Multistage Quadrature Sigma-Delta Modulators for Reconfigurable Multi-Band Analog-Digital Interface in Cognitive Radio Devices

Abstract

This article addresses the design, analysis, and parameterization of reconfigurable multi-band noise and signal transfer functions (NTF and STF), realized with multistage quadrature ΣΔ modulator (QΣΔM) concept and complex-valued in-phase/quadrature (I/Q) signal processing. Such multi-band scheme was already proposed earlier by the authors at a preliminary level, and is here developed further toward flexible and reconfigurable A/D interface for cognitive radio (CR) receivers enabling efficient parallel reception of multiple noncontiguous frequency slices. Owing to straightforward parameterization, the NTF and the STF of the multistage QΣΔM can be adapted to input signal conditions based on spectrum-sensing information. It is also shown in the article through closed-form response analysis that the so-called mirror-frequency-rejecting STF design can offer additional operating robustness in challenging scenarios, such as the presence of strong mirror-frequency blocking signals under I/Q imbalance, which is an unavoidable practical problem with quadrature circuits. The mirror-frequency interference stemming from these blockers is analyzed with a novel analytic closed-form I/Q imbalance model for multistage QΣΔMs with arbitrary number of stages. Concrete examples are given with three-stage QΣΔM, which gives valuable degrees of freedom for the transfer function design. High-order frequency asymmetric multi-band noise shaping is, in general, a valuable asset in CR context offering flexible and frequency agile adaptation capability to differing waveforms to be received and detected. As demonstrated by this article, multistage QΣΔMs can indeed offer these properties together with robust operation without risking stability of the modulator.

1. Introduction

Nowadays, a growing number of parallel wireless communication standards, together with ever-increasing traffic amounts, create a widely acknowledged need for novel radio solutions, such as emerging cognitive radio (CR) paradigm [1, 2]. On the other hand, transceiver implementations, especially in mobile terminals, should be small-sized, power efficient, highly integrable, and cheap [3–7]. Thus, it would be valuable to avoid implementing parallel transceiver units for separate communication modes. However, operating band of this kind of software defined radio (SDR) should be extremely wide (even GHz range), and dynamic range of the receiver should be high (several tens of dBs) [5–10]. In addition, the transceiver should be able to adapt to numerous different transmission schemes and waveforms [4–8, 10]. The SDR concept is considered as a physical layer foundation for CR [1], but these demands create a big challenge for transceiver design, especially for mobile devices.

Particularly, the analog-to-digital (A/D) interface has been identified as a key performance-limiting bottleneck [1, 3, 4, 8, 10–12]. For example, GSM reception demands high dynamic range, and WLAN and LTE bandwidths, in turn, can be up to 20 MHz. Combining this kind of differing radio characteristics set massive demands for the A/D converter (ADC) in the receiver. Traditional Nyquist ADCs (possibly with oversampling) divide the conversion resolution equally on all the frequencies, and thus, if 14-bit resolution is needed for one of the signals converted, then similar resolution is used over the whole band even if it would not be necessary [12]. At the same time, in wideband SDR receiver, the resolution demand might be even higher because of the increased dynamic range due to multiple waveforms with differing power levels entering the ADC. On the other hand, ΣΔ ADCs have inherent tradeoff between the sampling frequency and resolution [13]. With narrowband signals (such as GSM), e.g., 14-bit resolution can be achieved with 1-bit quantization because of high oversampling and digital filtering. At the same time, modulator structure can be reconfigured for reception of wideband waveforms to meet differing requirements set by, for example, WLAN or LTE standards [8, 14, 15].

Based on this, one promising solution for the receiver design in this kind of scenario is wideband direct-conversion or low-IF architecture [16] with a bandpass ΣΔ ADC [8, 14]. Additional degrees of freedom can be obtained by introducing quadrature ΣΔ modulator (QΣΔM) in the receiver, allowing efficient frequency asymmetric quantization noise shaping [17, 18]. Furthermore, a multi-band modulator aimed to CR receivers is preliminarily proposed in [19] and illustrated with receiver block diagram and principal spectra in Figure 1. This kind of multi-band design for QΣΔM offers frequency agile flexibility and reconfigurability based on spectrum-sensing information [20] together with capability of receiving multiple parallel frequency bands [19], which are considered essential when realizing A/D interface for CR solutions [1]. In practice, multiple noise-shaping notches can be created on independent, noncontiguous signal bands. In addition, the center frequencies of these noise notches can be tuned based on the spectrum-sensing information obtained in the receiver.

Figure 1

Block diagram of multi-band low-IF quadrature receiver, based on QΣΔM. Principal spectra, where the two light gray signals are the preferred ones, are illustrating the signal compositions at each stage.

Noise-shaping capabilities of a single-stage QΣΔM are limited by the order of the modulator [18]. However, the order of the overall noise transfer function (NTF) can be increased using cascaded multistage modulator [21–23]. Therein, the overall noise shaping is of the combined order of the stages. In a multistage QΣΔM, the noise notches of the stages can be placed independently, thus further increasing the flexibility of the ADC [21].

Unfortunately, implementing quadrature circuits brings always a challenge of matching the in-phase (I) and quadrature (Q) rails, which should ideally have symmetric component values. Inaccuracies in circuit implementation always shift the designed values, creating imbalance between the rails, known as I/Q imbalance [18, 24]. This mismatch induces image response of the input signal in addition to the original input, causing mirror-frequency interference (MFI) [18, 24]. This image response can be modeled mathematically with altered complex conjugate of the signal component. In QΣΔMs generally, the mismatches generate conjugate response for both the input signal and the quantization error [18, 19, 25], which is a clear difference to mirror-frequency problematics in more traditional receivers. Specifically, feedback branch mismatches have been highlighted as the most important MFI source [23, 26]. From the noise point of view, placing a NTF notch also on mirror frequency to cancel MFI was initially proposed in [18] and discussed further in [27]. This, however, wastes noise shaping performance from the desired signal point of view and restricts design freedom, especially in multi-band scenario. In addition, this does not take the mirroring of the input signal into account. In wideband SDR quadrature receiver, the MFI stemming from the input of the receiver is a crucial viewpoint because of possible blocking signals. Furthermore, alterations to analog circuitry have been proposed in [26, 28, 29] to minimize the interference. Sharing the components between the branches, however, degrades sampling properties of the modulator [28, 29]. On the other hand, additional components add to the circuit area and power dissipation of the modulator [26]. In [19], the authors found that mirror-frequency-rejecting signal transfer function (STF) design mitigates the input signal-originating MFI in case of mismatch in the feedback branch of a first-order QΣΔM. In [21], this idea is extended to cover multi-band design of [19] with a simple two-stage QΣΔM. The feedback I/Q imbalance effects and related digital calibration in two-stage QΣΔM are addressed also in [23], where only a frequency-flat STF is considered. In addition, the mirror-frequency-rejecting STF design has a benefit of not demanding additional components to the original QΣΔM structure.

In this article, an analytic closed-form model for QΣΔM I/Q imbalance effects is derived covering multistage modulators with arbitrary number of stages, extending the preliminary analysis with two first-order stages in [21]. Herein, the I/Q imbalance model for second-order QΣΔM presented by the authors in [30] is used for each of the stages. Furthermore, design of the transfer functions (STF and NTF) of the stages in such multistage QΣΔM is addressed in detail with emphasis on robust operation under I/Q mismatches. In [31, 32], QΣΔM STF designs are proposed for reducing the dynamics of the receiver and to filter adjacent channel signals for lowpass and quadrature bandpass modulators, respectively. However, adapting the STF based on spectrum-sensing information is not covered in case of the QΣΔM in [32]. In addition, NTF adaptation to frequency handoffs or multi-band reception is not considered in either [31] or [32]. Herein, frequency agile design of the STF and the NTF of an I/Q mismatched multistage QΣΔM is discussed taking both the input signal and the quantization noise-oriented MFI into account during multi-band reception.

The push for development of multi-channel ADCs for SDR and CR solutions has been acknowledged, e.g., in [11]. A multi-channel system with parallel ADCs is one possible solution, which, however, sets additional burden for size, cost, and power dissipation of the receiver implementation [11, 13]. On the other hand, quadrature ΣΔ noise shaping makes exploitation of whole quantization precision on the desired signal bands possible. Three-stage lowpass ΣΔ modulators have traditionally been used only for applications demanding very high resolution [33], but like shown in this article, the QΣΔM variant allows noncontiguous placement of the NTF zeros, and thus the quantization precision can be divided on multiple parallel frequency bands. A reconfigurable three-stage converter using lowpass ΣΔ stages together with a pipeline ADC is proposed in [15] for mobile terminals. In comparison, a three-stage QΣΔM discussed in this article offers more efficient noise shaping and additional degrees of freedom for the receiver design. These are essential characteristics when heading toward a frequency agile reconfigurable ADC for CR receivers. Thus, a multistage QΣΔM offers a competent platform for realizing flexible multi-band A/D conversion in CR devices.

The rest of the article is organized as follows. In Section 2, basics of quadrature ΣΔ modulation are reviewed, while Section 3 presents a closed-form model for I/Q imbalance effects in a second-order QΣΔM as a single stage of a multistage modulator and proposes a novel extension of the given model for multistage modulators with arbitrary number of stages. Parameterization and design of the modulator transfer functions in CR receivers in the presence of I/Q mismatches are discussed in Section 4. The receiver system level targets and QΣΔM performance are discussed in Section 5. Thereafter, Section 6 presents the results of the designs in the previous section with closed-form transfer function analysis and computer simulations. Finally, Section 7 concludes the article.

Short note on terminology and notations: term "order" refers in this article to the order of polynomial(s) in z-domain transfer functions, while term "stage" refers to individual QΣΔM block in a multistage converter where multiple QΣΔM blocks are interconnected. The z-domain representations of sequences x(k) and x*(k) are denoted as X[z] and X*[z*], respectively, where superscript (·)* denotes complex conjugation.

2. Basics of Quadrature ΣΔ Modulation

Quadrature variant of the ΣΔ modulator was originally presented in [18]. The concept is based on the modulator structure similar to the one used in real lowpass and bandpass modulators, but employing complex-valued input and output signals together with complex loop filters (integrators). This complex I/Q signal processing gives additional degree of freedom to response design, allowing for frequency-asymmetric STF and NTF. For analysis purposes, a linear model of the modulator is typically used. In other words, this means that quantization error is assumed to be additive and having no correlation with the input signal. Although not being exactly true, this allows analytic derivation of the transfer functions and has thus been applied widely, e.g., in [18, 33]. Now, the output of a single-stage QΣΔM, depicted in Figure 2, is defined as

Figure 2

Discrete-time linearized model of a P th-order QΣΔM with complex-valued signals and coefficients.

$$V^{\mathsf{\text{ideal}}}\left[z\right]=\mathsf{\text{STF}}\left[z\right]U\left[z\right]+\mathsf{\text{NTF}}\left[z\right]E\left[z\right],$$

(1)

where STF[z] and NTF[z] are generally complex-valued functions, and U [z] and E [z] denote z-transforms of the input signal and quantization noise, respectively.

The achievable NTF shaping and STF selectivity are defined by the order of the modulator. With P th-order modulator, it is possible to place P zeros and poles in both transfer functions. This is confirmed by derivation of the transfer functions for the structure presented in Figure 2. The NTF of the P th-order QΣΔM is given by

$$\text{NTF}[z]=\frac{1}{1-{\displaystyle {\sum }_{p=1}^P{R}_p}{\displaystyle {\prod }_{i=1}^p\frac{1}{z-M_i}}}$$

(2)

and, on the other hand, the corresponding STF is

$$\text{STF}[z]=\frac{A+{\displaystyle {\sum }_{p=1}^P{B}_p}{\displaystyle {\prod }_{i=1}^p\frac{1}{z-M_i}}}{1-{\displaystyle {\sum }_{p=1}^P{R}_p}{\displaystyle {\prod }_{i=1}^p\frac{1}{z-M_i}}},$$

(3)

where 1/(z − M _i ) terms are the transfer functions of the complex loop filters (integrators). Both transfer functions have common denominator and thus common poles. It can also be seen that in addition to the loop filters, only the feedback coefficients R _p (feeding the output to the loop filters) affect the noise shaping. Thus, input coefficients A (feeding the input to the quantizer) and B _p (feeding the input to the loop filters) can be used to tune the STF zeros independent of the NTF.

The NTF zeros are usually placed on the desired signal band(s) to create the noise-shaping effect. At the same time, the STF zeros can be used to attenuate out-of-band frequencies and thus include some of the receiver selectivity in the QΣΔM. The transfer function design for CR is discussed in more detail in Section 4. In the following subsections, multi-band and multistage principles will be presented. These are important concepts, considering reconfigurability in the A/D interface and frequency agile conversion with high-enough resolution in CR devices.

2.1. Multi-Band Quadrature ΣΔ ADC for CR

With QΣΔM of higher than first order, it is possible to place multiple NTF zeros on the conversion band [18]. Traditional way of exploiting this property has been making the noise-shaping notch wider, thus improving the resolution of the interesting information signal over wider bandwidths [18]. However, in CR-based systems, it is desirable to be able to receive more than one detached frequency bands - and signals - in parallel [1]. The multi-band scheme offers transmission robustness, e.g., in case of appearance of a primary user when the CR user has to vacant that frequency band [1]. In that case, the transmissions can be continued on the other band(s) in use. In addition, if the CR traffic is divided on multiple bands, then lower power levels can be used, and thus the interference generated for primary users is decreased [1].

Multi-band noise shaping without restriction to frequency symmetry is able to respond to this need with noncontiguous NTF notches. This reception scheme is illustrated graphically in Figure 3. The possible number of these notches is defined by the overall order of the modulator. With multistage QΣΔM this is the combined order of all the stages. In addition, the frequencies of the notches can be tuned straightforwardly, e.g., in case of frequency handoff. This tunability of the transfer functions allows also for adaptation to differing waveforms, center frequencies and bandwidths to be received. The resolution and bandwidth demands of the waveforms at hand can be taken into account and the response of the QΣΔM can be optimized for the scenario of the moment based on the spectrum-sensing information. Further details on design and parameterization of multi-band transfer functions are given in Section 4.

Figure 3

Principal illustration of complex multi-band QΣΔM scheme for cognitive radio devices. The light gray signals are assumed to be the preferred ones and principal total STF and NTF are illustrated with magenta dotted and black solid lines, respectively. Quantization noise is shaped away from preferred frequency bands and out-of-band signals are attenuated.

2.2. Multistage Quadrature ΣΔ ADC

Multistage ΣΔ modulators have been introduced to improve resolution, e.g., in case of wideband information signal, when attainable oversampling is limited. This principle was first proposed with lowpass modulator [33], but has thereafter been extended to quadrature bandpass modulator [23, 26]. The block diagram of L-stage quadrature ΣΔ ADC is given in Figure 4, where all the stages are of arbitrary order. The inputs u _l (k) of the L individual stages (1 ≤ l ≤ L, l ∈ ℤ) are defined in the following manner. The input of the first-stage (l = 1) is the overall input of the whole structure, i.e., u₁ (k) = u (k), and for the latter stages, the (ideal) input is the quantization error of the previous stage; thus, u _l (k) = e_l−1(k) when 2 ≤ l ≤ L.

Figure 4

Multistage QΣΔM with arbitrary-order noise shaping in all the individual stages. Filters $H_1^{\mathsf{\text{D}}}\left[z\right]$ to $H_L^{\mathsf{\text{D}}}\left[z\right]$ are implemented digitally.

The main goal in multistage QΣΔM is to digitize quantization error of the previous stage with the next stage and thereafter subtract it from the output of that previous stage. Owing to the noise shaping in the stages, the digitized error estimate must be filtered in the same way, in order to achieve effective cancelation. Similarly, the output of the first stage must be filtered with digital equivalent of the second-stage STF (e.g., to match the delays). These filters are depicted in Figure 4 with $H_1^{\mathsf{\text{D}}}\left[z\right]$ to $H_L^{\mathsf{\text{D}}}\left[z\right]$. Now, assuming ideal implementation, the final output becomes

$$V^{\mathsf{\text{ideal}}}\left[z\right]=\sum _{l=1}^L{\left(-1\right)}^{l+1}{H}_l^{\mathsf{\text{D}}}\left[z\right]V_l^{\mathsf{\text{ideal}}}\left[z\right],$$

(4)

where

$$V_l^{\mathsf{\text{ideal}}}\left[z\right]={\mathsf{\text{STF}}}_l^{\mathsf{\text{ideal}}}\left[z\right]U_l\left[z\right]+{\mathsf{\text{NTF}}}_l^{\mathsf{\text{ideal}}}\left[z\right]E_l\left[z\right],1\le l\le L,l\in \mathbb{Z}$$

(5)

and

$$H_l^{\mathsf{\text{D}}}\left[z\right]=\frac{H_1^{\mathsf{\text{D}}}\left[z\right]{\prod }_{l=1}^{L-1}{\mathsf{\text{NTF}}}_l^{\mathsf{\text{ideal}}}\left[z\right]}{{\prod }_{l=2}^L{\mathsf{\text{STF}}}_l^{\mathsf{\text{ideal}}}\left[z\right]},1\le l\le L,l\in \mathbb{Z},$$

(6)

to match the analog transfer functions and the digital filters. It is usually chosen that $H_1^{\mathsf{\text{D}}}\left[z\right]=\mathsf{\text{ST}}{\mathsf{\text{F}}}_2\left[z\right]$, thus giving $H_2^{\mathsf{\text{D}}}\left[z\right]=\mathsf{\text{NT}}{\mathsf{\text{F}}}_1\left[z\right]$ and $H_3^{\mathsf{\text{D}}}\left[z\right]=\mathsf{\text{NT}}{\mathsf{\text{F}}}_1\left[z\right]\mathsf{\text{NT}}{\mathsf{\text{F}}}_2\left[z\right]∕\mathsf{\text{ST}}{\mathsf{\text{F}}}_3\left[z\right]$, etc. With these selections, the quantization errors of the earlier stages are canceled (assuming ideal circuitry), and the overall output of the L-stage QΣΔM becomes (L ≥ 2)

$$V^{\mathsf{\text{ideal}}}\left[z\right]={\mathsf{\text{STF}}}_1^{\mathsf{\text{ideal}}}\left[z\right]{\mathsf{\text{STF}}}_2^{\mathsf{\text{D}}}\left[z\right]U\left[z\right]+\frac{{\prod }_{l=1}^L{\mathsf{\text{NTF}}}_l^{\mathsf{\text{ideal}}}\left[z\right]}{{\prod }_{l=3}^L{\mathsf{\text{STF}}}_l^{\mathsf{\text{ideal}}}\left[z\right]}{E}_L\left[z\right]={\mathsf{\text{STF}}}_{\mathsf{\text{TOT}}}^{\mathsf{\text{ideal}}}\left[z\right]U\left[z\right]+{\mathsf{\text{NTF}}}_{\mathsf{\text{TOT}}}^{\mathsf{\text{ideal}}}\left[z\right]E_L\left[z\right],$$

(7)

where only the quantization error of the last stage is present. It is observed that, if three or more stages are used, then special care should be taken in designing the STF of the third and the latter stages, which operate in the denominator of the noise-shaping term. However, the leakage of the quantization noise of the earlier stages might be limiting achievable resolution in practice because of nonideal matching of the digital filters [33]. One way to combat this phenomenon is to use adaptive filters [34, 35].

3. I/Q Imbalance in Multistage QΣΔMs

In this section, a closed-form transfer function analysis is carried out for a general multistage QΣΔM taking also the possible coefficient mismatches in complex I/Q signal processing into account. For mathematical tractability and notational convenience, second-order QΣΔM stages are assumed as individual building blocks (individual stages) in Figure 4, and the purpose is to derive a complete closed-form transfer function model for the overall multistage converter. Such analysis is missing from the existing state-of-the-art literature. For notational simplicity, the modulator coefficients are denoted in the following analysis as shown in the block diagram of Figure 5. With this structure, the ideal NTF for the l th stage is given by

Figure 5

Discrete-time-linearized model of the l th second-order QΣΔM stage in a multistage QΣΔM with complex-valued signals and coefficients.

$$\mathsf{\text{NT}}{\mathsf{\text{F}}}_l\left[z\right]=\frac{1-\left(M^{\left(l\right)}+N^{\left(l\right)}\right)z^{-1}+\left(M^{\left(l\right)}{N}^{\left(l\right)}\right)z^{-2}}{1-\left(M^{\left(l\right)}+N^{\left(l\right)}+R^{\left(l\right)}\right)z^{-1}+\left(M^{\left(l\right)}{N}^{\left(l\right)}+N^{\left(l\right)}{R}^{\left(l\right)}-S^{\left(l\right)}\right)z^{-2}}.$$

(8)

At the same time, the ideal STF for the l th stage is defined as

$$\mathsf{\text{ST}}{\mathsf{\text{F}}}_l\left[z\right]=\frac{A^{\left(l\right)}+\left(B^{\left(l\right)}-N^{\left(l\right)}{A}^{\left(l\right)}-M^{\left(l\right)}{A}^{\left(l\right)}\right)z^{-1}+\left(C^{\left(l\right)}-N^{\left(l\right)}{B}^{\left(l\right)}+M^{\left(l\right)}{N}^{\left(l\right)}{A}^{\left(l\right)}\right)z^{-2}}{1-\left(M^{\left(l\right)}+N^{\left(l\right)}+R^{\left(l\right)}\right)z^{-1}+\left(M^{\left(l\right)}{N}^{\left(l\right)}+N^{\left(l\right)}{R}^{\left(l\right)}-S^{\left(l\right)}\right)z^{-2}}.$$

(9)

The transfer functions of (8) and (9) are valid when I and Q rails of the QΣΔM are matched perfectly. With this perfect matching, (1) and (4) give the outputs for single-stage and multistage modulators, respectively.

3.1. I/Q Imbalance Effects on Individual QΣΔM Stage

Quadrature signal processing is, in practice, implemented with parallel real signals and coefficients. In Figure 6, this is demonstrated in case of a single second-order QΣΔM stage (parallel real I and Q signal rails) and taking possible mismatches in the coefficients into account. Deviation between coefficient values of the rails, which should ideally be the same, results in MFI. This interference can be presented mathematically with conjugate response of the signal and the noise components. Thus, image signal transfer function (ISTF) and image noise transfer function (INTF) are introduced, in addition to the traditional STF and NTF, to describe the output under I/Q imbalance. In the following, an analytic model is presented, first for individual stages of a multistage QΣΔM, and then for I/Q mismatched multistage QΣΔM, having arbitrary number of stages, as a whole. Such analysis has not been presented in the literature earlier.

Figure 6

Implementation structure of the l th second-order QΣΔM stage in a multistage QΣΔM with parallel real signals and coefficients taking possible mismatches into account.

The I/Q imbalance analysis for a single stage is based on the block diagram given in Figure 6. In this figure, real and imaginary parts of the coefficients of Figure 5 are marked with subscripts re and im, whereas nonideal implementation values of the signal rails are separated with subscripts 1 and 2. The independent coefficients of the stages are denoted with superscript l. Thus, to obtain the complex outputs $V_l\left[z\right]=V_{\mathsf{\text{I}},l}\left[z\right]+j{V}_{\mathsf{\text{Q}},l}\left[z\right]$ of the stages (l ∈ {1,L}), the I branch outputs can be first shown to be

$$V_{\mathsf{\text{I}},l}\left[z\right]=\frac{{\alpha }_{\mathsf{\text{I}}}^{\left(l\right)}\left[z\right]}{{\gamma }_{\mathsf{\text{I}}}^{\left(l\right)}\left[z\right]}{U}_{\mathsf{\text{I}},l}\left[z\right]-\frac{{\beta }_{\mathsf{\text{I}}}^{\left(l\right)}\left[z\right]}{{\gamma }_{\mathsf{\text{I}}}^{\left(l\right)}\left[z\right]}{U}_{\mathsf{\text{Q}},l}\left[z\right]+\frac{{\epsilon }_{\mathsf{\text{I}}}^{\left(l\right)}\left[z\right]}{{\gamma }_{\mathsf{\text{I}}}^{\left(l\right)}\left[z\right]}{E}_{\mathsf{\text{I}},l}\left[z\right]+\frac{{\eta }_{\mathsf{\text{I}}}^{\left(l\right)}\left[z\right]}{{\gamma }_{\mathsf{\text{I}}}^{\left(l\right)}\left[z\right]}{E}_{\mathsf{\text{Q}},l}\left[z\right]-\frac{{\rho }_{\mathsf{\text{I}}}^{\left(l\right)}\left[z\right]}{{\gamma }_{\mathsf{\text{I}}}^{\left(l\right)}\left[z\right]}{V}_{\mathsf{\text{Q}},l}\left[z\right],$$

(10)

where the auxiliary variables multiplying the signal components are defined by the coefficients (see Figure 6) in the following manner:

$$\begin{array}{c}{\alpha }_{\text{I}}^{(l)}[z]=a_{\text{re},1}^{(l)}+[b_{\text{re},1}^{(l)}-m_{\text{re},1}^{(l)}{a}_{\text{re},1}^{(l)}-n_{\text{re},1}^{(l)}{a}_{\text{re},1}^{(l)}+n_{\text{im},2}^{(l)}{a}_{\text{im},1}^{(l)}+m_{\text{im},2}^{(l)}{a}_{\text{im},1}^{(l)}]z^{-1}+[c_{\text{re},1}^{(l)}-n_{\text{re},1}^{(l)}{b}_{\text{re},1}^{(l)}\\ +n_{\text{re},1}^{(l)}{m}_{\text{re},1}^{(l)}{a}_{\text{re},1}^{(l)}-n_{\text{re},1}^{(l)}{m}_{\text{im},2}^{(l)}{a}_{\text{im},1}^{(l)}+n_{\text{im},2}^{(l)}{b}_{\text{im},1}^{(l)}-n_{\text{im},2}^{(l)}{m}_{\text{im},1}^{(l)}{a}_{\text{re},1}^{(l)}-n_{\text{im},2}^{(l)}{m}_{\text{re},2}^{(l)}{a}_{\text{im},1}^{(l)}]z^{-2},\end{array}$$

(11)

$$\begin{array}{c}{\beta }_{\text{I}}^{(l)}[z]=a_{\text{im},2}^{(l)}+[b_{\text{im},2}^{(l)}-n_{\text{re},1}^{(l)}{a}_{\text{im},2}^{(l)}-n_{\text{im},2}^{(l)}{a}_{\text{re},2}^{(l)}-m_{\text{re},1}^{(l)}{a}_{\text{im},2}^{(l)}-m_{\text{im},2}^{(l)}{a}_{\text{re},2}^{(l)}]z^{-1}+[c_{\text{im},2}^{(l)}-n_{\text{re},1}^{(l)}{b}_{\text{im},2}^{(l)}\\ +n_{\text{re},1}^{(l)}{m}_{\text{re},1}^{(l)}{a}_{\text{im}2,}^{(l)}+n_{\text{re},1}^{(l)}{m}_{\text{im},2}^{(l)}{a}_{\text{re},2}^{(l)}-n_{\text{im},2}^{(l)}{b}_{\text{re},2}^{(l)}-n_{\text{im},2}^{(l)}{m}_{\text{im},1}^{(l)}{a}_{\text{im},2}^{(l)}+n_{\text{im},2}^{(l)}{m}_{\text{re},2}^{(l)}{a}_{\text{re},2}^{(l)}]z^{-2},\end{array}$$

(12)

$${\epsilon }_{\mathsf{\text{I}}}^{\left(l\right)}\left[z\right]=1-\left[n_{\mathsf{\text{re}},1}^{\left(l\right)}+m_{\mathsf{\text{re}},1}^{\left(l\right)}\right]z^{-1}+\left[n_{\mathsf{\text{re}},1}^{\left(l\right)}{m}_{\mathsf{\text{re}},1}^{\left(l\right)}-n_{\mathsf{\text{im}},2}^{\left(l\right)}{m}_{\mathsf{\text{im}},1}^{\left(l\right)}\right]z^{-2},$$

(13)

$${\eta }_{\mathsf{\text{I}}}^{\left(l\right)}\left[z\right]=\left[n_{\mathsf{\text{im}},2}^{\left(l\right)}+m_{\mathsf{\text{im}},2}^{\left(l\right)}\right]z^{-1}-\left[n_{\mathsf{\text{re}},1}^{\left(l\right)}{m}_{\mathsf{\text{im}},2}^{\left(l\right)}+n_{\mathsf{\text{im}},2}^{\left(l\right)}{m}_{\mathsf{\text{re}},2}^{\left(l\right)}\right]z^{-2},$$

(14)

$${\rho }_{\mathsf{\text{I}}}^{\left(l\right)}\left[z\right]=\left[n_{\mathsf{\text{im}},2}^{\left(l\right)}+r_{\mathsf{\text{im}},2}^{\left(l\right)}+m_{\mathsf{\text{im}},2}^{\left(l\right)}\right]z^{-1}-\left[s_{\mathsf{\text{im}},2}^{\left(l\right)}-n_{\mathsf{\text{re}},1}^{\left(l\right)}{r}_{\mathsf{\text{im}},2}^{\left(l\right)}-n_{\mathsf{\text{im}},2}^{\left(l\right)}{r}_{\mathsf{\text{re}},2}^{\left(l\right)}-n_{\mathsf{\text{re}},1}^{\left(l\right)}{m}_{\mathsf{\text{im}},2}^{\left(l\right)}-n_{\mathsf{\text{im}},2}^{\left(l\right)}{m}_{\mathsf{\text{re}},2}^{\left(l\right)}\right]z^{-2}$$

(15)

$${\gamma }_{\mathsf{\text{I}}}^{\left(l\right)}\left[z\right]=1-\left[n_{\mathsf{\text{re}},1}^{\left(l\right)}+r_{\mathsf{\text{re}},1}^{\left(l\right)}+m_{\mathsf{\text{re}},1}^{\left(l\right)}\right]z^{-1}+\left[s_{\mathsf{\text{re}},1}^{\left(l\right)}-n_{\mathsf{\text{re}},1}^{\left(l\right)}{r}_{\mathsf{\text{re}},1}^{\left(l\right)}+n_{\mathsf{\text{im}},2}^{\left(l\right)}{r}_{\mathsf{\text{im}},1}^{\left(l\right)}-n_{\mathsf{\text{re}},1}^{\left(l\right)}{m}_{\mathsf{\text{re}},1}^{\left(l\right)}+n_{\mathsf{\text{im}},2}^{\left(l\right)}{m}_{\mathsf{\text{im}},1}^{\left(l\right)}\right]z^{-2}.$$

(16)

This follows directly from a step-by-step signal analysis of the implementation structure in Figure 6. Similarly, the real-valued Q branch outputs are given by

$$V_{\mathsf{\text{Q}},l}\left[z\right]=\frac{{\beta }_{\mathsf{\text{Q}}}^{\left(l\right)}\left[z\right]}{{\gamma }_{\mathsf{\text{Q}}}^{\left(l\right)}\left[z\right]}{U}_{\mathsf{\text{I}},l}\left[z\right]+\frac{{\alpha }_{\mathsf{\text{Q}}}^{\left(l\right)}\left[z\right]}{{\gamma }_{\mathsf{\text{Q}}}^{\left(l\right)}\left[z\right]}{U}_{\mathsf{\text{Q}},l}\left[z\right]+\frac{{\epsilon }_{\mathsf{\text{Q}}}^{\left(l\right)}\left[z\right]}{{\gamma }_{\mathsf{\text{Q}}}^{\left(l\right)}\left[z\right]}{E}_{\mathsf{\text{Q}},l}\left[z\right]-\frac{{\eta }_{\mathsf{\text{Q}}}^{\left(l\right)}\left[z\right]}{{\gamma }_{\mathsf{\text{Q}}}^{\left(l\right)}\left[z\right]}{E}_{\mathsf{\text{I}},l}\left[z\right]+\frac{{\rho }_{\mathsf{\text{Q}}}^{\left(l\right)}\left[z\right]}{{\gamma }_{\mathsf{\text{Q}}}^{\left(l\right)}\left[z\right]}{V}_{\mathsf{\text{I}},l}\left[z\right],$$

(17)

where

$$\begin{array}{c}{\alpha }_{\text{Q}}^{(l)}[z]=a_{\text{re},2}^{(l)}+[b_{\text{re},2}^{(l)}+n_{\text{im},1}^{(l)}{a}_{\text{im},2}^{(l)}-n_{\text{re},2}^{(l)}{a}_{\text{re},2}^{(l)}+m_{\text{im},1}^{(l)}{a}_{\text{im},2}^{(l)}-m_{\text{re},2}^{(l)}{a}_{\text{re},2}^{(l)}]z^{-1}+[c_{\text{re},2}^{(l)}-n_{\text{re},2}^{(l)}{b}_{\text{re},2}^{(l)}\\ -n_{\text{im},1}^{(l)}{m}_{\text{im},2}^{(l)}{a}_{\text{re},2}^{(l)}+n_{\text{im},1}^{(l)}{b}_{\text{im},2}^{(l)}-n_{\text{im},1}^{(l)}{m}_{\text{re},1}^{(l)}{a}_{\text{im},2}^{(l)}-n_{\text{re},2}^{(l)}{m}_{\text{im},1}^{(l)}{a}_{\text{im},2}^{(l)}+n_{\text{re},2}^{(l)}{m}_{\text{re},2}^{(l)}{a}_{\text{re},2}^{(l)}]z^{-2},\end{array}$$

(18)

$$\begin{array}{c}{\beta }_{\text{Q}}[z]=a_{\text{im},1}^{(l)}+[b_{\text{im},1}^{(l)}-n_{\text{im},1}^{(l)}{a}_{\text{re},1}^{(l)}-n_{\text{re},2}^{(l)}{a}_{\text{im},1}^{(l)}-m_{\text{im},1}^{(l)}{a}_{\text{re},1}^{(l)}-m_{\text{re},2}^{(l)}{a}_{\text{im},1}^{(l)}]z^{-1}+[c_{\text{im},1}^{(l)}-n_{\text{re},2}^{(l)}{b}_{\text{im},1}^{(l)}\\ -n_{\text{im},1}^{(l)}{m}_{\text{im},2}^{(l)}{a}_{\text{im},1}^{(l)}-n_{\text{im},1}^{(l)}{b}_{\text{re},1}^{(l)}+n_{\text{im},1}^{(l)}{m}_{\text{re},1}^{(l)}{a}_{\text{re},1}^{(l)}+n_{\text{re},2}^{(l)}{m}_{\text{im},1}^{(l)}{a}_{\text{re},1}^{(l)}+n_{\text{re},2}^{(l)}{m}_{\text{re},2}^{(l)}{a}_{\text{im},1}^{(l)}]z^{-2},\end{array}$$

(19)

$${\epsilon }_{\mathsf{\text{Q}}}^{\left(l\right)}\left[z\right]=1-\left[n_{\mathsf{\text{re}},2}^{\left(l\right)}+m_{\mathsf{\text{re}},2}^{\left(l\right)}\right]z^{-1}+\left[n_{\mathsf{\text{re}},2}^{\left(l\right)}{m}_{\mathsf{\text{re}},2}^{\left(l\right)}-n_{\mathsf{\text{im}},1}^{\left(l\right)}{m}_{\mathsf{\text{im}},2}^{\left(l\right)}\right]z^{-2},$$

(20)

$${\eta }_{\mathsf{\text{Q}}}^{\left(l\right)}\left[z\right]=\left[n_{\mathsf{\text{im}},1}^{\left(l\right)}+m_{\mathsf{\text{im}},1}^{\left(l\right)}\right]z^{-1}+\left[n_{\mathsf{\text{im}},1}^{\left(l\right)}{m}_{\mathsf{\text{re}},1}^{\left(l\right)}+n_{\mathsf{\text{re}},2}^{\left(l\right)}{m}_{\mathsf{\text{im}},1}^{\left(l\right)}\right]z^{-2},$$

(21)

$${\rho }_{\mathsf{\text{Q}}}^{\left(l\right)}\left[z\right]=\left[n_{\mathsf{\text{re}},2}^{\left(l\right)}+r_{\mathsf{\text{re}},2}^{\left(l\right)}+m_{\mathsf{\text{re}},2}^{\left(l\right)}\right]z^{-1}+\left[s_{\mathsf{\text{re}},2}^{\left(l\right)}+n_{\mathsf{\text{im}},1}^{\left(l\right)}{r}_{\mathsf{\text{im}},2}^{\left(l\right)}-n_{\mathsf{\text{re}},2}^{\left(l\right)}{r}_{\mathsf{\text{re}},2}^{\left(l\right)}+n_{\mathsf{\text{im}},1}^{\left(l\right)}{m}_{\mathsf{\text{im}},2}^{\left(l\right)}-n_{\mathsf{\text{re}},2}^{\left(l\right)}{m}_{\mathsf{\text{re}},2}^{\left(l\right)}\right]z^{-2},$$

(22)

$${\gamma }_{\mathsf{\text{Q}}}^{\left(l\right)}\left[z\right]=1-\left[n_{\mathsf{\text{im}},1}^{\left(l\right)}+r_{\mathsf{\text{im}},1}^{\left(l\right)}+m_{\mathsf{\text{im}},1}^{\left(l\right)}\right]z^{-1}+\left[s_{\mathsf{\text{im}},1}^{\left(l\right)}-n_{\mathsf{\text{im}},1}^{\left(l\right)}{r}_{\mathsf{\text{re}},1}^{\left(l\right)}-n_{\mathsf{\text{re}},2}^{\left(l\right)}{r}_{\mathsf{\text{im}},1}^{\left(l\right)}-n_{\mathsf{\text{im}},1}^{\left(l\right)}{m}_{\mathsf{\text{re}},1}^{\left(l\right)}-n_{\mathsf{\text{re}},2}^{\left(l\right)}{m}_{\mathsf{\text{im}},1}^{\left(l\right)}\right]z^{-2}.$$

(23)

In this way, the complex-valued output and the exact behavior of each transfer function can be solved analytically in different I/Q mismatch scenarios. As a result, the complex output of an individual stage with nonideal matching of the I and Q branches becomes

$$V_l\left[z\right]=V_{\mathsf{\text{I}},l}\left[z\right]+j{V}_{\mathsf{\text{Q}},l}\left[z\right]=\mathsf{\text{ST}}{\mathsf{\text{F}}}_l\left[z\right]U_l\left[z\right]+\mathsf{\text{IST}}{\mathsf{\text{F}}}_l\left[z\right]U_l^{*}\left[z^{*}\right]+\mathsf{\text{NT}}{\mathsf{\text{F}}}_l\left[z\right]E_l\left[z\right]+\mathsf{\text{INT}}{\mathsf{\text{F}}}_l\left[z\right]E_l^{*}\left[z^{*}\right],$$

(24)

where superscript asterisk (*) denotes complex conjugation, and the transfer functions are, based on (10) and (17) (omitting [z] from the modulator coefficient variables of (11)-(16) and (18)-(23) for notational convenience), given by

$$\mathsf{\text{ST}}{\mathsf{\text{F}}}_l\left[z\right]=\frac{{\gamma }_Q^{\left(l\right)}{\alpha }_{\mathsf{\text{I}}}^{\left(l\right)}+{\gamma }_{\mathsf{\text{I}}}^{\left(l\right)}{\alpha }_Q^{\left(l\right)}-{\rho }_Q^{\left(l\right)}{\beta }_{\mathsf{\text{I}}}^{\left(l\right)}-{\rho }_{\mathsf{\text{I}}}^{\left(l\right)}{\beta }_Q^{\left(l\right)}}{2\left({\gamma }_{\mathsf{\text{I}}}^{\left(l\right)}{\gamma }_Q^{\left(l\right)}+{\rho }_{\mathsf{\text{I}}}^{\left(l\right)}{\rho }_Q^{\left(l\right)}\right)}+j\frac{{\rho }_{\mathsf{\text{I}}}^{\left(l\right)}{\alpha }_Q^{\left(l\right)}+{\rho }_Q^{\left(l\right)}{\alpha }_{\mathsf{\text{I}}}^{\left(l\right)}+{\gamma }_Q^{\left(l\right)}{\beta }_{\mathsf{\text{I}}}^{\left(l\right)}+{\gamma }_{\mathsf{\text{I}}}^{\left(l\right)}{\beta }_Q^{\left(l\right)}}{2\left({\gamma }_{\mathsf{\text{I}}}^{\left(l\right)}{\gamma }_Q^{\left(l\right)}+{\rho }_{\mathsf{\text{I}}}^{\left(l\right)}{\rho }_Q^{\left(l\right)}\right)},$$

(25)

$$\mathsf{\text{IST}}{\mathsf{\text{F}}}_l\left[z\right]=\frac{{\gamma }_Q^{\left(l\right)}{\alpha }_I^{\left(l\right)}-{\gamma }_I^{\left(l\right)}{\alpha }_Q^{\left(l\right)}+{\rho }_Q^{\left(l\right)}{\beta }_I^{\left(l\right)}-{\rho }_I^{\left(l\right)}{\beta }_Q^{\left(l\right)}}{2\left({\gamma }_I^{\left(l\right)}{\gamma }_Q^{\left(l\right)}+{\rho }_I^{\left(l\right)}{\rho }_Q^{\left(l\right)}\right)}+j\frac{{\rho }_Q^{\left(l\right)}{\alpha }_I^{\left(l\right)}-{\rho }_I^{\left(l\right)}{\alpha }_Q^{\left(l\right)}+{\gamma }_I^{\left(l\right)}{\beta }_Q^{\left(l\right)}-{\gamma }_Q^{\left(l\right)}{\beta }_I^{\left(l\right)}}{2\left({\gamma }_I^{\left(l\right)}{\gamma }_Q^{\left(l\right)}+{\rho }_I^{\left(l\right)}{\rho }_Q^{\left(l\right)}\right)},$$

(26)

$$\mathsf{\text{NT}}{\mathsf{\text{F}}}_l\left[z\right]=\frac{{\gamma }_Q^{\left(l\right)}{\epsilon }_I^{\left(l\right)}+{\gamma }_I^{\left(l\right)}{\epsilon }_Q^{\left(l\right)}+{\rho }_I^{\left(l\right)}{\eta }_Q^{\left(l\right)}+{\rho }_Q^{\left(l\right)}{\eta }_I^{\left(l\right)}}{2\left({\gamma }_I^{\left(l\right)}{\gamma }_Q^{\left(l\right)}+{\rho }_I^{\left(l\right)}{\rho }_Q^{\left(l\right)}\right)}+j\frac{{\rho }_I^{\left(l\right)}{\epsilon }_Q^{\left(l\right)}+{\rho }_Q^{\left(l\right)}{\epsilon }_I^{\left(l\right)}-{\gamma }_Q^{\left(l\right)}{\eta }_I^{\left(l\right)}-{\gamma }_I^{\left(l\right)}{\eta }_Q^{\left(l\right)}}{2\left({\gamma }_I^{\left(l\right)}{\gamma }_Q^{\left(l\right)}+{\rho }_I^{\left(l\right)}{\rho }_Q^{\left(l\right)}\right)},$$

(27)

$$\mathsf{\text{INT}}{\mathsf{\text{F}}}_l\left[z\right]=\frac{{\gamma }_Q^{\left(l\right)}{\epsilon }_I^{\left(l\right)}-{\gamma }_I^{\left(l\right)}{\epsilon }_Q^{\left(l\right)}+{\rho }_I^{\left(l\right)}{\eta }_Q^{\left(l\right)}-{\rho }_Q^{\left(l\right)}{\eta }_I^{\left(l\right)}}{2\left({\gamma }_I^{\left(l\right)}{\gamma }_Q^{\left(l\right)}+{\rho }_I^{\left(l\right)}{\rho }_Q^{\left(l\right)}\right)}+j\frac{{\gamma }_Q^{\left(l\right)}{\eta }_I^{\left(l\right)}-{\gamma }_I^{\left(l\right)}{\eta }_Q^{\left(l\right)}+{\rho }_Q^{\left(l\right)}{\epsilon }_I^{\left(l\right)}-{\rho }_I^{\left(l\right)}{\epsilon }_Q^{\left(l\right)}}{2\left({\gamma }_I^{\left(l\right)}{\gamma }_Q^{\left(l\right)}+{\rho }_I^{\left(l\right)}{\rho }_Q^{\left(l\right)}\right)}.$$

(28)

In Section 3.2, the above analysis for the individual stages l ∈ {1, L} is combined to complete the closed-form overall model for the multistage QΣΔM.

Based on (24), the converter output consists of not only the (filtered) input signal and quantization noise but also their complex conjugates, which, in frequency domain, corresponds to spectral mirroring or imaging. Thus, based on (24), the so-called image rejection ratios (IRRs) of the l th stage are

$${\mathsf{\text{IRR}}}_{\mathsf{\text{STF}}}^{\left(l\right)}\left[e^{∕2\pi f{T}_{\mathsf{\text{s}}}}\right]=10{log}_{10}\left({\left|\mathsf{\text{ST}}{\mathsf{\text{F}}}_l\left[e^{j2\pi f{T}_{\mathsf{\text{s}}}}\right]\right|}^2∕{\left|\mathsf{\text{IST}}{\mathsf{\text{F}}}_l\left[e^{j2\pi f{T}_{\mathsf{\text{s}}}}\right]\right|}^2\right)$$

(29)

and

$${\mathsf{\text{IRR}}}_{\mathsf{\text{NTF}}}^{\left(l\right)}\left[e^{j2\pi f{T}_{\mathsf{\text{s}}}}\right]=10{log}_{10}\left({\left|\mathsf{\text{NT}}{\mathsf{\text{F}}}_l\left[e^{j2\pi f{T}_{\mathsf{\text{s}}}}\right]\right|}^2∕{\left|\mathsf{\text{INT}}{\mathsf{\text{F}}}_l\left[e^{j2\pi f{T}_{\mathsf{\text{s}}}}\right]\right|}^2\right),$$

(30)

where actual frequency-domain responses are attained with the substitution $z\leftarrow e^{j2\pi f{T}_{\mathsf{\text{S}}}}$ to the earlier transfer functions, where f is the frequency measured in Hertz and T_S is the sampling time. These IRR quantities describe the relation of the direct input signal and noise energy to the respective mismatch-induced MFI at the output signal. As an example, ${\mathsf{\text{IRR}}}_{\mathsf{\text{STF}}}^{\left(1\right)}\left(e^{j2\pi f_0{T}_{\mathsf{\text{S}}}}\right)=20\mathsf{\text{dB}}$ means that the power of the mismatch-induced (mirrored) conjugate input signal is 20 dB lower than the direct input signal at the frequency f₀. Similarly, ${\mathsf{\text{IRR}}}_{\mathsf{\text{NTF}}}^{\left(1\right)}\left(e^{j2\pi f_0{T}_{\mathsf{\text{S}}}}\right)=20\mathsf{\text{dB}}$ indicates that the nonconjugated quantization error level is 20 dB above the mirror image of the quantization error at the frequency f₀. Notice also that, in general, both IRRs are frequency-dependent functions.

3.2. Combined I/Q imbalance Effects of the Stages in Multistage QΣΔM

For multistage QΣΔM, as illustrated in Figure 4, the final output signal is defined as a difference of digitally filtered output signals of the stages [33]. Furthermore, like shortly discussed already, the first-stage input U₁[z] = U[z] while for l > 1, U _l [z] = E_l−1[z]. The output of the first stage, given by (24) with l = 1, is filtered with digital filter $H_1^{\mathsf{\text{D}}}\left[z\right]$ (usually matched to the STF of the second stage) and the output of the second stage, similarly given by (24) with l = 2, is filtered with $H_2^{\mathsf{\text{D}}}\left[z\right]$ (usually matched to the NTF of the first stage), and so on for l ∈ {1, L} . Thus, the final output in case of I/Q mismatches in all the stages can now be expressed as

$$V[z]={\displaystyle \sum _{l=1}^L(}-1{)}^{l+1}{H}_l^{\text{D}}[z]V_l[z].$$

(31)

Replacing V _l [z] in (31) with (24) for l ∈ {1, L} gives now an expression for the overall output as

$$V\left[z\right]=\sum _{l=1}^L{(-1)}^{l+1}{H}_l^{\mathsf{\text{D}}}\left[z\right]\left(\mathsf{\text{ST}}{\mathsf{\text{F}}}_l\left[z\right]U_l\left[z\right]+\mathsf{\text{IST}}{\mathsf{\text{F}}}_l\left[z\right]U_l^{*}\left[z^{*}\right]+\mathsf{\text{NT}}{\mathsf{\text{F}}}_l{E}_l\left[z\right]+\mathsf{\text{INT}}{\mathsf{\text{F}}}_l{E}_l^{*}\left[z^{*}\right]\right),$$

(32)

where the transfer functions are as defined in (25)-(28). Again, the digital filters are assumed matched to the analog transfer functions according to (6). As a concrete example, (32) can be evaluated for a three-stage (L = 3) QΣΔM, giving

$$\begin{array}{ll}\hfill V\left[z\right]& =H_1^{\mathsf{\text{D}}}\left[z\right]\left(\mathsf{\text{ST}}{\mathsf{\text{F}}}_1\left[z\right]U\left[z\right]+\mathsf{\text{IST}}{\mathsf{\text{F}}}_1\left[z\right]U^{*}\left[z^{*}\right]+\mathsf{\text{NT}}{\mathsf{\text{F}}}_1{E}_1\left[z\right]+\mathsf{\text{INT}}{\mathsf{\text{F}}}_1{E}_1^{*}\left[z^{*}\right]\right)\\ -H_2^{\mathsf{\text{D}}}\left[z\right]\left(\mathsf{\text{ST}}{\mathsf{\text{F}}}_2\left[z\right]E_1\left[z\right]+\mathsf{\text{IST}}{\mathsf{\text{F}}}_2\left[z\right]E_1^{*}\left[z^{*}\right]+\mathsf{\text{NT}}{\mathsf{\text{F}}}_2{E}_2\left[z\right]+\mathsf{\text{INT}}{\mathsf{\text{F}}}_2{E}_2^{*}\left[z^{*}\right]\right)\\ +H_3^{\mathsf{\text{D}}}\left[z\right]\left(\mathsf{\text{ST}}{\mathsf{\text{F}}}_3\left[z\right]E_2\left[z\right]+\mathsf{\text{IST}}{\mathsf{\text{F}}}_3\left[z\right]E_2^{*}\left[z^{*}\right]+\mathsf{\text{NT}}{\mathsf{\text{F}}}_3{E}_3\left[z\right]+\mathsf{\text{INT}}{\mathsf{\text{F}}}_3{E}_3^{*}\left[z^{*}\right]\right)\\ ={\mathsf{\text{STF}}}_2^{\mathsf{\text{D}}}\mathsf{\text{ST}}{\mathsf{\text{F}}}_1\left[z\right]U\left[z\right]+{\mathsf{\text{STF}}}_2^{\mathsf{\text{D}}}\mathsf{\text{IST}}{\mathsf{\text{F}}}_1\left[z\right]U^{*}\left[z^{*}\right]\\ +\left({\mathsf{\text{STF}}}_2^{\mathsf{\text{D}}}\left[z\right]\mathsf{\text{NT}}{\mathsf{\text{F}}}_1\left[z\right]-{\mathsf{\text{NTF}}}_1^{\mathsf{\text{D}}}\left[z\right]\mathsf{\text{ST}}{\mathsf{\text{F}}}_2\left[z\right]\right)E_1\left[z\right]+\left({\mathsf{\text{STF}}}_2^{\mathsf{\text{D}}}\left[z\right]\mathsf{\text{INT}}{\mathsf{\text{F}}}_1\left[z\right]+{\mathsf{\text{NTF}}}_1^{\mathsf{\text{D}}}\left[z\right]\mathsf{\text{IST}}{\mathsf{\text{F}}}_2\left[z\right]\right)E_1^{*}\left[z^{*}\right]\\ +\left(-{\mathsf{\text{NTF}}}_1^{\mathsf{\text{D}}}\left[z\right]\mathsf{\text{NT}}{\mathsf{\text{F}}}_2\left[z\right]+\left({\mathsf{\text{NTF}}}_1^{\mathsf{\text{D}}}\left[z\right]{\mathsf{\text{NTF}}}_2^{\mathsf{\text{D}}}\left[z\right]∕{\mathsf{\text{STF}}}_3^{\mathsf{\text{D}}}\left[z\right]\right)\mathsf{\text{ST}}{\mathsf{\text{F}}}_3\left[z\right]\right)E_2\left[z\right]\\ +\left(-{\mathsf{\text{NTF}}}_1^{\mathsf{\text{D}}}\left[z\right]\mathsf{\text{INT}}{\mathsf{\text{F}}}_2\left[z\right]+\left({\mathsf{\text{NTF}}}_1^{\mathsf{\text{D}}}\left[z\right]{\mathsf{\text{NTF}}}_2^{\mathsf{\text{D}}}\left[z\right]∕{\mathsf{\text{STF}}}_3^{\mathsf{\text{D}}}\left[z\right]\right)\mathsf{\text{IST}}{\mathsf{\text{F}}}_3\left[z\right]\right)E_2^{*}\left[z^{*}\right]\\ +\left({\mathsf{\text{NTF}}}_1^{\mathsf{\text{D}}}\left[z\right]{\mathsf{\text{NTF}}}_2^{\mathsf{\text{D}}}\left[z\right]\mathsf{\text{NT}}{\mathsf{\text{F}}}_3\left[z\right]∕{\mathsf{\text{STF}}}_3^{\mathsf{\text{D}}}\left[z\right]\right)E_3\left[z\right]+\left({\mathsf{\text{NTF}}}_1^{\mathsf{\text{D}}}\left[z\right]{\mathsf{\text{NTF}}}_2^{\mathsf{\text{D}}}\left[z\right]\mathsf{\text{INT}}{\mathsf{\text{F}}}_3\left[z\right]∕{\mathsf{\text{STF}}}_3^{\mathsf{\text{D}}}\left[z\right]\right)E_3^{*}\left[z^{*}\right]\\ =\mathsf{\text{ST}}{\mathsf{\text{F}}}_{\mathsf{\text{TOT}}}\left[z\right]U\left[z\right]+\mathsf{\text{IST}}{\mathsf{\text{F}}}_{\mathsf{\text{TOT}}}\left[z\right]U^{*}\left[z^{*}\right]+\mathsf{\text{NT}}{\mathsf{\text{F}}}_{\mathsf{\text{TOT}},1}\left[z\right]E_1\left[z\right]+\mathsf{\text{INT}}{\mathsf{\text{F}}}_{\mathsf{\text{TOT}},1}\left[z\right]E_1^{*}\left[z^{*}\right]\\ +\mathsf{\text{NT}}{\mathsf{\text{F}}}_{\mathsf{\text{TOT}},2}\left[z\right]E_2\left[z\right]+\mathsf{\text{INT}}{\mathsf{\text{F}}}_{\mathsf{\text{TOT}},2}\left[z\right]E_2^{*}\left[z^{*}\right]+\mathsf{\text{NT}}{\mathsf{\text{F}}}_{\mathsf{\text{TOT}},3}\left[z\right]E_3\left[z\right]+\mathsf{\text{INT}}{\mathsf{\text{F}}}_{\mathsf{\text{TOT}},3}\left[z\right]E_3^{*}\left[z^{*}\right]\end{array}$$

(33)

with digital filters $H_1^{\mathsf{\text{D}}}\left[z\right]={\mathsf{\text{STF}}}_2^{\mathsf{\text{D}}}\left[z\right],H_2^{\mathsf{\text{D}}}\left[z\right]={\mathsf{\text{NTF}}}_1^{\mathsf{\text{D}}}\left[z\right]$, and $H_3^{\mathsf{\text{D}}}\left[z\right]={\mathsf{\text{NTF}}}_1^{\mathsf{\text{D}}}\left[z\right]{\mathsf{\text{NTF}}}_2^{\mathsf{\text{D}}}\left[z\right]∕{\mathsf{\text{STF}}}_3^{\mathsf{\text{D}}}\left[z\right]$. It should be noted that STF_TOT[z]U[z] and NTF_TOT,3[z]E₃[z] correspond structurally to the ideal output given in (7). However, the responses of STF_TOT[z] and NTF_TOT,₃[z] can be altered when compared to ${\mathsf{\text{STF}}}_{\mathsf{\text{TOT}}}^{\mathsf{\text{ideal}}}\left[z\right]$ and ${\mathsf{\text{NTF}}}_{\mathsf{\text{TOT}}}^{\mathsf{\text{ideal}}}\left[z\right]$ because of possible common-mode errors in the modulator coefficients [25]. Consequently, the six additional terms in (33) are considered as mismatch-induced interference, which includes the leakage of the first- and second-stage noises and the corresponding MFI (conjugate) components. It should also be noticed that the first-stage quantization error terms ${\mathsf{\text{STF}}}_2^{\mathsf{\text{D}}}\left[z\right]\mathsf{\text{NT}}{\mathsf{\text{F}}}_1\left[z\right]E_1\left[z\right]$ and ${\mathsf{\text{NTF}}}_1^{\mathsf{\text{D}}}\left[z\right]\mathsf{\text{ST}}{\mathsf{\text{F}}}_2\left[z\right]E_1\left[z\right]$ do not reduce to zero because of noncommutativity of the complex transfer functions under I/Q imbalance [23]. On the other hand, second-stage quantization error vanishes if ${\mathsf{\text{NTF}}}_1^{\mathsf{\text{D}}}\left[z\right]\mathsf{\text{NT}}{\mathsf{\text{F}}}_2\left[z\right]$ and (${\text{(NTF}}_1^{\text{D}}[z]{\text{NTF}}_2^{\text{D}}[z]/{\text{STF}}_3^{\text{D}}[z]){\text{STF}}_3[z]$ are equal. This means that ${\mathsf{\text{NTF}}}_2^{\mathsf{\text{D}}}\left[z\right]$ and ${\mathsf{\text{STF}}}_3^{\mathsf{\text{D}}}\left[z\right]$ should be equal to their analog counterparts, which can realized with, e.g., adaptive digital filters [34, 35]. The matching can also be made more robust by designing the third stage to have unity signal response (STF₃ [z] = 1).

Now, based on (33), it is clear that filtered versions of the original and conjugate components of the input, the first-stage, the second-stage, and the third-stage quantization errors all contribute to the final output. In order to inspect the overall IRR of the complete multistage structure, the transfer functions of the original signals (the input and the errors) and their conjugate counterparts should be compared. Based on (33), this gives the following formulas for the three-stage case considered herein:

$$\mathsf{\text{IR}}{\mathsf{\text{R}}}_{\mathsf{\text{ST}}{\mathsf{\text{F}}}_{\mathsf{\text{TOT}}}}\left[e^{j2\pi f{T}_{\mathsf{\text{S}}}}\right]=10{log}_{10}\left({\left|\mathsf{\text{ST}}{\mathsf{\text{F}}}_{\mathsf{\text{TOT}}}\left[e^{j2\pi f{T}_{\mathsf{\text{S}}}}\right]\right|}^2∕{\left|\mathsf{\text{IST}}{\mathsf{\text{F}}}_{\mathsf{\text{TOT}}}\left[e^{j2\pi f{T}_{\mathsf{\text{S}}}}\right]\right|}^2\right),$$

(34)

$$\mathsf{\text{IR}}{\mathsf{\text{R}}}_{\mathsf{\text{NT}}{\mathsf{\text{F}}}_{\mathsf{\text{TOT,}}1}}\left[e^{j2\pi f{T}_{\mathsf{\text{S}}}}\right]=10{log}_{10}\left({\left|\mathsf{\text{NT}}{\mathsf{\text{F}}}_{\mathsf{\text{TOT}},1}\left[e^{j2\pi f{T}_{\mathsf{\text{S}}}}\right]\right|}^2∕{\left|\mathsf{\text{INT}}{\mathsf{\text{F}}}_{\mathsf{\text{TOT}},1}\left[e^{j2\pi f{T}_{\mathsf{\text{S}}}}\right]\right|}^2\right),$$

(35)

$$\mathsf{\text{IR}}{\mathsf{\text{R}}}_{\mathsf{\text{NT}}{\mathsf{\text{F}}}_{\mathsf{\text{TOT,}}2}}\left[e^{j2\pi f{T}_{\mathsf{\text{S}}}}\right]=10{log}_{10}\left({\left|\mathsf{\text{NT}}{\mathsf{\text{F}}}_{\mathsf{\text{TOT}},2}\left[e^{j2\pi f{T}_{\mathsf{\text{S}}}}\right]\right|}^2∕{\left|\mathsf{\text{INT}}{\mathsf{\text{F}}}_{\mathsf{\text{TOT}},2}\left[e^{j2\pi f{T}_{\mathsf{\text{S}}}}\right]\right|}^2\right),$$

(36)

$$\mathsf{\text{IR}}{\mathsf{\text{R}}}_{\mathsf{\text{NT}}{\mathsf{\text{F}}}_{\mathsf{\text{TOT,}}3}}\left[e^{j2\pi f{T}_{\mathsf{\text{s}}}}\right]=10{log}_{10}\left({\left|\mathsf{\text{NT}}{\mathsf{\text{F}}}_{\mathsf{\text{TOT}},3}\left[e^{j2\pi f{T}_{\mathsf{\text{S}}}}\right]\right|}^2∕{\left|\mathsf{\text{INT}}{\mathsf{\text{F}}}_{\mathsf{\text{TOT}},3}\left[e^{j2\pi f{T}_{\mathsf{\text{S}}}}\right]\right|}^2\right).$$

(37)

In addition to the above IRRs, the performance of a nonideal QΣΔM can be measured by the amount of total additional interference stemming from the implementation nonidealities. This can be expressed with interference rejection ratio Γ. In case of a three-stage QΣΔM, following from (33), the signal component (interference-free output) is defined as

$$\sigma \left(k\right)=\mathsf{\text{ST}}{\mathsf{\text{F}}}_{\mathsf{\text{TOT}}}\left(k\right)*u\left(k\right)+\mathsf{\text{NT}}{\mathsf{\text{F}}}_{\mathsf{\text{TOT}},3}\left(k\right)*e_3\left(k\right),$$

(38)

where impulse responses of the STF and third-stage NTF are convolving the overall input and third-stage quantization error, respectively. At the same time, the total interference component (total additional interference caused by the nonidealities) is defined as

$$\begin{array}{c}\tau \left(k\right)=\mathsf{\text{IST}}{\mathsf{\text{F}}}_{\mathsf{\text{TOT}}}\left(k\right)*u^{*}\left(k\right)+\mathsf{\text{NT}}{\mathsf{\text{F}}}_{\mathsf{\text{TOT}},1}\left(k\right)*e_1\left(k\right)+\mathsf{\text{INT}}{\mathsf{\text{F}}}_{\mathsf{\text{TOT}},1}\left(k\right)*e_1^{*}\left(k\right)\\ +\mathsf{\text{NT}}{\mathsf{\text{F}}}_{\mathsf{\text{TOT}},2}\left(k\right)*e_2\left(k\right)+\mathsf{\text{INT}}{\mathsf{\text{F}}}_{\mathsf{\text{TOT}},2}\left(k\right)*e_2^{*}\left(k\right)*+\mathsf{\text{INT}}{\mathsf{\text{F}}}_{\mathsf{\text{TOT}},3}\left(k\right)*e_3^{*}\left(k\right)*,\end{array}$$

(39)

where time-domain signal components are again convolved by respective transfer function impulse responses. It should be noted that, in case of ideal three-stage QΣΔM, (39) reduces to zero. Now, interference rejection ratio at any given useful signal band is given by the integrals of spectral densities $G_{\sigma }\left(e^{j2\pi f{T}_{\mathsf{\text{S}}}}\right)$ and $G_{\tau }\left(e^{j2\pi f{T}_{\mathsf{\text{S}}}}\right)$ of the above random signals σ(k) and τ(k), i.e.,

$${\Gamma }_1=\frac{{\int }_{f\in {\Omega }_{\mathsf{\text{C,}}1}}{G}_{\sigma }\left(e^{j2\pi f{T}_{\mathsf{\text{S}}}}\right)df}{{\int }_{f\in {\Omega }_{\mathsf{\text{C,}}1}}{G}_{\tau }\left(e^{j2\pi f{T}_{\mathsf{\text{S}}}}\right)df},$$

(40)

where integration is done over the desired signal band, defined as Ω_C,1 = {f_C,1 − W₁ /2, ..., f_C,1 + W₁ /2} (where W₁ is the bandwidth of the signal). If there are two parallel signals (two-band scenario), the interference rejection ratio of the second signal is calculated in similar manner:

$${\Gamma }_2=\frac{{\int }_{f\in {\Omega }_{\mathsf{\text{C,}}2}}{G}_{\sigma }\left(e^{j2\pi f{T}_{\mathsf{\text{S}}}}\right)df}{{\int }_{f\in {\Omega }_{\mathsf{\text{C,}}2}}{G}_{\tau }\left(e^{j2\pi f{T}_{\mathsf{\text{S}}}}\right)df},$$

(41)

where Ω_C,2 = { f_C,2 − W₂ /2, ..., f_C,2 + W₂/2}.

An example of interference rejection ratio analysis in receiver-dimensioning context is given in Section 5. In addition, the roles of the separate signal components are further illustrated with numerical results in Section 6.

4. QΣΔM Transfer Function Parametrization and Design for CR under I/Q Imbalance

In CR-type wideband receiver, signal dynamics can be tens of (even 50-60) dBs [5, 6]. With such signal composition, controlling linearity and image rejection of the receiver components is essential [5, 6, 9]. In this section, we concentrate on QΣΔM transfer function design under I/Q imbalance, having minimization of the input signal oriented MFI as the goal.

4.1. Transfer Function Parametrization for Reconfigurable CR Receivers

The NTF and STF of a QΣΔM can be designed by placing transfer function zeros and poles, parameterized and tuned (allowing reconfigurability) by the QΣΔM coefficients, inside the unit circle [18]. In the following, the design process is described for a second-order QΣΔM as a single-stage converter or an individual stage l of a multistage converter. This is then extended to multistage converters in Section 4.2.

Based on the numerator of (8), the NTF zeros of the second-order QΣΔM are defined by the loop-filter feedback coefficients, i.e.,

$${\phi }_{\mathsf{\text{NTF}},1}^{\left(l\right)}=M^{\left(l\right)}={\lambda }_{\mathsf{\text{NTF}},1}^{\left(l\right)}{e}^{j2\pi f_{\mathsf{\text{NTF,}}1}^{\left(l\right)}{T}_{\mathsf{\text{s}}}},$$

(42)

$${\phi }_{\mathsf{\text{NTF}},2}^{\left(l\right)}=N^{\left(l\right)}={\lambda }_{\mathsf{\text{NTF}},2}^{\left(l\right)}{e}^{j2\pi f_{\mathsf{\text{NTF,}}2}^{\left(l\right)}{T}_{\mathsf{\text{S}}}},$$

(43)

where ${\lambda }_{\mathsf{\text{NTF}},1}^{\left(l\right)}=\left|{\phi }_{\mathsf{\text{NTF}},1}^{\left(l\right)}\right|$ and ${\lambda }_{\mathsf{\text{NTF}},2}^{\left(l\right)}=\left|{\phi }_{\mathsf{\text{NTF}},2}^{\left(l\right)}\right|$, being usually set to unity for the zero-placement on the unit circle, and $f_{\mathsf{\text{NTF}},1}^{\left(l\right)}$ and $f_{\mathsf{\text{NTF}},2}^{\left(l\right)}$ are the frequencies of the two NTF notches. Thus, designing these complex gains tunable allows straightforward reconfigurability for NTF notch frequencies based on the spectrum-sensing information about the desired information signals. Common choice is to place NTF zeros on the desired signal band or in case of multi-band reception on those bands, generating the preferred noise-shaping effect. At the same time, the poles, which are common to the NTF and the STF, are solved based on the denominator of either (8) or (9), giving

$$\begin{array}{ll}\hfill {\psi }_{\mathsf{\text{common}},1}^{\left(l\right)}& =\frac{R^{\left(l\right)}+M^{\left(l\right)}+N^{\left(l\right)}+\sqrt{R^{{\left(l\right)}^2}+M^{{\left(l\right)}^2}+N^{{\left(l\right)}^2}+2R^{\left(l\right)}{N}^{\left(l\right)}-2R^{\left(l\right)}{M}^{\left(l\right)}-2M^{\left(l\right)}{N}^{\left(l\right)}+4S^{\left(l\right)}}}{2}\\ ={\lambda }_{\mathsf{\text{pole}},1}^{\left(l\right)}{e}^{j2\pi f_{\mathsf{\text{po}}1\mathsf{\text{e,}}1}^{\left(l\right)}{T}_{\mathsf{\text{S}}}},\end{array}$$

(44)

$$\begin{array}{ll}\hfill {\psi }_{\mathsf{\text{common}},2}^{\left(l\right)}& =\frac{R^{\left(l\right)}+M^{\left(l\right)}+N^{\left(l\right)}-\sqrt{R^{{\left(l\right)}^2}+M^{{\left(l\right)}^2}+N^{{\left(l\right)}^2}+2R^{\left(l\right)}{N}^{\left(l\right)}-2R^{\left(l\right)}{M}^{\left(l\right)}-2M^{\left(l\right)}{N}^{\left(l\right)}+4S^{\left(l\right)}}}{2}\\ ={\lambda }_{\mathsf{\text{po}}1\mathsf{\text{e}},2}^{\left(l\right)}{e}^{j2\pi f_{\mathsf{\text{po}}1\mathsf{\text{e,}}2}^{\left(l\right)}{T}_{\mathsf{\text{S}}}},\end{array}$$

(45)

where ${\lambda }_{\mathsf{\text{po}}1\mathsf{\text{e}},1}^{\left(l\right)}=\left|{\psi }_{\mathsf{\text{common}},1}^{\left(l\right)}\right|$ and ${\lambda }_{\mathsf{\text{po}}1\mathsf{\text{e}},2}^{\left(l\right)}=\left|{\psi }_{\mathsf{\text{common}},2}^{\left(l\right)}\right|$, which can be used to tune the magnitude of the poles and $f_{\mathsf{\text{po}}1\mathsf{\text{e}},1}^{\left(l\right)}$ and $f_{\mathsf{\text{po}}1\mathsf{\text{e}},2}^{\left(l\right)},$, are the frequencies of the poles. The coefficients M^(l)and N^(l)are already fixed according to (42), leaving R^(l)and S^(l)free to tune the pole placement. The poles can, e.g., be placed on the frequency bands of the desired signals to elevate the STF response and thus give gain for the desired signals. However, the pole placement elevates also the NTF response, and thus this kind of design is always a tradeoff between the noise-shaping and STF selectivity efficiencies.

On the other hand, the loop-filter coefficients (M^(l)and N^(l)) have also their effects on the STF zeros, which, however, can be further tuned with the input coefficients (A^(l), B^(l), and C^(l)) of the modulator. This is illustrated in case of second-order QΣΔM, based on (9), by the expressions

$$\begin{array}{ll}\hfill {\phi }_{\mathsf{\text{STF}},1}^{\left(l\right)}& =\left(1∕2A^{\left(l\right)}\right)\left(A^{\left(l\right)}{M}^{\left(l\right)}+A^{\left(l\right)}{N}^{\left(l\right)}-B^{\left(l\right)}\right)\\ +\left(1∕2A^{\left(l\right)}\right)\sqrt{B^{{\left(l\right)}^2}+A^{{\left(l\right)}^2}{M}^{{\left(l\right)}^2}+A^{{\left(l\right)}^2}{N}^{{\left(l\right)}^2}+2A^{\left(l\right)}{B}^{\left(l\right)}{M}^{\left(l\right)}-2A^{\left(l\right)}{B}^{\left(l\right)}{N}^{\left(l\right)}-2A^{\left(l\right)}{M}^{\left(l\right)}{N}^{\left(l\right)}-4A^{\left(l\right)}{C}^{\left(l\right)}}\\ ={\lambda }_{\mathsf{\text{STF}},1}^{\left(l\right)}{e}^{j2\pi f_{\mathsf{\text{STF,}}1}^{\left(l\right)}{T}_{\mathsf{\text{S}}}},\end{array}$$

(46)

$$\begin{array}{ll}\hfill {\phi }_{\mathsf{\text{STF}},2}^{\left(l\right)}& =\left(1∕2A^{\left(l\right)}\right)\left(A^{\left(l\right)}{M}^{\left(l\right)}+A^{\left(l\right)}{N}^{\left(l\right)}-B^{\left(l\right)}\right)\\ -\left(1∕2A^{\left(l\right)}\right)\sqrt{B^{{\left(l\right)}^2}+A^{{\left(l\right)}^2}{M}^{{\left(l\right)}^2}+A^{{\left(l\right)}^2}{N}^{{\left(l\right)}^2}+2A^{\left(l\right)}{B}^{\left(l\right)}{M}^{\left(l\right)}-2A^{\left(l\right)}{B}^{\left(l\right)}{N}^{\left(l\right)}-2A^{\left(l\right)}{M}^{\left(l\right)}{N}^{\left(l\right)}-4A^{\left(l\right)}{C}^{\left(l\right)}}\\ ={\lambda }_{\mathsf{\text{STF}},2}^{\left(l\right)}{e}^{j2\pi f_{\mathsf{\text{STF,}}2}^{\left(l\right)}{T}_{\mathsf{\text{S}}}},\end{array}$$

(47)

where ${\lambda }_{\mathsf{\text{STF}},1}^{\left(l\right)}=\left|{\phi }_{\mathsf{\text{STF}},1}^{\left(l\right)}\right|$ and ${\lambda }_{\mathsf{\text{STF}},2}^{\left(l\right)}=\left|{\phi }_{\mathsf{\text{STF}},2}^{\left(l\right)}\right|$. Thus, (46)-(47) clearly show that A^(l), B^(l), and C^(l)allow independent placement of the STF zeros. In proportion to the NTF zero analysis above, $f_{\mathsf{\text{STF}},1}^{\left(l\right)}$ and $f_{\mathsf{\text{STF}},2}^{\left(l\right)}$ are the frequencies of the two STF notches. The proposed way to design the STF includes setting $f_{\mathsf{\text{STF}},1}^{\left(l\right)}$ and $f_{\mathsf{\text{STF}},2}^{\left(l\right)}$ to be the mirror frequencies of the desired information signals (based on the spectrum-sensing information) to attenuate possible blockers on those critical frequency bands. More generally, these frequencies, and thus the STF zero locations, can be tuned to give preferred frequency-selective response for the STF. On the other hand, if frequency-flat STF design is preferred, then the zeros can be set to the origin by setting ${\lambda }_{\mathsf{\text{STF}},1}^{\left(l\right)}$ and ${\lambda }_{\mathsf{\text{STF}},2}^{\left(l\right)}$ to zero.

Usually, the first step in the QΣΔM NTF and STF design is to obtain the placements of the zeros and the poles as already discussed above. Thereafter, the modulator coefficient values realizing those zeros and poles should be found out. In the following, this procedure is explained for a second-order QΣΔM as the l th stage of a multistage QΣΔM. Practically, the goal is to find values for the input coefficients (A^(l), B^(l), and C^(l)), the loop-filter coefficients (M^(l)and N^(l)) and the feedback coefficients (R^(l)and S^(l)) that realize the STF zeros (${\phi }_{\mathsf{\text{STF}},1}^{\left(l\right)}$ and ${\phi }_{\mathsf{\text{STF}},2}^{\left(l\right)}$), the NTF zeros (${\phi }_{\mathsf{\text{NTF}},1}^{\left(l\right)}$ and${\phi }_{\mathsf{\text{NTF}},2}^{\left(l\right)}$), and the common poles (${\psi }_{\text{common},1}^{(l)}$ and ${\psi }_{\mathsf{\text{common}},2}^{\left(l\right)}$) fixed above based on the transfer function characteristics.

The numerator of the NTF, the numerator of the STF, and the denominator of both transfer functions are used to solve the coefficient values. To begin with, the loop-filter feedback coefficients M^(l), and N^(l), the numerator of the NTF can be expressed with the modulator coefficients of the respective stage, as in (8), or with the help of the respective zeros ${\phi }_{\mathsf{\text{NTF}},1}^{\left(l\right)}$ and ${\phi }_{\mathsf{\text{NTF}},2}^{\left(l\right)}$. Setting these expressions equal, i.e.,

$$1-\left(M^{\left(l\right)}+N^{\left(l\right)}\right)z^{-1}+\left(M^{\left(l\right)}{N}^{\left(l\right)}\right)z^{-2}=1-\left({\phi }_{\mathsf{\text{NTF}},1}^{\left(l\right)}+{\phi }_{\mathsf{\text{NTF}},2}^{\left(l\right)}\right)z^{-1}+\left({\phi }_{\mathsf{\text{NTF}},1}^{\left(l\right)}{\phi }_{\mathsf{\text{NTF}},2}^{\left(l\right)}\right)z^{-2},$$

(48)

allows for solving the coefficient values of the l th stage based on the zeros by setting the terms with similar delays equal. Thus,

$$M^{\left(l\right)}+N^{\left(l\right)}={\phi }_{\mathsf{\text{NTF}},1}^{\left(l\right)}+{\phi }_{\mathsf{\text{NTF}},2}^{\left(l\right)},$$

(49)

$$M^{\left(l\right)}{N}^{\left(l\right)}={\phi }_{\mathsf{\text{NTF}},1}^{\left(l\right)}{\phi }_{\mathsf{\text{NTF}},2}^{\left(l\right)},$$

(50)

giving

$$M^{\left(l\right)}={\phi }_{\mathsf{\text{NTF}},1}^{\left(l\right)},$$

(51)

$$N^{\left(l\right)}={\phi }_{\mathsf{\text{NTF,}}2}^{\left(l\right)}.$$

(52)

This result confirms that the NTF zeros are set by the complex-valued feedback gains of the loop integrators.

The input coefficients A^(l), B^(l), and C^(l)of the l th stage can be solved in similar manner, based on the STF numerator given in (9). Next, the numerator of (9) is set equal to the STF numerator presented with the respective zeros ${\phi }_{\mathsf{\text{STF}},1}^{\left(l\right)}$ and ${\phi }_{\mathsf{\text{STF}},2}^{\left(l\right)}$, i.e.,

$$\begin{array}{c}{A}^{\left(l\right)}+\left(B^{\left(l\right)}-N^{\left(l\right)}{A}^{\left(l\right)}-M^{\left(l\right)}{A}^{\left(l\right)}\right)z^{-1}+\left(C^{\left(l\right)}-N^{\left(l\right)}{B}^{\left(l\right)}+M^{\left(l\right)}{N}^{\left(l\right)}{A}^{\left(l\right)}\right)z^{-2}\\ =1-\left({\phi }_{\mathsf{\text{STF}},1}^{\left(l\right)}+{\phi }_{\mathsf{\text{STF}}.2}^{\left(l\right)}\right)z^{-1}+\left({\phi }_{\mathsf{\text{STF}},1}^{\left(l\right)}{\phi }_{\mathsf{\text{STF}},2}^{\left(l\right)}\right)z^{-2}.\end{array}$$

(53)

Now, A^(l), B^(l), and C^(l)can be solved setting the separate delay components equal. This gives

$$A^{\left(l\right)}=1,$$

(54)

$$B^{\left(l\right)}=N^{\left(l\right)}{A}^{\left(l\right)}+M^{\left(l\right)}{A}^{\left(l\right)}-\left({\phi }_{\mathsf{\text{STF}},1}^{\left(l\right)}+{\phi }_{\mathsf{\text{STF}},2}^{\left(l\right)}\right),$$

(55)

$$C^{\left(l\right)}=N^{\left(l\right)}{B}^{\left(l\right)}-M^{\left(l\right)}{N}^{\left(l\right)}{A}^{\left(l\right)}+{\phi }_{\mathsf{\text{STF}},1}^{\left(l\right)}{\phi }_{\mathsf{\text{STF}},2}^{\left(l\right)},$$

(56)

pronouncing that these coefficient can be used to tune the STF response. However, the NTF zeros should also be taken indirectly into account because they define the values of M^(l)and N^(l), as found out in (51)-(52).

At this point, only the feedback coefficients R^(l)and S^(l)of the l th stage remain unknown. Those can be solved using the common denominator of the NTF and the STF in (8) and (9). Again, the denominator of (8) and (9) is set equal to the denominator presented with the common poles of the transfer functions ${\psi }_{\text{common},1}^{(l)}$ and ${\psi }_{\mathsf{\text{common}},2}^{\left(l\right)}$. In other words,

$$\begin{array}{c}1-\left(M^{\left(l\right)}+N^{\left(l\right)}+R^{\left(l\right)}\right)z^{-1}+\left(M^{\left(l\right)}{N}^{\left(l\right)}+N^{\left(l\right)}{R}^{\left(l\right)}-S^{\left(l\right)}\right)z^{-2}\\ =1-\left({\psi }_{\mathsf{\text{common}},1}^{\left(l\right)}+{\psi }_{\mathsf{\text{common}},2}^{\left(l\right)}\right)z^{-1}+\left({\psi }_{\mathsf{\text{common}},1}^{\left(l\right)}{\psi }_{\mathsf{\text{common}},2}^{\left(l\right)}\right)z^{-2}.\end{array}$$

(57)

Again, setting the separate delay components equal gives solutions for the feedback coefficients:

$$R^{\left(l\right)}=-M^{}$$