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
(8)
At the same time, the ideal STF for the l th stage is defined as
(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.
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 of the stages (l ∈ {1,L}), the I branch outputs can be first shown to be
(10)
where the auxiliary variables multiplying the signal components are defined by the coefficients (see Figure 6) in the following manner:
(11)
(12)
(13)
(14)
(15)
(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
(17)
where
(18)
(19)
(20)
(21)
(22)
(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
(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
(25)
(26)
(27)
(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
(29)
and
(30)
where actual frequency-domain responses are attained with the substitution to the earlier transfer functions, where f is the frequency measured in Hertz and TS 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, 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, 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] = El−1[z]. The output of the first stage, given by (24) with l = 1, is filtered with digital filter (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 (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
(31)
Replacing V
l
[z] in (31) with (24) for l ∈ {1, L} gives now an expression for the overall output as
(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
(33)
with digital filters , and . It should be noted that STFTOT[z]U[z] and NTFTOT,3[z]E3[z] correspond structurally to the ideal output given in (7). However, the responses of STFTOT[z] and NTFTOT,3[z] can be altered when compared to and 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 and 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 and ( are equal. This means that and 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:
(34)
(35)
(36)
(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
(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
(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 and of the above random signals σ(k) and τ(k), i.e.,
(40)
where integration is done over the desired signal band, defined as ΩC,1 = {fC,1 − W1 /2, ..., fC,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:
(41)
where ΩC,2 = { fC,2 − W2 /2, ..., fC,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.