Patent Publication Number: US-2021194746-A1

Title: Image rejection compensation method for i/q down-conversion in radio frequency receivers, corresponding circuit, radio frequency receiver device and computer program product

Description:
BACKGROUND 
     Technical Field 
     The description relates to a method of signal processing for wideband radio communication receivers and receiver architectures. Specifically, the description relates to compensating interference generated by receiver I/Q imbalance, in particular for the reception of wideband signals. 
     One or more embodiments may be applied, for instance, to terrestrial broadcast radio receivers, satellite broadcast radio receivers, GNSS receivers, etc. 
     Description of the Related Art 
     The reception of radio frequency channels by usage of heterodyne architectures is affected by the interference of the so-called image signal. 
     For instance, radio receiver architectures, especially those based on a heterodyne technique and employing an I/Q mixer in a Radio Frequency Front-End (briefly, RFFE) to tune to a Radio Frequency (briefly, RF) channel, may be affected by a parasitic effect: due to crosstalk generated by I/Q imbalance, an attenuated replica of the so-called image band may overlap onto the desired signal band, resulting in in-band levels of the image replica that may degrade performance beyond an acceptable level. 
     RFFEs are known devices that may have internal circuitry elements configured to generate and/or process so-called in-phase I and quadrature Q signals in response to the receipt of an RF (radio frequency) signal x RF (t). RFFEs configured for down-converting signal components whose spectrum is transmitted by modulating a carrier placed at the RF frequency ω RF , possibly related to the nominal frequency of the desired channel within the RF signal x RF (t), to a convenient Intermediate Frequency (IF) ω IF , which can either be positive, negative or, in direct down-conversion applications, can also be 0 Hz. 
     For instance, a conventional RFFE receiver arrangement  10 , as exemplified in  FIG. 1 , may comprise:
         an input node x RF  configured to receive an RF signal x RF (t), e.g., carrying multiple channels of (user-)data,   an impedance matching element  11 , coupled to a low-noise amplifier (briefly, LNA)  12 , and   an analog I/Q down-converter  13 , comprising:
           a) a first down-converter branch  13 I, comprising a first RF-IF mixer  14 I and a first IF low-pass filter  16 I, the first RF-IF mixer  14 I coupled to the first IF low-pass filter  16 I,   b) a second down-converter branch  13 Q, comprising a second RF-IF mixer  14 Q and a second IF low-pass filter  16 Q, the second RF-IF mixer  14 Q coupled to the second IF low-pass filter  16 Q.   
               

     The first  13 I and second  13 Q down-converter branches, which may comprise respective mixing and filtering stages, may be configured to apply mixing processing  14 I, resp.  14 Q, to the RF signal x RF (t) and a local oscillator (LO) signal x LO (t) component I, resp. Q, the mixing processing  14 I,  14 Q followed by low-pass filtering processing  16 I, resp.  16 Q. 
     As mentioned, such local oscillator (LO) signal x LO (t) may be modeled as split into respective I/Q components having a frequency ω LO =ω RF −ω IF , which may be expressed in time-domain as, respectively: 
         x   LO,I ( t )=Re[ x   LO ( t )] 
         x   LO,Q ( t )=Im[ x   LO ( t )] 
     As a result of mixing processing and low-pass filtering, a down-converted signal x IF (t) may be generated, the down-converted signal x IF (t) having respective I and Q components which may be expressed as: 
         x   IF,I ( t )=Re[ x   IF ( t )] 
         x   IF,Q ( t )=Im[ x   IF ( t )] 
     It is noted that the oscillator frequency ω LO  can either be lower (low-side injection), higher (high-side injection) or equal (direct down-conversion) with respect to the RF frequency ω RF , according to the value used for the intermediate frequency ω IF . 
     An RFFE receiver  10  is hence tuned to collect the RF signal x RF (t). However, the analog down-converter branches  13 I,  13 Q may have differential DC offset, gain, and quadrature phase errors. For instance, when mismatches exist between:
         the gains of the two local oscillator signals for the RF-IF mixers  14 I and  14 Q of the branches  13 I,  13 Q of the (analog) down-converter  13 , and/or   their phases with respect to the quadrature condition, and/or   any of the IF elements  16 I (amplifier, low-pass filter, analog portion of analog-to-digital converter) of the first down-converter branch  13 I with respect to the respective elements  16 Q of the second down-converter branch  13 Q, IF and baseband signals may be corrupted.       

     Existing solutions to such an imbalance problem may comprise applying specific training sequences at wideband receiver inputs and running calibration algorithms to the receiver before its startup/operation. Such solutions may be anyway difficult to implement, e.g., due to high implementation costs, and may facilitate solely sub-optimum results. For instance, calibration parameters evaluation may employ a Digital Signal Processor (DSP) and, also, it may be rather difficult to find and generate the proper training sequence. 
     As mentioned, solutions employing training signals may partially calibrate the device since, usually, it is difficult to obtain optimal parameters setting. 
     Another disadvantage of solutions employing a training sequence is the employ of such sequences at the Electrical Wafer Sorting (EWS) phase, in order to properly calibrate the receiver or during final test of the device, at the application level before operation. Calibration operations may be time consuming, increasing the cost of EWS or final test stages. Customers may rather avoid applying calibration phase at application level. 
     BRIEF SUMMARY 
     In an embodiment, a method comprises: receiving an input signal comprising at least one sequence of input data samples separated by a sampling period therebetween, the input signal comprising a desired signal component and an interfering signal component superimposed thereon; applying interfering component estimation processing to said input signal obtaining as a result a filtered signal comprising a sequence of filtered data samples; and subtracting said filtered signal from said input signal and obtaining as a result an output signal comprising a sequence of output data samples. The interfering component estimation processing comprises: applying conjugating processing to said input signal, providing a conjugated version of said input signal; computing of at least one adaptive signal processing coefficient value; and applying adaptive signal processing to said conjugated version of said input signal using at least one adaptive processing coefficient. The computing at least one adaptive signal processing coefficient value comprises: performing correlation processing between said sequence of output data samples of the output signal and a conjugated version of said sequence of output data samples of the output signal and obtaining as a result a sequence of estimates of residual correlation; and applying integration processing to said sequence of estimates of residual correlation provided, said integration processing using an integration step parameter and at least one starting point parameter, obtaining at least one computed adaptive signal processing coefficient as a result of applying said integration processing. In an embodiment, applying adaptive signal processing comprises applying processing selected out of: adaptive multiplication processing having at least one multiplication factor equal to said at least one computed adaptive signal processing coefficient; and adaptive finite impulse response, FIR, filtering processing, said adaptive FIR filtering processing comprising computing a weighted sum of a conjugated version of data samples in said sequence of input data samples comprised in said input signal, using said at least one adaptive processing coefficient as weights of said weighted sum. In an embodiment, the adaptive FIR filtering processing comprises applying said at least one adaptive processing coefficient to a subset of elements belonging to an input delay line of the FIR at a related subset of delay values; and performing correlation processing and applying integration processing respectively comprise calculating and integrating estimates of residual correlation at delay values of said related subset of delay values. In an embodiment, the adaptive FIR filtering coefficients are applied to a comb of elements belonging to said input delay line of the FIR, the comb comprising a number N d  of delay elements, the delay elements having a distance value multiple of said sampling period by a factor d. In an embodiment, the distance value is in excess of ten times said sampling period. In an embodiment, selecting between said adaptive multiplication processing and said adaptive FIR filtering processing in applying said adaptive signal processing comprises: providing a first adaptive signal processing configuration register configured to store a first value or a second value; and selecting between applying one of said adaptive multiplication processing or said adaptive FIR filtering processing as a function of said first value or said second value stored in said first adaptive signal processing configuration register. In an embodiment, the method comprises: providing at least one configuration register configured for storing indexes of delay elements in the set of delay elements of the delay line; and selecting a subset of delay elements in the set of delay elements of the delay line as a function of said indexes stored in the at least one configuration register. In an embodiment, applying integration processing comprises applying loop filter processing to said sequence of estimates of residual correlation provided. In an embodiment, applying integration processing comprises: providing a first integration parameter register configured to store a value of said integration step parameter; providing a second integration parameter register configured to store a value of said at least one starting point parameter and a third integration parameter register configured to store a value indicating whether to activate said integration processing to use said value of said at least one starting point parameter; or combinations thereof. In an embodiment, computing at least one value of said at least one adaptive processing coefficient comprises applying automatic gain control, AGC, processing to said output signal. In an embodiment, performing correlation processing between said sequence of output data samples of the output signal and a conjugated version of said sequence of output data samples of the output signal comprises performing block-like correlation of a number of adjacent data samples forming a block having block-length L of data samples, wherein said block-length L is selected as a function of a length value stored in a configuration register. In an embodiment, the adaptive FIR filtering processing comprises second order (or three tap) adaptive FIR filtering processing. 
     In an embodiment, a circuit, comprises: an input node configured to receive an input signal comprising at least one sequence of input data samples, the input signal comprising a desired signal component and an interfering signal component superimposed thereon, wherein input data samples in the sequence of input data samples are separated by a sampling period therebetween; and signal processing circuitry coupled to the input node, wherein the signal processing circuitry, in operation: applies interfering component estimation processing to said input signal obtaining as a result a filtered signal comprising a sequence of filtered data samples; and subtracts said filtered signal from said input signal, obtaining an output signal comprising a sequence of output data samples, wherein said interfering component estimation processing comprises: applying conjugating processing to said input signal, providing a conjugated version of said input signal; computing an adaptive signal processing coefficient value; and applying adaptive signal processing to said conjugated version of said input signal using the adaptive processing coefficient, wherein said computing the adaptive signal processing coefficient value comprises: performing correlation processing between said sequence of output data samples of the output signal and a conjugated version of said sequence of output data samples of the output signal and obtaining as a result a sequence of estimates of residual correlation; applying integration processing to said sequence of estimates of residual correlation, said integration processing using an integration step parameter and at least one starting point parameter, obtaining the adaptive signal processing coefficient as a result of applying said integration processing. In an embodiment, the signal processing circuitry, in operation, selectively applies one of: adaptive multiplication processing having a multiplication factor equal to said computed adaptive signal processing coefficient; and adaptive finite impulse response, FIR, filtering processing, said adaptive FIR filtering processing comprising computing a weighted sum of a conjugated version of data samples in said sequence of input data samples comprised in said input signal, using said adaptive processing coefficient as a weight of said weighted sum. In an embodiment, the adaptive FIR filtering processing comprises applying the adaptive processing coefficient to a subset of elements belonging to an input delay line of the FIR at a related subset of delay values; and performing correlation processing and applying integration processing respectively comprise calculating and integrating estimates of residual correlation at delay values of said related subset of delay values. In an embodiment, adaptive FIR filtering coefficients are applied to a comb of elements belonging to said input delay line of the FIR, the comb comprising a number N d  of delay elements, the delay elements having a distance value multiple of said sampling period by a factor d. In an embodiment, said distance value is in excess ten times the sampling period. In an embodiment, the signal processing circuitry, in operation, selects between said adaptive multiplication processing and said adaptive FIR filtering processing by: providing a first adaptive signal processing configuration register configured to storing a first value or a second value in a first adaptive signal processing configuration register; and applying one of said adaptive multiplication processing or said adaptive FIR filtering processing as a function of said first value or said second value stored in said first adaptive signal processing configuration register. In an embodiment, the signal processing circuitry, in operation, stores indexes of delay elements in the set of delay elements of the delay line in a configuration register; and selects a subset of delay elements in the set of delay elements of the delay line as a function of said indexes. In an embodiment, the integration processing comprises applying loop filter processing to said sequence of estimates of residual correlation provided. In an embodiment, the integration processing comprises: storing a value of said integration step parameter; storing a value of said at least one starting point parameter and a control parameter to activate said integration processing to use said value of said at least one starting point parameter; or combinations thereof. In an embodiment, the signal processing circuitry, in operation, applies automatic gain control, AGC, processing to the output signal. In an embodiment, performing correlation processing between said sequence of output data samples of the output signal and a conjugated version of said sequence of output data samples of the output signal comprises performing block-like correlation of a number of adjacent data samples forming a block having block-length L of data samples, wherein said block-length L is selected as a function of a length value stored in a configuration register. In an embodiment, the adaptive FIR filtering processing comprises second order (or three-tap) adaptive FIR filtering processing. In an embodiment, said signal processing circuitry comprises a complex down-converter circuit including: a first mixer branch, having a common input node configured to receive at least one signal, a first digital mixer, a first digital low pass filter, a first baseband decimator with decimation factor M and a first interfacing node, the first interfacing node configured to provide a first signal component; a second mixer branch, having a common input node configured to receive at least one signal, a second digital mixer, a second digital low pass filter, a second baseband decimator with decimation factor M and a second interfacing node, the second interfacing node configured to provide a second signal component; and an image rejection correction (IRC) control loop configured to apply interfering component removal processing to said first signal component, and to said second signal component. 
     In an embodiment, a radio frequency receiver comprises: an antenna configured to receive an RF signal; a radio frequency front-end coupled to said antenna, the radio frequency front-end configured to receive said RF signal at an input node and to apply down-conversion processing to said radio frequency signal, the radio frequency front-end having a first output node and configured for providing an intermediate frequency, IF, signal as a result of said down-conversion processing at said first output node; an analog-to-digital converter, ADC, having an input node coupled to said first output node and a second output node, the ADC configured to receive said IF signal from the first output node, to apply signal sampling thereto and to provide a sampled sequence of data samples of said IF signal at said second output node; and signal processing circuitry coupled to the second output node, wherein the signal processing circuitry, in operation: applies interfering component estimation processing to said input signal, obtaining a filtered signal comprising a sequence of filtered data samples; and subtracts said filtered signal from said input signal, obtaining an output signal comprising a sequence of output data samples, wherein said interfering component estimation processing comprises: applying conjugating processing to said input signal, generating a conjugated version of said input signal; computing an adaptive signal processing coefficient value; and applying adaptive signal processing to said conjugated version of said input signal using the adaptive processing coefficient, wherein said computing the adaptive signal processing coefficient value comprises: performing correlation processing between said sequence of output data samples of the output signal and a conjugated version of said sequence of output data samples of the output signal, obtaining a sequence of estimates of residual correlation; and applying integration processing to said sequence of estimates of residual correlation, said integration processing using an integration step parameter and at least one starting point parameter, obtaining the adaptive signal processing coefficient. In an embodiment, the signal processing circuitry comprises a complex down-converter circuit including: a first mixer branch, having a common input node configured to receive at least one signal, a first digital mixer, a first digital low pass filter, a first baseband decimator with decimation factor M and a first interfacing node, the first interfacing node configured to provide a first signal component; a second mixer branch, having a common input node configured to receive at least one signal, a second digital mixer, a second digital low pass filter, a second baseband decimator with decimation factor M and a second interfacing node, the second interfacing node configured to provide a second signal component; and an image rejection correction (IRC) control loop configured to apply interfering component removal processing to said first signal component, and to said second signal component. In an embodiment, the interfering component estimation processing comprises: computing a plurality of adaptive signal processing coefficient values; and applying adaptive signal processing to said conjugated version of said input signal using the plurality of adaptive processing coefficients values. 
     In an embodiment, a non-transitory computer readable medium&#39;s contents configure a receiver to perform a method, the method comprising: applying interfering component estimation processing to an input signal, obtaining a filtered signal comprising a sequence of filtered data samples; and subtracting the filtered signal from the input signal, obtaining an output signal comprising a sequence of output data samples, wherein the interfering component estimation processing comprises: applying conjugating processing to said input signal, generating a conjugated version of the input signal; computing one or more adaptive signal processing coefficients; and applying adaptive signal processing to the conjugated version of the input signal using the one or more adaptive processing coefficients, wherein the computing the adaptive signal processing coefficients comprises: performing correlation processing between the sequence of output data samples of the output signal and a conjugated version of the sequence of output data samples of the output signal, obtaining a sequence of estimates of residual correlation; applying integration processing to the sequence of estimates of residual correlation, said integration processing using an integration step parameter and at least one starting point parameter, obtaining the adaptive signal processing coefficients. In an embodiment, the contents comprise instructions, which, when executed by a processor of the receiver, cause the receiver to perform the method. 
     One or more embodiments may comprise a tailored modified Symmetric Adaptive Decorrelation algorithm, facilitating to solve the problems of SAD algorithm for wideband receivers, where the I/Q imbalance can have strong variations versus frequency. 
     One or more embodiments may comprise an image rejection correction loop. 
     One or more embodiments may comprise an adaptive filter stage and detector/correlator stage. 
     One or more embodiments may comprise at least one adjustable tap-delay line. 
     Specifically, both adaptive filter stage and detector/correlator stage may employ such an adjustable tap-delay, facilitating to mitigate side effects due to the decorrelation time of clean source signals. 
     In one or more embodiments, such an approach can be used likewise in loop-implementations (e.g., in steady-state conditions) as well as in closed form implementations (e.g., to speed-up the initial convergence). 
     One or more embodiments may, advantageously:
         facilitating avoiding a chip/receiver calibration phase,   facilitating self-converging to optimal parameter values;   facilitating receiver operations in both narrowband or wideband receiver architectures   reducing application management resources,   facilitating adjustability and adaptiveness, e.g., using a Finite-duration Impulse Response (FIR) filter and a correlator comprising tap-delays, increasing flexibility;   improving receiver performances and sensitivity with strong image interferer.       

    
    
     
       BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS 
       One or more embodiments will now be described, by way of non-limiting example only, with reference to the annexed Figures, wherein: 
         FIG. 1  is discussed in the foregoing; 
         FIGS. 1A and 1B  are exemplary of model signal spectra; 
         FIG. 2  is an exemplary diagram of an RF front-end architecture and following processing, for instance up to baseband signals; 
         FIGS. 2A, 2B, 2C , are exemplary diagrams of possible behaviors of signal spectra in one or more embodiments; 
         FIG. 3  is an exemplary diagram of a method as per the present disclosure, in double-branch IRC structures; 
         FIG. 4  is an exemplary diagram of a method as per the present disclosure, in single-branch IRC structures; 
         FIGS. 3A, 3B, 4A, 4B and 4C  are exemplary diagrams of possible behaviors of signal spectra in one or more embodiments; 
         FIG. 5  is exemplary of a control loop arrangement as per the present disclosure; 
         FIGS. 6 and 7  are exemplary of portions of the control loop arrangement of  FIG. 5 ; 
         FIGS. 8A and 8B  are exemplary of possible behaviors of signals in one or more embodiments; and 
         FIG. 9  is exemplary of a receiver arrangement as per the present disclosure. 
     
    
    
     DETAILED DESCRIPTION 
     In the ensuing description, one or more specific details are illustrated, aimed at providing an in-depth understanding of examples of embodiments of this description. The embodiments may be obtained without one or more of the specific details, or with other methods, components, materials, etc. In other cases, known structures, materials, or operations are not illustrated or described in detail so that certain aspects of embodiments will not be obscured. 
     Reference to “an embodiment” or “one embodiment” in the framework of the present description is intended to indicate that a particular configuration, structure, or characteristic described in relation to the embodiment is comprised in at least one embodiment. Hence, phrases such as “in an embodiment” or “in one embodiment” that may be present in one or more points of the present description do not necessarily refer to one and the same embodiment. 
     Moreover, particular conformations, structures, or characteristics may be combined in any adequate way in one or more embodiments. 
     The references used herein are provided merely for convenience and hence do not define the extent of protection or the scope of the embodiments. 
     The drawings are in simplified form and are not to precise scale. For the sake of simplicity, directional (up/down, etc.) or motional (forward/back, etc.) terms may be used with respect to the drawings. The term “couple” and similar terms do not necessarily denote direct and immediate connections, but also include connections through intermediate elements or devices. 
     Also, throughout this description, certain circuit nodes and the signals at these nodes will be indicated with same reference for simplicity and ease of explanation. 
       FIG. 2  may be used to provide an equivalent model of an RFFE  20  and following processing, for instance up to baseband signals, comprising for instance:
         an input node x RF  configured to receive an RF signal,   a first down-conversion stage  21  from RF to IF (briefly, RF-IF stage), for instance an (analog) RF-IF down-converter, and   a second down-conversion stage  22  from IF to baseband (briefly, IF-BB stage), for instance a digital IF-BB down-converter.       

     In a single down-conversion chain  21 , for instance, the first down-conversion stage  21  may provide input to the subsequent IRC processing stages via interfacing node x in . 
     In a double down-conversion chain  20 , for instance, both the first down-conversion stage  21  and the second down-conversion stage  22  are coupled together and may provide inputs to subsequent IRC processing stages via interfacing nodes s in , i in . 
     For instance, the first down-conversion stage  21  may comprise an analog I/Q down-converter  13  comprising RF-IF mixers  14 I,  14 Q coupled to the input node x RF , IF low-pass filters  16 I,  16 Q coupled to corresponding mixers, an intermediate frequency analog-to-digital converter (briefly, ADC)  18 , for instance having sampling period T, an interfacing node x in . 
     For instance, as exemplified in  FIG. 2 , the first down-conversion stage  21  may virtually comprise a combiner  17 , for instance a two-real to complex converter, modeling the combination of I/Q signal components interposed between low-pass filters  16 I,  16 Q in respective branches of the down-converter  13 , wherein the combiner  17  may be configured to provide an intermediate frequency signal x IF  to the ADC  18 . The ADC in turn may sample the signal and provide it at the interfacing node x in  and/or to the second down-conversion stage  22 . 
     For instance, the second down-conversion stage  22  may be configured for receiving a time-sampled IF signal and to apply processing thereto. 
     For instance, the second down-conversion stage  22  may comprise:
         a first branch  22 A, comprising a first digital IF-BB mixer  24 A, a first digital baseband low-pass filter (briefly, BB-LPF)  26 A, a first baseband decimator  28 A with decimation factor M, a first interfacing node s in ,   a second branch  22 B, comprising a second digital IF-BB mixer  24 B, a second digital baseband low-pass filter (briefly, BB-LPF)  26 B, a second baseband decimator  28 B with decimation factor M, a second interfacing node i in .       

     In the example considered, the IF-BB mixers  24 A,  24 B may comprise a pair of complex mixers, typically implemented as numerically controlled oscillators (briefly, NCOs), configured to provide signals for converting to zero-frequency, respectively, the desired signal and the image signal. 
     In the following, a continuous-time signal may be indicated as x(t), the related spectrum is indicated as X(ω), assuming the following definition of the Fourier transform: 
     
       
         
           
             
               X 
                
               
                 ( 
                 ω 
                 ) 
               
             
             = 
             
               
                 ∫ 
                 
                   - 
                   ∞ 
                 
                 ∞ 
               
                
               
                 
                   x 
                    
                   
                     ( 
                     t 
                     ) 
                   
                 
                  
                 
                   e 
                   
                     
                       - 
                       j 
                     
                      
                     
                         
                     
                      
                     ω 
                      
                     
                         
                     
                      
                     t 
                   
                 
                  
                 
                   dt 
                   . 
                 
               
             
           
         
       
     
     In the following, a discrete-time signal, which may be possibly obtained by sampling with period T a continuous-time signal x a (t), may be indicated as x(k)=x a (kT), where k is the discrete-time index. The related spectrum may be indicated as X(ω), in the hypothesis of defining the discrete-time Fourier transform as follows, applying a suitable scaling: 
     
       
         
           
             
               X 
                
               
                 ( 
                 ω 
                 ) 
               
             
             = 
             
               T 
                
               
                 
                   ∑ 
                   
                     k 
                     = 
                     
                       - 
                       ∞ 
                     
                   
                   ∞ 
                 
                  
                 
                   
                     x 
                      
                     
                       ( 
                       n 
                       ) 
                     
                   
                    
                   
                     
                       e 
                       
                         
                           - 
                           j 
                         
                          
                         
                             
                         
                          
                         ω 
                          
                         
                             
                         
                          
                         kT 
                       
                     
                     . 
                   
                 
               
             
           
         
       
     
     In the following, modulation of a carrier placed at ω 0  may indicate the conceptual mixing operation applied on a baseband equivalent x(t) in order to transmit an RF signal x RF (t)=2 Re[x(t) e jω     0     t ]. 
     In the following, a spectral contribution X(ω−ω 0 ), related to a term x(t) e jω     0     t , may be indicated as contribution X, centered around ω 0 . 
     A Radio Frequency (RF) signal x RF (t) may be expressed in terms of its components. For instance, considering only the components of interest, without loss of generality, the RF signal may be built in the following way: 
     
       
         
           
             
               
                 
                   
                     
                       x 
                       RF 
                     
                      
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                     
                    
                   
                     2 
                      
                     
                       Re 
                       [ 
                       
                         
                           
                             s 
                              
                             
                               ( 
                               t 
                               ) 
                             
                           
                            
                           
                             e 
                             
                               
                                 j 
                                  
                                 
                                   ( 
                                   
                                     
                                       ω 
                                       LO 
                                     
                                     + 
                                     
                                       ω 
                                       IF 
                                     
                                   
                                   ) 
                                 
                               
                                
                               t 
                             
                           
                         
                         + 
                         
                           
                             i 
                              
                             
                               ( 
                               t 
                               ) 
                             
                           
                            
                           
                             e 
                             
                               
                                 j 
                                  
                                 
                                   ( 
                                   
                                     
                                       ω 
                                       LO 
                                     
                                     - 
                                     
                                       ω 
                                       IF 
                                     
                                   
                                   ) 
                                 
                               
                                
                               t 
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       
                         [ 
                         
                           
                             
                               s 
                                
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                              
                             
                               e 
                               
                                 j 
                                  
                                 
                                     
                                 
                                  
                                 
                                   ω 
                                   IF 
                                 
                                  
                                 t 
                               
                             
                           
                           + 
                           
                             
                               i 
                                
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                              
                             
                               e 
                               
                                 
                                   - 
                                   j 
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   ω 
                                   IF 
                                 
                                  
                                 t 
                               
                             
                           
                         
                         ] 
                       
                        
                       
                         e 
                         
                           j 
                            
                           
                               
                           
                            
                           
                             ω 
                             
                               LO 
                               t 
                             
                           
                         
                       
                     
                     + 
                   
                 
               
             
             
               
                 
                     
                    
                   
                     
                       [ 
                       
                         
                           
                             
                               s 
                               * 
                             
                              
                             
                               ( 
                               t 
                               ) 
                             
                           
                            
                           
                             e 
                             
                               
                                 - 
                                 j 
                               
                                
                               
                                   
                               
                                
                               
                                 ω 
                                 
                                   IF 
                                   t 
                                 
                               
                             
                           
                         
                         + 
                         
                           
                             
                               i 
                               * 
                             
                              
                             
                               ( 
                               t 
                               ) 
                             
                           
                            
                           
                             e 
                             
                               j 
                                
                               
                                   
                               
                                
                               
                                 ω 
                                 IF 
                               
                                
                               t 
                             
                           
                         
                       
                       ] 
                     
                      
                     
                       e 
                       
                         
                           - 
                           j 
                         
                          
                         
                             
                         
                          
                         
                           ω 
                           LO 
                         
                          
                         t 
                       
                     
                   
                 
               
             
           
         
       
     
     where:
         s(t) indicates the complex baseband equivalent of the desired signal, transmitted in RF by modulating a carrier placed at ω RF =ω LO +ω IF ;   i(t) indicates the complex baseband equivalent of the image signal, transmitted in RF by modulating a carrier placed at ω LO −ω IF ;   the asterisk in apex (*) indicates complex conjugation, defined as: x*=Re(x)−jIm(x).       

     Applying the suitable substitution x(t)=s(t) e jω     IF     t +i(t)e −jω     IF     t , the RF signal can be expressed in the simplified form: 
         x   RF ( t )=2 Re[ x ( t ) e   jω     LO     t ]= x ( t ) e   jω     LO     t   +x *( t ) e   −jω     LO     t    
       FIG. 2B  may be exemplary of a diagram of RF signal contributions S, I, S* and I* in the domain of frequency (or spectral domain), obtained from the baseband equivalents S and I of  FIG. 2A , respectively related to desired and image signals. 
     When analyzing the I/Q imbalance problem, three cases may be of interest:
         an “ideal” balanced case, for instance where no I/Q imbalance is present,   a first imbalance case where I/Q imbalance may be considered frequency-independent, for instance constant in a certain frequency range,   a second imbalance case where I/Q imbalance may be considered frequency-dependent, for instance varying as a function of frequency.       

     In the “ideal” balanced case, a (ideal) local oscillator (LO) signal x LO (t) may be expressed as: 
         x   LO ( t )= x   LO,I ( t )+ jx   LO,Q ( t )= e   −jω     LO     t    
     where:
         x LO,I (t) indicates a first LO signal component, which may be expressed as x LO,I (t)=cos(ω LO  t);   x LO,Q (t) indicates a second LO signal component, which may be expressed as x LO,Q (t)=sin(ω LO  t).       

     In the “ideal” balanced case, generated LO signal components have same amplitudes and are in quadrature therebetween, that is one is separated in phase by 90° (or π/2) with respect to the other. 
     Still in this “ideal” balanced case, the filter stages  16 I,  16 Q have respective impulse responses h IF,I (t), h IF,Q (t) (and transfer functions H IF,I (ω), H IF,Q (ω)) which may be modeled as a nominal intermediate frequency low pass filter (briefly, IF-LPF) having likewise impulse response h IF (t) (and transfer function H IF (ω)) on both I and Q signal paths. 
     Therefore, such filter impulse responses (and respective transfer functions) may be expressed as: 
         h   IF,I ( t )= h   IF,Q ( t )= h   IF ( t )  H   IF,I (ω)= H   IF,Q (ω)= H   IF (ω).
 
     In the considered exemplary “ideal” balanced case, the IF complex signal x IF (t) may be expressed as: 
         x   IF ( t )= h   IF ( t )*[ x   RF ( t ) x   LO ( t )]= x ( t )= s ( t ) e   jω     IF     t   +i ( t ) e   −jω     IF     t    
     where the asterisk symbol (*) indicates the convolution between two continuous-time signals, which may be expressed as: 
         h ( t )* x ( t )=∫ −∞   ∞   h (τ) x ( t −τ) dτ.  
 
     For instance, in the exemplary ideal case considered herein, as exemplified in  FIG. 1A :
         contributions S, I of the RF signal x RF (t) and the oscillator signal x LO (t) may have frequency spectra |X RF (ω)| dB , |X LO (ω)| dB , as shown in portion a) of  FIG. 1A ,   the I/Q down-converter  13  may provide a down-converted signal x IF (t) having a frequency spectrum |X IF (ω)| dB , as exemplified in portion b) of  FIG. 1A , wherein contributions S and I are placed at positive and negative IF frequency, respectively, ideally without any mutual overlap.       

     As a result of baseband mixing  24 A,  24 B and filtering  26 A,  26 B, performed in the discrete-time domain (after ADC conversion  18 ) in the second down-conversion stage  22 , two baseband signals s BB  and i BB  (placed around zero-frequency) are obtained, wherein the signal component and the image component do not interfere therebetween. 
     Such baseband signals s BB  and i BB  may be expressed in the discrete-time domain k as: 
         s   BB ( k )= h   BB ( k )*[ x   IF ( kT ) x   NCO ( k )]= s ( kT ) 
         i   BB ( k )= h   BB ( k )*[ x   IF ( kT ) x*   NCO ( k )]= i ( kT ) 
     where:
         X NCO (k)=ee jω     IF     t  is the NCO signal,   h BB (k) is the impulse response of the baseband low-pass filter (BB-LPF)  26 A,  26 B, having transfer function H BB (ω) in the frequency domain, and   the asterisk symbol (*) indicates the convolution between two discrete-time signals, defined as:       

     
       
         
           
             
               
                 h 
                  
                 
                   ( 
                   k 
                   ) 
                 
               
               * 
               
                 x 
                  
                 
                   ( 
                   k 
                   ) 
                 
               
             
             = 
             
               
                 ∑ 
                 
                   n 
                   = 
                   
                     - 
                     ∞ 
                   
                 
                 ∞ 
               
                
               
                 
                   h 
                    
                   
                     ( 
                     n 
                     ) 
                   
                 
                  
                 
                   
                     x 
                      
                     
                       ( 
                       
                         k 
                         - 
                         n 
                       
                       ) 
                     
                   
                   . 
                 
               
             
           
         
       
     
     In a more realistic case, for instance in the first imbalance case, I/Q imbalance may be present and may be constant in frequency. 
     In such an exemplary first imbalance case, the local oscillator signal x LO (t) may rather be expressed for instance as: 
     
       
         
           
             
               
                 
                   x 
                   
                     LO 
                     , 
                     I 
                   
                 
                  
                 
                   ( 
                   t 
                   ) 
                 
               
               = 
               
                 cos 
                  
                 
                   ( 
                   
                     
                       ω 
                       LO 
                     
                      
                     t 
                   
                   ) 
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 x 
                 
                   LO 
                   , 
                   Q 
                 
               
                
               
                 ( 
                 t 
                 ) 
               
             
             = 
             
               
                 - 
                 g 
               
                
               
                   
               
                
               
                 sin 
                  
                 
                   ( 
                   
                     
                       
                         ω 
                         LO 
                       
                        
                       t 
                     
                     + 
                     φ 
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 x 
                 LO 
               
                
               
                 ( 
                 t 
                 ) 
               
             
             = 
             
               
                 
                   x 
                   
                     LO 
                     , 
                     I 
                   
                 
                  
                 
                   ( 
                   t 
                   ) 
                 
               
               + 
               j 
             
           
         
       
       
         
           
             
               
                 x 
                 
                   LO 
                   , 
                   Q 
                 
               
                
               
                 ( 
                 t 
                 ) 
               
             
             = 
             
               
                 
                   K 
                   1 
                 
                  
                 
                   e 
                   
                     
                       - 
                       j 
                     
                      
                     
                         
                     
                      
                     
                       ω 
                       LO 
                     
                      
                     t 
                   
                 
               
               + 
               
                 
                   K 
                   2 
                 
                  
                 
                   e 
                   
                     j 
                      
                     
                         
                     
                      
                     
                       ω 
                       LO 
                     
                      
                     t 
                   
                 
               
             
           
         
       
     
     wherein: 
     
       
         
           
             
               K 
               1 
             
             = 
             
               
                 1 
                 + 
                 
                   ge 
                   
                     
                       - 
                       j 
                     
                      
                     
                         
                     
                      
                     φ 
                   
                 
               
               2 
             
           
         
       
       
         
           
             
               K 
               2 
             
             = 
             
               
                 1 
                 - 
                 
                   ge 
                   
                     
                       j 
                        
                       
                           
                       
                        
                       φ 
                     
                      
                     
                         
                     
                   
                 
               
               2 
             
           
         
       
     
     where g is the gain imbalance, ϕ is the phase imbalance and K 1 , K 2  are the mismatch coefficients: in the “ideal” case, K 1 =1, K 2 =0; typically, |K 2 |&lt;&lt;|K 1 |≅1. 
     As a result, at the output of the RF-IF stage  21  of the receiver  10 , the IF signal x IF  may be expressed as a function of time t as: 
     
       
         
           
             
               
                 x 
                 IF 
               
                
               
                 ( 
                 t 
                 ) 
               
             
             = 
             
               
                 
                   
                     h 
                     IF 
                   
                    
                   
                     ( 
                     t 
                     ) 
                   
                 
                 * 
                 
                   [ 
                   
                     
                       
                         x 
                         RF 
                       
                        
                       
                         ( 
                         t 
                         ) 
                       
                     
                      
                     
                       
                         x 
                         LO 
                       
                        
                       
                         ( 
                         t 
                         ) 
                       
                     
                   
                   ] 
                 
               
               = 
               
                 
                   
                     
                       K 
                       1 
                     
                      
                     
                       x 
                        
                       
                         ( 
                         t 
                         ) 
                       
                     
                   
                   + 
                   
                     
                       K 
                       2 
                     
                      
                     
                       
                         x 
                         * 
                       
                        
                       
                         ( 
                         t 
                         ) 
                       
                     
                   
                 
                 = 
                 
                   
                     
                       
                         K 
                         1 
                       
                        
                       
                         [ 
                         
                           
                             
                               s 
                                
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                              
                             
                               e 
                               
                                 j 
                                  
                                 
                                     
                                 
                                  
                                 
                                   ω 
                                   IF 
                                 
                                  
                                 t 
                               
                             
                           
                           + 
                           
                             
                               i 
                                
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                              
                             
                               e 
                               
                                 
                                   - 
                                   j 
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   ω 
                                   IF 
                                 
                                  
                                 t 
                               
                             
                           
                         
                         ] 
                       
                     
                     + 
                     
                       
                         K 
                         2 
                       
                        
                       
                         [ 
                         
                           
                             
                               
                                 s 
                                 * 
                               
                                
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                              
                             
                               e 
                               
                                 
                                   - 
                                   j 
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   ω 
                                   IF 
                                 
                                  
                                 t 
                               
                             
                           
                           + 
                           
                             
                               
                                 i 
                                 * 
                               
                                
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                              
                             
                                 
                             
                              
                             
                               e 
                               
                                 j 
                                  
                                 
                                     
                                 
                                  
                                 
                                   ω 
                                   IF 
                                 
                                  
                                 t 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                   = 
                   
                     
                       
                         [ 
                         
                             
                         
                          
                         
                           
                             
                               K 
                               1 
                             
                              
                             
                               s 
                                
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                           
                           + 
                           
                             
                               K 
                               2 
                             
                              
                             
                               
                                 i 
                                 * 
                               
                                
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                           
                         
                         ] 
                       
                        
                       
                           
                       
                        
                       
                         e 
                         
                           j 
                            
                           
                               
                           
                            
                           
                             ω 
                             IF 
                           
                            
                           t 
                         
                       
                     
                     + 
                     
                       
                         [ 
                         
                             
                         
                          
                         
                           
                             
                               K 
                               1 
                             
                              
                             
                               i 
                                
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                           
                           + 
                           
                             
                               K 
                               2 
                             
                              
                             
                               
                                 s 
                                 * 
                               
                                
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                           
                         
                         ] 
                       
                        
                       
                         
                           e 
                           
                             
                               - 
                               j 
                             
                              
                             
                                 
                             
                              
                             
                               ω 
                               IF 
                             
                              
                             t 
                           
                         
                         . 
                       
                     
                   
                 
               
             
           
         
       
     
     For instance, in the exemplary first imbalance case as considered herein, as exemplified in  FIG. 1A :
         contributions S, I of the RF signal x RF (t) and the oscillator signal x LO (t) may have frequency spectra |X RF (ω)| dB , |X LO (ω)| dB , as shown in portion a) of  FIG. 1B ,   as exemplified in portion b) of  FIG. 1B , the down-converter mixer  13  may provide a down-converted IF signal x IF (t) having a frequency spectrum |X IF (ω)| dB , wherein around ω IF  a first contribution K 1  S may be overlapped with a second contribution K 2  I*; vice versa, around −ω IF , a third contribution K 1  I may be overlapped with a fourth contribution K 2  S*.       

     As a consequence, in the exemplary first imbalance case as discussed herein, the desired signal s(t) and the image signal i(t) may be mixed indistinguishably, introducing unwanted cross-interference. 
     Such an interference may propagate and be observed also in the baseband signals s BB , i BB , produced by the IF-BB down-conversion stage  22 . For instance, baseband signals s BB , i BB  may be expressed as: 
         s   BB ( k )= h   BB ( k )*[ x   IF ( kT ) X   NCO ( k )]= K   1   s ( kT )+ K   2   i *( kT ) 
         i   BB ( k )= h   BB ( k )*[ x   IF ( kT ) x*   NCO ( k )]= K   1   i ( kT )+ K   2   s *( kT ) 
     I/Q imbalance may lead to crosstalk between wanted signal and image signal, whose amount can be quantified by coefficients ratio K 2 /K 1 *. 
     As mentioned, in the considered example of the first imbalance case, such I/Q imbalance here considered is frequency-independent, because this ratio is related only to the (constant) coefficients K 1  and K 2 . 
     In the second imbalance case, filter impulse responses h IF,I (t), h IF,Q (t) (or transfer functions H IF,I (ω), H IF,Q (ω), in frequency domain) of the respective I/Q signal paths  16 I and  16 Q may have a mismatch with respect to the impulse response h IF (t) (or, similarly, the related transfer function H IF (ω)) of the nominal IF-LPF. 
     For instance, in the second imbalance case, such filter impulse responses (and respective transfer functions) may be expressed as: 
         h   IF,I ( t )= h   I ( t )* h   IF ( t )  h   IF,Q ( t )= h   Q ( t )* h   IF ( t ) 
         H   IF,I (ω)= H   I (ω) H   IF (ω) H   IF,Q (ω) H   Q (ω) H   IF (ω)
 
     where h I (t), h Q (t) represent, in time domain, the effect of mismatch on I and Q branches, with respect to the nominal IF-LPF, and H I (ω), H Q (ω) represent like mismatch expressed as a function of frequency. 
     In this second imbalance case, as a result of down-conversion  13 , the IF signal components x IF,I (t), X IF,Q (t) may be expressed as: 
     
       
         
           
             
               
                 x 
                 
                   IF 
                   , 
                   I 
                 
               
                
               
                 ( 
                 t 
                 ) 
               
             
             = 
             
               
                 
                   
                     h 
                     
                       IF 
                       , 
                       I 
                     
                   
                    
                   
                     ( 
                     t 
                     ) 
                   
                 
                 * 
                 
                   [ 
                   
                     
                       
                         x 
                         RF 
                       
                        
                       
                         ( 
                         t 
                         ) 
                       
                     
                      
                     
                       
                         x 
                         
                           LO 
                           , 
                           I 
                         
                       
                        
                       
                         ( 
                         t 
                         ) 
                       
                     
                   
                   ] 
                 
               
               = 
               
                 
                   
                     h 
                     I 
                   
                    
                   
                     ( 
                     t 
                     ) 
                   
                 
                 * 
                 
                   
                     
                       x 
                        
                       
                         ( 
                         t 
                         ) 
                       
                     
                     + 
                     
                       
                         x 
                         * 
                       
                        
                       
                         ( 
                         t 
                         ) 
                       
                     
                   
                   2 
                 
               
             
           
         
       
       
         
           
             
               
                 x 
                 
                   IF 
                   , 
                   Q 
                 
               
                
               
                 ( 
                 t 
                 ) 
               
             
             = 
             
               
                 
                   
                     h 
                     
                       IF 
                       , 
                       Q 
                     
                   
                    
                   
                     ( 
                     t 
                     ) 
                   
                 
                 * 
                 
                   [ 
                   
                     
                       
                         x 
                         RF 
                       
                        
                       
                         ( 
                         t 
                         ) 
                       
                     
                      
                     
                       
                         x 
                         
                           LO 
                           , 
                           Q 
                         
                       
                        
                       
                         ( 
                         t 
                         ) 
                       
                     
                   
                   ] 
                 
               
               = 
               
                 
                   
                     gh 
                     Q 
                   
                    
                   
                     ( 
                     t 
                     ) 
                   
                 
                 * 
                 
                   
                     
                       
                         x 
                          
                         
                           ( 
                           t 
                           ) 
                         
                       
                        
                       
                         e 
                         
                           
                             - 
                             j 
                           
                            
                           
                               
                           
                            
                           φ 
                         
                       
                     
                     - 
                     
                       
                         
                           x 
                           * 
                         
                          
                         
                           ( 
                           t 
                           ) 
                         
                       
                        
                       
                         e 
                         
                           j 
                            
                           
                               
                           
                            
                           φ 
                         
                       
                     
                   
                   
                     2 
                      
                     j 
                   
                 
               
             
           
         
       
     
     and their combination, in the combiner  17 , may be expressed as: 
     
       
         
           
             
               
                 x 
                 IF 
               
                
               
                 ( 
                 t 
                 ) 
               
             
             = 
             
               
                 
                   
                     x 
                     
                       IF 
                       , 
                       I 
                     
                   
                    
                   
                     ( 
                     t 
                     ) 
                   
                 
                 + 
                 
                   
                     jx 
                     
                       IF 
                       , 
                       Q 
                     
                   
                    
                   
                     ( 
                     t 
                     ) 
                   
                 
               
               = 
               
                 
                   
                     
                       
                         k 
                         1 
                       
                        
                       
                         ( 
                         t 
                         ) 
                       
                     
                     * 
                     
                       x 
                        
                       
                         ( 
                         t 
                         ) 
                       
                     
                   
                   + 
                   
                     
                       
                         k 
                         2 
                       
                        
                       
                         ( 
                         t 
                         ) 
                       
                     
                     * 
                     
                       
                         x 
                         * 
                       
                        
                       
                         ( 
                         t 
                         ) 
                       
                     
                   
                 
                 = 
                 
                   
                     
                       
                         k 
                         1 
                       
                        
                       
                         ( 
                         t 
                         ) 
                       
                     
                     * 
                     
                       [ 
                       
                         
                           
                             s 
                              
                             
                               ( 
                               t 
                               ) 
                             
                           
                            
                           
                             e 
                             
                               j 
                                
                               
                                   
                               
                                
                               
                                 ω 
                                 IF 
                               
                                
                               t 
                             
                           
                         
                         + 
                         
                           
                             i 
                              
                             
                               ( 
                               t 
                               ) 
                             
                           
                            
                           
                             e 
                             
                               
                                 - 
                                 j 
                               
                                
                               
                                   
                               
                                
                               
                                 ω 
                                 IF 
                               
                                
                               t 
                             
                           
                         
                       
                       ] 
                     
                   
                   + 
                   
                     
                       
                         k 
                         2 
                       
                        
                       
                         ( 
                         t 
                         ) 
                       
                     
                     * 
                     
                       [ 
                       
                         
                           
                             
                               s 
                               * 
                             
                              
                             
                               ( 
                               t 
                               ) 
                             
                           
                            
                           
                             e 
                             
                               
                                 - 
                                 j 
                               
                                
                               
                                   
                               
                                
                               
                                 ω 
                                 IF 
                               
                                
                               t 
                             
                           
                         
                         + 
                         
                           
                             
                               i 
                               * 
                             
                              
                             
                               ( 
                               t 
                               ) 
                             
                           
                            
                           
                             e 
                             
                               j 
                                
                               
                                   
                               
                                
                               
                                 ω 
                                 IF 
                               
                                
                               t 
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
     where 
     
       
         
           
             
               
                 k 
                 1 
               
                
               
                 ( 
                 t 
                 ) 
               
             
             = 
             
               
                 
                   
                     h 
                     1 
                   
                    
                   
                     ( 
                     t 
                     ) 
                   
                 
                 + 
                 
                   
                     ge 
                     
                       
                         - 
                         j 
                       
                        
                       
                           
                       
                        
                       φ 
                     
                   
                    
                   
                     
                       h 
                       Q 
                     
                      
                     
                       ( 
                       t 
                       ) 
                     
                   
                 
               
               2 
             
           
         
       
       
         
           
             
               
                 k 
                 2 
               
                
               
                 ( 
                 t 
                 ) 
               
             
             = 
             
               
                 
                   
                     h 
                     1 
                   
                    
                   
                     ( 
                     t 
                     ) 
                   
                 
                 - 
                 
                   
                     ge 
                     
                       j 
                        
                       
                           
                       
                        
                       φ 
                     
                   
                    
                   
                     
                       h 
                       Q 
                     
                      
                     
                       ( 
                       t 
                       ) 
                     
                   
                 
               
               2 
             
           
         
       
     
     represent the mismatch impulse responses. 
     A similar analysis for this exemplary second imbalance case may be performed in the frequency domain, yielding the following expression for IF signal spectrum X IF : 
     
       
         
           
             
               
                 X 
                 IF 
               
                
               
                 ( 
                 ω 
                 ) 
               
             
             = 
             
               
                 
                   
                     
                       K 
                       1 
                     
                      
                     
                       ( 
                       ω 
                       ) 
                     
                   
                    
                   
                       
                   
                    
                   
                     X 
                      
                     
                       ( 
                       ω 
                       ) 
                     
                   
                 
                 + 
                 
                   
                     
                       K 
                       2 
                     
                      
                     
                       ( 
                       ω 
                       ) 
                     
                   
                    
                   
                       
                   
                    
                   
                     
                       X 
                       * 
                     
                      
                     
                       ( 
                       
                         - 
                         ω 
                       
                       ) 
                     
                   
                 
               
               = 
               
                 
                   
                     
                       K 
                       1 
                     
                      
                     
                       ( 
                       ω 
                       ) 
                     
                   
                    
                   
                       
                   
                   [ 
                   
                     
                       S 
                        
                       
                         ( 
                         
                           ω 
                           - 
                           
                             ω 
                             IF 
                           
                         
                         ) 
                       
                     
                     + 
                     
                       I 
                        
                       
                         ( 
                         
                           ω 
                           - 
                           
                             ω 
                             IF 
                           
                         
                         ) 
                       
                     
                   
                   ] 
                 
                 + 
                 
                   
                     
                       K 
                       2 
                     
                      
                     
                       ( 
                       ω 
                       ) 
                     
                   
                    
                   
                       
                   
                   [ 
                   
                     
                       
                         S 
                         * 
                       
                        
                       
                         ( 
                         
                           
                             - 
                             ω 
                           
                           - 
                           
                             ω 
                             IF 
                           
                         
                         ) 
                       
                     
                     + 
                     
                       
                         I 
                         * 
                       
                        
                       
                         ( 
                         
                           ω 
                           - 
                           
                             ω 
                             IF 
                           
                         
                         ) 
                       
                     
                   
                   ] 
                 
               
             
           
         
       
     
     wherein 
     
       
         
           
             
               
                 K 
                 1 
               
                
               
                 ( 
                 ω 
                 ) 
               
             
             = 
             
               
                 
                   
                     
                       
                         H 
                         I 
                       
                        
                       
                         ( 
                         ω 
                         ) 
                       
                     
                     + 
                     
                       g 
                        
                       
                           
                       
                        
                       
                         e 
                         
                           
                             - 
                             j 
                           
                            
                           
                               
                           
                            
                           φ 
                         
                       
                        
                       
                           
                       
                        
                       
                         
                           H 
                           Q 
                         
                          
                         
                           ( 
                           ω 
                           ) 
                         
                       
                     
                   
                   2 
                 
                  
                 
                     
                 
                  
                 
                   
                     K 
                     2 
                   
                    
                   
                     ( 
                     ω 
                     ) 
                   
                 
               
               = 
               
                 
                   
                     
                       H 
                       I 
                     
                      
                     
                       ( 
                       ω 
                       ) 
                     
                   
                   - 
                   
                     g 
                      
                     
                         
                     
                      
                     
                       e 
                       
                         j 
                          
                         
                             
                         
                          
                         φ 
                       
                     
                      
                     
                         
                     
                      
                     
                       
                         H 
                         Q 
                       
                        
                       
                         ( 
                         ω 
                         ) 
                       
                     
                   
                 
                 2 
               
             
           
         
       
     
     represent mismatch transfer functions, which may vary as a function of frequency. 
     A ratio K 2 (ω)/K* 1 (ω) involving mismatch transfer functions K 1 (ω), K 2 (ω) may be indicative of the amount of introduced cross-talk. This ratio K 2 (ω)/K* 1 (−ω) may vary, in general, as a function of frequency. The I/Q imbalance here considered may be, in general, frequency-dependent. 
     It is noted that in a special case wherein mismatch transfer functions are equal, e.g., H I (ω)=H Q (ω), it is possible to retrieve a frequency-independent I/Q imbalance, characterized by K 2 (ω)/K* 1 (−ω)=K 2 /K* 1 , as in the case with nominal H IF (ω) (a difference lying in that mismatches with respect to the nominal H IF (ω) could be responsible, in any case, of small distortions). 
     A cause of frequency-dependent I/Q imbalance, as in the second imbalance case scenario, involves time delay i of “Q” signal path  16 Q, for example, with respect to “I” signal path  16 I. Such a time delay is related to signal path mismatches which may be expressed by the respective impulse responses h I (t), h Q (t) and the corresponding transfer functions H I (ω), H Q (ω) as follows: 
         h   I ( t )=δ( t )  h   Q ( t )=δ( t −τ)
 
         H   I (ω)=1  h   Q (ω)= e   −jωτ 
 
     wherein δ(t) is the Dirac delta function. 
     Still in the considered exemplary second imbalance case scenario, processing mismatched signals in the IF-BB down-conversion stage  22  may yield baseband signal spectra s BB (ω), I BB (ω) which, for instance for |ωT|≤π, may expressed as: 
         s   BB (ω)= H   BB (ω) X   IF (ω+ω IF )= K   1 (ω+ω IF ) S (ω)+ K   2 (ω+ω IF ) I *(−ω)
 
         I   BB (ω)= H   BB (ω) X   IF (ω−ω IF )= K   1 (ω−ω IF ) I (ω)+ K   2 (ω−ω IF ) S *(−ω).
 
     As mentioned, solutions for improving detection and correction of such I/Q imbalance errors may facilitate to achieve a satisfactory level of performance of a signal processing system comprising a wideband receiver  20 . 
     As mentioned, the ratio K 2 /K* 1  may be used to quantify the crosstalk phenomenon, in terms of leakage amount. Other indicators based on the mismatch coefficients K 1 , K 2  can be used as well. 
     For instance, the so-called Image Rejection Ratio (briefly, IRR) is a relevant figure of merit of such receiver RFFE architectures  20 , providing an indication of the amount of attenuation achieved inside the front-end, relatively to the crosstalk phenomenon. 
     Such an IRR parameter may be evaluated at the output of the RF-IF down-conversion stage  21  and expressed as follows:
         in the first imbalance case (frequency-independent):       

     
       
         
           
             IRR 
             = 
             
               10 
                
               
                   
               
                
               
                 log 
                 10 
               
                
               
                 
                   
                      
                     
                       K 
                       1 
                     
                      
                   
                   2 
                 
                 
                   
                      
                     
                       K 
                       2 
                     
                      
                   
                   2 
                 
               
             
           
         
       
         
         
           
             in the second imbalance case (frequency-dependent): 
           
         
       
    
     
       
         
           
             
               IRR 
                
               
                 ( 
                 ω 
                 ) 
               
             
             = 
             
               10 
                
               
                   
               
                
               
                 log 
                 10 
               
                
               
                 
                   
                     
                        
                       
                         
                           K 
                           1 
                         
                          
                         
                           ( 
                           ω 
                           ) 
                         
                       
                        
                     
                     2 
                   
                   
                     
                        
                       
                         
                           K 
                           2 
                         
                          
                         
                           ( 
                           ω 
                           ) 
                         
                       
                        
                     
                     2 
                   
                 
                 . 
               
             
           
         
       
     
     For the sake of simplicity, exemplary embodiments are discussed in the following mainly with respect to the case wherein I/Q imbalance is constant in frequency (first imbalance case scenario), being otherwise understood that such a discussion is purely exemplary and in no way limiting. 
     One or more embodiments may be particularly suited to deal with I/Q imbalance which varies with respect to frequency, for instance in the second imbalance case scenario. 
     In such an exemplary first imbalance case scenario, as exemplified in portion b) of  FIG. 1B , one or more embodiments may facilitate extracting a desired signal S from the received RF signal, for instance using decorrelation techniques such as symmetric adaptive decorrelation (briefly, SAD) or blind source separation, as discussed in the following. 
     Known analog techniques aiming to remove or attenuate such a parasitic effect may facilitate obtaining values of IRR parameter approximately of the order of 30 to 40 dB. 
     Modern radio applications may highly benefit from values of IRR of about 80 dB or more, which may be achieved thanks to digital algorithms. 
     For instance, in heterodyne radio receivers, applying digital decorrelation algorithms after the analog front-end, may reduce the effects of I/Q imbalance for narrowband receivers. 
     Specifically, a Symmetric Adaptive Decorrelation (SAD) algorithm, for instance as disclosed in S. Van Gerven and D. Van Compernolle, “Signal Separation by Symmetric Adaptive Decorrelation: Stability, Convergence, and Uniqueness”,  IEEE Trans. Signal Processing , vol. 43, no. 7, pp. 1602-1612, July 1995, may be found suitable for use in narrowband receivers. 
     A SAD technique may hardly be a feasible for use with wideband receivers, due to lack of efficiency and effectiveness, e.g., due to the presence of a strongly high frequency-dependent I/Q imbalance. 
     As mentioned in the foregoing, for narrowband receivers the I/Q imbalance can be considered constant versus frequency, as discussed with respect to the first imbalance case, while in a wideband radio receiver the I/Q imbalance can have strong variations as a function of frequency, as discussed with respect to the second (frequency-dependent) imbalance case. 
       FIGS. 2A, 2B and 2C  are exemplary diagrams of possible frequency spectra which may be processed as exemplified herein, for instance in a first case of (frequency-independent) I/Q imbalance. 
     For instance:
           FIG. 2A  comprises two portions a) and b), wherein:       

     portion a) is an exemplary diagram of a baseband equivalent spectrum S of a desired signal S(ω) (expressed in time domain as s(t)), and 
     portion b) is an exemplary diagram of a baseband equivalent spectrum I of an image signal I(ω) (expressed in time domain as i(t));
           FIG. 2B  is an exemplary diagram of an RF signal spectrum X RF  of a received RF signal X RF (ω) (expressed in time domain as x RF (t)), wherein the RF signal spectrum X RF  comprises:       

     a first pair of desired signal spectral contributions S, S*, placed around ±(ω LO +ω IF ), respectively, 
     a second pair of image signal spectral contributions I, I*, placed around ±(ω LO −ω IF ), respectively;
           FIG. 2C  is an exemplary diagram of an IF signal spectrum X IF  of an IF signal X IF (ω) (expressed in time domain as x IF (t)) which may be obtained as a result of signal processing  13 ,  21 , wherein the IF signal spectrum X IF  comprises:   a first IF signal spectrum portion which may be placed around a first “positive” IF frequency ω IF , the first IF signal spectrum portion comprising a superposition of a first spectral contribution K 1  S and a second spectral contribution K 2  I*, wherein the first spectral contribution K 1  S may be a result of the product of a first mismatch coefficient K 1  with a first desired signal spectral contribution S, and wherein the second spectral contribution K 2  I* may be a result of the product of a second mismatch coefficient K 2  with a second image signal spectral contribution I*;   a second IF signal spectrum portion which may be placed around a second “negative” IF frequency ω IF  opposed said first “positive” frequency, the second IF signal spectrum portion comprising a superposition of a third spectral contribution K 1  I and a fourth spectral contribution K 2  S*, wherein the third spectral contribution K 1  I may be a result of the product of a first mismatch coefficient K 1  with a first image signal spectral contribution I, and wherein the fourth spectral contribution K 2  S* may be a result of the product of a second mismatch coefficient K 2  with a second desired signal spectral contribution S*.       

     In general, in the second imbalance case wherein I/Q imbalance may vary with frequency such as in wideband receivers, applying SAD technique may yield unsatisfactory performances limited by decorrelation time of clean source signals (desired and image signals, without crosstalk). Specifically, SAD may be inefficient when the loop operates at a high sampling rate as in the case of wideband receivers, wherein the decorrelation time could be higher than the sampling period. 
     As discussed herein, “narrowband” may refer to a receiver in which the bandwidth of interest within the RF signal (e.g., suitable bandwidth for successive baseband processing, after down-conversion to IF and analog-to-digital conversion) is (significantly) lower (e.g., by a factor of at least 10) with respect to the analog bandwidth of the system, limited by the IF stage  21 . Otherwise, the receiver is considered “wideband”. 
       FIG. 3  is exemplary of a method for image rejection correction (briefly, IRC), comprising an IRC control loop stage  36 , configured to adjust at least one complex coefficient value w(k), in order to get to an expected value E[s out (k) i out (k)]=0, as discussed in the following. 
     Specifically, one or more embodiments, as exemplified in  FIG. 3 , may be useful when a non-zero intermediate frequency is used, e.g., ω IF ≠0 Hz, and when IRC is performed on baseband signals. As a result, as discussed in the following, an IRC processing stage may be coupled to baseband stages, which follow a second down-conversion stage  22 . 
     Unless otherwise discussed in the following, in  FIG. 3  (and in  FIGS. 4 and 5  as well) parts or elements like parts or elements already discussed in the foregoing are indicated with like reference/numerals, so that a corresponding detailed description will not be repeated here for brevity. 
     In one or more embodiments as exemplified in  FIG. 3 , a (double-branch) crosstalk correction system  30  may comprise:
         an RF input node x RF , configured to receive an RF signal x RF (t), e.g., carrying multiple channels of data desired by a user,   a first down-conversion stage  21 , e.g., an analog down-conversion stage, coupled to the RF input node x RF  and configured to receive therefrom the RF signal, to multiply the RF signal x RF (t) with the components x LO,I (t), X LO,Q (t) of a local oscillator (LO), having a frequency ω w , and low-pass filter the resulting signals, hence generating respective x IF,I (t) and X IF,Q (t) signal components;   a second down-conversion stage  22 , e.g., a digital down-conversion stage, coupled to the first down-conversion stage  21  and configured to receive therefrom sampled versions x IF,I (kT) and X IF,Q  (kT) of the respective signal components and to apply digital complex processing thereto, thus providing complex baseband signals s BB (k), i BB (k),   an image rejection correction (IRC) control loop  36  coupled to the second down-conversion stage  22  and configured to receive a pair of (sampled) signals s in , i in , for instance expressed as s in (k)=s BB (kM), i in (k)=i BB (kM), where M is a suitable decimation factor, the IRC control loop  36  configured to apply image rejection correction processing  36  to such pair of (sampled) signals s in , i in , facilitating decorrelating a desired signal component s out (k) and an image signal component i out (k) therebetween, as discussed in the following.       

     As mentioned, in one or more embodiments, the components in  FIG. 3  may operate partially in the analog domain and partially in the digital domain. 
     In one or more embodiments, the ADC stage  18  in the first down-conversion stage  21  may comprise a first ADC coupled to the I signal path filter  16 I and a second ADC coupled to the Q signal path filter  16 Q and configured to convert from analog to digital the respective I and Q signal components of the down-converted signal x IF (kT). 
     In one or more embodiments as exemplified in  FIG. 3 , crosstalk parasitic effects, originating from the analog down-conversion implemented in the first down-conversion stage  21 , may be spectrally shifted to baseband in the second down-conversion stage  22 , the latter employing digital techniques for the related down-conversion processing of complex signals and, therefore, without introducing further crosstalk contributions. 
     Alternatively, in one or more embodiments as exemplified in  FIG. 4  and discussed in the foregoing, crosstalk parasitic effects, originating from a direct down-conversion implemented in the first down-conversion stage  21 , may directly affect baseband. 
       FIG. 3A  is a diagram exemplary of possible input signal spectra s in , I in  for the (double-branch) IRC control loop  36 , as discussed in the following. 
     As mentioned, due to I/Q imbalance generated in the first (analog) down-conversion stage  21 , output signals from the second down-conversion stage  22 , as exemplified in  FIG. 3A , may comprise a superposition of a first signal proportional to the expected/desired signal, e.g., K 1  S, as well as a second signal proportional to a conjugated version of an other/image signal, e.g., K 2  I*. 
     Specifically:
         portion a) of  FIG. 3A  is exemplary of a first input (complex) signal spectrum s in  comprising a superposition of a first spectral contribution K 1  S and a second spectral contribution K 2  I*, wherein the first spectral contribution K 1  S may be a result of the product of a first mismatch coefficient K 1  with a first desired signal spectral contribution S, and wherein the second spectral contribution K 2  I * may be a result of the product of a second mismatch coefficient K 2  with a second image signal spectral contribution I*;   portion b) of  FIG. 3A  is exemplary of a second input (complex) signal spectrum I in  comprising a superposition of a third spectral contribution K 1  I and a fourth spectral contribution K 2  S*, wherein the third spectral contribution K 1  I may be a result of the product of a first mismatch coefficient K 1  with a first image signal spectral contribution I, and wherein the fourth spectral contribution K 2  S* may be a result of the product of a second mismatch coefficient K 2  with a second desired signal spectral contribution S*.       

     In one or more embodiments as exemplified in  FIG. 3 , the (double-branch) IRC control loop  36  may comprise:
         a first input node s in  and a second input node i in , wherein the first input node s in  is configured to receive a first complex signal s in (k) (for example, decimated by a factor M) from the second down-conversion stage  22 , for instance a first signal expressed as s in (k)=s BB (kM); the second input node i in  is configured to receive a second complex signal i in (k) (for example, decimated by a factor M) from the second down-conversion stage  22 , for instance a second signal expressed as i in (k)=i BB (kM);   a first conjugating stage  360 A and a second conjugating stage  360 B, the first conjugating stage  360 A being coupled to first input node s in  and configured to provide a first conjugated complex signal s* in (k), and the second conjugating stage  360 B being coupled to the second input node i in  and configured to provide a second conjugated complex signal i* in (k);   a first adaptive multiplier (or filter)  362 A and a second adaptive multiplier (or filter)  362 B having multiplication/filtering coefficients as discussed in the foregoing, the first adaptive multiplier  362 A being coupled to first conjugating stage  360 A and the second adaptive multiplier  362 B being coupled to second conjugating stage  360 B;   a first adder stage  364 A and a second adder stage  364 B, the first adder stage  364 A being coupled to the input node s in  and to the output of the first multiplier stage  362 A, and the second adder stage  364 B being coupled to the second input node i in  and to the output of the second multiplier stage  364 B,   a feedback branch  368 ,  369  coupled to the first and second adder stages  364 A,  364 B and to the first and second adaptive multiplier stages  362 A,  362 B, the feedback branch  368 ,  369  configured to vary or adapt multiplier stage coefficients as discussed in the following,   a first output node s out  and a second output node i out , the first output node s out  being coupled to the output node of the first adder stage  364 A and the second output node i out  being coupled to the output node of the second adder stage  364 B.       

     In one or more embodiments, the first and second adder stages  364 A,  364 B may comprise circuitry configured to add a plurality of input signals therebetween and to output respective (combined) output signals at respective output nodes s out , i out . 
     In one or more embodiments, the feedback branch  368 ,  369  may be configured to provide coefficients to the first and/or second multiplier stages  362 A,  362 B as discussed in the following so that a suitable signal may be injected in the signal paths in order to compensate and cancel out unwanted signals due to parasitic cross-interference or I/Q imbalance. 
     In one or more embodiments, the feedback branch  368 - 369  may comprise circuitry, logic, and/or code configured to determine at least one coupling coefficient w indicative of coupling between received signals. 
     In one or more embodiments, the at least one complex coefficient w of the multiplier stages  362 A,  362 B may be computed in the IRC control loop  36 , as discussed in the following. 
     For instance, in one or more embodiments as exemplified in  FIG. 3 , the feedback branch  368 ,  369  may comprise:
         a correlator stage  368 , for instance a multiplier operable as correlator, coupled to the output of the first adder  364 A and to the output of the second adder  364 B, and   a loop filter stage  369  configured to determine the complex coupling coefficients, to provide such coefficients to the first and second multipliers  362 A,  362 B, in order to configure them to generate output signals that may cancel superimposed signals (due to cross-interference) when subtracted to the signals input at the adders  364 A,  364 B.       

     In one or more embodiments, processing  36  may be performed in the assumption that the desired signal and the conjugated of the image signal (or, equivalently, the image signal and the conjugated of the desired signal) are uncorrelated, which may be expressed as: 
         E [ s ( t )( i *( t ))*]= E [ i ( t )( s *( t ))*]= E [ s ( t ) i ( t )]=0. 
     As mentioned, mixing and filtering operations in the first down-conversion stage  21  may cause spectral overlap between desired signal and image signal components s(t), i(t), resulting in non-zero cross-interference, while mixing in the second down-conversion stage  22  may shift these effects to baseband, for instance on signals s BB  (k), i BB (k), which may be applied to IRC control loop  36  input nodes s in , i in . 
     For instance, in one or more embodiments before starting compensation processing, the feedback branch correlator/multiplier  368  may receive:
         the desired signal K 1  s(kMT), corrupted by a superimposed version of the image signal K 2  i*(kMT);   the image signal K 1  i(kMT), corrupted by a superimposed version of the desired signal K 2  S*(kMT).       

     In one or more embodiments, the IRC control loop stage  36  may decorrelate the desired signal K 1  s(kMT), corrupted by a conjugated version of the image signal K 2  i*(kMT), from the image signal K 1  i(kMT), corrupted by a conjugated version of the desired signal K 2  S*(kMT), such that the estimates at the IRC outputs are uncorrelated, in order to get to an expected value E[s t (k) i out (k)]=0. 
     In one or more embodiments, the at least one complex coupling coefficient w between signals may be determined utilizing decorrelation techniques, and the desired signal may be extracted using the determined coefficient w. 
     In one or more embodiments as exemplified by the related spectra in  FIG. 3A , signals at the respective first and second input nodes s in , i in  of the IRC control loop  36  may be expressed as a function of discrete-time index k as: 
         s   in ( k )= s   BB ( kM )= K   1   s ( kMT )+ K   2   i *( kMT ) 
         i   in ( k )= i   BB ( kM )= K   1   i ( kMT )+ K   2   s *( kMT ) 
     where M is a suitable decimation factor, in particular in case baseband bandwidth may be substantially lower than IF bandwidth. 
     In one or more embodiments, applying IRC processing  36  to such input signals s in (k), i in (k) may be expressed with the following set of equations: 
         s   out ( k )= s   in ( k )− w ( k ) i*   in ( k )
 
         i   out ( k )= i   in ( k )− w ( k ) s*   in ( k )
 
     wherein w(k) is an expression in the discrete-time domain for the at least one coupling coefficient w whose value may vary, being computed and updated in time as a result of the feedback branch  368 ,  369  in the IRC control loop stage  36 . For instance, the at least one coupling coefficient w may have its value updated according to an update function which may be expressed in discrete-time domain as: 
         w ( k+ 1)= w ( k )+μ s   out ( k ) i   out ( k )
 
     wherein μ is a multiplication factor which may be set in the loop filter  389 . 
     In one or more embodiments, the update function may depend on an estimated residual correlation between output signals at a given discrete-time moment, for instance output signals s out (k) and i* out k) (or, equivalently, i out (k) and s* out (k)). For instance, such residual correlation may be estimated as a result of integrating increments generated from a product of output signals, for instance s out (k) i out (k). 
     In one or more embodiments, a steady-state condition may be reached for a given expected value condition, e.g., E[s out (k) i out  (k)]=0, wherein such condition may yield a satisfactory coefficient value equal to the ratio between mismatch coefficients, for instance w=K 2 /K* 1 . 
     Correspondingly, output signals s out , i out  may be expressed in discrete-time domain k as: 
     
       
         
           
             
               
                 s 
                 out 
               
                
               
                 ( 
                 k 
                 ) 
               
             
             = 
             
               
                 K 
                 1 
               
                
               
                   
                 
                   
                     
                       ( 
                       
                         1 
                         - 
                         
                           
                             
                                
                               
                                 K 
                                 2 
                               
                                
                             
                             2 
                           
                           
                             
                                
                               
                                 K 
                                 1 
                               
                                
                             
                             2 
                           
                         
                       
                       ) 
                     
                      
                     
                       s 
                        
                       
                         ( 
                         kMT 
                         ) 
                       
                     
                      
                     
                         
                     
                      
                     
                       
                         i 
                         out 
                       
                        
                       
                         ( 
                         k 
                         ) 
                       
                     
                   
                   = 
                   
                     
                       
                         K 
                         1 
                       
                        
                       
                         ( 
                         
                           1 
                           - 
                           
                             
                               
                                  
                                 
                                   K 
                                   2 
                                 
                                  
                               
                               2 
                             
                             
                               
                                  
                                 
                                   K 
                                   1 
                                 
                                  
                               
                               2 
                             
                           
                         
                         ) 
                       
                     
                      
                     
                       
                         i 
                          
                         
                           ( 
                           kMT 
                           ) 
                         
                       
                       . 
                     
                   
                 
               
             
           
         
       
     
       FIG. 3B  is an exemplary diagram of possible frequency spectra of output signals which may be found at output nodes s out , i out , for instance while IRC control loop processing  36  is ongoing. 
     Specifically:
         portion a) of  FIG. 3B  is a diagram of an exemplary frequency spectrum of a first output signal s out (ω) (expressed as s out (k) in discrete-time domain), comprising a superposition of a first attenuated spectral contribution α K 1  S and a second attenuated spectral contribution ε K 2  I*, and   portion b) of  FIG. 3B  is a diagram of an exemplary frequency spectrum of a second output signal I out (ω) (expressed as i out (k) in discrete-time domain), comprising a superposition of a third attenuated spectral contribution α K 1  I and a fourth attenuated spectral contribution ε K 2  S*,
 
wherein attenuation factors ε, α values may be indicative of the IRC control loop processing  36  facilitating reaching a steady-state condition in which attenuation factors ε, α tend to reach zero and one, respectively, for instance expressed as:
       

     
       
         
           
             ɛ 
             = 
             
               
                 0 
                  
                 
                     
                 
                  
                 α 
               
               = 
               
                 
                   1 
                   - 
                   
                     
                       
                          
                         
                           K 
                           2 
                         
                          
                       
                       2 
                     
                     
                       
                          
                         
                           K 
                           1 
                         
                          
                       
                       2 
                     
                   
                 
                 ≅ 
                 1. 
               
             
           
         
       
     
     In one or more embodiments as exemplified in  FIG. 4 , a single-branch crosstalk correction system  30 ′ may comprise:
         an input node x RF  configured to receive an RF signal x RF (t), for instance the RF signal may be expressed as:       

         x   RF =2 Re[ s ( t ) e   jω     LO     t ]= s ( t ) e   jω     LO     t   +s *( t ) e   −jω     LO     t            a (analog) down-conversion stage  21 , for instance a direct down-conversion stage  21  configured to shift the desired signal component to an intermediate frequency ω IF  which may be zero, for instance ω IF ≅0 Hz; for instance the intermediate frequency IF signal may be expressed as:       
         x   IF   =h   IF ( t )*[ x   RF ( t ) x   LO ( t )]= K   1   s ( t )+ K   2   s *( t )         a single-branch IRC control loop  36 ′, discussed in the following, coupled to the down-conversion stage  21  and configured to receive a first input signal x in (k) therefrom, the input signal having the form:       
         x   in ( k )= x   IF ( kT )= K   1   x ( kT )+ K   2   x *( kT ) 
     wherein x(kT) is a sampled version of a signal x(t) which may be expressed as x(t)=s(t). 
       FIG. 4A  is an exemplary diagram of a frequency spectrum X RF  of an RF signal X RF (ω) (expressed in time domain as x RF (t)) which may be received at the input node x RF  in the special case with ω IF =0, wherein such a frequency spectrum comprises a pair of desired signal contributions S, S* placed around frequencies ±ω LO , respectively. 
     In one or more embodiments, the single-branch IRC control loop  36 ′ may comprise:
         a first input node x in  configured to receive a first input signal x in (k) from the down-conversion stage  21 , wherein the input signal x in (k)=x IF (kT) may be expressed in the form:       

         x   in ( k )= x   IF ( kT )= K   1   x ( kT )+ K   2   x *( kT ) 
     wherein x(kT) is a sampled version of a signal x(t) which may be expressed as x(t)=s(t)e jω     IF     t +i(t)e −jω     IF     t ;
         a first conjugating stage  360 ′ coupled to the first input node x in  and configured to provide a first conjugated complex signal x* in (k),   a first adaptive multiplier (or filter)  362 ′ having multiplication coefficient w, the first adaptive multiplier  362 ′ being coupled to first conjugating stage  360 ′,   a first adder stage  364 ′ being coupled to the input node x in  and to the output node of the first multiplier stage  362 ′,   a feedback branch  368 ′,  369 ′ coupled to the first adder stages  364  and to the first adaptive multiplier stage  362 , the feedback branch  368 ′,  369 ′ configured to vary or adapt multiplier stage coefficients as discussed in the following,   a first output node x out  coupled to the output node of the first adder stage  364 ′.       

     In one or more embodiments as exemplified in  FIG. 4 , the output signal x out (k) may be expressed as: 
         x   out ( k )= x   in ( k )− w ( k ) x*   in ( k )
 
     wherein w(k) indicates at least one weight coefficient w expressed as a function of (discrete) time. 
     In one or more embodiments as exemplified in  FIG. 4 , feedback loop  368 ′,  369 ′ in the single-branch IRC control loop  36 ′ in the single-branch circuit  30 ′ may be configured to provide at least one weight coefficient w to configure the multiplier stage  362 ′, to apply to the input signals such a coefficient whose value may vary in time according to the following expression: 
         w ( k+ 1)= w ( k )+μ x   out ( k ) 2  
 
     wherein μ is a parameter which may be given in the loop filter  369 ′ and which is used to multiply a squared output signal factor x out (k) 2 . 
     In one or more embodiments, for instance as exemplified in  FIG. 4 :
         a conjugating stage  360 ′ may generate a conjugated signal x* in (k),   a correlator stage  368 ′ may compute the squared output signal factor x out (k) 2 ,   a loop filter stage  369 ′ may be coupled to the correlator stage  368 ′ and configured to integrate such computed factor x out (k) 2 ,   a multiplier stage  362 ′ may receive the at least one weight coefficient w(k) and the conjugated signal x* in (k) to compute a product term w(k) x* in (k) of the equation of the output signal expression,   and adder node  364 ′ may receive the product term w(k) x* in (k) and the input signal x in (k), the first being subtracted from the latter and provided as an output of the adder node  364 ′.       

     One or more embodiments may employ a single-branch IRC control loop  36 ′, wherein a single complex signal may be used for the update of the coefficients, estimating residual correlation between the output signal x out  (k) and it conjugated version x* out (k), thus as a function of integrating increments originated from x out  (k) 2 . 
     In one or more embodiments, a steady-state condition may be reached for a given expected value condition, e.g., E [x out (k) 2 ]=0, which may yields, satisfactorily, the at least one weight coefficient w having a value equal to the ratio of the mismatch coefficients, e.g., w=K 2 /K* 1 . 
     Correspondingly, the output signal x out  may be expressed in discrete-time domain k as: 
     
       
         
           
             
               
                 x 
                 out 
               
                
               
                 ( 
                 k 
                 ) 
               
             
             = 
             
               
                 
                   K 
                   1 
                 
                  
                 
                   ( 
                   
                     1 
                     - 
                     
                       
                         
                            
                           
                             K 
                             2 
                           
                            
                         
                         2 
                       
                       
                         
                            
                           
                             K 
                             1 
                           
                            
                         
                         2 
                       
                     
                   
                   ) 
                 
               
                
               
                 
                   x 
                    
                   
                     ( 
                     kT 
                     ) 
                   
                 
                 . 
               
             
           
         
       
     
       FIG. 4B  is an exemplary diagram of possible frequency spectra of signals which may be processed in a (single-branch) IRC loop processing circuit  36 ′. 
     Specifically, for instance in the special case where IF frequency is zero, ω IF =0:
         portion a) of  FIG. 4B  is a diagram of an exemplary frequency spectrum of a possible input signal X in (ω) (expressed as x 1  (k) in discrete-time domain) which may be received at the input node x in  of the IRC control loop  36 ′, comprising a superposition of a first spectral contribution K 1  S and a second spectral contribution K 2  S*, and   portion b) of  FIG. 4B  is a diagram of an exemplary frequency spectrum of a first output signal X out (ω) (expressed as x out (k) in discrete-time domain), comprising a superposition of a first attenuated spectral contribution α K 1  S and a second attenuated spectral contribution ε K 2  S*, wherein attenuation factors E, a values may be are indicative of the IRC control loop processing  36 ′ facilitating reaching a steady-state condition in which attenuation factors E, a tend to reach zero and one, respectively, for instance expressed as:       

     
       
         
           
             ɛ 
             = 
             
               
                 0 
                  
                 
                     
                 
                  
                 α 
               
               = 
               
                 
                   1 
                   - 
                   
                     
                       
                          
                         
                           K 
                           2 
                         
                          
                       
                       2 
                     
                     
                       
                          
                         
                           K 
                           1 
                         
                          
                       
                       2 
                     
                   
                 
                 ≅ 
                 1. 
               
             
           
         
       
     
       FIG. 4C  is a further exemplary diagram of a possible frequency spectrum X out  (a)) of an output signal X out (ω) (expressed as x out (k) in discrete-time domain) which may be present at the output node x out  while the IRC processing stage  36 ′ may be ongoing, for instance as a result of processing the signal x(t)=s(t) e jω     IF     t +i(t) e −jω     IF     t , wherein the output signal X out (ω) may comprise:
         a first signal spectrum portion which may be placed around a first “positive” IF frequency ω IF , the first IF signal spectrum portion comprising a superposition of a first attenuated spectral contribution α K 1  S and a second attenuated spectral contribution ε K 2  I*,   a second signal spectrum portion which may be placed around a second “negative” IF frequency −ω IF  opposed said first “positive” frequency, the second IF signal spectrum portion comprising a superposition of a third attenuated spectral contribution α K 1  I and a fourth attenuated spectral contribution ε K 2  S*, wherein attenuation factors ε, α values may be are indicative of the IRC control loop processing  36 ′ facilitating reaching a steady-state condition in which attenuation factors ε, α tend to reach zero and one, respectively, for instance expressed as:       

     
       
         
           
             ɛ 
             = 
             
               
                 0 
                  
                 
                     
                 
                  
                 α 
               
               = 
               
                 
                   1 
                   - 
                   
                     
                       
                          
                         
                           K 
                           2 
                         
                          
                       
                       2 
                     
                     
                       
                          
                         
                           K 
                           1 
                         
                          
                       
                       2 
                     
                   
                 
                 ≅ 
                 1. 
               
             
           
         
       
     
     In one or more embodiments, IRC control circuit  36 ,  36 ′,  36 ″ may comprise multi-tap processing, which may be particularly suited for compensating imbalance which may vary with frequency (see, for instance, the second imbalance case discussed in the foregoing). 
     It is noted that, for the sake of simplicity, one or more embodiments using multi-tap processing may be discussed mainly with respect to a single-branch IRC control loop  36 ′, being otherwise understood that such a discussion is purely exemplary and in no way limiting. 
     For instance, in the single-branch IRC control loop  36 ′, an input signal to be processed may be provided as x in (k)=x IF  (kT), having a corresponding frequency spectrum which may be expressed as: 
         X   in (ω)= X   IF (ω)= K   1 (ω) X (ω)+ K   2 (ω) X *(−ω) |ω T|&lt;π.  
 
     In one or more embodiments, double-branch IRC control loop  36  processing may also comprise multi-tap processing, wherein input signals treated may be provided as s in (k)=s BB (kM) and i in (k)=i BB (kM), wherein corresponding signal spectra may be expressed as: 
         s   in (ω)= s   BB (ω)= K   1 (ω+ω IF ) S (ω)+ K   2 (ω+ω IF ) I *(−ω) |ω MT|≤π 
 
         I   in (ω)= I   BB (ω)= K   1 (ω−ω IF ) I (ω)+ K   2 (ω−ω IF ) S *(−ω) |ω MT|≤π.  
 
     In one or more embodiments as exemplified in  FIG. 5 , an IRC processing pipeline  36 ″ may be configured to receive an input signal X in  which may be expressed as a function of (discrete) time k and frequency co, respectively, as: 
         x   in ( k )= x   IF ( kT )=[ k   1 ( t )* x ( t )] t=kT +[ k   2 ( t )* x *( t )] t=kT    
         X   in (ω)= X   IF (ω)= K   1 (ω) X (ω)+ K   2 (ω) X *(−ω) |ω T|≤π.  
 
     One or more embodiments as exemplified in  FIG. 5 , may be particularly advantageous to deal with such an input signal x in (k), facilitating to deal with a (potentially) frequency-dependent I/Q imbalance. 
     In one or more embodiments as exemplified in  FIG. 5 , the multi-tap IRC control loop  36 ″ may comprise:
         a first conjugating stage  360 ″ coupled to the input node x in  and configured to generate a conjugated input signal, for instance x* in (k),   an adaptive filter stage  362 ″, for instance a Finite-duration Impulse Response (FIR) filter, coupled to the first conjugating stage  360  and configured to receive the conjugated input signal x* in (k) therefrom and to compute an estimation of a signal interference term x* 2 (k) which may be expressed as: x* 2 (k)=[k 2 (t)*x*(t)] t=kT )   an adder stage  364 , coupled to the input node, to the adaptive filter stage  362 ″ and to an output node x out ,   a feedback network  368 ″,  369 ″ coupled to the output node and to the FIR filter stage  362 ″.       

     In one or more embodiments as exemplified in  FIG. 5 , the feedback network  368 ″,  369 ″ may comprise:
         a correlator stage  368 ″, coupled to the output node and to the loop filter  369 ″, the correlator stage  368 ″ configured to compute an estimate of residual correlation R n (k), for instance a product of an output signal at a given time and an output signal after a given delay n, product which can be expressed as: R n (k)=x out (k) x out (k−n), 0≤n&lt;N,   a loop filter  369 ″ coupled to the correlator stage  368 ″ and to the FIR filter  362 ″, the loop filter  369 ″ configured to integrate such an estimate of residual correlation R and to update the set of coefficients W accordingly, as discussed in the following.       

     In one or more embodiments, a Finite-duration Impulse Response (FIR) filter  362 ″ may comprise a filter whose impulse response (that is, the output in response to a Kronecker delta, or Dirac impulse, input) or response to any finite-length input, has a finite duration which may settle to zero in a limited/finite time. 
     For instance, a FIR filter of order N−1 may have an impulse response which may have a duration equal to a number N of (signal) samples before settling to zero, wherein samples N may be counted from first non-zero element through last non-zero element, with both included. 
     In one or more embodiments, the FIR filter  362 ″ may comprise input delay line elements, facilitating to delay the input signal by a given number of samples. 
     In one or more embodiments employing a set W of N weight coefficients w 0 , w 1 , w 2 , . . . , w N-1 , an output signal x out (k), at the output node x out  of the IRC control loop  36 ″, may be computed as a result of applying adaptive FIR filtering  362 ″, imposing decorrelation for N consecutive values of delay: E[x out (k)x out (k−n)]=0, 0≤n&lt;N. 
     Specifically, the output signal x out (k) may be expressed as: 
     
       
         
           
             
               
                 x 
                 out 
               
                
               
                 ( 
                 k 
                 ) 
               
             
             = 
             
               
                 
                   x 
                   in 
                 
                  
                 
                   ( 
                   k 
                   ) 
                 
               
               - 
               
                 
                   ∑ 
                   
                     n 
                     = 
                     0 
                   
                   
                     N 
                     - 
                     1 
                   
                 
                  
                 
                     
                 
                  
                 
                   
                     
                       w 
                       n 
                     
                      
                     
                       ( 
                       k 
                       ) 
                     
                   
                    
                   
                       
                   
                    
                   
                     
                       x 
                       in 
                       * 
                     
                      
                     
                       ( 
                       
                         k 
                         - 
                         n 
                       
                       ) 
                     
                   
                 
               
             
           
         
       
     
     whereas an update function of coefficients in the set of weight coefficients W may be expressed as: 
         w   n ( k+ 1)= w   n ( k )+μ x   out ( k ) x   out ( k−n ) 0≤ n&lt;N.  
 
     In one or more embodiments as exemplified in  FIG. 5 , coefficients W may be applied to a subset of elements belonging to the delay line of the FIR  362 ″, and decorrelation may be imposed solely for the related delay values. 
     For instance, the set of coefficients W may be applied to a comb of delay elements, the comb comprising a number N d  of delay elements, the delay elements having distance d therebetween, wherein the distance d may be indicated as tap-delay distance or, simply, tap-delay. 
     In the example considered, imposing decorrelation may be expressed as: 
         E [ x   out ( k ) x   out ( k−d n )]=0, 0≤ n&lt;N   d .
 
     Still in the considered example, the output signal x out  may be expressed as: 
     
       
         
           
             
               
                 x 
                 out 
               
                
               
                 ( 
                 k 
                 ) 
               
             
             = 
             
               
                 
                   x 
                   in 
                 
                  
                 
                   ( 
                   k 
                   ) 
                 
               
               - 
               
                 
                   ∑ 
                   
                     n 
                     = 
                     0 
                   
                   
                     
                       N 
                       d 
                     
                     - 
                     1 
                   
                 
                  
                 
                     
                 
                  
                 
                   
                     
                       w 
                       n 
                     
                      
                     
                       ( 
                       k 
                       ) 
                     
                   
                    
                   
                       
                   
                    
                   
                     
                       x 
                       in 
                       * 
                     
                      
                     
                       ( 
                       
                         k 
                         - 
                         
                           d 
                            
                           
                               
                           
                            
                           n 
                         
                       
                       ) 
                     
                   
                 
               
             
           
         
       
     
     and the update function of the set of coefficients W may be expressed as: 
         w   n ( k+ 1)= w   n ( k )+μ x   out ( k ) x   out ( k−d n ) 0≤ n&lt;N   d .
 
     In one or more embodiments as exemplified in  FIG. 5 , the set of coefficients W may comprise three coefficients w 0 , w 1 , w 2 . For instance, such a number of coefficients may be particularly suited in satellite radio applications. 
     In one or more embodiments as exemplified in  FIG. 5 , the comb of delay elements may comprise a number N d =3 of delay elements, whereas the distance d may have a configurable, controllable or variable (tap-delay) value. 
     In one or more embodiments as exemplified in  FIG. 5 , the FIR filter  362 ″ may compute a weighted input signal value  (k), indicative of an estimation of the interference term defined as x* 2 (k)=[k 2 (t)*x*(t)] t=kT , estimation which may be expressed as: 
     
       
         
           
             
                
               
                 ( 
                 k 
                 ) 
               
             
             = 
             
               
                 ∑ 
                 
                   n 
                   = 
                   0 
                 
                 
                   
                     N 
                     d 
                   
                   - 
                   1 
                 
               
                
               
                   
               
                
               
                 
                   
                     w 
                     n 
                   
                    
                   
                     ( 
                     k 
                     ) 
                   
                 
                  
                 
                     
                 
                  
                 
                   
                     
                       x 
                       in 
                       * 
                     
                      
                     
                       ( 
                       
                         k 
                         - 
                         
                           d 
                            
                           
                               
                           
                            
                           n 
                         
                       
                       ) 
                     
                   
                   . 
                 
               
             
           
         
       
     
     In one or more embodiments as exemplified in  FIG. 5 , such a weighted signal value F, for instance F(k)= (k), may be provided to the adder stage  364 , wherein it may be subtracted to the input signal x in , providing the output signal x out  as difference between such quantities x in , F, for instance as x out (k)=x in (k)− (k). 
     In one or more embodiments, the single-branch IRC structure  36 ′,  36 ″ may comprise a detector/correlator stage  368 ′,  368 ″, which may perform an estimation of a residual correlation between an output signal x out (k) and its complex conjugate version x out (k). 
     In one or more embodiments, the double-branch IRC structure  36  may comprise a detector/correlator stage  368  which may perform an estimation of a residual correlation between a first output signal s out (k) and a conjugated version of a second output signal i* out (k), or between a second output signal i out (k) and a conjugated version of a first output signal s* out (k). 
     In one or more embodiments, such correlation estimations can be performed, for instance:
         using a first approach based on the “last available data”, for instance computing and evaluating sample by sample the value of the estimate of residual correlation R n (k), which may be expressed as R n (k)=x out (k) x out (k−d n), 0≤n&lt;N d , or,   alternatively, using a second approach based on collecting a set of (sampled) data (points) and applying a “block-based” computation approach.       

     For instance, in such a second “block-based” computational approach, a second form R′ n (k) for the estimate of residual correlation may be computed as: 
     
       
         
           
             
               
                 R 
                 n 
                 ′ 
               
                
               
                 ( 
                 k 
                 ) 
               
             
             = 
             
               
                 
                   1 
                   L 
                 
                  
                 
                   
                     ∑ 
                     
                       l 
                       = 
                       0 
                     
                     
                       L 
                       - 
                       1 
                     
                   
                    
                   
                       
                   
                    
                   
                     
                       
                         x 
                         out 
                       
                        
                       
                         ( 
                         
                           
                             k 
                              
                             
                                 
                             
                              
                             L 
                           
                           + 
                           l 
                         
                         ) 
                       
                     
                      
                     
                         
                     
                      
                     
                       
                         x 
                         out 
                       
                        
                       
                         ( 
                         
                           
                             k 
                              
                             
                                 
                             
                              
                             L 
                           
                           + 
                           l 
                           - 
                           
                             d 
                              
                             
                                 
                             
                              
                             n 
                           
                         
                         ) 
                       
                     
                      
                     
                         
                     
                      
                     0 
                   
                 
               
               ≤ 
               n 
               &lt; 
               
                 N 
                 d 
               
             
           
         
       
     
     wherein:
         k is a block index, and   L is a (possibly configurable) block-length.       

     In one or more embodiments using this second approach, an update function for the coefficients may also comprise a “block-wise” implementation in the computation of the values of the coefficients in the set of coefficients W. 
     For instance, in one or more embodiments employing a block-based approach, an output signal x out  may be expressed as: 
     
       
         
           
             
               
                 x 
                 out 
               
                
               
                 ( 
                 
                   
                     k 
                      
                     
                         
                     
                      
                     L 
                   
                   + 
                   l 
                 
                 ) 
               
             
             = 
             
               
                 
                   
                     x 
                     in 
                   
                    
                   
                     ( 
                     
                       
                         k 
                          
                         
                             
                         
                          
                         L 
                       
                       + 
                       l 
                     
                     ) 
                   
                 
                 - 
                 
                   
                     ∑ 
                     
                       n 
                       = 
                       0 
                     
                     
                       
                         N 
                         d 
                       
                       - 
                       1 
                     
                   
                    
                   
                       
                   
                    
                   
                     
                       
                         w 
                         n 
                       
                        
                       
                         ( 
                         k 
                         ) 
                       
                     
                      
                     
                         
                     
                      
                     
                       
                         x 
                         in 
                         * 
                       
                        
                       
                         ( 
                         
                           
                             k 
                              
                             
                                 
                             
                              
                             L 
                           
                           + 
                           l 
                           - 
                           
                             d 
                              
                             
                                 
                             
                              
                             n 
                           
                         
                         ) 
                       
                     
                      
                     
                         
                     
                      
                     0 
                   
                 
               
               ≤ 
               l 
               &lt; 
               L 
             
           
         
       
     
     while an update function for the coefficients in the set of coefficients W may be expressed as: 
     
       
         
           
             
               
                 w 
                 n 
               
                
               
                 ( 
                 
                   k 
                   + 
                   1 
                 
                 ) 
               
             
             = 
             
               
                 
                   
                     w 
                     n 
                   
                    
                   
                     ( 
                     k 
                     ) 
                   
                 
                 + 
                 
                   
                     μ 
                     L 
                   
                    
                   
                     
                       ∑ 
                       
                         l 
                         = 
                         0 
                       
                       
                         L 
                         - 
                         1 
                       
                     
                      
                     
                         
                     
                      
                     
                       
                         
                           x 
                           out 
                         
                          
                         
                           ( 
                           
                             
                               k 
                                
                               
                                   
                               
                                
                               L 
                             
                             + 
                             l 
                           
                           ) 
                         
                       
                        
                       
                           
                       
                        
                       
                         
                           x 
                           out 
                         
                          
                         
                           ( 
                           
                             
                               k 
                                
                               
                                   
                               
                                
                               L 
                             
                             + 
                             l 
                             - 
                             
                               d 
                                
                               
                                   
                               
                                
                               n 
                             
                           
                           ) 
                         
                       
                        
                       
                           
                       
                        
                       0 
                     
                   
                 
               
               ≤ 
               n 
               &lt; 
               
                 
                   N 
                   d 
                 
                 . 
               
             
           
         
       
     
     It is noted that the first “last available data” approach may be a special case of the second “block-based” approach in the case wherein block-length L is configured to be unitary, for instance L=1. 
     In one or more embodiments, the inventors have observed that in a frequency spectrum of a signal output by the IRC control loop  36 ′,  36 ″, the residual correlation between the positive spectrum and a mirrored version of the negative spectrum, both containing contributions placed around ω IF  and derived from S and I*, may be intrinsically combined with the residual correlation between the negative spectrum and a mirrored version of the positive spectrum, both containing contributions placed around −ω IF  and derived from I and S*, taking advantage from the fact that both the correlation terms facilitate to converge to the same values of the coefficients. 
     As mentioned, in one or more embodiments as exemplified in  FIG. 5 , the result of the computed correlation R, R′ may be integrated via a loop filter  369 ″, extending from the detector/correlator  368 ″ to the adaptive filter  362 ″. 
     As mentioned, in one or more embodiments, residual image crosstalk may be estimated and rejected, for instance with DSP techniques operating in the digital domain, by using a correlator stage  368 ″ and an adaptive filter stage  362 ″, respectively. 
     Specifically, in one or more embodiments the FIR filter  362 ″ may output a filtered signal  (k) which may be subtracted to the IRC input signal x in (k), in order to reduce (and virtually remove), on the resulting IRC output signal x out (k), the crosstalk effect due to I/Q imbalance in the RFFE  10 . 
     In one or more embodiments, the IRC control loop  36 ″ may converge to an improved set of filter coefficients W, e.g., optimal in the sense of Least Mean Squares (LMS). 
     Specifically, the set of coefficients W of the FIR filter  362 ″ may be updated using the information coming from the loop filter  369 ″, as a function of signals received from the correlator  368 ″. 
     One or more embodiments may comprise an automatic gain controller (AGC) stage  366 , which may facilitate maintaining a fixed signal level at the input of the detector/correlator  368 ″, in order to have convergence with an almost stable time constant, independently from the input signal power condition. 
     As mentioned, performances of known techniques may be limited by the decorrelation time of each of the “clean” source signals s(t), i(t) (indicative of desired and image signals, respectively, without crosstalk), in particular when the loop operates at high sampling rate, either due to intentional oversampling or to particular spectral conditions. Specifically, such a limitation may be present when the decorrelation time is much higher than the sampling period. 
     As mentioned, wideband receivers may be affected more critically by such a problem and its related effects due to the presence of a strongly frequency-dependent I/Q imbalance. 
     In one or more embodiments as exemplified in  FIGS. 5, 6 and 7 , it may be possible to configure the IRC control loop  36 ″ to compute coefficients W in different way depending on the application, advantageously increasing flexibility. 
     For instance, the detector/correlator stage  368 ″ in the IRC control loop  36 ″ may comprise:
         a first input TapSel configured to be coupled to a first tap-selection configuration register  3680 , wherein the detector/correlator stage  368 ″ may be operated with a single-tap or multi-tap IRC implementation as a function of a first value stored in the first tap-selection configuration register  3680 ,   a second input DtcDly configured to be coupled to a second tap-delay configuration register  3682 , wherein the detector/correlator stage  368 ″ may be operated with a tap-delay value d (in number of samples) as a function of a second value stored in the second tap-delay configuration register  3682 , and   optionally, a third input BlkLen configured to be coupled to a third configuration register  3684 , wherein the detector/correlator stage  368 ″ may be operated with a block-length L value selected as a function of a third value stored in the third block-length configuration register  3684 .       

     In one or more embodiments, the possibility to use the second tap-delay configuration register  3682  to select a convenient tap-delay distance d value may facilitate performing correlation processing solely for a subset of delay elements, which may be selected accordingly. 
     In one or more embodiments as exemplified in  FIG. 5 , the FIR filter stage  362 ″ in the IRC control loop  36 ″ may comprise, for instance:
         a first input TapSel configured to be coupled to the first tap-selection configuration register  3680 , wherein the FIR filter stage  362 ″ may be operated with a single-tap or multi-tap implementation as a function of the first value stored in the first tap-selection configuration register  3680 ,   a second input DtcDly configured to be coupled to the second tap-delay configuration register  3682 , wherein the FIR filter stage  362 ″ may be operated with a tap-delay value d (in number of samples) as a function of the second value stored in the second tap-delay configuration register  3682 .       

     In one or more embodiments, for instance, the adaptive filter stage  362 ″ in the IRC loop  36 ″ may comprise a 3-tap type FIR filter having a “comb” implementation with tap-delay d selected as a function of the value in the second tap-delay configuration register  3682 , after activating the FIR implementation of the filter stage  362 ″ as a function of the value in the first tap-selection configuration register  3680 . 
     In one or more embodiments as exemplified in  FIG. 5 , the optional block-length configuration register  3684  can be used to configure block-wise calculation of correlation and update of coefficients W, facilitating to increase flexibility of the processing system  36 ,  36 ′,  36 ″. 
     In one or more embodiments, the loop filter  369 ″ may comprise a set of registers  3690 ,  3692 ,  3694  storing respective loop filter parameters μ, PS, Winit. 
     In one or more embodiments, for instance:
         a first configuration register  3690  may comprise, e.g., an adequate integration coefficient μ, and/or   a second configuration register  3692  may be used to select a “pre-set values” functionality PS, and/or   a third configuration register (or set of registers)  3694  may comprise optional pre-set coefficient values Winit, which may be pre-stored in the dedicated register memory locations  3694  and may be used to initialize coefficients W for the FIR filter  362 ′ when the pre-set values option PS is selected from  3692 .       

     In one or more embodiments, the loop filter  369 ″ in the IRC control loop  36 ″ may comprise, for instance:
         a first input mu configured to be coupled to such a first configuration register  3690  which may comprise, e.g., an adequate integration coefficient and/or   a second input Preset configured to be coupled to such a second configuration register  3692  which may be used to select a “pre-set values” functionality PS, and/or   a third (set of) input(s) Winit configured to be coupled to such a third (set of) configuration register(s)  3694 , comprising optional pre-set coefficient values Winit, which may be pre-stored in the dedicated register memory locations  3694  and may be used to initialize coefficients W for the FIR filter  362 ′ when the pre-set values option PS is selected from  3692 ,   an additional input TapSel configured to be coupled to the tap-selection configuration register  3680 , wherein the loop filter  369 ″ may be operated with a single-tap or multi-tap IRC implementation as a function of the value stored in tap-selection configuration register  3680 .       

     In one or more embodiments, such a solution  36 ″ may facilitate avoiding to select a subset of elements which are separated by a short delay, advantageously, mitigating side effects due to the decorrelation time of (clean) source signals s(t), i(t). 
     One or more embodiments as exemplified in  FIGS. 8A and 8B , may be related to a QPSK application adopting the IRC control loop  36 ,  36 ′,  36 ″, wherein the expected (optimum) value for a first FIR coefficient w 1  and a second FIR coefficient w 2  in the set of coefficients W may be zero. 
     In one or more embodiments as exemplified in  FIG. 8A , when a short tap-delay is used (e.g., d=1), the loop  36  may converge to sub-optimal values, e.g., due to the side effects related to the decorrelation time of the clean sources, in the detector/correlator  368 . As a result, a first QPSK constellation as exemplified in  FIG. 8A  may have a “slightly confused” distribution, due to the poor signal-to-interferer ratio, generating a high bit error rate. 
     In one or more embodiments as exemplified in  FIG. 8B , when a long tap-delay is used (e.g., d=12), side effects due to the decorrelation time of the clean sources are mitigated and the loop  36  may converge to the satisfactory value, improving the quality of the QPSK constellation, due to a higher signal-to-interferer ratio and, consequently, resulting in a negligible bit error rate. 
     As exemplified in  FIG. 8B , the IRC loop  36  may facilitate recovering full system functionality by attenuating the crosstalk effects and providing output signal x out (k) having values close to those that may be obtained in the ideal case (see, for instance, the discussion in the foregoing with respect to ideal case). 
     One or more embodiments as exemplified in  FIG. 9 , may comprise a communication device  90 , equipped with circuitry configured to perform an image rejection correction method as discussed herein. 
     As exemplified in  FIG. 9 , in one or more embodiments such a communication device  90 , e.g., an RF receiver, may comprise:
         an antenna  92 , configured to sense electromagnetic signals,   a radio frequency front-end (RFFE) module  93 , coupled to the antenna  92  and configured to receive the RF signal therefrom and to process it,   an analog-to-digital converter (ADC)  94 , configured to receive the processed signal from the RFFE module  93 ;   an automatic gain controller (AGC) stage  95 , which may couple the ADC module  94  with the RFFE module  93 ,   processing circuitry  96 , configured to correct I/Q imbalance problem, according to one or more embodiments, and   a baseband processor  98 , configured to further process the imbalance-corrected signal, received from the processing circuitry  96 .       

     In one or more embodiments, the communication device  90  may be in a transceiver, for example, and may be utilized for receiving satellite, terrestrial or cable television or radio signals, or any RF signal requiring a frequency down-conversion through analog I/Q mixer. 
     In one or more embodiments, the receiver  90  may be operable to receive satellite, terrestrial or cable television or radio signals, down-convert and process the signals for communication to a display device and/or a set of loudspeakers. 
     For instance, the RFFE  93  may comprise one or more RF receive (Rx) and transmit (Tx) paths for receiving signals from a satellite system, cable TV head-end, and/or terrestrial TV antennas, for example. 
     In one or more embodiments, the RFFE may further comprise impedance matching elements, low-noise amplifiers (LNAs), power amplifiers, variable gain amplifiers, and filters, for example. In one or more embodiments the RFFE stage  93  may thus be operable to receive, amplify, and filter RF signals before communicating them to, e.g., the baseband processor  98 . 
     In one or more embodiments, the ADC  94  may comprise a wideband ADC and may be operable to convert received analog signals to digital signals. 
     The processing circuitry  96  may comprise a compensation circuit  30 ,  30 ′ comprising an IRC loop circuit  36 ,  36 ′,  36 ″ as discussed in the foregoing, for instance with respect to  FIG. 5 , or it may comprise general processing means and a memory configured to perform the method of imbalance correction  36 ,  36 ′,  36 ″ according to one or more embodiments, e.g., via software. 
     In one or more embodiments, the process may be performed entirely in the digital circuitry without using complex analog circuitry. 
     In one or more embodiments, the communication device processing circuitry  96  may comprise a memory, e.g., a programmable memory module that may be operable to store software and data, for example, for the operation of the receiver device  90 . 
     For instance, the memory may store the adaptive filter coefficients computed in the IRC control loop in the compensation circuit. 
     One or more embodiments may be employed in any communication receiver  90  in which signal processing employs a frequency down-conversion stage based on an analog I/Q mixer, regardless of the information content of the received signal (broadcasting services, networks, astronomy, etc.), the modulation technique that is adopted (AM, FM, OFDM, QPSK, etc.) and the physical nature of the transmission channel, which can be based on a radio link (terrestrial or via satellite) or on a medium-guided approach (metal cable, waveguide, optical fiber, etc.). 
     As mentioned, some examples of such communication receivers may be used for broadcasting services (television, radio, positioning systems: GNSS and others, etc.) or for network services (point-to-point, point-to-multipoint, switched networks for telephonic signals, digital data, etc.). 
     One or more embodiments may improve performances of, among others, communication receivers for terrestrial radio wideband receivers, satellite radio wideband receivers, satellite GNSS receivers. 
     As exemplified herein, a method (for instance,  30 ,  30 ′) may comprise:
         receiving an input signal (for instance, x in , X in ) comprising at least one sequence of input data samples separated by a sampling period (for instance, T) therebetween, the input signal comprising a desired signal component and an interfering signal component superimposed thereon,   applying interfering component estimation processing (for instance,  36 ,  36 ′,  36 ″) to said input signal obtaining as a result a filtered signal (for instance, F) comprising a sequence of filtered data samples,   subtracting (for instance,  364 A,  364 B,  364 ′,  364 ″) said filtered signal from said input signal and obtaining as a result an output signal (for instance, x out , X out ) comprising a sequence of output data samples,       

     wherein said interfering component estimation processing comprises: 
     a) applying conjugating processing (for instance,  360 A,  360 B,  360 ′,  360 ″) to said input signal, providing a conjugated version of said input signal, 
     b) computing of at least one adaptive signal processing coefficient (for instance, w; w 0 , w 1 , w 2 ) value, 
     c) applying adaptive signal processing (for instance,  362 A,  362 B,  362 ′,  362 ″) to said conjugated version of said input signal using said at least one adaptive processing coefficient, 
     wherein said computing at least one adaptive signal processing coefficient (w; w 0 , w 1 , w 2 ) value comprises: 
     i) performing correlation processing (for instance,  368 ,  368 ′,  368 ″) between said sequence of output data samples of the output signal (x out , X out ) and a conjugated version of said sequence of output data samples of the output signal (x out , X out ) and obtaining as a result a sequence of estimates of residual correlation, 
     ii) applying integration processing (for instance,  369 ,  369 ′,  369 ″) to said sequence of estimates of residual correlation provided ( 368 ,  368 ′,  368 ″), said integration processing comprising an integration step parameter (p) and at least one starting point parameter (Winit), 
     iii) obtaining at least one computed adaptive signal processing coefficient (w; w 0 , w 1 , w 2 ) as a result of applying said integration processing ( 369 ,  369 ′,  369 ″). 
     As exemplified herein, said applying adaptive signal processing may comprise applying processing selected (for instance,  3680 ,  3682 ,  3684 ) out of:
         adaptive multiplication processing (for instance,  362 A,  362 B,  362 ′) having at least one multiplication factor equal to said at least one computed adaptive signal processing coefficient (for instance, w),   adaptive finite impulse response, FIR, filtering processing (for instance,  362 ″), such as second order or 3-tap adaptive FIR filtering processing, said adaptive FIR filtering processing (for instance,  362 ″) comprising computing a weighted sum of a conjugated version of data samples in said sequence of input data samples comprised in said input signal using said at least one adaptive processing coefficient as weights of said weighted sum.       

     As exemplified herein:
         adaptive FIR filtering processing may comprise applying said at least one adaptive processing coefficient to a subset of elements belonging to an input delay line of the FIR at a related subset of delay values, and   performing correlation processing and applying integration processing respectively may comprise calculating and integrating estimates of residual correlation at delay values of said related subset of delay values.       

     As exemplified herein, adaptive FIR filtering coefficients may be applied to a comb of elements belonging to said input delay line of the FIR, the comb comprising a number N d  of delay elements having a distance value multiple of said sampling period by a factor d. 
     As exemplified herein, said distance value may be in excess ten times with respect to said sampling period. 
     As exemplified herein, selecting between said adaptive multiplication processing and said adaptive FIR filtering processing in applying said adaptive signal processing may comprise:
         providing a first adaptive signal processing configuration register (for instance,  3680 ) configured to store a first value or a second value,   selecting between applying one of said adaptive multiplication processing or said adaptive FIR filtering processing as a function of said first value or said second value stored in said first adaptive signal processing configuration register, respectively.       

     As exemplified herein, the method may comprise:
         providing at least one second configuration register (for instance,  3682 ) configured for storing indexes of delay elements in the set of delay elements of the delay line, and   selecting a subset of delay elements in the set of delay elements of the delay line as a function of said indexes stored in the at least one second configuration register.       

     As exemplified herein, applying integration processing may comprise applying loop filter processing ( 369 ) to said sequence of estimates of residual correlation provided. 
     As exemplified herein, applying integration processing may comprise at least one of:
         providing a first integration parameter register (for instance,  3690 ) configured to store a value of said integration step parameter (for instance, μ),   providing at least one second integration parameter register (for instance,  3694 ) configured to store a value of said at least one starting point parameter (for instance, Winit) and a third integration parameter register (for instance,  3692 ) configured to activate (for instance, PS,  3692 ) said integration processing to use said value of said at least one starting point parameter (for instance, Winit).       

     As exemplified herein, computing at least one value of said at least one adaptive processing coefficient (for instance, w; w 0 , w 1 , w 2 ) may comprise applying automatic gain control, AGC, processing (for instance,  366 ) to said output signal. 
     As exemplified herein, performing correlation processing between said sequence of output data samples of the output signal and a conjugated version of said sequence of output data samples of the output signal may comprise performing block-like correlation of a number of adjacent data samples forming a block having block-length L of data samples, for example, said block-length L is selected as a function of a length value stored in a configuration register (for instance,  3684 ). 
     As exemplified herein, a circuit may comprise:
         an input node (for instance, x in , X in ) configured to receive at least one input signal (for instance, x in , X in ) comprising at least one sequence of input data samples, the input signal comprising a desired signal component and an interfering signal component superimposed thereon, wherein input data samples in the sequence of input data samples are separated by a sampling period (for instance, T) therebetween,   signal processing circuitry (for instance,  30 ,  30 ′,  96 ) configured to apply interfering component removal processing (for instance,  36 ,  36 ′,  36 ″) to the received at least one input signal (for instance, x in , X in ) with the method as exemplified herein.       

     As exemplified herein, said signal processing circuitry may further comprise a complex down-converter circuit (for instance,  22 ) comprising:
         a first mixer branch (for instance,  22 A), comprising a common input node (for instance, x in , X in ) configured to receive at least one signal (for instance, x in , X in ), a first digital mixer (for instance,  24 A), a first digital low pass filter (for instance,  26 A), a first baseband decimator (for instance,  28 A) with decimation factor M and a first interfacing node (for instance, s in ), the first interfacing node configured to provide a first signal component (for instance, s in , and   a second mixer branch (for instance,  22 B), comprising a common input node (for instance, x in , X in ) configured to receive at least one signal a second digital mixer (for instance,  24 B), a second digital low pass filter (for instance,  26 B), a second baseband decimator (for instance,  28 B) with decimation factor M and a second interfacing node (for instance, the second interfacing node configured to provide a second signal component (for instance,       

     As exemplified herein, a radio frequency receiver device (for instance,  90 ) may comprise:
         an antenna (for instance,  92 ) configured to receive an RF signal,   a radio frequency front-end (for instance,  93 ) coupled to said antenna, the radio frequency front-end configured to receive said RF signal at an input node (for instance, x RF ) and to apply down-conversion processing to said radio frequency signal, the radio frequency front-end having a first output node (for instance, x IF ) and configured for providing an intermediate frequency, IF, signal as a result of said down-conversion processing at said first output node (for instance, x IF ),   an analog-to-digital converter, ADC, stage (for instance,  94 ) having an input node coupled to said first output node (x IF ) and a second output node, the ADC stage configured to receive said IF signal from the first output node, to apply signal sampling thereto and to provide a sampled sequence of data samples of said IF signal at said second output node,   a circuit (for instance,  96 ) as exemplified herein, having said input node coupled to said second output node of said ADC stage, the circuit configured to receive said IF signal as an input signal.       

     As exemplified herein, a computer program product loadable into the memory of at least one processing circuit (for instance,  96 ) may comprise software code portions for executing the steps of the method as exemplified herein when the product is run on at least one processing circuit. 
     It will be otherwise understood that the various individual implementing options exemplified throughout the figures accompanying this description are not necessarily intended to be adopted in the same combinations exemplified in the figures. One or more embodiments may thus adopt these (otherwise non-mandatory) options individually and/or in different combinations with respect to the combination exemplified in the accompanying figures. 
     Without prejudice to the underlying principles, the details and embodiments may vary, even significantly, with respect to what has been described by way of example only, without departing from the extent of protection. 
     Some embodiments may take the form of or comprise computer program products. For example, according to one embodiment there is provided a computer readable medium comprising a computer program adapted to perform one or more of the methods or functions described above. The medium may be a physical storage medium, such as for example a Read Only Memory (ROM) chip, or a disk such as a Digital Versatile Disk (DVD-ROM), Compact Disk (CD-ROM), a hard disk, a memory, a network, or a portable media article to be read by an appropriate drive or via an appropriate connection, including as encoded in one or more barcodes or other related codes stored on one or more such computer-readable mediums and being readable by an appropriate reader device. 
     Furthermore, in some embodiments, some or all of the methods and/or functionality may be implemented or provided in other manners, such as at least partially in firmware and/or hardware, including, but not limited to, one or more application-specific integrated circuits (ASICs), digital signal processors, discrete circuitry, logic gates, standard integrated circuits, controllers (e.g., by executing appropriate instructions, and including microcontrollers and/or embedded controllers), field-programmable gate arrays (FPGAs), complex programmable logic devices (CPLDs), etc., as well as devices that employ RFID technology, and various combinations thereof. 
     The various embodiments described above can be combined to provide further embodiments. Aspects of the embodiments can be modified, if necessary to employ concepts of the various patents, applications and publications to provide yet further embodiments.