Frequency-Agile Multiband Quadrature Sigma-Delta Modulator for Cognitive Radio: Analysis, Design and...

2222 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 31, NO. 11, NOVEMBER 2013

Frequency-Agile Multiband QuadratureSigma-Delta Modulator for Cognitive Radio:Analysis, Design and Digital Post-Processing

Jaakko Marttila, Student Member, IEEE, Markus Allén, Student Member, IEEE, andMikko Valkama, Member, IEEE

Abstract—A quadrature ΣΔ analog-to-digital converter(ADC) is a promising solution for intermediate frequency digi-tizing software defined cognitive radio (CR) receivers because of,e.g., multiband capability and power efficiency. However, inher-ent coefficient mismatches between the in-phase and quadraturerails can severely damage the performance of such receiver bycreating mirror-frequency interference (MFI). In this article, anovel frequency-agile and reconfigurable transfer function de-sign, allowing digital post-compensation of the MFI, is proposed.The design is based on a novel closed-form transfer functionmodel for higher-order quadrature ΣΔ modulators (QΣΔMs)under implementation inaccuracies, proposed herein. By doingthe compensation in digital domain, it is possible to take intoaccount all error-sources of the receiver chain at once, including,e.g., a quadrature mixer before the ADC. This capability isobtained by preserving the mirror-band signal information andusing the noise transfer function of a QΣΔM to remove quan-tization noise from therein. This is demonstrated in a multibandscenario aimed for CR receivers, where a number of frequencychannels can be received and detected in parallel. Practicalexamples of the transfer function analysis under implementationinaccuracies and the post-compensation performance are givenwith a two-stage QΣΔM, having stage-orders of four, allowingeighth-order noise shaping.

Index Terms—Analog-to-digital conversion, complex filters,cognitive radio, digital radio, I/Q imbalance, mirror-frequencyinterference, radio receivers, sigma-delta modulation.

I. INTRODUCTION

A. Research Motivation

ASOFTWARE-DEFINED radio (SDR) based on direct-conversion or wideband intermediate frequency (IF) dig-

itization has been proposed as a solution for the cognitive(CR) physical layer [1]–[4]. Therein, analog-to-digital con-verters (ADCs) using bandpass ΣΔ modulator have receivedconsiderable attention [1], [2], [4]. Furthermore, quadratureΣΔ modulators (QΣΔMs) have been proven to have severalperformance benefits over the real counterparts [5], [6], mak-ing this a very interesting choice for development of future

Manuscript received April 19, 2012; revised August 31, 2012.The authors are with the Department of Communications Engineering,

Tampere University of Technology, Korkeakoulunkatu 1, 33720 Tampere,Finland (e-mail: {jaakko.marttila, markus.allen, mikko.e.valkama}@tut.fi).

This work was supported by the Finnish Funding Agency for Technologyand Innovation (Tekes) under the project “Enabling Methods for DynamicSpectrum Access and Cognitive Radio,” the Academy of Finland under project251138 “Digitally-Enhanced RF for Cognitive Radio Devices,” the AustrianCompetence Center in Mechatronics (ACCM), the TUT Graduate School, theHPY Research Foundation, and the Nokia Foundation.

Digital Object Identifier 10.1109/JSAC.2013.131101

CR receivers [7]. This receiver principle is illustrated in Fig. 1,where the components of focus in this article are highlightedwith grey background.

Traditional Nyquist ADCs (possibly with oversampling)divide the conversion precision equally on all the frequenciesand thus, similar resolution is used over the whole band even ifunnecessary. Furthermore, in wideband SDR receiver the res-olution demand might be even higher because of the increaseddynamic range due to numerous waveforms with differingpower levels entering the ADC [8]. Opposed to Nyquist con-verters, ΣΔ converters have different noise transfer function(NTF) and signal transfer function (STF) and inherent tradeoffbetween the sampling frequency and resolution [6], [9], al-lowing sophisticated allocation of the quantization precision.Thus, also the power used for sampling and quantizing canbe optimized, when only the interesting frequency bands arecleared of the quantization noise. With multiband QΣΔM, thisreconfigurability has even more degrees-of-freedom, allowingfrequency-agile reception of multiple narrowband signals or acouple of wideband waveforms in parallel [7].

Unfortunately, exploiting complex-valued signal processingwith quadrature circuits brings the problem of matching in-phase (I) and quadrature (Q) rails [6], [10]. This is usu-ally known as I/Q imbalance and generates mirror-frequencyinterference (MFI) via spectral mirroring. Especially underchallenging signal conditions typical for SDR and CR re-ceivers, the MFI can degrade the quality of the received signalssignificantly [7], [11], [12]. Specific to the QΣΔM, also thequantization noise generated inside the modulator experiencesspectral mirroring [6], [10]. This article proposes a noveltransfer function design for QΣΔMs combined with digitalpost-compensation of exactly this interference, taking intoaccount the mirror-frequency interference originating fromboth the input signal and quantization noise. The compensationblock is shown at principal level also in Fig. 1. Other possibleerror-sources, such as sampling jitter or modulator nonlinearityare not considered in this work.

B. State-of-the-Art in Quadrature ΣΔ Modulator Design andMirror-Frequency Interference Compensation

A single-stage quadrature ΣΔ converter for single-band IFdigitization was proposed in [6]. Therein, the STF is usedto attenuate out-of-band signal content in the input of theQΣΔM. In addition, one NTF zero is used to create a noise

0733-8716/13/$31.00 c© 2013 IEEE

MARTTILA et al.: FREQUENCY-AGILE MULTIBAND QUADRATURE SIGMA-DELTA MODULATOR FOR COGNITIVE RADIO: ANALYSIS, DESIGN. . . 2223

f

f

f0

DSP post-compen-sation forhardware

nonidealities

IntegratedRF ampand filter

Analog DigitalI

Q

�/2

Multiband

ADC

Quadrature

��

Filteringand

AGC

LO

0

f

f

fRF IF

Spectrumsensing

Final down-conversion,decimation

anddetection

( ). *

Fig. 1. Block diagram of a multiband low-IF quadrature receiver, based onfrequency-agile QΣΔM. Principal spectra, where the two light grey signalsare the desired ones, are illustrating the signal compositions at each stage.Post-compensation scheme is illustrated in simplified format.

shaping notch on the mirror-band, in order to reduce the MFIoriginating from the quantization noise. However, this doesnot take into account the mirroring of the input signal. Thisis very important because in a CR system, the power level ofthe mirror-band signal can be tens of dBs higher compared tothe desired signal [11]. The MFI might be originating fromeither the analog radio front-end or the QΣΔM itself. If theerror source, such as an I/Q mixer, is located early in thereceiver chain, the filtering before the ADC cannot mitigatethis problem. Thus, it is desirable to compensate the MFI ofthe whole receiver in the digital domain.

For the digital interference compensation as such, there aremultiple solutions in the literature, e.g., [13]–[17]. In general,it is common for the digital compensation that some informa-tion of the original interference sources should be availablein the digital domain. Precisely this information is lost ifthe mirror-frequency signals are filtered out in the analogdomain. In this article, a multiband STF design preservingthe mirror-band information, preliminarily proposed by theauthors in [18], is further developed and analyzed in realisticconditions. This is done with the help of a novel transferfunction analysis method under implementation inaccuraciesproposed in Section III. The analysis method is one of the keycontributions of this article and allows derivation of the NTFand the STF, giving also the image noise transfer function(INTF) and the image signal transfer function (ISTF) of anI/Q mismatched general multistage QΣΔM having arbitrarynumber of stages of arbitrary order. The multiband design ofQΣΔM transfer functions has been discussed in [7], [19]–[21].In [20], the basic multiband concept was proposed, being thendeveloped further for CR devices in [7], [19]. Moreover, inaddition to [6], designs mitigating the MFI generated insidethe QΣΔM are proposed in [7], [19], [22]–[27]. However,none of these designs is able to mitigate MFI appearingbefore the actual ΣΔ modulator (e.g., the I/Q mixer as anerror source) [7], [19], [26] or to avoid inserting calibrationsignals [23], additional analog circuitry [24], [25] or impairingthe sampling properties by sharing the capacitors [27]. Theselimitations are overcome by the proposed QΣΔM solutions inthis article.

In demonstrating the overall receiver concept and perfor-mance, the blind MFI suppression algorithm published in [13]

v k( )

e k( )

R(2)

u k( )

R( )P

B(1)

A

R(1)

...

...

...

B( )P

B(2)

additivequantization

noise

M( )P

z-1

M(2)

z-1

M(1)

z-1

Fig. 2. Discrete-time linearized model of a P th-order QΣΔM with complex-valued signals and coefficients.

is deployed. This frequency-independent algorithm is chosenbased on the inherent oversampling present in the ΣΔ con-version (assuming mild frequency selectivity inside a singlesubband) and the simplicity of the blockwise compensator.

The rest of this article is organized as follows. In Section II,QΣΔM basics are reviewed whereas Section III proposes anovel transfer function analysis method for an I/Q mismatchedmultistage QΣΔM. The transfer function design facilitatingthe digital post-compensation of MFI and the compensation it-self are proposed in Section IV. Thereafter, Section V presentsthe results of the design and compensation scheme usingboth closed-form transfer function analysis and computersimulations. Finally, Section VI concludes the article.

II. QUADRATURE ΣΔ MODULATION FOR COGNITIVERADIO

A. Basics of Quadrature ΣΔ Modulation

The analytical presentation of a QΣΔM is based on [6].The structure of a P th order QΣΔM, employing complex-valued input u(k), quantization noise e(k) and output v(k) ispresented in Fig. 2. The ΣΔ modulator architecture consid-ered in this article is a complex-valued version of cascadedintegrators with distributed feedback and input [6], [28]. Thisallows the use of frequency-asymmetric transfer functions(NTF and STF). By adopting a linear model of the modulator[6], the output of an ideal modulator (without implementationinaccuracies) is given in z-domain by

V id[z] = STF [z]U [z] +NTF [z]E[z]. (1)

The P th order structure of Fig. 2 allows placing P NTFand STF zeros on the conversion band, together with P polescommon to both of the transfer functions. The NTF of theQΣΔM is given by

NTF [z] =1

1−P∑

p=1R(p)

p∏i=1

z−1

1−M(i)z−1

(2)

and the corresponding STF is defined as

STF [z] =

A+P∑

p=1B(p)

p∏i=1

z−1

1−M(i)z−1

1−P∑

p=1R(p)

p∏i=1

z−1

1−M(i)z−1

, (3)

where the z−1/(1−M (i)z−1) terms are the transfer functionsof the complex loop filters (integrators).


It is a common choice to place the NTF zeros (defined byM (1) to M (P ) in Fig. 2 and (2) on the desired signal band, inorder to shape the noise towards out-of-band frequencies [6].At the same time, the STF zeros (defined by A and B(1) toB(P ) in Fig. 2 and (3)) can be used to implement part of thereceiver selectivity filtering inside the QΣΔM [6], [7], [19].The exact locations of the common poles (defined by R(1) toR(P ) in Fig. 2 and (2)-(3)) are, on the other hand, a questionof design optimization. One well-documented choice for thepoles is to use Butterworth positions around the NTF zeros [6].The transfer function design aspects unden implementationinaccuracies will be further discussed in Section IV.

B. Multiband Transfer Function Design

A multiband transfer function design for QΣΔM was orig-inally introduced in [20] and targeted for CR devices in [19]with further discussions in [7] and [21]. It is an efficient toolaiming towards a flexible and reconfigurable A/D interfacefor CR devices. Herein, the basics of the concept are reviewedshortly and further details can be checked from, e.g., [7], [19].

With quadrature ΣΔ modulation, the P zeros placed on theconversion band do not need to be contiguous. This propertycan be used, e.g., to receive multiple noncontiguous frequencybands and information signals. In this kind of scenario, TheNTF zeros are divided to remove the noise from those bands.At the same time, the STF is used to attenuate the out-of-bandsignal content, preserving only the desired signals.

This transfer function design principle would be suitablefor an ideal receiver. However, the effects of implemen-tation inaccuracies in the receiver hardware set additionalchallenges, which should also be taken into account in thedesign. The novel multiband reception combined with digitalpost-compensation of interference induced by the precedingradio front-end or the QΣΔM itself is discussed further inSection IV.

C. Multistage Quadrature ΣΔ Modulation

Multistage ΣΔ modulation is an efficient way to improvethe noise shaping efficiency of a ΣΔ ADC without impracticalincrease in the order of a single modulator stage. In [7], ageneral transfer function model for an ideal QΣΔM having Lstages was introduced. This is reviewed here as a basis for thefollowing analysis, taking also implementation inaccuracies ofthe arbitrary order stages into account (Section III).

The output of an ideal L-stage QΣΔM, depicted in Fig. 3,is given by

V id[z] =L∑

l=1

(−1)l+1HDl [z]V id

l [z], (4)

where

V idl [z] = STF id

l [z]Ul[z] +NTF idl [z]El[z],

l ∈ {1, 2, ..., L} (5)

and

v k( )

-

H z2( )

H z1( )

u k( )v k1( )

e k1( )

v k2( )

e k2( )

-

u k e k2 1( )= ( )

D

DFirst-stage

Q M��

Second-stage

Q M��

-

H zL( )v kL( ) DLth-stage

Q M��

e kL( )

-

u k e kL L-( )= ( )1

u k u k1( )= ( )

Fig. 3. Multistage QΣΔM with arbitrary-order noise shaping in all theindividual stages. Filters HD

1 [z] to HDL [z] are implemented digitally.

HDl [z] =

HD1 [z]

L−1∏l=1

NTF idl [z]

L∏l=2

STF idl [z]

, l ∈ {1, 2, ..., L}. (6)

The digital filters HDl [z] are used to match the nonshaped

digital noise estimate to the NTF-shaped noise at the outputof the previous stage. In addition, (4) can be rewritten as

V id[z] =STF id1 [z]STFD

2 [z]U [z] +

L∏l=1

NTF idl [z]

L∏l=3

STF idl [z]

EL[z]

=STF idTOT[z]U [z] +NTF id

TOT[z]EL[z],

(7)

from where it is clearly visible that the STF of the first stageis shaping the signal component, being thereafter filtered withthe STF of the second stage and on top of this appears thequantization noise of the last stage shaped by the product ofall the stage NTFs (assuming the STFs for stages l ≥ 3 to beunity).

III. CLOSED-FORM TRANSFER FUNCTION ANALYSIS FORARBITRARY-ORDER MULTISTAGE QΣΔM UNDER

IMPLEMENTATION INACCURACIES

In this section, a novel matrix algebraic method is pro-posed for transfer function analysis of I/Q mismatched mul-tistage quadrature ΣΔ modulators. Such analysis is missingin the current state-of-the-art literature. The proposed analysismethod has the advantage of being easily applicable to coverarbitrary-order QΣΔMs. This is demonstrated by derivingthe four transfer functions (STF, ISTF, NTF and INTF) fora P th order modulator. First, the matrix representation ofan I/Q mismatched complex multiplication and division areintroduced, forming the basis for the transfer function analysisproposed in the following.


are,1

are,2

aim,2

u kI( ) v kI,mm( )

v kQ,mm( )u kQ( ) -

aim,1

Au k( ) v k( )

Fig. 4. Complex multiplication v(k) = Au(k) implemented with idealcomplex-valued (left) and nonideal real-valued (right) signal processing.

A. Matrix Representation of Mismatched Complex Multiplica-tion and Division

Realizing a complex multiplication is an essential part ofthe QΣΔM implementation. This is because the input of themodulator is a complex-valued signal and modulator coeffi-cients are complex-valued (see Fig. 2). Thus, the P th orderQΣΔM of Fig. 2 has generally 3P+1 complex multiplications(P + 1 inputs, P loop filters and P feedbacks). The matrixrepresentation of a complex number [29], [30] is used here todescribe those multiplications.

A complex multiplication is implemented with four realmultiplications as shown in Fig. 4. Mathematically this isformulated as

v(k) = vI(k) + jvQ(k)

= Au(k) = (are + jaim)(uI(k) + juQ(k))

= areuI(k)− aimuQ(k) + jaimuI(k) + jareuQ(k),

(8)

assuming ideal matching of the real-valued coefficients(are,1 = are,2 = are and aim,1 = aim,2 = aim). Introducingmismatches to the multiplying coefficients (see Fig. 4), due toinherent implementation inaccuracies, gives

vmm(k) =are,1uI(k)− aim,2uQ(k)

+ jaim,1uI(k) + jare,2uQ(k).(9)

From (9), the real and imaginary parts of the result canbe solved to be vI,mm(k) = are,1uI(k) − aim,2uQ(k) andvQ,mm(k) = aim,1uI(k) + are,2uQ(k).

Now, according to basic matrix algebra, a 1 × 2 vectorconsisting of the real valued outputs (vI,mm(k) and vQ,mm(k))of the mismatched complex multiplication can be solved bysingle matrix multiplication. This holds when the multipliercoefficients (are,1, aim,1, are,2 and aim,2) are used to form amatrix A. This A is used to multiply a 1×2 vector consistingof the real valued inputs (uI(k) and uQ(k)) to give the outputs,i.e.,

[vI,mm(k) vQ,mm(k)

]=[uI(k) uQ(k)

]A

=[uI(k) uQ(k)

] [ are,1 aim,1

−aim,2 are,2

]=[are,1uI(k)− aim,2uQ(k) aim,1uI(k) + are,2uQ(k)

].

(10)

Now, by comparing (9) to (10), it is evident that the resultsare equal.

Furthermore, a division of two complex-valued coefficientsA = are + jaim and B = bre + jbim, being ideally defined as

A

B=

AB∗

|B|2 =arebre + aimbim + j(arebim − aimbre)

b2re + b2im, (11)

can also be represented in matrix format, taking inherentcoefficient mismatches into account. This is done by right-multiplying A, defined above, with the inverse of the matrixB, defined similar to A. This is justified by assuming thecoefficient matrices invertible because of the 2 × 2 structurewhere the real part (diagonal) elements share the same signand the imaginary part (cross diagonal) elements have oppositesigns, making the determinant nonzero in practical cases. Thisresults in

AB−1 =

[are,1 aim,1−aim,2 are,2

] [bre,1 bim,1−bim,2 bre,2

]−1

=1

bre,1bre,2 + bim,1bim,2×[

are,1bre,2 + aim,1bim,2 −are,1bim,1 + aim,1bre,1

are,2bim,2 − aim,2bre,2 are,2bre,2 + aim,2bim,1

],

(12)

giving the real and imaginary parts of the mismatched scalarcomplex division, opposed to (11) where ideal coefficients areassumed.

Although the complex division operation is not specifi-cally implemented in ΣΔ structure, it is an essential partof the transfer function analysis for such system. A simpleexample of this is the transfer function of a single loop-filter, H [z] = 1/(z − M). Now, the four transfer functioncomponents induced by the implementation inaccuracies [10]can be verified to be

[Hre,1[z] Him,1[z]−Him,2[z] Hre,2[z]

]= (Iz −M)

−1

=

[z −mre,1 −mim,1

mim,2 z −mre,2

]−1

=1

(z −mre,1)(z −mre,2) +mim,1mim,2×[

z −mre,2 mim,1−mim,2 z −mre,1

],

(13)

where M is defined in a corresponding manner to A and Babove and I is a 2×2 identity matrix. Assuming ideal matchingof the real coefficients, (13) simplifies to

[Hre[z] Him[z]−Him[z] Hre[z]

]=[z−mre

(z−mre)2+m2

im

mim(z−mre)

2+m2im−mim

(z−mre)2+m2

im

z−mre(z−mre)

2+m2im

],

(14)

giving finally


H [z] = Hre[z] + jHim[z] =1

z −mre − jmim=

1

z −M.

(15)

From the above analysis, it can be concluded that by replac-ing the ideal complex multipliers by the matrix multiplicationof (10), the effects of implementation inaccuracies can beincluded in a straightforward manner. In addition, the right-multiplication with the inverse of the coefficient matrix can beused to replace the complex division in the transfer functionanalysis, as shown in (13)-(15).

B. Arbitrary-Order Multistage QΣΔM Transfer FunctionAnalysis with Matrix Representation

Now, the matrix representation of a complex multiplica-tion is used to derive the output and the transfer functionsof I/Q mismatched arbitrary-order (order P ) multistage (Lstages) QΣΔM. The derivation is based on the ideal transferfunctions introduced in Section II. The scalar multiplicationsin (1)–(7) are replaced by matrix multiplications applyingthe possibly mismatched real-valued modulator coefficients.These coefficients are illustrated in Fig. 5 for the lth stageof a multistage QΣΔM, having order P . In this way, thework of manually analyzing individual signal paths of sucha complicated structure can be avoided.

To start with, the coefficient matrices are defined in thefollowing manner. The input coefficients in the branch feedingthe quantizer of the lth stage are defined by

A(l) =

[a(l)re,1 a

(l)im,1

−a(l)im,2 a

(l)re,2

]. (16)

In a similar manner, the rest of the input coefficients of thelth stage are defined by

B(p,l) =

[b(p,l)re,1 b

(p,l)im,1

−b(p,l)im,2 b

(p,l)re,2

], (17)

where p ∈ {1, 2, ..., P}, going through the branches. Likewise,the loop filter and feedback coefficients are defined by

M(p,l) =

[m

(p,l)re,1 m

(p,l)im,1

−m(p,l)im,2 m

(p,l)re,2

](18)

and

R(p,l) =

[r(p,l)re,1 r

(p,l)im,1

−r(p,l)im,2 r

(p,l)re,2

], (19)

having p ∈ {1, 2, ..., P} again.Now, inserting these matrices into (2) in place of the (ideal)

scalar coefficients results in

[NTF

(l)re,1[z] NTF

(l)im,1[z]

−NTF(l)im,2[z] NTF

(l)re,2[z]

]=

(I−

P∑p=1

R(p,l)

p∏i=1

Iz−1(I−M(i,l)z−1

)−1)−1

,

(20)

where NTF(l)re,1[z], NTF

(l)im,1[z], NTF

(l)im,2[z] and NTF

(l)re,2[z]

are the parallel real components of the widely-linear I/Qmismatched lth stage noise transfer function NTF (l)[z]. Theactual lth stage complex NTF and INTF can be solved by

NTFl[z] =NTF

(l)re,1[z] +NTF

(l)re,2[z]

2

+ jNTF

(l)im,1[z] +NTF

(l)im,2[z]

2

(21)

and

INTFl[z] =NTF

(l)re,1[z]−NTF

(l)re,2[z]

2

+ jNTF

(l)im,1[z]−NTF

(l)im,2[z]

2,

(22)

respectively [10].In parallel, the STF and the ISTF of the lth stage can be

solved by replacing the scalar coefficients of (3) with thematrices of (16)-(19). This results in the real valued STFcomponents given by

[STF

(l)re,1[z] STF

(l)im,1[z]

−STF(l)im,2[z] STF

(l)re,2[z]

]=

(A(l) +

P∑p=1

B(p,l)

p∏i=1


)−1)×

(I−

P∑p=1

R(p,l)

p∏i=1


)−1)−1

.

(23)

Now, the STF and the ISTF can be solved [10] by

STFl[z] =STF

(l)re,1[z] + STF

(l)re,2[z]

2

+ jSTF

(l)im,1[z] + STF

(l)im,2[z]

2

(24)

and

ISTFl[z] =STF

(l)re,1[z]− STF

(l)re,2[z]

2

+ jSTF

(l)im,1[z]− STF

(l)im,2[z]

2,

(25)

respectively.The actual output of the lth stage can be derived either

using real-valued V(l)

I [z] and V(l)

Q [z] or going for the complex-valued output signal Vl[z] The real valued output signals areobtained using the transfer function matrices of (20) and (23)to multiply the in-phase and quadrature components of thequantization noise and the input signal as 1 × 2 vectors,respectively. This gives


e kI ( )

e kQ( )

-

are,1

are,2

aim,2

aim,1

v kI ( )

v kQ( )

-

additivequantization

noise

z-1

mre,1

mre,2

mim,1

mim,2

-

-

-

bre,2

bim,2 bre,1

bim,1

rre,1

rre,2

rim,2

rim,1

z-1

-z

-1

mre,1

mre,2

mim,1

mim,2

-

-

-

bre,2

bim,2 bre,1

bim,1

rre,1

rre,2

rim,2

rim,1

z-1

-

(2, )l(2, )l

(2, )l(2, )l

(2, )l

(2, )l

(1, )l (1, )l

(1, )l(1, )l

(1, )l

(1, )l

( )l( )l

( )l( )l(1, )l

(1, )l(1, )l(1, )l

(1, )l (1, )l

(2, )l

(2, )l(2, )l

(2, )l

(2, )l(2, )l

z-1

mre,1

mre,2

mim,1

mim,2

-

-

-

bre,2

bim,2 bre,1

bim,1

rre,1

rre,2

rim,2

rim,1

z-1

-

u kI ( )

u kQ( )

( , )P l( , )P l

( , )P l

( , )P l

( , )P l

( , )P l

( , )P l

( , )P l( , )P l

( , )P l

( , )P l( , )P l

......

...

...

...

...

...

...

( )l

( )l

( )l

( )l

( )l

( )l

Fig. 5. Implementation structure of the lth QΣΔM stage of order P in a multistage QΣΔM with parallel real signals and coefficients taking possiblemismatches into account.

V[z] =U[z]

[STF

(l)re,1[z] STF

(l)im,1[z]

−STF(l)im,2[z] STF

(l)re,2[z]

]

+E[z]

[NTF

(l)re,1[z] NTF

(l)im,1[z]

−NTF(l)im,2[z] NTF

(l)re,2[z]

],

(26)

where

V[z] =[V

(l)I [z] V

(l)Q [z]

], (27)

U[z] =[U

(l)I [z] U

(l)Q [z]

], (28)

and

E[z] =[E

(l)I [z] E

(l)Q [z]

]. (29)

On the other hand, the four complex-valued transfer functions,given by (21), (22), (24) and (25), can be used to analyze thefiltering of the quantization noise El[z], the complex conjugateof the quantization noise E∗

l [z∗], the input signal Ul[z] and the

complex conjugate of the input signal U∗l [z

∗], resulting in

Vl[z] = STFl[z]U [z] + ISTFl[z]U∗l [z

∗]+NTFl[z]El[z] + INTFl[z]E

∗l [z

∗].(30)

Now, the overall output of a multistage QΣΔM is given by

V [z] =

L∑l=1

(−1)l+1

HDl [z]Vl[z], (31)

which can be written, using the quantization noise of theprevious stage as an input for the following one, as

V [z] =HD1 [z]

(STF1[z]U [z] + ISTF1[z]U

∗[z∗]

+NTF1[z]E1[z] + INTF1[z]E∗1 [z

∗])

+

L∑l=2

(−1)l+1

HDl [z]

(STFl−1[z]El−1[z]

+ ISTFl−1[z]E∗l−1[z

∗] +NTFl[z]El[z]

+ INTFl[z]E∗l [z

∗]).

(32)

From (32), it is finally possible to deduce the overall transferfunctions for the input signal and the quantization noise of allL stages and respective conjugate components, given as

STFTOT[z] = HD1 [z]STF1[z],

ISTFTOT[z] = HD1 [z]ISTF1[z],

NTFTOT,l[z] = HDl [z]NTFl[z]−HD

l+1[z]STFl+1[z],

l ∈ {1, 2, ..., L− 1},INTFTOT,l[z] = HD

l [z]INTF [z] +HDl+1[z]ISTFl+1[z],

l ∈ {1, 2, ..., L− 1},NTFTOT,L[z] = HD

L [z]NTFL[z],

INTFTOT,L[z] = HDL [z]INTFL[z].

(33)

Considering now a practical example of two-stage QΣΔMwith L = 2, application of (32) gives


V [z] =HD1 [z]

(STF1[z]U [z] + ISTFl[z]U

∗[z∗]


∗])

−HD2 [z]

(STF2[z]E1[z] + ISTF2[z]E

∗1 [z

∗]+NTF2[z]E2[z] + INTF2[z]E

∗2 [z

∗])

=STFD2 STF1[z]U [z] + STFD

2 [z]ISTF1[z]U∗[z∗]

+(STFD

2 [z]NTF1[z]

−NTFD1 [z]STF2[z]

)E1[z]

+(STFD

2 [z]INTF1[z]

+NTFD1 [z]ISTF2[z]

)E∗

1 [z∗]

−NTFD1 [z]NTF2[z]E2[z]

−NTFD1 [z]INTF2[z]E

∗2 [z

∗]=STFTOT[z]U [z] + ISTFTOT[z]U

∗[z∗]+NTFTOT,1[z]E1[z] + INTFTOT,1[z]E

∗1 [z

∗]+NTFTOT,2[z]E2[z] + INTFTOT,2[z]E

∗2 [z

∗],(34)

with digital filters HD1 [z] = STFD

2 [z] and HD2 [z] =

NTFD1 [z]. It should be noted that STFTOT[z]U [z] and

NTFTOT,2[z]E2[z] correspond structurally to the ideal outputgiven in (7). However, the responses of STFTOT[z] andNTFTOT,2[z] can differ from STF id

TOT[z] and NTF idTOT,2[z] due

to possible common-mode errors in the modulator coefficients[26]. Furthermore, the four additional terms in (34) are iden-tified as mismatch induced interference, which includes theMFI (conjugate) component of the input signal, the leakageof the first-stage noise and its conjugate component and theconjugate of the second-stage noise.

Now, the interference rejection ratio introduced in [7] canbe used to measure the power of the I/Q imbalance inducedinterference components in relation to the power of the idealoutput for the signal band Ωs in multiband scenario consistingof a total of S signals to be received. This is given as

Γs =

∫f∈Ωs

Gσ(ej2πfT )df∫

f∈Ωs

Gτ (ej2πfT )df, s ∈ {1, 2, ..., S}, (35)

where Gσ(ej2πfT ) and Gτ (e

j2πfT ) are the spectral densitiesof the ideal output signal and the I/Q imbalance inducedinterference (f being the frequency measured in hertz andT the sampling interval of the ADC in seconds), respectively,given by

σ(k) = hTOTSTF (k) ∗ u(k) + hTOT,L

NTF (k) ∗ eL(k) (36)

and

τ(k) =hISTFTOT (k) ∗ u∗(k) + hTOT,L

INTF (k) ∗ e∗L(k)

+

L−1∑l=1

(hTOT,l

NTF (k) ∗ el(k) + hTOT.lINTF (k) ∗ e∗l (k)

).

(37)

Here, the impulse responses correspond to inverse z-transforms of the total transfer functions defined in (33). Theintegration is done over the bandwidth of the signal s, given

by Ωs = {fC,s −Ws/2, ..., fC,s +Ws/2}, where Ws is thebandwidth of the respective signal band.

IV. TRANSFER FUNCTION DESIGN AND DIGITALPOST-COMPENSATION FOR MIRROR-FREQUENCY

INTERFERENCE IN QΣΔM-BASED MULTICHANNELCOGNITIVE RADIO RECEIVERS

First, in this section, post-compensation challenges specificfor receivers utilizing QΣΔM are discussed. Thereafter, de-signing the transfer functions (STF and NTF) of the modulator,to facilitate the digital interference compensation and to com-bat the mentioned challenges, is proposed. For example, formirror-frequency interference compensation, it is essential tomaintain information about the mirror-band content throughthe receiver front-end and A/D interface to digital domain.

A. QΣΔM Related Challenges in Post-Compensation

The traditional way of designing QΣΔM transfer functionshas been to shape the quantization noise away from the desiredsignal band (place the NTF zeros on the band) and to eitheruse a simple frequency-flat signal transfer function or filter theout-of-band signal content partially away [6]. Combining thiskind of noise shaping and STF filtering is an efficient solutionwhen considering an ideal receiver having quantization noiseas the dominant error source. However, the distortion andinterference generated by the implementation inaccuracies ofboth the preceding radio front-end and the QΣΔM itself mustbe taken into account when considering true circuits. Suchtreatment is lacking in the current literature.

Lately, there have been significant advances in digital com-pensation of these so called dirty-RF impairments, such as themirror-frequency interference originating from the mismatchesof quadrature circuits [11], [31]–[33]. These compensationalgorithms typically need access to the original interferencesources in the digital domain. Thus, in the case of I/Qimbalance considered in this article, the mirror-band signal,being the source of mirror-frequency interference, should beconverted into the digital domain with reasonable precision.This is where the above-described traditional NTF design (pos-sibly combined with the STF filtering) becomes unsuitable.If all the NTF zeros are placed on the desired signal band,the noise load of the mirror-band is potentially very high,resulting in a very low signal-to-noise ratio (SNR). This is, ofcourse, even more emphasized if the STF is attenuating themirror-band. In [6], a mirror-band NTF notch was proposedto avoid mirroring of the high noise levels. However, theSTF was used to filter the mirror-band input signal contentaway, impeding the actual post-compensation of input signaloriginating mirror-frequency interference.

The mirror-frequency interference inside the QΣΔM (thefeedback branch I/Q mismatches being most crucial) canbe controlled by mirror-frequency rejecting STF design [7],[19]. However, problems arise when the I/Q mismatches ofthe preceding receiver components (and also other QΣΔMcoefficients) are taken into account. For example, when theI/Q mixer used to bring the desired signal down to IF createsmirror-frequency interference and the original mirror-bandsignal is filtered out inside the QΣΔM, the interference


remains on top of the desired signal but the information ofthe interference source is removed, making post-compensationa troublesome task. The signal and noise transfer functiondesigns solving this problem are proposed in the followingsubsection.

B. Transfer Function Design to Facilitate Post-Compensationand Frequency Agile Operation

In this section, a transfer function design targeted to becombined with digital post-compensation of the overall mirror-frequency interference induced by the receiver front-end andthe QΣΔM is proposed. The design is based on a single-band NTF design with one mirror-band NTF zero of [6],being extended to the multiband concept of [7], [19]. Now,the possible mirror-band blockers are preserved, emphasizingthe usability of such information for digital compensationpurposes, by creating STF passbands also on the mirrorbands. Although allowing the mirror band signals to enter themodulator increases the input dynamics, it brings a significantbenefit for compensation of the mirror-frequency interferencegenerated in the receiver front-end. Without the knowledgeabout these signals, the compensation task of such interferencebecomes very challenging (or even impossible). In addition,in the highly dynamic conditions typical for cognitive radio(considering both power levels and operating frequencies),avoiding mirror-frequency interference in the receiver front-end becomes nearly impossible, because it is very challengingor practically impossible to design RF filters that would havesufficient tunability and selectivity to get rid of the othersignal content. The QΣΔM coefficients are derived in similarmanner as in [7] and the straightforward reconfigurability, e.g.,in case of frequency hand-off, discussed in Section II and[7] is maintained. Thus, the easy implementation of all theQΣΔM coefficients as functions of the desired signals’ centerfrequencies and bandwidths enables efficient frequency-agileoperation.

This design is demonstrated in a practical scenario with atwo-stage QΣΔM having both the stages of fourth order. Thus,the overall NTF order, being the combined order of the stages,is eight, allowing eight noise notches to be placed on the con-version band. This capability allows flexible reconfiguration ofthe frequency bands to be received and detected, in a similarmanner as discussed in [7] in case of a three-stage QΣΔM. Onone hand, the NTF zeros can be placed inside one contiguousband for reception of wideband waveforms (such as LTE with20 MHz bandwidth), or on the other hand, noncontiguousNTF design allows reception of multiple distinct informationsignals on separate frequency bands. Tuning of the QΣΔMcoefficients is in practice implemented with digital control,having 6-8 bit resolution, resulting in sufficient dynamics andcoefficient accuracy when operating on, for example, TV whitespaces, where channel spacing and center frequency resolutionis 8 MHz. The actual implementation details of the controlare, however, not in the scope of this article, but remain as animportant future work topic.

In the following, a concrete transfer function design ex-ample is given, having the NTF zeros divided for the simul-taneous reception of two desired signals on noncontiguous

frequency bands by placing three NTF zeros on both bands.The remaining two zeros are placed on the mirror-bands ofthe desired signals to improve the signal-to-noise ratio of thesignals possibly appearing therein. On the other hand, the mainprinciple in the STF design is to allow the desired signalsand their mirror-band signals to be converted with reasonableresolution. In simplest, this means STF being equal to unity.However, the STF zeros of the first stage can be used to filterout other noninteresting signal content possibly limiting thedynamics of the converter. The second-stage STF is designedto be frequency flat. The frequency-flat design is chosenbecause of its minimal delay and because the second stageprocesses only the quantization error of the first stage, not thesignal itself. Different design options for the second-stage STFare discussed further in [34] by the authors. It is important tonotice that both transfer functions can be straightforwardlyreconfigured using information available about the transmis-sion center frequencies, bandwidths, resolution demands andspectrum sensing information about the strongest blockingsignals possibly present within the overall conversion band.

The exact placement of the NTF zeros inside the desiredsignal band is based on [35]. This is because the analysis forlowpass ΣΔ modulators can also be exploited for QΣΔMs,the noise shaping differing only by the translation of thecenter frequency to an IF. Since the noise shaping order perone desired signal band is three, as described above, thefour first stage zeros are placed on the edges of the bands,i.e., ϕ

(1)NTF,1 = ej2π(fC,1+foffset,1)T , ϕ

(1)NTF,2 = ej2π(fC,1−foffset,1)T ,

ϕ(1)NTF,3 = ej2π(fC,2+foffset,2)T and ϕ

(1)NTF,3 = ej2π(fC,2−foffset,2)T ,

where fC,1 and fC,2 are the desired signal center frequenciesand foffset,1 = W1

√3/5/2 and foffset,2 = W2

√3/5/2 (follow-

ing [35]), with W1 and W2 being the bandwidths of the desiredsignals in hertz. The second-stage NTF zeros are placed inthe middle of the desired signal bands and on the mirrorfrequencies, giving ϕ

(2)NTF,1 = ej2πfC,1T , ϕ

(2)NTF,2 = ej2πfC,2T ,

ϕ(2)NTF,3 = e−j2πfC,1T and ϕ

(2)NTF,3 = e−j2πfC,2T . The poles of the

stages are placed in Butterworth positions around the NTFzeros in order to secure the stability of the modulators [6], asshown in Fig. 6 in the example case of fC,1 = 40.28 MHzand W1 = W2 = 10 MHz (assuming sampling frequency of128 MHz). The stability is improved by limiting the out-of-band gain of the NTF, which is one of the most importantfactors in this aspect [28]. The STF zeros of the first stageare placed on the center frequencies of the possible addi-tional, noninteresting, signals in order to reduce the overallsignal dynamics at the quantizer input. In the example ofFig. 6, these zero locations are ϕ

(1)STF,1 = ej2πfint,1T = 1,

ϕ(1)STF,2 = ej2πfint,2T = −1, ϕ

(1)STF,3 = ej2πfint,3T = ej2π0.2139

and ϕ(1)STF,4 = ej2πfint,4T = e−j2π0.2139. The second-stage STF

is designed as frequency-flat, resulting in the zero locationsoverlapping with the Butterworth poles of the second-stageNTF, as illustrated in Fig. 6. Overall, by tuning the locationsof these zeros and poles, it is straightforward to reconfigurethe QΣΔM response, thus allowing frequency-agile operation.

For the stability of the modulator, [28] gives two criteria; forsingle-bit modulators is required that maxω

∣∣NTF (ejω)∣∣ <

1.5 (being, however, neither necessary nor sufficient condition)and for multi-bit modulators maxn |u(n)| ≤ Mstep + 2 −


−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

Real Part

Imag

inar

y P

art

Unit circle

NTF zeros, stage 1

STF zeros, stage 1

Poles, stage 1

NTF zeros, stage 2

STF zeros, stage 2

Poles, stage 2

−64 −32 0 32 64−80

−60

−40

−20

0

20

Frequency [MHz]

Gai

n [d

B]

STF, stage 1

NTF, stage 1

STF, stage 2

NTF, stage 2

Fig. 6. Zero-pole plot (left) of the overall NTF and STF of a two-stage QΣΔM having fourth order individual stages (overall noise shaping order of eight)and frequency responses of the transfer functions of the stages (right). Multiband reception of two information signals with center frequencies of 40.28 MHzand 14.49 MHz is assumed. These bands to be detected are marked with vertical black lines in the right plot.

∑∞n |hNTF(n)| (sufficient), where Mstep is the number of quan-

tizer steps. For the design at hand, maxω∣∣NTF1(e

jω)∣∣ = 1.89

and maxω∣∣NTF2(e

jω)∣∣ = 2.36 and thus 1-bit implemen-

tation cannot be expected to be fully stable. On the otherhand, if 3-bit quantizers are chosen, the latter criterion givesmaxn |u(n)| = Mstep,1 + 2−∑∞

n |hNTF,1(n)| = 5.28, makingstable operation guaranteed with 66 % of the quantizer full-scale range. For the second stage of the design, the corre-sponding figures, with 3-bit quantizer, are maxn |u2(n)| =Mstep,2+2−∑∞

n |hNTF,2(n)| = 6.07 and 76 % of the full-scalerange. Thus, 3-bit quantizers and feedback digital-to-analogconverters (DACs) are chosen for both the stages.

Based on the above design, it is noted that the second stageNTF is actually symmetric (the zeros are complex conjugatepairs) and thus real valued and could be realized with a realbandpass ΣΔ modulator stage. However, this is only a specialcase with exactly two distinct signals to be received. In thechallenging and dynamic conditions, typical for the cognitiveradio solutions, the flexibility of the quadrature (complex)modulator is appreciated for its frequency agile operation anddegrees-of-freedom and thus maintained in the general design.With QΣΔM stages at hand, the NTF can be configuredfor the reception of one to four separate signal bands. Thisvaluable capability would be partly lost with a real bandpassΣΔ modulator as the second stage.

C. Removing Mirror-Frequency Interference with DigitalPost-Processing

The mirror-frequency interference experienced by the de-sired signals is generated in both the receiver radio front-endand the QΣΔM considered for the A/D conversion, becauseof the inherent implementation inaccuracies resulting in I/Qimbalance between the I and Q signal branches. In the radiofront-end, the imbalance source can be, e.g., a quadraturemixer having imperfect phase and amplitude matching. Thus,the input signal (in z-domain U [z]) fed to the QΣΔM isalready a combination of the ideal signal, say X [z], andits complex conjugate X∗[z∗], scaled by certain, generallyfrequency selective, gains or responses, i.e.,

U [z] = STFFE[z]X [z] + ISTFFE[z]X∗[z∗], (38)

where STFFE[z] ISTFFE[z] denote the signal transfer func-tion and image signal transfer functions of the receiver front-end, which are assumed to be unknown.

When this nonideal signal (including the conjugate term) isused as an input for the I/Q mismatched QΣΔM, both com-ponents experience the STF and the ISTF of the modulator.In addition, the noise and image noise components are added,shaped by the NTF and the INTF, respectively. In this scenario,the final output of a single-stage QΣΔM is given by

V [z] = STFSD[z]U [z] + ISTFSD[z]U∗[z] +NTF [z]E[z]

+ INTF [z]E∗[z∗]= STFSD[z](STFFE[z]X [z] + ISTFFE[z]X

∗[z∗])+ ISTFSD[z](ISTFFE[z]X [z] + STFFE[z]X

∗[z∗])+NTF [z]E[z] + INTF [z]E∗[z∗]

= STFFE+SD[z]X [z] + ISTFFE+SD[z]X∗[z∗]

+NTF [z]E[z] + INTF [z]E∗[z∗],(39)

where STFFE+SD and ISTFFE+SD are the combined STF andthe ISTF of the front-end and the QΣΔM, respectively. Foran L-stage QΣΔM the corresponding output is given by

V [z] =HD1 [z]

(STFFE+SD1[z]X [z] + ISTFFE+SD1[z]X

∗[z∗]


∗])

+

L∑l=2

(−1)l−1

HDl [z]

(STFl[z]El−1[z]

+ ISTFl[z]E∗l−1[z

∗] +NTFl[z]El[z]

+ INTFl[z]E∗l [z

∗]),

(40)

where STFFE+SD1[z] and ISTFFE+SD1[z] are the combinedSTF and ISTF of the receiver front-end and the first ΣΔstage. Similarly as in (33), the total effective signal transferfunctions, including now the effects of both the analog radiofront-end and the QΣΔM converter structure, can be writtenas


STFTOT, FE+SD1[z] = HD1 [z]STFFE+SD1[z]

ISTFTOT,FE+SD1[z] = HD1 [z]ISTFFE+SD1[z].

(41)

The noise transfer functions, in turn, are not dependent on theanalog radio front-end, and are thus identical to those in (33).

Now, in the case of two-stage QΣΔM as a practicalexample, the overall interference rejection ratio of (35) iscalculated using the signal components ideally present at theoutput,

σ(k) = hTOT,FE+SD1STF (k) ∗ x(k) + hTOT,2

NTF (k) ∗ e2(k), (42)

and the I/Q imbalance induced interference components,

τ (k) =hTOT,FE+SD1ISTF (k) ∗ x∗(k) + hTOT,1

NTF (k) ∗ e1(k)+ hTOT,1

INTF (k) ∗ e∗1(k) + hTOT,2INTF (k) ∗ e∗2(k),

(43)

where the STF and the ISTF components include now thecontribution of the whole receiver. Thus, the interferencerejection ratio of the whole receiver, for the signal band Ωs,is given by

Γs =

∫f∈Ωs

Gσ(ej2πfT )df∫

f∈Ωs

Gτ (ej2πfT )df, s ∈ {1, 2, ..., S}. (44)

In the special case of an ideal receiver before the QΣΔM, (44)reduces to (35) because then σ(k) = σ(k) and τ (k) = τ(k).In Section V, this measure Γs is used to verify, quantify andanalyze the efficiency of the digital MFI post-compensationdiscussed in the following.

In case of the ΣΔ modulation, the frequency-selectivebehavior of the signal transfer functions can be assumedmild inside a single subband because of the inherent highoversampling. Thus, a simple frequency-independent compen-sation algorithm [13], per subband, can be used to demonstratethe efficiency of digital post-compensation, facilitated by theabove discussed transfer function designs. Possible deviationsat the level of the interference between separate signals andrespective frequency bands are taken into account by doingthe compensation separately for each of the desired informa-tion signals, i.e., separately for the subbands, as illustratedin Fig. 7. At the same time, the transfer function designwould also allow use of more sophisticated, e.g., frequency-selective digital post-compensation algorithms, such as [17].This option should be considered especially for reception ofvery wideband waveforms, when there might be significantfrequency-selective behavior also inside the individual sub-bands, because of, e.g., the I/Q mismatch properties of theanalog front-end. This is an interesting topic for future work.

With the previous narrowband assumption and assuminga two-stage QΣΔM again as a practical example, the com-pensator output signals ys(k) (s ∈ {1, 2, ..., S}), depicted inFig. 7, are given by

( )*

v k( )

��

( )* �

�

y k1( )

y k2( )

Filtering stage Compensation stage

F k1( )

F k2( )

F k2( )

F k1( )*

*

v k1( )_

_

_

v k2( )

( )* �S

y kS( )F kS( )

F kS( )*

v kS( )

Fig. 7. Digital MFI post-compensation structure for multiband reception,processing S parallel information signals to be received and detected. Thecompensation algorithm used per signal band is published in [13]. The filtersF1(k) to FS(k) and F ∗

1 (k) to F ∗S (k) are used to pick the desired signals

and the related mirror-bands, respectively.

ys(k) =vs(k) + ωsv∗s (k)

≈K1,xs xs(k) +K2,xs x∗s(k) +K1,e1,s e1,s(k)

+K2,e1,s e∗1,s(k) +K1,e2,s e2,s(k) +K2,e2,s e

∗2,s(k)

+ ωs

[K∗

1,xsx∗s(k) +K∗

2,xsxs(k) +K∗

1,e1,se∗1,s(k)

+K∗2,e1,s

e1,s(k) +K∗1,e2,s

e∗2,s(k) +K∗2,e2,s

e2,s(k)],

(45)

where the signal and noise variables with the bar and thesubscript s are the corresponding inband components afterthe subband filters Fs(k) for the desired signals and F ∗

s (k)for the image bands, respectively. In addition, the modelcoefficients are the complex scaling factors for the signal andnoise components, defined as

K1,xs = STFTOT, FE+SD1[ej2πfC,sT ],

K2,xs = ISTFTOT,FE+SD1[ej2πfC,sT ],

K1,e1,s = NTFTOT,1[ej2πfC,sT ],

K2,e1,s = INTFTOT,1[ej2πfC,sT ],

K1,e2,s = NTFTOT,2[ej2πfC,sT ],

K2,e2,s = INTFTOT,2[ej2πfC,sT ],

(46)

where the transfer functions are evaluated at the desired signalcenter frequencies and T is the sampling interval in seconds.Further, by assuming practical (clearly above 0 dB) inbandsignal-to-quantization noise ratio and thus neglecting the lowerpower noise components (also the conjugate noise componentis assumed to be controlled by the mirror-band NTF notch),(45) can be simplified as

ys(k) ≈K1,xs xs(k) +K2,xs x∗s(k)

+ ωs[K∗1,xs

x∗s(k) +K∗

2,xsxs(k)].

(47)

This approximation simplifies finding the mirror-frequencycompensator coefficients ωs significantly.

Stemming from the signal model structure in (47), thecompensator coefficients ωs can be estimated based on thecircularity restoring mirror-frequency compensation principle


proposed in [13]. The circularity is defined as the comple-mentary autocorrelation function cys(τ) of ys(k) being zeroat zero lag, i.e., cys(0) = cys = E

[ys(k)

2]= 0, meaning

that real and imaginary parts of the compensated outputs areequally powerful and instantaneously mutually uncorrelated.Similarly, traditional autocorrelation function γys (τ) at zerolag is given by γys(0) = γys = E

[|ys(k)|2

]. Now, as shown

in [13], the optimal compensator coefficient for signal subbands, restoring the circularity of the compensated signal, can bewritten as a function of the observed subband signal second-order statistics, cvs and γvs , as

ωs = − cvs

γvs +√γ2vs −

∣∣c2vs∣∣ . (48)

In above, s ∈ {1, 2} for two-band reception and generallys ∈ {1, 2, ..., S}, for S signal bands to be received. In practice,sample estimates of the second-order statistics, cvs and γvs , areused, calculated over a block of received data. This estimation-compensation approach is deployed in the following computersimulations. The results are illustrated in Section V.

V. RESULTS OF TRANSFER FUNCTION OPTIMIZATION ANDPOST-COMPENSATION

A. Closed-Form Two-Stage QΣΔM Transfer Function Analy-sis

In this section, the transfer function model proposed inSection III is used to calculate the four transfer functions ofthe two-band QΣΔM design discussed in Section IV. Twodetached frequency channels are assumed for the desired infor-mation signals, with center frequencies of fC,1 = 40.28 MHz,fC,2 = −14.49 MHz and bandwidths W1 = W2 = 10 MHz.The sampling frequency is 128 MHz, giving T = 1/fs =7.8125 ns. The transfer functions of the QΣΔM are designedaccording to the proposed design flow of Section IV. Theoverall noise shaping order is eight and two zeros are placedon the mirror-bands of the desired signals, leaving three zerosper signal band to be used for the actual noise shaping. Thedigital filters HD

1 [z] and HD2 [z] are assumed to be matched

to the ideal analog transfer functions.The transfer functions are evaluated with randomly deviated

real gain values in the modulator coefficients in order to modelthe implementation inaccuracies and evaluate their effects. Thegain values are drawn from a uniform distribution of ±1 %

around the ideal value. In other words, the realized value r(1,1)re,1

of ideal r(1,1)re,1 is uniformly distributed between 0.99r(1,1)re,1 and

1.01r(1,1)re,1 and similarly for the other coefficients.

First, the case of QΣΔM feedback mismatch is considered,being mentioned as the most crucial MFI source inside themodulator [24]. This is because the feedback errors are notshaped by the NTF as the forward branch errors. In Fig. 8,the results for 50 independent draws from the above describeddistribution for the feedback coefficients are illustrated. Theresults show the image rejection of the input signal (STFresponse of 10 dB versus the ISTF response of −40 dB) to bearound 50 dB on the frequency channel around 14.49 MHzand 40 dB (STF response of 10 dB versus the ISTF response of−40 dB) around 40.28 MHz. The second stage quantization

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Fig. 8. Two-stage QΣΔM STF and ISTF (top) together with NTF andINTF for first- (middle) and second-stage (bottom) quantization noise. 50independent random realizations in real gain values of feedback branches ofboth stages are considered and multiband reception of two information signalswith center frequencies of 40.28 MHz and −14.49 MHz is assumed. Thesebands to be detected are marked with vertical black lines in the plots.

noise (being ideally the only quantization noise present) isattenuated between 15 dB and 30 dB inside the desired signalbands and the related conjugate response (INTFTOT,2[z]) hasmore than 50 dB attenuation, remaining clearly below thenonconjugate response. The mirror band zeros push the noiselevels down also on those frequencies. From the middle plotof Fig. 8, it is clearly visible that the quantization error ofthe first stage is not cancelled completely. This is because ofthe noncommutativity of the complex transfer functions underI/Q imbalance [26]. The conjugate component of the first-stagenoise (INTFTOT,1[z]) has 30 to 40 dBs of attenuation on thenegative frequency channel and 15 to 35 dBs on the positivefrequency channel. The nonconjugate noise (NTFTOT,1[z]) isattenuated on average by 40 dB on both the channels. Thisconfirms that the transfer functions work as designed in case ofthe modulator feedback coefficient mismatch. However, thereremains the risk of heavy MFI if strong mirror-band signalsare present at the input signal. The post-compensation of thisinterference is demonstrated in the following subsection.

In addition, Fig. 9 gives the transfer function analysis re-sults assuming implementation inaccuracies in all the QΣΔMcoefficients in both the stages of the previously described two-stage modulator used as a practical example. The findingsin Fig. 9 have certain significant differences to the ones inFig. 8. For example, the ISTF maximum response has risenby 10 (positive frequency channel) to 20 (negative frequencychannel) dBs. In addition, out-of-band peaking is clearlyvisible in the ISTFs. At the same time, the second stage noisebehavior on the desired bands remains similar. However, theconjugate response is elevated on the mirror-bands, having


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Fig. 9. Two-stage QΣΔM STF and ISTF (top) together with NTF andINTF for first- (middle) and second-stage (bottom) quantization noise. 50independent random realizations in real gain values of all the coefficients ofboth stages are considered and multiband reception of two information signalswith center frequencies of 40.28 MHz and −14.49 MHz is assumed. Thesebands to be detected are marked with vertical black lines in the plots.

still around 20 dB attenuation. From the first stage noisepoint-of-view, the most visible difference is that conjugateattenuation on the negative frequency channel has decreasedby around 10 dB. Overall, the results indicate that, whilethe feedback mismatches might be the most important errorsource, the mismatches of the feed-forward coefficients cannotbe excluded either and have a pronounced effect on the imagerejection of the whole QΣΔM.

B. Verification Simulations on Digital Post-Compensation ofMirror-Frequency Interference

Herein, the QΣΔM design and MFI compensation of Sec-tion IV are simulated using realistic communication wave-forms to be received (16-QAM and QPSK modulated signals),together with mirror-frequency blocking signals as interfer-ence sources. Two separate scenarios are presented in the fol-lowing. In the first case, QΣΔM feedback I/Q mismatches areassumed to be the only implementation inaccuracy. The lattercase, on the other hand, assumes all the QΣΔM coefficientsto be nonideally implemented and, in addition, the frequencyindependent image rejection ratio of the preceding receiverchain (mainly I/Q mixer) is assumed to be 30 dB. Whennonideally implemented QΣΔM coefficients are considered,the realized real-valued coefficient values are drawn froma uniform distribution of ±1 % around the ideal value asdescribed also in the above case of transfer function analysis.

An example of the assumed input signal in the computersimulations is illustrated in Fig. 10. The ideal overall transferfunctions are included in the figure, in order to illustrate

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Fig. 10. An example spectrum of input signal used in the simulations togetherwith the ideal designed transfer functions (overall response). The desiredinformation signals are located around center frequencies of 40.28 MHzand −14.49 MHz with 16-QAM and QPSK waveforms, respectively. Theblocking signals are located on the mirror frequencies of the desired signals.

the noise shaping and the STF passbands. In addition to thesignal content, a white noise floor is included at the receiverinput, giving 18 dB signal-to-noise ratio for the 16-QAMwaveform around fC,1 = 40.28 MHz and 10 dBs for the QPSKwaveform around fC,2 = −14.49 MHz. The difference is dueto the 8 dB higher power level for the 16-QAM signal. Themirror-frequency blockers consist of band-limited Gaussiannoise, with a bandwidth of 3 MHz, and are simulated on therelative power level from 0 dB to +60 dB compared to therespective desired signal. In this way, with the highest blockerlevel, the mirror-band signal across the 16-QAM signal is68 dB stronger than the weaker QPSK waveform, representingan extremely challenging scenario. A true 3-bit quantizer issimulated in both the QΣΔM stages (and separately for theI and Q branches) for feasible quantization noise levels. TheDACs of the stages are assumed ideal.

In both simulation cases, 150 independent realizationsof the QΣΔM coefficients are drawn from the describeddistributions. The interference rejection ratio Γs values areevaluated for each realization of the modulator coefficients bysubtracting the output of an ideal QΣΔM from the output ofa mismatched QΣΔM, obtaining thus an estimate of the dis-tortion component. The signal length used is 218 samples andthe blockwise post-compensation (when applicable) is doneusing these samples. The optimization of the compensationblock-length is not considered in this article.

The amplitudes of the real and imaginary parts of the overallreceived input signal are limited by the receiver automaticgain control mechanism to be equal to or less than 0.7 inorder to avoid quantizer clipping (quantizer full scale rangefrom -1 to 1), i.e., |uI,1(k)| ≤ 0.7 and |uQ,1(k)| ≤ 0.7 forall k. This limitation is maintained also when increasing theblocking signal power levels, which means that with increasingblocker input power, the useful signals are scaling down andbecome more and more sensitive to, e.g., quantization noise.The stability of the modulator within the amplitude range hasbeen confirmed by extensive simulations.

The interference rejection ratio simulation results are illus-trated in the following figures. In addition to the proposed STFdesign, also frequency flat and mirror-frequency rejecting [7]STF designs are simulated for reference. In Fig. 11, the results


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Fig. 11. Interference rejection ratios for a 16-QAM signal with two-stageQΣΔM, using 3-bit quantizers at each stage, as a function of blocker signalpower. The proposed STF, frequency-flat (“Flat STF”) and mirror-frequencyrejecting STF [7] (“Reject STF”) designs are simulated. Implementationinaccuracies are considered only in the feedback branch coefficients of boththe stages.

are given for the 16-QAM waveform, considering first I/Qmismatches only in the feedbacks of the QΣΔM stages. Thegain achieved with the digital post-compensation is clearlyvisible, the compensated interference ratio reaching up to3 dB gain with both frequency flat and the proposed STFdesigns with higher relative blocker powers. However, it isclear that in this special case, the mirror-frequency rejectingSTF design of [7] has better performance than either theproposed or the flat STF designs combined with digital post-compensation, exceeding the results of the proposed designand compensation by around 5 dB. This is because themirror-frequency rejecting STF design combats exactly theeffects of the feedback branch I/Q mismatches. The digitalcompensation does not bring additional improvement with thismirror-frequency rejecting STF design because the mirror-band information is lost before the conversion. However, oneshould note that this first simulation scenario is unrealisticsince only feedback branch I/Q mismatches are consideredwhile the other coefficients of the ΣΔ modulator as well asthe preceding radio receiver I/Q matching are assumed ideal.

In Fig. 12, similar results are given for the QPSK signal andfeedback branch I/Q mismatches only. The biggest differencetherein is the drop in the post-compensation gain. This happensbecause the QPSK signal has a lower power level comparedto the 16-QAM signal and thus smaller SNR making theinput noise components dominant error sources over the inputsignal originating MFI. The good performance of the mirror-frequency rejecting STF is again clearly visible, exceeding theother simulated options by up to 20 dB at relative blocker levelof +20 to +30 dB.

Fig. 13 and Fig. 14 illustrate then the corresponding resultsof a much more realistic scenario with I/Q mismatches inall the QΣΔM coefficients and also 30 dB image rejectionratio for the preceding receiver chain for 16-QAM and QPSKwaveforms, respectively. This is indeed the realistic scenario

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Fig. 12. Interference rejection ratios for a QPSK signal with two-stageQΣΔM, using 3-bit quantizers at each stage, as a function of blocker signalpower. The proposed STF, frequency-flat (“Flat STF”) and mirror-frequencyrejecting STF [7] (“Reject STF”) designs are simulated. Implementationinaccuracies are considered only in the feedback branch coefficients of boththe stages.

for which the proposed transfer function design was developedand combined with the digital post-compensation of the MFIof the whole receiver. As expected, there is significant differ-ence between the results in Fig. 13 (16-QAM and mismatchesin the whole receiver) and Fig. 11 (16-QAM and only feed-back mismatches). The performance of the mirror-frequencyrejecting STF design has dropped by more than 20 dB whenconsidering the relative blocker level range up from +20 dB.At the same time, while significant additional error-sourceshave been introduced compared to the previous scenario,the performance of the proposed STF design combined withthe digital post-compensation has dropped only by 3 dB.This is because the post-compensation, when combined withproposed transfer function optimization, is able to mitigatethe MFI independent of the exact location of the error-source. The gain achieved by this post-compensation withthe proposed STF design is from 6 to 14 dB on the relativeblocker level range from +20 to +50 dB. In parallel, also thefrequency-flat STF design has around 6 dB gain as a resultof the post-compensation. With the highest blocker powerlevels, the quantization noise and its conjugate componentbecome eventually the dominating error-source because of theeffective down-scaling of the desired waveforms. The digitalcompensation applied is not able to mitigate this interferenceas efficiently as the input signal originating MFI. Overall,it is anyway clear that the combination of the proposedSTF design and the digital compensation algorithm has thebest performance in this realistic and challenging scenarioconsidering the whole simulated blocker level range.

The results for the QPSK signal in Fig. 14 show similarcharacteristics compared to the 16-QAM results. The cleardrop in the performance of the mirror-frequency rejecting STFdesign is visible as well as increased post-compensation gainfor the proposed STF design due to the increased role ofthe input signal originating MFI because of the additional


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Fig. 13. Interference rejection ratios for a 16-QAM signal with two-stageQΣΔM, using 3-bit quantizers at each stage, as a function of blocker signalpower. The proposed STF, frequency-flat (“Flat STF”) and mirror-frequencyrejecting STF [7] (“Reject STF”) designs are simulated. Implementationinaccuracies are considered in all the coefficients of both the stages togetherwith image rejection ratio of 30 dB for the preceding receiver chain.

mismatches and nonideal image rejection of the receiver front-end. The achieved gain is now between 4 dB and 6 dB, thelatter being with higher blocker powers. These results supportthe conclusion that in this more realistic scenario, consideringpractical error sources, the proposed STF design combinedwith the digital post-compensation outperforms the earliersolutions, taking the whole simulated relative blocker powerrange into account.

Overall, when considering the practical simulation scenariowith mismatches in the whole receiver, the results showsignificant increase in the interference rejection ratio withthe proposed transfer function design complemented with thedigital MFI compensation discussed earlier. The gains are mostevident in good SNR conditions. With low SNR, the noisecomponents (nonconjugated and conjugated) become the dom-inant error source. Although the mirror-frequency rejectingSTF design copes well with the feedback branch mismatches,its performance drops significantly, e.g., by 20 dB in theprevious 16-QAM example, when also other mismatches areintroduced. In parallel, this drop for the proposed STF designcombined with post-compensation was only few dBs, showingparticular robustness in the realistic reception scenario.

VI. CONCLUSION

In this article, a novel frequency-agile multiband trans-fer function design for single- and multistage QΣΔMs wasproposed, combined with digital post-compensation of MFIoriginating from implementation inaccuracies of the precedingradio front-end and the QΣΔM itself. This allows the compen-sation to be done independently of the location of the actualerror source in the receiver chain. This is essential for practicalimplementations of such a receiver, especially in challeng-ing signal conditions typical for CR systems. The computersimulations showed significant interference rejection gain,achieved by simple post-compensation algorithm presented in

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Fig. 14. Interference rejection ratios for a QPSK signal with two-stageQΣΔM, using 3-bit quantizers at each stage, as a function of blocker signalpower. The proposed STF, frequency-flat (“Flat STF”) and mirror-frequencyrejecting STF [7] (“Reject STF”) designs are simulated. Implementationinaccuracies are considered in all the coefficients of both the stages togetherwith image rejection ratio of 30 dB for the preceding receiver chain.

the literature, proving the digital post-compensation approachto be feasible also with QΣΔM (in addition to the traditionalADCs without noise shaping). This is an important steptowards robust receiver implementations using quadrature ΣΔADC, allowing the promising multiband and reconfigurabilityproperties to be exploited in practice. The proposed designcan be implemented without additional analog componentsand is straightforwardly reconfigurable in dynamic conditions,e.g., in case of frequency hand-off because of a primaryuser appearance. In addition, facilitating the digital post-compensation of the receiver front-end originating interferenceeases the demands for the matching of these circuits.

In addition, a novel closed-form transfer function analysismethod was presented, based on the matrix-representationof I/Q mismatched QΣΔM coefficients. In this way, thefour transfer functions (NTF, STF, INTF and ISTF) of theQΣΔM can be solved for any modulator order or number ofstages. Furthermore, a closed-form equation for the output ofsuch modulator under coefficient mismatches was presented.This allows the designers to examine the effects of inherentimplementation inaccuracies in more detail already beforemeasuring the actual implementation.

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Jaakko Marttila (S’10) was born in Tampere, Fin-land, on March 30, 1982. He received the M.Sc.degree in signal processing and communicationsengineering from Tampere University of Technology(TUT), Tampere, Finland, in 2010.

He is currently working towards the Ph.D. de-gree at TUT, Department of Communications En-gineering, as a Researcher. At present, his mainresearch topic is quadrature sigma-delta analog-to-digital (AD) conversion. Generally, his researchinterests include AD techniques for software defined

and cognitive radios, and related interference mitigation algorithms.

Markus Allén (S’10) was born in Ypäjä, Finland,on October 28, 1985. He received the B.Sc. andM.Sc. degrees in signal processing and commu-nications engineering from Tampere University ofTechnology (TUT), Finland, in 2008 and 2010,respectively.

He is currently with the Department of Communi-cations Engineering at TUT as a Researcher, headingtowards the Ph.D. degree. His research interests in-clude cognitive radios, analog-to-digital converters,receiver front-end nonlinearities, and their digital

mitigation algorithms.

Mikko Valkama (S’00, M’02) was born in Pirkkala,Finland, on November 27, 1975. He received theM.Sc. and Ph.D. degrees (both with honours) inelectrical engineering (EE) from Tampere Universityof Technology (TUT), Finland, in 2000 and 2001,respectively. In 2002, he received the Best Ph.D.Thesis award from the Finnish Academy of Scienceand Letters for his dissertation entitled “AdvancedI/Q signal processing for wideband receivers: Mod-els and algorithms.”

In 2003, he worked as a visiting researcher withthe Communications Systems and Signal Processing Institute at SDSU, SanDiego, CA. Currently, he is a Full Professor and Department Head at theDepartment of Communications Engineering at TUT, Finland. He has beeninvolved in organizing conferences, such as IEEE SPAWC’07 (PublicationsChair) held in Helsinki, Finland. His general research interests includecommunications signal processing, estimation and detection techniques, signalprocessing algorithms for software defined flexible radios, cognitive radio,digital transmission techniques such as different variants of multicarrier mod-ulation methods and OFDM, radio localization methods, and radio resourcemanagement for ad-hoc and mobile networks.

Frequency-Agile Multiband Quadrature Sigma-Delta Modulator for Cognitive Radio: Analysis, Design and...

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