed to realize digital modulators. The final part of this chapter will review switches, phase shifters, and M-ary phase and QAM modulators based on phase shifters and Gilbert cell mixer topologies.

Introduction

The second half of the twentieth century experienced an explosive growth in information technology, including data transmission, processing, and computation. This trend will continue at an even faster pace in the twenty-first century. Radios and televisions started in the 1920s and 1940s respectively, and involved transmission from a single transmitter to multiple receivers using AM and FM modulations. Baseband analog telephony, starting in the 1900s, was originally suited only for local area person-to-person communication. It became possible to have long-distance communication after using cascades of regeneration repeaters based on digital PCM modulation. Various digital modulations with and without coding, across microwave, satellite, and optical fiber links, allowed the explosive transmissions of data around the world starting in the 1950s–1960s. The emergence of Ethernet, local area net, and, finally, the World Wide Web in the 1980s–1990s allowed almost unlimited communication from any computer to another computer. In the first decade of the twenty-first century, by using wireless communication technology, we have achieved cellular telephony and instant/personal data services for humans, and ubiquitous data collection and transmission using ad hoc and sensor networks. By using cable, optical fibers, and direct satellite communications, real-time on-demand wideband data services in offices and homes are feasible.

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.

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.

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.

Introduction

In this chapter, we present on-chip filtering techniques based on active blocker cancellation that can potentially allow the removal of SAW filters in receivers. Active blocker cancellation can be in two major forms: feedforward blocker cancellation and feedback blocker cancellation. Both feedforward and feedback blocker cancellations can generate on-chip high-Q bandpass filters with the center frequency controlled precisely by the clock. Receivers in [15, 16] are two examples of feedforward blocker cancellation. The feedback blocker cancellation is used in [17, 18]. Although feedforward blocker cancellation is simpler architecturally than feedback blocker cancellation and has no stability issue, the gain and phase of the feedforward path must be well matched to those of the main receiver path. On the other hand, feedback blocker cancellation eliminates the tight gain and phase control requirements of feedforward blocker cancellation but introduces stability concerns. Both feedforward and feedback active blocker cancellations use the frequency translation technique to construct a high-Q bandpass filter using two identical low-Q baseband filters. To do so, a complex downconversion mixer clocked by the corresponding LO clocks of the zero-IF receiver, frequency shifts the incoming signal to the IF. The desired signal, which is centered around the LO, is downconverted to around DC, whereas the strong downconverted blocker sits at an IF that is equal to the separation between the blocker and the desired signal. The complex IF signal is passed through the two baseband bandpass filters, in which the strong blocker signal is located in the passband.

Introduction

In the previous chapter, we saw that a simple passive mixer driven by 50% duty-cycle clocks converts a low-Q baseband impedance to a high-Q bandpass impedance through frequency translation. The center of this high-Q bandpass impedance is controlled precisely by the clock frequency, making it very attractive for reconfigurable receivers in which it is desirable to have high-Q bandpass filters with centers that can be tuned over a wide range of frequencies. Being implemented with just switches and capacitors, the resulting high-Q filter is exceptionally linear, and because the switches carry no DC, there is no major flicker noise issue. We saw, however, that this filter has the problem of image folding and cannot be useful in its current format. In this chapter, this filter evolves to a new high-Q bandpass filter that no longer folds the image. The resulting filter is still controlled by the clock frequency and is still composed of only switches and capacitors [23, 53, 54, 57–62]. The only complication of this evolved filter is the need for four arms of switches in series with baseband impedances plus a more complicated clocking scheme. The four-phase high-Q BPF requires four nonoverlapped 25% duty-cycle clocks that are progressively phase-shifted by 90°.

The four-phase filter offers more flexibility in the choice of four baseband impedances. For example, if all four baseband impedances are replaced with a single complex baseband impedance, the switching system would frequency-shift the complex baseband impedance to the LO frequency, resulting in a high-Q bandpass filter having a center that is offset from the LO clock by an amount that is dictated by the complex filter.

Introduction

M-phase filters offer high-Q filtering by frequency translation of low-Q baseband impedances to the clock frequency. These filters are placed typically at high-impedance RF nodes, and the quality factor of the resulting bandpass filtering at the RF is proportional approximately to this impedance. Low-impedance RF nodes cannot benefit from these M-phase filters for the following two reasons: (a) for a given quality factor, the required baseband capacitors (and, therefore, the filter size) is inversely proportional to this impedance; (b) the attenuation of the filter at far-out frequencies is equal to the ratio of this impedance to the switch resistance. For low-impedance nodes, the far-out attenuation becomes insignificant.

The well-known duality theorem in circuit theory, however, suggests that the low and high impedances of any given electrical network are mapped to high and low impedances, respectively. This phenomenon suggests that because the M-phase filter is beneficial for high-impedance nodes, its dual must be useful for low-impedance nodes. Therefore, in this chapter, we visit briefly the widely recognized concept of duality in electrical networks. Following that, we will describe the dual of the conventional M-phase filter and its variations and will explain how, for low-impedance nodes, these new filters can offer high-Q filtering with centers controlled by clocks.

Introduction

With the ever-increasing demand for instant access to data over wideband communication channels, the quest for a universal mobile terminal capable of delivering the ultimate user experience has become imperative. Over the last decade, researchers were exploring the possibility of having a universal radio that can be programmed and reconfigured through software to operate on any bands, channel bandwidths, and modulations. Such a universal radio was named software-defined radio (SDR) [1–6]. The SDRs face unique challenges because their targeted applications are mostly in mobile handheld devices. They must be small and affordable, and must last longer between charges. The design of such a low-cost, low-power, and flexible radio that meets the tough requirements of individual standards is enormously challenging and was and still is a hot topic of research for circuit designers as well as system and hardware engineers. One common yet relatively simple example of an SDR is a 3G cell phone, which can support as many as 17 bands in three modes of operation, namely GSM, EDGE, and WCDMA/HSPA.

The most aggressive SDR architecture was proposed by Mitola in 1995 [1], and is shown in Fig. 1.1(a). The only analog blocks in the receiver and the transmitter are an ADC and a DAC, respectively. Such a transceiver provides maximum flexibility through the digital signal processor (DSP), and it is even capable of simultaneously detecting several standards.