1. Introduction 1.1. Background and Motivation "Every day we are moving closer to having almost as many mobile-cellular subscriptions as people on earth" [1]. This statement of the director of the ITU Telecommunication Development Bureau in the ICT report for the year 2013 reflects the growing importance of wireless communication systems in modern societies. The annual report of the ITU recorded a number of 6.4 Billion mobile cellular subscriptions worldwide. The mobile-broadband market shows an average annual growth rate of 40% reaching a number of 2.1 Billion subscriptions worldwide [1]. The rapid development of new wireless communication technologies that support higher data transfer rates such as HSPA, LTE, LTE-Advanced, etc. bring new challenges to the wireless communication devices. The demand of multi-functional smart phones and integration of several communication technologies in various systems for different applications increase the need for adaptive, efficient, low-power and low-cost transceivers [2]. Frequency conversion is an essential operation in modern transceivers. The adoption of frequency conversion in communication systems dates back to the „heterodyne receiver” [3]. In contrary to the „tuned radio-frequency receiver (TRF)”, where for the detection of signals at different carrier frequencies several tuned filter stages are needed [4], a single tuned filter stage in the heterodyne receiver is used. This property of the heterodyne receiver is realized by converting the frequency of the relevant signal down to a fixed frequency (downconversion). Since the so-called „intermediate frequency (IF)”, to which the received signal is converted, is constant, a single tuned filter circuitry is sufficient for the detection of signals at different carrier frequencies. The frequency conversion is realized using a so-called „radio frequency (RF) mixer” [5, 6]. The function of the RF mixer is to convert the frequency of the received signal with an auxiliary sinusoidal signal generated by a local oscillator circuit within the receiver. The frequency of this local oscillator signal (LO signal) is tuned according to the received signal to be equal to the difference between the carrier frequency and the intermediate frequency. Shown in Figure 1.1 is the RF Front-end of a receiver consisting of a „Low noise amplifier (LNA)” for a low noise filtration and amplification, followed by an RF mixer to perform a frequency conversion of the amplified signal with the help of the LO signal. The signal is then filtered, amplified and forwarded to the demodulator [5, 6]. The increasing demand for the digitalization of the analog front-end and the rise of „Software Defined Radio (SDR)” in the last decade [7] gave the direct-conversion receiver a greater importance than ever before. The direct-conversion (known also as
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1. Introduction
Figure 1.1.: RF Front-end of a receiver homodyne) receiver directly converts the received signal into the baseband. It hereby relaxes the requirements set to the amplifier and the filter circuitry of the receiver. Direct-conversion receivers enable therefore a higher level of monolithic integration and exhibit a lower power consumption compared to heterodyne receivers. Furthermore, the classical problem of image-rejection in heterodyne receivers is extenuated [8, 9]. However, the direct-conversion receiver entails some challenges that do not exist in a heterodyne receiver. The receiver component that is mostly affected by these challenges is the RF mixer, since the requirements set to the mixer change according to the type of frequency conversion used. The design of direct-conversion mixers must consider several design problems including DC offset, third-order and even-order distortion, as well as flicker noise and LO leakage [9, 10].
1.2. Problem Formulation The electrical/electronic implementation of frequency conversion exhibits several undesired phenomena, that degrade the quality of the performed operation. These ”Realization non-idealities” are caused mainly by the nonlinearity ”deterministic process” and noise ”stochastic process” of the used devices. While noise may spread over the whole frequency spectrum [11], undesired nonlinearity leads to spurious spectral components, that lie on different frequencies in the output spectrum [12] (see Figure 1.2). The term ”undesired” was chosen explicitly to differentiate from the ”desired” nonlinear behaviour needed to realize the operation of multiplication, which is a core operation in realizing frequency conversion. The undesired nonlinearity of a mixer refers to the nonlinear behaviour of the circuit with respect to the RF input signal. Noise in mixer circuits is specified by the noise figure (N F ), which corresponds to the ratio of the signal-to-noise ratio (SN R) at the input to that at the output of the circuit [5]. The undesired nonlinearity of the mixer is specified by characterizing the
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1.2. Problem Formulation
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Figure 1.2.: Nonlinearity leads to spurious spectral components (red-colored) in mixer’s output, while noise (grey-colored) spreads over the whole spectrum
influence of each spurious spectral component resulting from the undesired nonlinearity of a mixer separately [5, 13, 14, 6]. This implies that the number of nonlinearity specification parameters needed for a mixer is equal to the number of spurious spectral components in the mixer’s output. Consequently the designer requires to consider several nonlinearity specification parameters depending on the design case. For the application case of heterodyne mixing, where the input signal is converted to/from an intermediate frequency fIF , two major spurious spectral components arise due to the mixer’s nonlinearity: an intermodulation term of 3rd order and a compression term [6]. All other spurious spectral components due to the mixer’s nonlinearity are usually either outside the bandwidth of interest or negligibly small. Therefore, most mixer designers need to consider two nonlinearity specification parameters: the input-intercept point third order IIP3 and the 1dB-Compression Point P1dB for the intermodulation term of third order and a compression term, respectively [5, 13, 15] (see section 3.2). However, due to the increased application of direct-conversion mixing in emerging technologies [7] the mixer designer requires a larger number of specification parameters for characterizing the nonlinear behaviour of the mixer. Since a direct-conversion mixer converts the input signal to DC, the mixer designer has to consider two further spurious spectral components resulting from the mixer’s nonlinear behaviour, namely an intermodulation term of second order and a DC offset. Consequently the mixer designer requires at least four nonlinearity specification parameters (IIP3 , IIP2 , P1dB and DC offset) [5, 9, 10, 13, 15] (see section 3.2). Even though the spurious spectral components, that are characterized by these specification parameters, are equally undesired, the use of these specification parameters leads the designer to consider each spurious spectral component separately. Hence, the designer is confronted with a multi-dimensional specification parameter space for characterizing the mixer’s nonlinearity. The existence of a multi-dimensional specification parameter space complicates the design problem, since simultaneous fulfilling of all specification parameters is difficult. In
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1. Introduction order to manage the design problem with a multi-dimensional specification parameter space, designers usually prioritize the design specifications. This prioritization of the design specifications translates the design problem from a single multi-dimensional problem (with respect to the specification parameter space) to several one-dimensional problems. An example for a design flow with a prioritization of the specification parameters is the power-constrained-noise-optimization (PCNO) presented in [16] for the low noise amplifier. Applying the same prioritization scheme on mixer design, one may start the design optimization with a focus on minimizing the nonlinearity and then proceed to other design specifications. However, unlike noise, which is conventionally specified by a single parameter (noise figure N F ), mixer’s nonlinearity is characterized using several specification parameters. Consequently, the designer has to prioritize the nonlinearity specification parameters, in order to consider them sequentially. The mixer designer considers the specification parameter of highest priority at first. He/she tunes the circuit to fulfill the requirements set to the first nonlinearity specification parameter and then proceeds to the second one, and so on. Nevertheless, a proportionality between each nonlinearity specification parameter with respect to the circuit parameters is dependent on the circuit’s architecture. This means that a variation of the design parameters for design optimization may improve a certain nonlinearity specification parameter while degrading another one. Such a „Trade-off” further complicates the design problem, since the designer will need to perform several design optimization loops to find a suitable design point. Furthermore, the required design time increases proportionally with the number of specification parameters [17]. A further disadvantage of the multi-dimensional specification parameter space exists at the task of choosing a mixer circuit. A comparison of different mixer architectures or mixer designs with respect to the nonlinearity behaviour requires the separate comparison of each one of the nonlinearity specification parameters [18, 19].
1.3. Solution Proposal The mixer designer considers all spurious spectral components that lie within the bandwidth of interest as equally undesired, independent from their position, as long as they are equally eliminable by filtering. Consequently, there is no need for specifying each spurious spectral component, that lies within the bandwidth of interest, separately. It is therefore expedient to define a single specification parameter to characterize the nonlinear behaviour of a mixer. In this thesis a specification parameter for mixer design is presented. This specification parameter is proposed for characterizing the nonlinear behaviour of mixers. For illustration, consider the mixer design space shown in Figure 1.3. The dimensions of this design space are the mixer’s specification parameters. Therefore, this design space is also referred in this thesis to as the specification parameter space. It is assumed in Figure 1.3 that m specification parameters exist, such that an m-dimensional design
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1.3. Solution Proposal Mixer Design Space A/T1
Rm A/T2
d(x,y)
A/T3
Ideal Mixer
Figure 1.3.: Each mixer design is a point in the m-dimensional design space of an architecture/technology combination A/Tx . m corresponds to the number of specification parameters. The proposed specification parameter indicates the deviation from an ideal mixer. space is formed. Each mixer design corresponds to a design point in the design space of a certain architecture/technology combination A/Tx within the mixer design space. An ideal mixer with no spurious spectral components in its output spectrum is practically unrealizable. Therefore, the ideal mixer is represented by a design point outside the reachable mixer design space. The new specification parameter proposed in this thesis indicates the deviation of the behaviour of a real mixer from that of an ideal mixer. From a geometrical point of view, this specification parameter measures the distance d(x, y) between the points x and y in the mixer design space, where x and y represent the design point of a real mixer and an ideal mixer, respectively. Since the ideal mixer corresponds to a fixed point in the design space, this specification parameter indicates the relative distance. In contrary, to the conventional nonlinearity specification parameters which characterize each spurious spectral component resulting from the mixer’s nonlinearity, the proposed metric considers all spurious spectral components lying within the signal’s bandwidth in the mixer output. Thereby, it serves as a single nonlinearity specification parameter. Replacing the conventional specification parameters by a single specification parameter for nonlinearity has the advantage of reducing the dimension of the specification parameter space to one. This reduction of the specification parameter space yields
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1. Introduction several benefits to the mixer design flow. Since all spurious spectral components are simultaneously considered by a single parameter, the separate consideration of each spectral component within the mixer design process is no more needed. Consequently, a prioritization of the spurious spectral components in order to consider them successively for mixer design is also unnecessary. Furthermore, the interrelation between the spectral components with respect to the circuit parameters is irrelevant for the mixer design process. The designer’s task is now to design and optimize a mixer circuit, such that only a single nonlinearity specification parameter is satisfied. The use of this new nonlinearity specification parameter leads thereby to a simplification of the mixer design process and a reduction of the required design time. Moreover, the use of the new specification parameter simplifies the comparison of different mixer circuits, since merely the comparison of the values of the specification parameter is sufficient. Since the new specification parameter indicates the deviation from the behaviour of the ideal mixer, the mixer design problem is to find a design point within the mixer design space, where the distance d(x, y) to the ideal mixer is a minimum (compare Figure 1.3), i.e. the condition for design optimization is d(x, y) = min!. Since the design point that fulfills the condition d(x, y) = 0 does not exist, the optimization problem here is similar to an approximation problem, e.g. least squares method.
1.4. Work Contributions This work handles the systematic design of mixers with a focus on the characterization of nonlinearity. The thesis is based on considering multiplication as the core operation for realizing a frequency conversion. Therefore, in order to understand the roots of nonlinearity in mixers, a study on the electrical/electronic implementation of multiplication is conducted. In the context of this work the following original contributions to the field of design of multipliers and mixers are presented: 1. A new nonlinearity specification parameter for multipliers: On the basis of a simple mathematical model, a specification parameter for the characterization of the nonlinearity of multipliers is introduced. Unlike the Total Harmonic Distortion (T HD) which considers the harmonic distortion only [12], the proposed specification parameter considers all spurious components in the output of the multiplier, that result from the nonlinearity of the multiplier. This specification parameter quantifies the purity of the multiplication performed by a system. 2. A new nonlinearity specification parameter for mixers: The specification parameter proposed for characterizing the nonlinear behaviour of multipliers is introduced and redefined for mixers. While the idea of the conventional nonlinearity specification parameters for the mixer is to quantify each spurious spectral component separately, the approach proposed in this work is to use a single specification parameter to quantify the nonlinearity of the mixer. This approach for
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1.4. Work Contributions specifying the undesired nonlinear behaviour of the mixer is more appropriate for the design process, because the designer considers all spurious components lying within the signal’s bandwidth as equally undesired. Therefore, there is no need for specifying each spurious spectral component separately, as done by the conventional nonlinearity specification parameters. Consequently, the dimension of the specification parameter space of mixers is reduced. This reduction simplifies the mixer design process. 3. A method for the calculation of spectral components in weakly nonlinear circuits with large-signal/small-signal excitation: This method is based on the Volterra series [20] and the theory of nonlinearly controlled currents [21]. The work in [12] presented a method for the calculation of the spectral components in nonlinear time-invariant networks. However, this method is only appropriate for time-invariant networks with small-signal excitations and therefore inappropriate for analyzing mixer circuits with a switching behaviour. In this work, this method is extended for nonlinear time-invariant networks with a large-signal excitation. The resulting method allows the derivation of symbolical expressions for the spectral components of the state variables of weakly nonlinear networks with a large-signal/small-signal excitation in dependency of design and input variables. Applied to mixer design, this enables the designer to analyze the generation of spurious spectral components in the mixer. Since the presented method is element-wise, the input-output description of the circuit is not required. 4. A systematic design flow for mixers: This design flow is based on the new nonlinearity specification parameter introduced in this work. The proposed design flow enables the comparison of different circuits with respect to the specification parameters in order to find the most appropriate mixer circuit for a certain application. In the first phase of the design flow different candidate mixer circuits are compared with respect to the given specification parameters. Here, semi-symbolical mathematical models for the specification parameters in dependency on the design parameters of each candidate mixer circuit are used. The n-dimensional design parameter space DP0 of each candidate mixer circuit is reduced by considering the conditions set to each specification parameter (see Figure 1.4). The constriction of the design parameter space leads to a sub-space DPopt , where the conditions set to all specification parameters are fulfilled. Candidate mixer circuits with DPopt = ∅ are then compared with respect to their nonlinearity behaviour. The candidate mixer circuit with the best performance is selected for the second design phase, where design optimization within the subdesign space DPopt is performed. Here, higher device models for calculating the specification parameters of the circuit are employed.
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1. Introduction
Rn
Design Parameter Space, DP0
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