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

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.

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.

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

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.

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.

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.

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].

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

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.

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.