Patent Publication Number: US-8537925-B2

Title: Apparatus and methods for compensating for signal imbalance in a receiver

Description:
CROSS-REFERENCE TO OTHER APPLICATIONS 
     This is a divisional application of U.S. patent application Ser. No. 11/938,848, entitled “APPARATUS AND METHODS FOR COMPENSATING FOR SIGNAL IMBALANCE IN A RECEIVER”, filed on Nov. 13, 2007, and issued as U.S. Pat. No. 8,090,043, which is a nonprovisional of the following U.S. Provisional Applications, all of which are hereby incorporated by reference herein in their entireties: U.S. Provisional Application No. 60/866,532, entitled, “PACKET AGGREGATION”, filed on Nov. 20, 2006, U.S. Provisional Application No. 60/866,527, entitled, “RETRANSMISSION IN A COORDINATED HOME NETWORK”, filed on Nov. 20, 2006, U.S. Provisional Application No. 60/866,519, entitled, “IQ IMBALANCING USING TWO-TONE TRANSMISSION IN MULTI-CARRIER RECEIVER”, filed on Nov. 20, 2006, U.S. Provisional Application No. 60/907,111, “SYSTEM AND METHOD FOR AGGREGATION OF PACKETS FOR TRANSMISSION THROUGH A COMMUNICATIONS NETWORK”, filed on Mar. 21, 2007, U.S. Provisional Application No. 60/907,126, entitled, “MAC TO PHY INTERFACE APPARATUS AND METHODS FOR TRANSMISSION OF PACKETS THROUGH A COMMUNICATIONS NETWORK”, filed on Mar. 22, 2007, U.S. Provisional Application No. 60/907,819, entitled, “SYSTEMS AND METHODS FOR RETRANSMITTING PACKETS OVER A NETWORK OF COMMUNICATION CHANNELS”, filed on Apr. 18, 2007, and U.S. Provisional Application No. 60/940,998, entitled “MOCA AGGREGATION”, filed on May 31, 2007. 
    
    
     FIELD OF THE INVENTION 
     The present invention relates generally to information networks and specifically to transmitting information such as media information over communication lines such as coax, thereby to form a communications network. 
     BACKGROUND OF THE INVENTION 
     Many structures, including homes, have networks based on coaxial cable (“coax”). 
     The Multimedia over Coax Alliance (“MoCA™”), provides at its website (www.mocalliance.org) an example of a specification (viz., that available under the trademark MoCA, which is hereby incorporated herein by reference in its entirety) for networking of digital video and entertainment information through coaxial cable. The specification has been distributed to an open membership. 
     Technologies available under the trademark MoCA, other specifications and related technologies (“the existing technologies”) often utilize unused bandwidth available on the coax. For example, coax has been installed in more than 70% of homes in the United States. Some homes have existing coax in one or more primary entertainment consumption locations such as family rooms, media rooms and master bedrooms. The existing technologies allow homeowners to utilize installed coax as a networking system and to deliver entertainment and information programming with high quality of service (“QoS”). 
     The existing technologies may provide high speed (270 mbps), high QoS, and the innate security of a shielded, wired connection combined with state of the art packet-level encryption. Coax is designed for carrying high bandwidth video. Today, it is regularly used to securely deliver millions of dollars of pay per view and premium video content on a daily basis. Networks based on the existing technologies can be used as a backbone for multiple wireless access points to extend the reach of wireless service in the structure. 
     Existing technologies provide throughput through the existing coaxial cables to the places where the video devices are located in a structure without affecting other service signals that may be present on the cable. The existing technologies provide a link for digital entertainment, and may act in concert with other wired and wireless networks to extend entertainment throughout the structure. 
     The existing technologies work with access technologies such as asymmetric digital subscriber lines (“ADSL”), very high speed digital subscriber lines (“VDSL”), and Fiber to the Home (“FTTH”), which provide signals that typically enter the structure on a twisted pair or on an optical fiber, operating in a frequency band from a few hundred kilohertz to 8.5 MHz for ADSL and 12 MHz for VDSL. As services reach such a structure via any type of digital subscriber line (“xDSL”) or FTTH, they may be routed via the existing technologies and the coax to the video devices. Cable functionalities, such as video, voice and Internet access, may be provided to the structure, via coax, by cable operators, and use coax running within the structure to reach individual cable service consuming devices in the structure. Typically, functionalities of the existing technologies run along with cable functionalities, but on different frequencies. 
     The coax infrastructure inside the structure typically includes coax, splitters and outlets. Splitters typically have one input and two or more outputs and are designed to transmit signals in the forward direction (input to output), in the backward direction (output to input), and to isolate outputs from different splitters, thus preventing signals from flowing from one coax outlet to another. Isolation is useful in order to a) reduce interference from other devices and b) maximize power transfer from Point Of Entry (“POE”) to outlets for best TV reception. 
     Elements of the existing technologies are specifically designed to propagate backward through splitters (“insertion”) and from output to output (“isolation”). One outlet in a structure can be reached from another by a single “isolation jump” and a number of “insertion jumps.” Typically isolation jumps have an attenuation of 5 to 40 dB and each insertion jump attenuates approximately 3 dB. MoCA™-identified technology has a dynamic range in excess of 55 dB while supporting 200 Mbps throughput. Therefore MoCA™-identified technology can work effectively through a significant number of splitters. 
     Managed network schemes, such as MoCAT™-identified technology, are specifically designed to support streaming video with minimal packet loss between outlets. 
     When a network-connected device receives a data signal from the network, which may be a network such as that described above, the signal is often decomposed into in-phase (“I”) and quadrature (“Q”) portions during down-conversion to device base-band frequency. When the I and Q portions are recombined for data decryption, they are often imbalanced with respect to amplitude, phase or both. Rebalancing I and Q portions may involve calculating compensation factors based on frequency-domain signatures of the carrier frequency and the I and Q portions. In the presence of carrier frequency uncertainty, the frequency-domain signatures of received signals may be difficult to resolve using digital computation methods. It would therefore be desirable to provide systems and methods for compensating signals, in the presence of carrier frequency uncertainty, using digital computation methods. 
     SUMMARY OF THE INVENTION 
     A system and/or method for compensating for an I/Q imbalance at a node on a communication network, substantially as shown in and/or described in connection with at least one of the figures, as set forth more completely in the claims. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The above and other features of the present invention, its nature and various advantages will be more apparent upon consideration of the following detailed description, taken in conjunction with the accompanying drawings, and in which: 
         FIG. 1  shows an illustrative schematic diagram of an illustrative single or multi-chip device that may be used in accordance with principles of the invention; 
         FIG. 2  shows an illustrative schematic diagram of a portion of a receiver in accordance with the principles of the invention; 
         FIG. 3  shows another illustrative schematic diagram of a portion of a receiver in accordance with the principles of the invention; 
         FIG. 4  shows an illustrative schematic diagram of a circuit in accordance with the principles of the invention; 
         FIG. 5  shows another illustrative schematic diagram of a circuit in accordance with the principles of the invention; 
         FIG. 6  shows an illustrative flow chart in accordance with the principles of the invention; 
         FIG. 7  shows, in abridged form, an illustrative data packet that may be processed in accordance with the principles of the invention; 
         FIG. 8  shows an illustrative portion of a discrete-valued frequency spectrum associated with signal processing in accordance with the principles of the invention; 
         FIG. 9  shows an illustrative schematic diagram of a another circuit in accordance with the principles of the invention; 
         FIG. 10  shows a schematic memory configuration in accordance with the principles of the invention; 
         FIG. 11  shows an illustrative Energy Loss of Image Figure as a function of the number of bins used; 
         12 A shows an illustrative simulation minimization and  FIG. 12B  shows various illustrative plots; 
         FIG. 13  shows an illustrative maximization plot; 
         FIG. 14  shows an illustrative plot that summarizes simulation results; 
         FIG. 15  shows an illustrative plot that summarizes simulation results; 
         FIG. 16  shows an illustrative plot of a frequency input Ideal Channel, with no AWGN after a first iteration; 
         FIG. 17  shows an illustrative plot of a frequency input Ideal Channel, with no AWGN after two iterations; 
         FIG. 18  shows a slicer input Ideal Channel, with no AWGN after three iterations; 
         FIG. 19  shows a slicer input Ideal Channel, with no AWGN after four iterations; 
         FIG. 20  shows a slicer input Ideal Channel, no AWGN no I/Q imbalance; 
         FIG. 21  shows a slicer input MoCA 10408 Channel, 15 [dB] AWGN SNR before cancellation; 
         FIG. 22  shows a frequency plot MoCA 10408 Channel, 15 [dB] AWGN SNR after first iteration; 
         FIG. 23  shows a slicer input MoCA 10408 Channel, 15 [dB] AWGN SNR after first iteration; 
         FIG. 24  shows a frequency plot MoCA 10408 Channel, 15 [dB] AWGN SNR after second iteration; 
         FIG. 25  shows a slicer input MoCA 10408 Channel, 15 [dB] AWGN SNR after second iteration; 
         FIG. 26  shows a frequency plot MoCA 10408 Channel, 15 [dB] AWGN SNR after third iteration; 
         FIG. 27  shows a slicer input MoCA 10408 Channel, 15 [dB] AWGN SNR after third iteration; 
         FIG. 28  shows slicer input MoCA 10408 channel 15 [dB] AWGN SNR no I/Q imbalance; 
         FIG. 29  shows slicer input ideal channel, under RX and TX I/Q imbalance; 
         FIG. 30  shows frequency plot ideal channel no AWGN, under RX and TX I/Q imbalance after third iteration; and 
         FIG. 31  shows slicer input ideal channel no AWGN, under RX and TX I/Q imbalance after third iteration. 
     
    
    
     DETAILED DESCRIPTION OF EMBODIMENTS 
     Apparatus and methods for compensating for an I/Q imbalance are provided in accordance with the principles of the invention. The methods may include compensating for an imbalance between a first component of a data signal and a second component of the data signal. The data signal may be modulated by a carrier signal having a frequency error. The first component may be characterized by at least one parameter. The method may include receiving the data and carrier signals; selecting a value for the parameter such that a frequency domain energy is reduced, the frequency domain energy corresponding to a negative frequency; and modifying at least one of the components based on the selected value. 
     The apparatus may include a circuit operative to record signal values corresponding to frequency components of a received signal. The signal may be one that carries at least one orthogonal frequency division multiplexing (“OFDM”) symbol. The signal values may correspond to a carrier frequency having a frequency error; a first tone; and a second tone. 
     The apparatus may include a system for compensating for an imbalance between a first component of a data signal and a second component of the data signal. The data signal may be modulated by a carrier signal having a frequency error. The first component may be characterized by at least one parameter. The system may include a hardware module configured to quantify a signal value corresponding to one of the data and carrier signals; and a software module configured to receive the signal value from the hardware. 
     The first and second tones may be transmitted in the context of a MoCA protocol probe2 transmission as set forth in the aforementioned MoCA specification. 
     Illustrative features of the invention are described below with reference to  FIGS. 1-8  and Appendices A-E. 
       FIG. 1  shows a single or multi-chip module  102  according to the invention, which can be one or more integrated circuits, in an illustrative data processing system  100  according to the invention. Data processing system  100  may include one or more of the following components: I/O circuitry  104 , peripheral devices  106 , processor  108  and memory  110 . These components may be coupled together by a system bus or other interconnections  112  and are disposed on a circuit board  120  in an end-user system  130 . Elements of module  102  may perform tasks involved in I/Q imbalance compensation. 
     In some embodiments, I/Q imbalance compensation may be performed during MoCA Probe2 burst reception. Probe2 is a two-tone signal which can be used for I/Q imbalance calculations or other RF calibrations in the receiver. A PHY layer performs bin selection and recording and the result is uploaded to the CPU for the I/Q compensation parameters calculations. 
       FIG. 2  shows a partial schematic diagram of illustrative receiver  200 . Receiver  200  may include radio frequency (“RF”) processing module  202 , time domain processing module  204  and frequency domain processing module  206 . RF signal  208  is received and gain-adjusted at gain  210 . Signal  208  is down-converted to base band (“BB”) frequency at  212 . Intentional frequency error  213  is added to signal  208  at  212 . Analog-to-digital converter  214  converts signal  208  to a digital signal sampled at the analog-to-digital sampling rate and passes signal  208  to imbalance compensation module  218 . I/Q imbalance compensation module  218  may be configured to carry out steps associated herein with I/Q compensation. I/Q imbalance compensation module  218  outputs signal  209 , which corresponds to Equation 1 (below). 
     Signal  209  passes to variable rate interpolator  224 , which resamples signal  209  to an appropriate sampling rate. 
     The variable rate interpolator  224  may receive timing signal  237  from numerically controlled oscillator (“NCO”) timing generator  236 . Timing signal  237  may be based on carrier frequency offset estimate (“CFOE”)  241 , from preamble processor  240 . CFOE  241  may be based on a preamble processor  240  estimate. Interpolator  224  outputs signal  225 , which may then pass through high pass filter (“HPF”)  228  to reject direct current (“DC”) signal components. 
     Carrier recovery loop  229  may be present to perform frequency compensation for intentional frequency error  213 . Carrier recovery loop  229  may receive input from NCO frequency generator  234 , which may be controlled by receiver controller  232 . NCO frequency generator  234  may receive carrier frequency offset estimate  241  from preamble processor  240 . A cyclic prefix may be removed from signal  225  at CP remover  246 . 
     Fast Fourier transform module  298  may be present in frequency domain processing module  206  to transform signal  225  into frequency domain information (“FFT output”) that may be stored in memory  299  and may be communicated to probe2 software processing routine  250 , which may output correction parameters  252  for return to I/Q imbalance compensation module  218 . 
       FIG. 3  shows a partial schematic diagram of illustrative receiver  300 . Receiver  300  may include radio frequency (“RF”) processing module  302 , time domain processing module  304  and frequency domain processing module  306 . RF signal  308  is received and gain-adjusted at gain  310 . Signal  308  is down-converted to base band frequency at  312 . Intentional frequency error  313  is added to signal  308  at  312 . Analog-to-digital converter  314  converts signal  308  to a digital signal and passes signal  308  to 100 MHz FIFO (“first in, first out”) buffer  316 . Buffer  316  passes signal  308  to I/Q imbalance compensation module  318 . I/Q imbalance compensation module  318  may be configured to carry out steps associated herein with I/Q compensation. I/Q imbalance compensation module  318  outputs signal  309 , which corresponds to Equation 1 (below). 
     Signal  309  passes to baseband-mode demixer  320 . Receiver  300  may include automatic gain controller  322 , which may provide feedback to gain  310  based on signal  309 . From demixer  320 , signal  309  may pass to Farrow interpolator  324 , which resamples 100 MHz signal  309  at a lower rate. 
     Farrow interpolator  324  may receive timing signal  337  from numerically controlled oscillator (“NCO”) timing generator  336 . Timing signal  337  may be based on carrier frequency offset estimate  341 , from preamble control processor  340 . Carrier frequency offset estimate  341  may be based on the output of TD phase rotator  330  (discussed below), via preamble processor  340 . In some embodiments, interpolator  324  outputs signal  325  at  100  MHz. Signal  325  may be synchronized to a transmitter clock (not shown) via a timing recover loop (not shown). Signal  325  may be down-sampled by a factor of 2, via half band filter decimator (“HB DEC 2→1”)  326 , to 50 MHz. Signal  325  may then pass through high pass filter (“HPF”)  328  to reject direct current (“DC”) signal components. 
     Time domain (“TD”) phase rotator  330  may be present to perform frequency compensation for intentional frequency error  313 . TD phase rotator may receive input from NCO frequency generator  334 , which may be controlled by receiver controller  332 . NCO frequency generator  334  may receive carrier frequency offset estimate  341  from preamble processor  340 . Signal  325  may then pass to delay buffer  342 . A cyclic prefix may be removed at sub-circuit  346 . In some embodiments, sub-circuit  346  may perform receiver windowing to reduce damage from narrow band interference noise that might otherwise leak into adjacent tones. 
     Fast Fourier transform module  398  may be present in frequency domain processing module  306  to transform signal  325  into frequency domain information that may be communicated to probe2 calculator  350 , which may output probe2 result  352 , for transmission to I/Q compensation module  318 . 
     Some embodiments include a bypass mode, in which signal input is routed to output around I/Q imbalance compensation module  318 . 
     In some embodiments, I/Q compensation is accomplished by digital signal analysis and processing. In those embodiments, ζ, ρ &amp; Scale_Q are I/Q compensation parameters that have to be estimated during Probe2. 
     Equation 1 shows compensated real and imaginary portions of a compensated signal that would be output from the I/Q imbalance compensation. module (see  FIG. 2 ). 
     
       
         
           
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       
                         Y 
                         ~ 
                       
                       real 
                     
                     = 
                     
                       { 
                       
                         
                           
                             
                               
                                 
                                   Bypass 
                                   == 
                                   1 
                                 
                               
                               
                                 
                                   Y 
                                   real 
                                 
                               
                             
                             
                               
                                 
                                   Bypass 
                                   == 
                                   0 
                                 
                               
                               
                                 
                                   { 
                                   
                                     
                                       
                                         
                                           Scale_Q 
                                           == 
                                           0 
                                         
                                       
                                       
                                         
                                           ϛ 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           
                                             Y 
                                             real 
                                           
                                         
                                       
                                     
                                     
                                       
                                         
                                           Scale_Q 
                                           == 
                                           1 
                                         
                                       
                                       
                                         
                                           Y 
                                           real 
                                         
                                       
                                     
                                   
                                 
                               
                             
                           
                           ⁢ 
                           
                             
 
                           
                           ⁢ 
                           
                             
                               Y 
                               ~ 
                             
                             imag 
                           
                         
                         = 
                         
                           { 
                           
                             
                               
                                 
                                   Bypass 
                                   == 
                                   1 
                                 
                               
                               
                                 
                                   Y 
                                   imag 
                                 
                               
                             
                             
                               
                                 
                                   Bypass 
                                   == 
                                   0 
                                 
                               
                               
                                 
                                   { 
                                   
                                     
                                       
                                         
                                           Scale_Q 
                                           = 
                                           0 
                                         
                                       
                                       
                                         
                                           
                                             
                                               Y 
                                               ~ 
                                             
                                             imag 
                                           
                                           = 
                                           
                                             
                                               Y 
                                               imag 
                                             
                                             + 
                                             
                                               ρ 
                                               ⁢ 
                                               
                                                   
                                               
                                               ⁢ 
                                               
                                                 Y 
                                                 real 
                                               
                                             
                                           
                                         
                                       
                                     
                                     
                                       
                                         
                                           Scale_Q 
                                           = 
                                           1 
                                         
                                       
                                       
                                         
                                           
                                             ϛ 
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             
                                               Y 
                                               imag 
                                             
                                           
                                           + 
                                           
                                             ρ 
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             
                                               Y 
                                               real 
                                             
                                           
                                         
                                       
                                     
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   1 
                 
               
             
           
         
       
     
       FIG. 4  shows illustrative circuit  400 , which may be included in a device for implementing the compensation set forth in Equation 1. 
       FIG. 5  shows illustrative circuit  500 , which may be included in a device for implementing the compensation set forth in Equation 1. Appendix A sets forth the theoretical basis for the compensation set forth in Equation 1. Appendix B shows exemplary imbalance and compensation measurements that have been made in connection with the apparatus and methods described herein. 
       FIG. 6  shows illustrative process  600  for compensating I/Q imbalance. Process  600  involves both hardware (“HW”) and software (“SW”) operations. The output of initial hardware operation  602  is a data array Z(k,m), which is the output of a Fast Fourier Transform (“FFT”) at bin k corresponding to Probe2 OFDM symbol m. 
     Initial hardware operation  602  may include numerically controlled oscillator (“NCO”) phase reset  604 . The phase of the first sample of an FFT window that results from Time Domain Unit (“TDU”) frequency compensation is determined. For this purpose the NCO Phase of the Phase Rotator in the Receiver TDU shall be reset to zero anytime after fine frequency compensation has been computed. The number of samples (number of phase accumulations) between the reset of the NCO and the first sample of the FFT window denoted as Δn shall be computed and sent to the SW routine. Zero phase accumulations (i.e., Δn=0) is most desirable since it reduces complexity of the SW routine. For the setting Δn=0, NCO phase accumulator  335  (in NCO frequency generator  334 —see  FIG. 3 ) should be reset once the first sample of the 356 point FFT window propagates through TD phase rotator  330  (see  FIG. 3 ) (and thus the first sample would be multiplied by unity). 
       FIG. 7  shows packet  700 , NCO reset, Δn and the start of the FFT window. 
     In some embodiments, bin selection  606  (see  FIG. 6 ) may be performed as a floating point computation, in which i 1  and i 2  are frequency bin indices computed as shown in Equation 2. 
                       i   1     =     -     round   ⁡     (       N   ·   CFO     π     )           ⁢     
     ⁢       i   2     =       i   1     +       sign   ⁡     (   CFO   )       ·     sign   ⁡     (            i   1          -            N   ·   CFO     π            )                     Equation   ⁢           ⁢   2               
Wherein CFO/(2π) is the estimated carrier frequency offset between a transmitter and the receiver and N is the number of FFT bins (e.g.,  256 ).
 
     In some embodiments, bin selection  606  (see  FIG. 6 ) may be performed as a fixed point computation. In those embodiments, CFO is a 17 bit signed integer, where ‘1’=2 14 . The computation of i 1  and i 2  may be done via comparison to fixed thresholds. The value of the FFT grid in fixed point representation is given by Equation 3: 
                       F   k     =     round   ⁡     (         2   ⁢   π   ⁢           ⁢   k     N     ·     2     Freq_bits   -   1         )         ,     
     ⁢       for   ⁢           ⁢   k     =     [       -   3     ,   3     ]       ,           Equation   ⁢           ⁢   3               
in which Freq_bits may be set to 14 or any other suitable number. Indices i 1  and i 2  are selected by finding the two FFT bins closest to 2CFO.
 
       FIG. 8  shows decision areas (absolute values only due to symmetry) corresponding to Equation 3. 
       FIG. 9  shows an illustrative hardware (“HW”) implementation for decision area boundary selection. Table 1 shows illustrative boundary values. 
     
       
         
           
               
               
               
               
               
               
               
             
               
                 TABLE 1 
               
               
                   
               
             
            
               
                 |Frequency Boundaries| 
                 
                   
                     
                       
                         
                           
                             F 
                             0 
                           
                           + 
                           
                             F 
                             1 
                           
                         
                         2 
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           
                             F 
                             1 
                           
                           + 
                           
                             F 
                             2 
                           
                         
                         2 
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           
                             F 
                             2 
                           
                           + 
                           
                             F 
                             3 
                           
                         
                         2 
                       
                     
                   
                 
                   F 1   
                   F 2   
                   F 3   
               
               
                   
               
               
                 Fixed point 
                 804 
                 2413 
                 4012 
                 1608 
                 3217 
                 4825 
               
               
                 value 
               
               
                   
               
            
           
         
       
     
     Equation 4 sets forth a definition for the sign operation. 
     
       
         
           
             
               
                 
                   
                     sign 
                     ⁡ 
                     
                       ( 
                       x 
                       ) 
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           1 
                         
                         
                           
                             x 
                             ≥ 
                             0 
                           
                         
                       
                       
                         
                           
                             - 
                             1 
                           
                         
                         
                           
                             x 
                             &lt; 
                             0 
                           
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   4 
                 
               
             
           
         
       
     
     In some embodiments, bin recording (at step  408 , see  FIG. 4 ) may involve 16 bit FFT outputs at bins k 1 , −k 1 +i 1 , −k 1 +i 2 , k 2 , −k 2 +i 1 , −k 2 +i 2 , which are then recorded for each of L OFDM symbols. It will be understood that there may be any suitable number of bits at the FFT output. The addresses in MoCA FFT that correspond to the bins are set forth in Table 2. 
     
       
         
           
               
               
               
             
               
                 TABLE 2 
               
               
                   
               
             
            
               
                 ADDR: k 1   
                 Z [k 1 , m] 
                 k 1  ∈ [146, 186] 
               
               
                 ADDR: k 2   
                 Z [k 2 , m] 
                 k 2  ∈ [217, 249] 
               
               
                 ADDR: 256 − 
                 Z [−k 1  + i 1 , m], 
                 −k 1  + i 1  = 256 − k 1  + i 1  ∈ [67, 113] 
               
               
                 k 1  + i 1   
               
               
                 ADDR: 256 − 
                 Z [−k 1  + i 2 , m], 
                 −k 1  + i 2  = 256 − k 1  + i 2  ∈ [66, 112] 
               
               
                 k 1  + i 2   
               
               
                 ADDR: 256 − 
                 Z [−k 2  + i 1 , m], 
                 −k 2  + i 1  = 256 − k 2  + i 1  ∈ [4, 36] 
               
               
                 k 2  + i 1   
               
               
                 ADDR: 256 − 
                 Z [−k 2  + i 2 , m] 
                 −k 2  + i 2  = 256 − k 2  + i 2  ∈ [3, 37] 
               
               
                 k 2  + i 2   
               
               
                   
               
            
           
         
       
     
       FIG. 10  shows an illustrative memory map at the end of a probe2 burst. 
     A CFO estimate is recorded at step  608  (shown in  FIG. 6 ). The estimate may be a 17-bit estimate. 
     In some embodiments, residual frequency error {circumflex over (ε)} estimation  610  (see  FIG. 6 ) may be performed by a software module. In some embodiments, residual frequency error estimation may be performed by a hardware module. In some embodiments, residual frequency error estimation may be computed as shown in Equation 5. 
     
       
         
           
             
               
                 
                   
                     
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                       ⁢ 
                       
                           
                       
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                               &gt; 
                               
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                                   k 
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                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   5 
                 
               
             
           
         
       
     
     In some embodiments, residual frequency error compensation and time averaging may be computed in accordance with Equations 6, which depend on {circumflex over (ε)} and whose derivations are set forth in Appendix A. 
     
       
         
           
             
               
                 
                   
                     
                       
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                           = 
                           o 
                         
                         
                           L 
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           Z 
                           ⁡ 
                           
                             [ 
                             
                               
                                 k 
                                 1 
                               
                               , 
                               m 
                             
                             ] 
                           
                         
                         ⁢ 
                         
                           ⅇ 
                           
                             
                               - 
                               j2π 
                             
                             ⁢ 
                             
                               
                                 ɛ 
                                 ^ 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   N 
                                   + 
                                   
                                     N 
                                     CP 
                                   
                                 
                                 ) 
                               
                             
                             ⁢ 
                             m 
                           
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       
                         Z 
                         _ 
                       
                       
                         k 
                         2 
                       
                     
                     = 
                     
                       
                         ∑ 
                         
                           m 
                           = 
                           o 
                         
                         
                           L 
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           Z 
                           ⁡ 
                           
                             [ 
                             
                               
                                 k 
                                 2 
                               
                               , 
                               m 
                             
                             ] 
                           
                         
                         ⁢ 
                         
                           ⅇ 
                           
                             
                               - 
                               j2π 
                             
                             ⁢ 
                             
                               
                                 ɛ 
                                 ^ 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   N 
                                   + 
                                   
                                     N 
                                     CP 
                                   
                                 
                                 ) 
                               
                             
                             ⁢ 
                             m 
                           
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       
                         Z 
                         _ 
                       
                       
                         
                           - 
                           
                             k 
                             1 
                           
                         
                         + 
                         
                           i 
                           1 
                         
                       
                     
                     = 
                     
                       
                         ∑ 
                         
                           m 
                           = 
                           o 
                         
                         
                           L 
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           Z 
                           ⁡ 
                           
                             [ 
                             
                               
                                 
                                   - 
                                   
                                     k 
                                     1 
                                   
                                 
                                 + 
                                 
                                   i 
                                   1 
                                 
                               
                               , 
                               m 
                             
                             ] 
                           
                         
                         ⁢ 
                         
                           ⅇ 
                           
                             
                               - 
                               
                                 j 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       2 
                                       ⁢ 
                                       CFO 
                                     
                                     + 
                                     
                                       2 
                                       ⁢ 
                                       π 
                                       ⁢ 
                                       
                                         ɛ 
                                         ^ 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                             ⁢ 
                             
                               ( 
                               
                                 N 
                                 + 
                                 
                                   N 
                                   CP 
                                 
                               
                               ) 
                             
                             ⁢ 
                             m 
                           
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       
                         Z 
                         _ 
                       
                       
                         
                           - 
                           
                             k 
                             2 
                           
                         
                         + 
                         
                           i 
                           1 
                         
                       
                     
                     = 
                     
                       
                         ∑ 
                         
                           m 
                           = 
                           o 
                         
                         
                           L 
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           Z 
                           ⁡ 
                           
                             [ 
                             
                               
                                 
                                   - 
                                   
                                     k 
                                     2 
                                   
                                 
                                 + 
                                 
                                   i 
                                   1 
                                 
                               
                               , 
                               m 
                             
                             ] 
                           
                         
                         ⁢ 
                         
                           ⅇ 
                           
                             
                               - 
                               
                                 j 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       2 
                                       ⁢ 
                                       CFO 
                                     
                                     + 
                                     
                                       2 
                                       ⁢ 
                                       π 
                                       ⁢ 
                                       
                                         ɛ 
                                         ^ 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                             ⁢ 
                             
                               ( 
                               
                                 N 
                                 + 
                                 
                                   N 
                                   CP 
                                 
                               
                               ) 
                             
                             ⁢ 
                             m 
                           
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       
                         Z 
                         _ 
                       
                       
                         
                           - 
                           
                             k 
                             1 
                           
                         
                         + 
                         
                           i 
                           2 
                         
                       
                     
                     = 
                     
                       
                         ∑ 
                         
                           m 
                           = 
                           o 
                         
                         
                           L 
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           Z 
                           ⁡ 
                           
                             [ 
                             
                               
                                 
                                   - 
                                   
                                     k 
                                     1 
                                   
                                 
                                 + 
                                 
                                   i 
                                   2 
                                 
                               
                               , 
                               m 
                             
                             ] 
                           
                         
                         ⁢ 
                         
                           ⅇ 
                           
                             
                               - 
                               
                                 j 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       2 
                                       ⁢ 
                                       CFO 
                                     
                                     + 
                                     
                                       2 
                                       ⁢ 
                                       π 
                                       ⁢ 
                                       
                                         ɛ 
                                         ^ 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                             ⁢ 
                             
                               ( 
                               
                                 N 
                                 + 
                                 
                                   N 
                                   CP 
                                 
                               
                               ) 
                             
                             ⁢ 
                             m 
                           
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       
                         Z 
                         _ 
                       
                       
                         
                           - 
                           
                             k 
                             2 
                           
                         
                         + 
                         
                           i 
                           2 
                         
                       
                     
                     = 
                     
                       
                         ∑ 
                         
                           m 
                           = 
                           o 
                         
                         
                           L 
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           Z 
                           ⁡ 
                           
                             [ 
                             
                               
                                 
                                   - 
                                   
                                     k 
                                     2 
                                   
                                 
                                 + 
                                 
                                   i 
                                   2 
                                 
                               
                               , 
                               m 
                             
                             ] 
                           
                         
                         ⁢ 
                         
                           ⅇ 
                           
                             
                               - 
                               
                                 j 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       2 
                                       ⁢ 
                                       CFO 
                                     
                                     + 
                                     
                                       2 
                                       ⁢ 
                                       π 
                                       ⁢ 
                                       
                                         ɛ 
                                         ^ 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                             ⁢ 
                             
                               ( 
                               
                                 N 
                                 + 
                                 
                                   N 
                                   CP 
                                 
                               
                               ) 
                             
                             ⁢ 
                             m 
                           
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   6 
                 
               
             
           
         
       
     
     Equations 7 may be used to evaluate an I/Q imbalance phasor estimate, which may be computed using Equation 8. 
     
       
         
           
             
               
                 
                   
                     B 
                     n 
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             
                               
                                 CFO 
                                 ≠ 
                                 0 
                               
                             
                             
                               
                                 
                                   
                                     
                                       
                                         
                                           1 
                                           2 
                                         
                                         ⁢ 
                                         
                                           ( 
                                           
                                             
                                               sin 
                                               ⁡ 
                                               
                                                 ( 
                                                 
                                                   2 
                                                   · 
                                                   CFO 
                                                   · 
                                                   N 
                                                 
                                                 ) 
                                               
                                             
                                             
                                               CFO 
                                               + 
                                               
                                                 
                                                   π 
                                                   · 
                                                   
                                                     i 
                                                     n 
                                                   
                                                 
                                                 N 
                                               
                                             
                                           
                                           ) 
                                         
                                       
                                       + 
                                     
                                   
                                 
                                 
                                   
                                     
                                       
                                         j 
                                         2 
                                       
                                       ⁢ 
                                       
                                         ( 
                                         
                                           
                                             
                                               cos 
                                               ⁡ 
                                               
                                                 ( 
                                                 
                                                   2 
                                                   · 
                                                   CFO 
                                                   · 
                                                   N 
                                                 
                                                 ) 
                                               
                                             
                                             - 
                                             1 
                                           
                                           
                                             CFO 
                                             + 
                                             
                                               
                                                 π 
                                                 · 
                                                 
                                                   i 
                                                   n 
                                                 
                                               
                                               N 
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   CFO 
                                   = 
                                   0 
                                 
                                 , 
                                 
                                   
                                     i 
                                     n 
                                   
                                   = 
                                   0 
                                 
                               
                             
                             
                               N 
                             
                           
                           
                             
                               
                                 
                                   CFO 
                                   = 
                                   0 
                                 
                                 , 
                                 
                                   
                                     i 
                                     n 
                                   
                                   ≠ 
                                   0 
                                 
                               
                             
                             
                               0 
                             
                           
                         
                         ⁢ 
                         
                           
 
                         
                         ⁢ 
                         
                           C 
                           1 
                         
                       
                       = 
                       
                         
                           
                             
                               ⅇ 
                               
                                 j2 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   CFO 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       Δ 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       n 
                                     
                                     ) 
                                   
                                 
                               
                             
                             N 
                           
                           ⁢ 
                           
                             
                               
                                 ( 
                                 
                                   
                                     
                                        
                                       
                                         B 
                                         1 
                                       
                                        
                                     
                                     2 
                                   
                                   + 
                                   
                                     
                                        
                                       
                                         B 
                                         2 
                                       
                                        
                                     
                                     2 
                                   
                                 
                                 ) 
                               
                               · 
                               
                                 
                                   Z 
                                   _ 
                                 
                                 
                                   k 
                                   1 
                                 
                               
                             
                             
                               
                                 
                                   
                                     B 
                                     1 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         Z 
                                         _ 
                                       
                                       
                                         
                                           - 
                                           
                                             k 
                                             1 
                                           
                                         
                                         + 
                                         
                                           i 
                                           1 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                                 * 
                               
                               + 
                               
                                 
                                   
                                     B 
                                     2 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         Z 
                                         _ 
                                       
                                       
                                         
                                           - 
                                           
                                             k 
                                             1 
                                           
                                         
                                         + 
                                         
                                           i 
                                           2 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                                 * 
                               
                             
                           
                           ⁢ 
                           
                             
 
                           
                           ⁢ 
                           
                             C 
                             2 
                           
                         
                         = 
                         
                           
                             
                               
                                 ⅇ 
                                 
                                   j2 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     CFO 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         Δ 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         n 
                                       
                                       ) 
                                     
                                   
                                 
                               
                               N 
                             
                             ⁢ 
                             
                               
                                 
                                   ( 
                                   
                                     
                                       
                                          
                                         
                                           B 
                                           1 
                                         
                                          
                                       
                                       2 
                                     
                                     + 
                                     
                                       
                                          
                                         
                                           B 
                                           2 
                                         
                                          
                                       
                                       2 
                                     
                                   
                                   ) 
                                 
                                 · 
                                 
                                   
                                     Z 
                                     _ 
                                   
                                   
                                     k 
                                     2 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       B 
                                       1 
                                     
                                     ⁡ 
                                     
                                       ( 
                                       
                                         
                                           Z 
                                           _ 
                                         
                                         
                                           
                                             - 
                                             
                                               k 
                                               2 
                                             
                                           
                                           + 
                                           
                                             i 
                                             1 
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                   * 
                                 
                                 + 
                                 
                                   
                                     
                                       B 
                                       2 
                                     
                                     ⁡ 
                                     
                                       ( 
                                       
                                         
                                           Z 
                                           _ 
                                         
                                         
                                           
                                             - 
                                             
                                               k 
                                               2 
                                             
                                           
                                           + 
                                           
                                             i 
                                             2 
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                   * 
                                 
                               
                             
                             ⁢ 
                             
                               
 
                             
                             ⁢ 
                             C 
                           
                           = 
                           
                             
                               1 
                               2 
                             
                             ⁢ 
                             
                               ( 
                               
                                 
                                   C 
                                   1 
                                 
                                 + 
                                 
                                   C 
                                   2 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   7 
                 
               
             
             
               
                 
                   
                     
                       g 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         ⅇ 
                         
                           - 
                           jθ 
                         
                       
                     
                     _ 
                   
                   = 
                   
                     
                       C 
                       - 
                       1 
                     
                     
                       C 
                       - 
                       1 
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   8 
                 
               
             
           
         
       
     
     I/Q imbalance compensation parameters ξ,ρ and Scale_Q (see, e.g., Equation 1) may then be computed in accordance with Equation 9. 
     
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             real 
                             ⁢ 
                             
                               { 
                               
                                 
                                   g 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     ⅇ 
                                     
                                       - 
                                       jθ 
                                     
                                   
                                 
                                 _ 
                               
                               } 
                             
                           
                           ≥ 
                           1 
                         
                       
                       
                         
                           ScaleQ 
                           = 
                           1 
                         
                       
                       
                         
                           
                             
                               ξ 
                               ^ 
                             
                             = 
                             
                               1 
                               
                                 real 
                                 ⁢ 
                                 
                                   { 
                                   
                                     
                                       g 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         ⅇ 
                                         
                                           - 
                                           jθ 
                                         
                                       
                                     
                                     _ 
                                   
                                   } 
                                 
                               
                             
                           
                           , 
                         
                       
                       
                         
                           
                             ρ 
                             ^ 
                           
                           = 
                           
                             - 
                             
                               
                                 imag 
                                 ⁢ 
                                 
                                   { 
                                   
                                     
                                       g 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         ⅇ 
                                         
                                           - 
                                           jθ 
                                         
                                       
                                     
                                     _ 
                                   
                                   } 
                                 
                               
                               
                                 real 
                                 ⁢ 
                                 
                                   { 
                                   
                                     
                                       g 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         ⅇ 
                                         
                                           - 
                                           jθ 
                                         
                                       
                                     
                                     _ 
                                   
                                   } 
                                 
                               
                             
                           
                         
                       
                     
                     
                       
                         otherwise 
                       
                       
                         
                           ScaleQ 
                           = 
                           0 
                         
                       
                       
                         
                           
                             
                               ξ 
                               ^ 
                             
                             = 
                             
                               real 
                               ⁢ 
                               
                                 { 
                                 
                                   
                                     g 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       ⅇ 
                                       
                                         - 
                                         jθ 
                                       
                                     
                                   
                                   _ 
                                 
                                 } 
                               
                             
                           
                           , 
                         
                       
                       
                         
                           
                             ρ 
                             ^ 
                           
                           = 
                           
                             
                               - 
                               imag 
                             
                             ⁢ 
                             
                               { 
                               
                                 
                                   g 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     ⅇ 
                                     
                                       - 
                                       jθ 
                                     
                                   
                                 
                                 _ 
                               
                               } 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   9 
                 
               
             
           
         
       
     
     Equation 9 avoids saturation at the receiver since is always smaller or equal than unity, thus attenuating the stronger I/Q signal rather than amplifying the weaker I/Q signal. In some embodiments, the above computations can be carried out in an iterative fashion over several probe2 transmissions. Equations 10 show how new phasor estimates may be used to update previous estimates.
 
(    ge   −jθ   ) i   ACC =(1−μ i )(    ge   −jθ   ) i-1   ACC +μ i (    ge   −jθ   ) i-1   ACC ·(    ge   −jθ   )
 
(    gee   −jθ   ) 0   ACC =1  Equations 10
 
     In Equations 10, (  ge −jθ   ), is the phasor estimate computed during the i′th probe2 transmission. Some embodiments may include an update routine that may use a first order loop with a loop gain of μ i ε[0,1]. The loop gain may provide a tradeoff between convergence speed and noise filtering by controlling the loop bandwidth (“BW”). A gear-shifting approach may be used in which the loop BW is dynamically changed during convergence. For fast convergence during the first two/three iterations, a high loop BW may be used. For consecutive probe2 transmissions, a small loop BW may be used. Equation 11 sets forth values that may be used for μ i . i denotes the probe2 burst index number. 
     
       
         
           
             
               
                 
                   
                     μ 
                     i 
                   
                   = 
                   
                     { 
                     
                       
                         
                           1 
                         
                         
                           
                             i 
                             = 
                             1 
                           
                         
                       
                       
                         
                           0.75 
                         
                         
                           
                             i 
                             = 
                             2 
                           
                         
                       
                       
                         
                           0.5 
                         
                         
                           
                             i 
                             = 
                             3 
                           
                         
                       
                       
                         
                           0.25 
                         
                         
                           
                             i 
                             = 
                             4 
                           
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   11 
                 
               
             
           
         
       
     
     Equations 12 set forth I/Q compensation parameters that may be used during the reception of the i′th probe2. 
     
       
         
           
             
               
                 
                     
                   
                     { 
                     
                       
                         
                           
                             
                               real 
                               ⁢ 
                               
                                 { 
                                 
                                   
                                     ( 
                                     
                                       
                                         g 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           ⅇ 
                                           
                                             - 
                                             jθ 
                                           
                                         
                                       
                                       _ 
                                     
                                     ) 
                                   
                                   
                                     i 
                                     - 
                                     1 
                                   
                                   ACC 
                                 
                                 } 
                               
                             
                             ≥ 
                             1 
                           
                         
                         
                           
                             
                               
                                 
                                   ξ 
                                   ^ 
                                 
                                 i 
                               
                               = 
                               
                                 1 
                                 
                                   real 
                                   ⁢ 
                                   
                                     { 
                                     
                                       
                                         ( 
                                         
                                           
                                             g 
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             
                                               ⅇ 
                                               
                                                 - 
                                                 jθ 
                                               
                                             
                                           
                                           _ 
                                         
                                         ) 
                                       
                                       
                                         i 
                                         - 
                                         1 
                                       
                                       ACC 
                                     
                                     } 
                                   
                                 
                               
                             
                             , 
                           
                         
                         
                           
                             
                               
                                 ρ 
                                 i 
                               
                               ^ 
                             
                             = 
                             
                               - 
                               
                                 
                                   imag 
                                   ⁢ 
                                   
                                     { 
                                     
                                       
                                         ( 
                                         
                                           
                                             g 
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             
                                               ⅇ 
                                               
                                                 - 
                                                 jθ 
                                               
                                             
                                           
                                           _ 
                                         
                                         ) 
                                       
                                       
                                         i 
                                         - 
                                         1 
                                       
                                       ACC 
                                     
                                     } 
                                   
                                 
                                 
                                   real 
                                   ⁢ 
                                   
                                     { 
                                     
                                       
                                         ( 
                                         
                                           
                                             g 
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             
                                               ⅇ 
                                               
                                                 - 
                                                 jθ 
                                               
                                             
                                           
                                           _ 
                                         
                                         ) 
                                       
                                       
                                         i 
                                         - 
                                         1 
                                       
                                       ACC 
                                     
                                     } 
                                   
                                 
                               
                             
                           
                         
                       
                       
                         
                           otherwise 
                         
                         
                           
                             
                               
                                 
                                   ξ 
                                   ^ 
                                 
                                 i 
                               
                               = 
                               
                                 real 
                                 ⁢ 
                                 
                                   { 
                                   
                                     
                                       ( 
                                       
                                         
                                           g 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           
                                             ⅇ 
                                             
                                               - 
                                               jθ 
                                             
                                           
                                         
                                         _ 
                                       
                                       ) 
                                     
                                     
                                       i 
                                       - 
                                       1 
                                     
                                     ACC 
                                   
                                   } 
                                 
                               
                             
                             , 
                           
                         
                         
                           
                             
                               
                                 ρ 
                                 i 
                               
                               ^ 
                             
                             = 
                             
                               
                                 - 
                                 imag 
                               
                               ⁢ 
                               
                                 { 
                                 
                                   
                                     ( 
                                     
                                       
                                         g 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           ⅇ 
                                           
                                             - 
                                             jθ 
                                           
                                         
                                       
                                       _ 
                                     
                                     ) 
                                   
                                   
                                     i 
                                     - 
                                     1 
                                   
                                   ACC 
                                 
                                 } 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   12 
                 
               
             
           
         
       
     
     Three to four iterations (which may correspond to 3 to 4 probe2 transmissions) are often sufficient to compensate for I/Q imbalance. 
     Appendix C sets forth pseudo-code for a fixed point implementation of the compensation. 
     Appendix D sets forth parameters for a hardware-software interface in a system for I/Q imbalance compensation. 
     A network node may acquire an estimate of signal to noise ratio (“SNR”) at each tone and carrier frequency offset (relative to an associated network coordinator (“NC”)) when the node processes one or more probe  1  bursts from the NC. The node may use the SNR estimates to inform the NC which two frequency bins to use for probe2 transmission to the node. The node may use the CFO estimate to calculate and communicate to the NC the number of OFDM symbols and the cyclic prefix (“CP”) length during probe2 transmission. 
     Appendix E sets forth illustrative pseudocode for computation of frequency offset introduction, CP and selection of a number of OFDM symbols. In some embodiments, the Probe2, CP and L algorithms set forth in Appendix E may be performed before sending a MoCA™ probe2 report and after a receiver RF generator introduces any required, necessary or intentional carrier offset. 
     For the sake of clarity, the foregoing description, including specific examples of parameters or parameter values, is sometimes specific to certain protocols such as those identified with the name MoCA™ and/or Ethernet protocols. However, this is not intended to be limiting and the invention may be suitably generalized to other protocols and/or other packet protocols. The use of terms that may be specific to a particular protocol such as that identified by the name MoCA™ or Ethernet to describe a particular feature or embodiment is not intended to limit the scope of that feature or embodiment to that protocol specifically; instead the terms are used generally and are each intended to include parallel and similar terms defined under other protocols. 
     It will be appreciated that software components of the present invention including programs and data may, if desired, be implemented in ROM (read only memory) form, including CD-ROMs, EPROMs and EEPROMs, or may be stored in any other suitable computer-readable medium such as but not limited to discs of various kinds, cards of various kinds and RAMs. Components described herein as software may, alternatively, be implemented wholly or partly in hardware, if desired, using conventional techniques. 
     Thus, systems and methods for compensating for I/Q imbalance have been described. Persons skilled in the art will appreciate that the present invention can be practiced using embodiments of the invention other than those described, which are presented for purposes of illustration rather than of limitation. The present invention is limited only by the claims which follow. 
     APPENDIX A 
     Probe2 Theory 
     The I/Q imbalance can be modeled as a multiplicative gain factor applied on one of the I/Q components as well a relative phase difference. During probe2 reception MoCA specifies that a receiver must introduce a frequency error during RF down conversion we shall denote this shift as φ. The converted signal is given by:
 
 z   i   [n]=s   i   [n ] cos(2 πφn )− s   q   [n ] sin(2 πφn )+ w   i   [n] 
 
 z   q   [n]=gs   i   [n ] sin(2 πφn −θ)+ gs   q   [n ] cos(2 πφn −θ)+ w   q   [n] 
 
     Some algebra shows that the above can be expressed as
 
 z[n]=K   1   s[n]e   j2πφn   +K   2   s*[n]e   9−j2πφn   +w[n] 
 
 K   1 =½(1 +ge   −jθ )
 
 K   2 =½(1 −ge   jθ )
 
     At the receiver I/Q compensation is performed the signal after I/Q compensation is given by:
 
 z′[n]={real ( K   1   s[n]e   j2πφn   +K   2   s*[n]e   −j2πφn   +w[n ])}
 
+ jξ{imag ( K   1   s[n]e   j2πφn   +K   2   s*[n]e   −j2πφn   +w[n ])}
 
+ jρ{real ( K   1   s[n]e   j2πφn   +K   2   s*[n]e   −j2πφn   +w[n ])}
 
     Assuming transmission of a single frequency at frequency bin k, after some algebra the compensated signal is given by
 
 z′[n]={|h|cos( 2 πn ( k/N +φ)+∠ h )}
 
+ jξ{g|h |sin(2 πn ( k/N +φ)+∠ h −θ)}
 
+ jρ{|h |cos(2 πn ( k/N +φ)+∠ h )}+[ w   r   [n]+j (ρ· w   r   [n]+ξ·w   i   [n ])]
 
     The signal over goes frequency compensation and then is transformed into the frequency domain via the FFT operation. After some algebra the frequency domain signals at bins k and −k are given by: 
     
       
         
           
             
               
                 Z 
                 ^ 
               
               ⁡ 
               
                 [ 
                 k 
                 ] 
               
             
             = 
             
               
                 
                    
                   h 
                    
                 
                 ⁢ 
                 
                   ⅇ 
                   
                     j∠ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     h 
                   
                 
                 ⁢ 
                 
                   N 
                   ⁡ 
                   
                     [ 
                     
                       
                         
                           1 
                           2 
                         
                         ⁢ 
                         
                           ( 
                           
                             1 
                             + 
                             
                               ξ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 gⅇ 
                                 
                                   - 
                                   jθ 
                                 
                               
                             
                           
                           ) 
                         
                       
                       + 
                       
                         
                           ρ 
                           2 
                         
                         ⁢ 
                         
                           ⅇ 
                           
                             j 
                             ⁢ 
                             
                               π 
                               2 
                             
                           
                         
                       
                     
                     ] 
                   
                 
               
               + 
               
                 
                    
                   h 
                    
                 
                 ⁢ 
                 
                   ⅇ 
                   
                     
                       - 
                       j∠ 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     h 
                   
                 
                 ⁢ 
                 
                   ⅇ 
                   
                     - 
                     
                       j 
                       ⁡ 
                       
                         ( 
                         
                           2 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             π 
                             ⁡ 
                             
                               ( 
                               
                                 N 
                                 - 
                                 1 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 k 
                                 N 
                               
                               + 
                               φ 
                             
                             ) 
                           
                         
                         ) 
                       
                     
                   
                 
                 ⁢ 
                 
                   
                     
                       sin 
                       ⁡ 
                       
                         ( 
                         
                           2 
                           ⁢ 
                           
                             π 
                             ⁡ 
                             
                               ( 
                               
                                 k 
                                 + 
                                 
                                   N 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   φ 
                                 
                               
                               ) 
                             
                           
                         
                         ) 
                       
                     
                     
                       sin 
                       ⁡ 
                       
                         ( 
                         
                           2 
                           ⁢ 
                           
                             π 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   k 
                                   N 
                                 
                                 + 
                                 φ 
                               
                               ) 
                             
                           
                         
                         ) 
                       
                     
                   
                   ⁡ 
                   
                     [ 
                     
                       
                         
                           1 
                           2 
                         
                         ⁢ 
                         
                           ( 
                           
                             1 
                             - 
                             
                               ξ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               g 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 ⅇ 
                                 jθ 
                               
                             
                           
                           ) 
                         
                       
                       + 
                       
                         
                           ρ 
                           2 
                         
                         ⁢ 
                         
                           ⅇ 
                           
                             j 
                             ⁢ 
                             
                               π 
                               2 
                             
                           
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
       
         
           
             
               
                 Z 
                 ^ 
               
               ⁡ 
               
                 [ 
                 
                   - 
                   k 
                 
                 ] 
               
             
             = 
             
               
                  
                 h 
                  
               
               ⁢ 
               
                 ⅇ 
                 
                   
                     - 
                     j∠ 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   h 
                 
               
               ⁢ 
               
                 ⅇ 
                 
                   - 
                   
                     j 
                     ⁡ 
                     
                       ( 
                       
                         2 
                         ⁢ 
                         
                           πφ 
                           ⁡ 
                           
                             ( 
                             
                               N 
                               - 
                               1 
                             
                             ) 
                           
                         
                       
                       ) 
                     
                   
                 
               
               ⁢ 
               
                 
                   sin 
                   ⁡ 
                   
                     ( 
                     
                       2 
                       ⁢ 
                       πφ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       N 
                     
                     ) 
                   
                 
                 
                   sin 
                   ⁡ 
                   
                     ( 
                     
                       2 
                       ⁢ 
                       πφ 
                     
                     ) 
                   
                 
               
               ⁢ 
               
                 { 
                 
                   
                     
                       1 
                       2 
                     
                     ⁢ 
                     
                       ( 
                       
                         1 
                         - 
                         
                           ξ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           g 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             ⅇ 
                             jθ 
                           
                         
                       
                       ) 
                     
                   
                   + 
                   
                     
                       ρ 
                       2 
                     
                     ⁢ 
                     
                       ⅇ 
                       
                         j 
                         ⁢ 
                         
                           π 
                           2 
                         
                       
                     
                   
                 
                 } 
               
             
           
         
       
     
     In a system without I/Q imbalance, the energy at the negative bin is zero. The energy at the negative bin due to I/Q imbalance is given by 
     
       
         
           
             
               
                  
                 
                   
                     Z 
                     ^ 
                   
                   ⁡ 
                   
                     [ 
                     
                       - 
                       k 
                     
                     ] 
                   
                 
                  
               
               2 
             
             = 
             
               
                 
                   
                      
                     h 
                      
                   
                   2 
                 
                 ⁢ 
                 
                   
                     ( 
                     
                       
                         sin 
                         ⁡ 
                         
                           ( 
                           
                             2 
                             ⁢ 
                             πφ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             N 
                           
                           ) 
                         
                       
                       
                         sin 
                         ⁡ 
                         
                           ( 
                           
                             2 
                             ⁢ 
                             πφ 
                           
                           ) 
                         
                       
                     
                     ) 
                   
                   2 
                 
                 ⁢ 
                 
                   { 
                   
                     
                       
                         1 
                         2 
                       
                       ⁢ 
                       
                         ( 
                         
                           1 
                           - 
                           
                             ξ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             g 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               ⅇ 
                               
                                 j 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 θ 
                               
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                       
                         ρ 
                         2 
                       
                       ⁢ 
                       
                         ⅇ 
                         
                           j 
                           ⁢ 
                           
                             π 
                             2 
                           
                         
                       
                     
                   
                   } 
                 
                 ⁢ 
                 
                   { 
                   
                     
                       
                         1 
                         2 
                       
                       ⁢ 
                       
                         ( 
                         
                           1 
                           - 
                           
                             ξ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             g 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               ⅇ 
                               
                                 
                                   - 
                                   j 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 θ 
                               
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                       
                         ρ 
                         2 
                       
                       ⁢ 
                       
                         ⅇ 
                         
                           
                             - 
                             j 
                           
                           ⁢ 
                           
                             π 
                             2 
                           
                         
                       
                     
                   
                   } 
                 
               
               = 
               
                 
                   1 
                   4 
                 
                 ⁢ 
                 
                   
                      
                     h 
                      
                   
                   2 
                 
                 ⁢ 
                 
                   
                     ( 
                     
                       
                         sin 
                         ⁡ 
                         
                           ( 
                           
                             2 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             πφ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             N 
                           
                           ) 
                         
                       
                       
                         sin 
                         ⁡ 
                         
                           ( 
                           
                             2 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             π 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             φ 
                           
                           ) 
                         
                       
                     
                     ) 
                   
                   2 
                 
                 ⁢ 
                 
                   { 
                   
                     1 
                     + 
                     
                       
                         ξ 
                         2 
                       
                       ⁢ 
                       
                         g 
                         2 
                       
                     
                     + 
                     
                       ρ 
                       2 
                     
                     - 
                     
                       2 
                       ⁢ 
                       ξ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       g 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       cos 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       θ 
                     
                     - 
                     
                       2 
                       ⁢ 
                       ξ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       g 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       ρ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         sin 
                         ⁡ 
                         
                           ( 
                           θ 
                           ) 
                         
                       
                     
                   
                   } 
                 
               
             
           
         
       
     
     Thus our target is to minimize the energy of bin −k by using ρ, ξ. Minimizing using the Lagrange multipliers method gives the following equations 
                 ∂              Z   ^     ⁡     [     -   k     ]            2         ∂   ξ       =         ∂     (     1   +       ξ   2     ⁢     g   2       +     ρ   2     -     2   ⁢   ξ   ⁢           ⁢   g   ⁢           ⁢   cos   ⁢           ⁢   θ     -     2   ⁢   ξ   ⁢           ⁢   g   ⁢           ⁢   ρ   ⁢           ⁢     sin   ⁡     (   θ   )           )         ∂   ξ       =   0                     ∂              Z   ^     ⁡     [     -   k     ]            2         ∂   ρ       =         ∂     (     1   +       ξ   2     ⁢     g   2       +     ρ   2     -     2   ⁢   ξ   ⁢           ⁢   g   ⁢           ⁢   cos   ⁢           ⁢   θ     -     2   ⁢   ξ   ⁢           ⁢   g   ⁢           ⁢   ρ   ⁢           ⁢     sin   ⁡     (   θ   )           )         ∂   ρ       =   0           2 ξg   2 −2 g  cos θ−2 ρg  sin(θ)=0
 
2ρ−2 ξg  sin(θ)=0
 
Solving the above yields
 
     
       
         
           
             
               
                 [ 
                 
                   
                     
                       
                         g 
                         2 
                       
                     
                     
                       
                         
                           - 
                           g 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           sin 
                           ⁡ 
                           
                             ( 
                             θ 
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           - 
                           g 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           sin 
                           ⁡ 
                           
                             ( 
                             θ 
                             ) 
                           
                         
                       
                     
                     
                       1 
                     
                   
                 
                 ] 
               
               ⁡ 
               
                 [ 
                 
                   
                     
                       ξ 
                     
                   
                   
                     
                       ρ 
                     
                   
                 
                 ] 
               
             
             = 
             
               
                 
                   [ 
                   
                     
                       
                         
                           g 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           cos 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           θ 
                         
                       
                     
                     
                       
                         0 
                       
                     
                   
                   ] 
                 
                 ⁢ 
                 
                   
 
                 
                 [ 
                 
                   
                     
                       ξ 
                     
                   
                   
                     
                       ρ 
                     
                   
                 
                 ] 
               
               = 
               
                 
                   
                     1 
                     
                       
                         ( 
                         
                           g 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           cos 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           θ 
                         
                         ) 
                       
                       2 
                     
                   
                   ⁡ 
                   
                     [ 
                     
                       
                         
                           
                             g 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             θ 
                           
                         
                       
                       
                         
                           
                             
                               g 
                               2 
                             
                             ⁢ 
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             θcos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             θ 
                           
                         
                       
                     
                     ] 
                   
                 
                 = 
                 
                   [ 
                   
                     1 
                     
                       
                         
                           
                             g 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             θ 
                           
                         
                       
                       
                         
                           
                             tan 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             θ 
                           
                         
                       
                     
                   
                   ] 
                 
               
             
           
         
       
     
     It is easy to show that such a selection actually brings the energy at bin −k to 0 and thus completely cancels the I/Q imbalance effects. Our goal now is to estimate the I/Q imbalance parameters from probe2 transmissions. 
     I/Q Parameter Estimation 
     Since I/Q imbalance corrupts the incoming signal it results in corrupted carrier frequency estimation as well as corrupted channel estimation. The channel estimation under I/Q imbalance is given by; 
     
       
         
           
             
               
                 h 
                 ^ 
               
               k 
             
             = 
             
               
                 
                   
                     A 
                     k 
                     * 
                   
                   
                     N 
                     ⁢ 
                     
                       
                          
                         
                           A 
                           k 
                         
                          
                       
                       2 
                     
                   
                 
                 ⁢ 
                 
                   
                     Z 
                     ^ 
                   
                   ⁡ 
                   
                     [ 
                     k 
                     ] 
                   
                 
               
               = 
               
                 
                   
                      
                     h 
                      
                   
                   ⁢ 
                   
                     ⅇ 
                     
                       j∠ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       h 
                     
                   
                   ⁢ 
                   
                     K 
                     1 
                   
                 
                 + 
                 
                   e 
                   k 
                 
               
             
           
         
       
     
     The FFT output at bins k and −k without I/Q compensation but after frequency compensation assuming a frequency estimation error of ε is given by 
               Z   ⁡     [   k   ]       =            h        ⁢     ⅇ     ∠   ⁢           ⁢   h       ⁢     K   1     ⁢     ⅇ     j   ⁢           ⁢   π   ⁢           ⁢     ɛ   ⁡     (     N   -   1     )           ⁢       sin   ⁡     (     π   ⁢           ⁢   ɛ   ⁢           ⁢   N     )         sin   ⁡     (     π   ⁢           ⁢   ɛ     )           +          h        ⁢     ⅇ       -   ∠     ⁢           ⁢   h       ⁢     K   2     ⁢     ⅇ       -   j     ⁢           ⁢   2   ⁢           ⁢     π   ⁡     (     N   -   1     )       ⁢     (       k   N     +   φ   -       1   2     ⁢   ɛ       )         ⁢       sin   ⁡     (     2   ⁢   π   ⁢           ⁢     N   ⁡     (       k   N     +   φ   -       1   2     ⁢   ɛ       )         )         sin   ⁡     (     2   ⁢           ⁢     π   ⁡     (       k   N     +   φ   -       1   2     ⁢   ɛ       )         )           +     W   k                     Z   ⁡     [     -   k     ]       =            h        ⁢     ⅇ       -   ∠     ⁢           ⁢   h       ⁢     K   2     ⁢       ⅇ       -   j     ⁢           ⁢   2   ⁢           ⁢     π   ⁡     (     φ   -       1   2     ⁢   ɛ       )       ⁢     (     N   -   1     )         ⁡     (       sin   ⁡     (     2   ⁢           ⁢   π   ⁢           ⁢     N   ⁡     [     φ   -       1   2     ⁢   ɛ       ]         )         sin   ⁡     (     2   ⁢           ⁢     π   ⁡     [     φ   -       1   2     ⁢   ɛ       ]         )         )         +          h        ⁢     ⅇ     ∠   ⁢           ⁢   h       ⁢     K   1     ⁢       ⅇ       -   j     ⁢           ⁢   2   ⁢           ⁢     π   ⁡     (       k   N     +       1   2     ⁢   ɛ       )       ⁢     (     N   -   1     )         ⁡     (       sin   ⁡     (     2   ⁢           ⁢   π   ⁢           ⁢     N   ⁡     [       k   N     +       1   2     ⁢   ɛ       ]         )         sin   ⁡     (     2   ⁢           ⁢     π   ⁡     [       k   N     +       1   2     ⁢   ɛ       ]         )         )         +     W     -   k               
Effects of Carrier Frequency Offset Greater than 50 Khz
 
     In the absence of carrier frequency error the image component resulting from the I/Q imbalance appears exactly at the mirror digital frequency (−k/N) of the transmitted tone. Under carrier frequency error (which is mandatory during probe II) the I/Q image appears at a digital frequency of (−k/N−2φ), where φ is the normalized carrier frequency error which is φ=Carrier Frequency Error/SymbolRate=Δfc/fs. The carrier frequency error can be as large as ±200 ppm of 1.5e9 Hz=300 kkHz. While the OFDM tone spacing is 50e6/256=195.3 kHz. Thus the image component can fall somewhere between [−k−3, k+3] interval in the frequency domain. The FFT output for bin −k+i is given by 
     
       
         
           
             
               Z 
               ⁡ 
               
                 [ 
                 
                   
                     - 
                     k 
                   
                   + 
                   i 
                 
                 ] 
               
             
             = 
             
               
                 
                    
                   h 
                    
                 
                 ⁢ 
                 
                   ⅇ 
                   
                     
                       - 
                       ∠ 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     h 
                   
                 
                 ⁢ 
                 
                   K 
                   2 
                 
                 ⁢ 
                 
                   
                     ⅇ 
                     
                       
                         - 
                         j 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       2 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         π 
                         ⁡ 
                         
                           ( 
                           
                             φ 
                             - 
                             
                               
                                 1 
                                 2 
                               
                               ⁢ 
                               ɛ 
                             
                             + 
                             
                               1 
                               
                                 2 
                                 ⁢ 
                                 N 
                               
                             
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         ( 
                         
                           N 
                           - 
                           1 
                         
                         ) 
                       
                     
                   
                   ⁡ 
                   
                     ( 
                     
                       
                         
                           
                             ( 
                             
                               - 
                               1 
                             
                             ) 
                           
                           i 
                         
                         ⁢ 
                         
                           sin 
                           ⁡ 
                           
                             ( 
                             
                               2 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               π 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 N 
                                 ⁡ 
                                 
                                   [ 
                                   
                                     φ 
                                     - 
                                     
                                       
                                         1 
                                         2 
                                       
                                       ⁢ 
                                       ɛ 
                                     
                                   
                                   ] 
                                 
                               
                             
                             ) 
                           
                         
                       
                       
                         sin 
                         ⁡ 
                         
                           ( 
                           
                             2 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               π 
                               ⁡ 
                               
                                 [ 
                                 
                                   φ 
                                   - 
                                   
                                     
                                       1 
                                       2 
                                     
                                     ⁢ 
                                     ɛ 
                                   
                                   + 
                                   
                                     1 
                                     
                                       2 
                                       ⁢ 
                                       N 
                                     
                                   
                                 
                                 ] 
                               
                             
                           
                           ) 
                         
                       
                     
                     ) 
                   
                 
               
               + 
               
                 
                    
                   h 
                    
                 
                 ⁢ 
                 
                   ⅇ 
                   
                     ∠ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     h 
                   
                 
                 ⁢ 
                 
                   K 
                   1 
                 
                 ⁢ 
                 
                   
                     ⅇ 
                     
                       
                         - 
                         j 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         π 
                         ⁡ 
                         
                           ( 
                           
                             
                               
                                 
                                   2 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   k 
                                 
                                 - 
                                 i 
                               
                               N 
                             
                             + 
                             ɛ 
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         ( 
                         
                           N 
                           - 
                           1 
                         
                         ) 
                       
                     
                   
                   ⁡ 
                   
                     ( 
                     
                       
                         sin 
                         ⁡ 
                         
                           ( 
                           
                             π 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               N 
                               ⁡ 
                               
                                 [ 
                                 
                                   
                                     
                                       
                                         2 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         k 
                                       
                                       - 
                                       i 
                                     
                                     N 
                                   
                                   + 
                                   ɛ 
                                 
                                 ] 
                               
                             
                           
                           ) 
                         
                       
                       
                         sin 
                         ⁡ 
                         
                           ( 
                           
                             π 
                             ⁡ 
                             
                               [ 
                               
                                 
                                   
                                     
                                       2 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       k 
                                     
                                     - 
                                     i 
                                   
                                   N 
                                 
                                 + 
                                 ɛ 
                               
                               ] 
                             
                           
                           ) 
                         
                       
                     
                     ) 
                   
                 
               
               + 
               
                 W 
                 
                   
                     - 
                     k 
                   
                   + 
                   i 
                 
               
             
           
         
       
     
     And so due to the fact that the compensated frequency error φ results in a shift of 2φ in the location of the image, we need to collect the image energy from the interval [−k−3, . . . , k+3]. Pragmatically since we know the frequency error φ (up to ε) we know that the image will appear at a digital frequency of 
     
       
         
           
             
               - 
               
                 ( 
                 
                   
                     
                       k 
                       + 
                       M 
                     
                     N 
                   
                   + 
                   r 
                 
                 ) 
               
             
             , 
             
               
                 2 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 φ 
               
               = 
               
                 
                   
                     M 
                     N 
                   
                   + 
                   
                     r 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     where 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     M 
                   
                 
                 = 
                 
                   
                     
                       ⌊ 
                       
                         2 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         φ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         N 
                       
                       ⌋ 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     and 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                        
                       r 
                        
                     
                   
                   &lt; 
                   
                     
                       1 
                       N 
                     
                     . 
                   
                 
               
             
           
         
       
     
     The loss of image energy in [dB] with respect to the image energy is a function of the number of bins used to collect energy and given by: 
     
       
         
           
             Loss 
             = 
             
               
                 10 
                 ⁢ 
                 
                   log 
                   10 
                 
                 ⁢ 
                 
                   N 
                   2 
                 
               
               - 
               
                 10 
                 ⁢ 
                 
                   
                     log 
                     10 
                   
                   ⁡ 
                   
                     ( 
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           
                             
                               - 
                               M 
                             
                             - 
                             L 
                           
                         
                         
                           
                             - 
                             M 
                           
                           + 
                           L 
                         
                       
                       ⁢ 
                       
                         
                           ( 
                           
                             
                               sin 
                               ⁡ 
                               
                                 ( 
                                 
                                   π 
                                   ⁡ 
                                   
                                     [ 
                                     
                                       M 
                                       + 
                                       Nr 
                                     
                                     ] 
                                   
                                 
                                 ) 
                               
                             
                             
                               sin 
                               ⁡ 
                               
                                 ( 
                                 
                                   π 
                                   ⁡ 
                                   
                                     [ 
                                     
                                       
                                         
                                           M 
                                           + 
                                           i 
                                         
                                         N 
                                       
                                       + 
                                       r 
                                     
                                     ] 
                                   
                                 
                                 ) 
                               
                             
                           
                           ) 
                         
                         2 
                       
                     
                     ) 
                   
                 
               
             
           
         
       
     
     The worst case loss is experienced when the image falls midway between bins (r=1/(2N)). Using just one bin which is closest to the image results in a worst case loss of 3.9223[dB] using two bins results in a loss of 0.9120[dB]  FIG. 11  summarizes the loss as a function of the number of bins used. 
     We shall use 2 bins seems like a reasonable trade-off between complexity and performance. 
     FFT Processing of Probe2 (Single OFDM Symbol) 
     For simplicity consider a single OFDM symbol the extension to multi OFDM symbols will be given shortly after. We have shown that the FFT outputs at bins k and −k are given by 
     
       
         
           
             
               Z 
               ⁡ 
               
                 [ 
                 k 
                 ] 
               
             
             = 
             
               
                 
                   
                     A 
                     1 
                   
                   ⁢ 
                   
                     hK 
                     1 
                   
                 
                 
                   ︷ 
                   
                     signal 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     term 
                   
                 
               
               + 
               
                 
                   
                     B 
                     1 
                   
                   ⁢ 
                   
                     h 
                     * 
                   
                   ⁢ 
                   
                     K 
                     2 
                   
                 
                 
                   ︸ 
                   
                     
                       ICl 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       from 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       bin 
                     
                     - 
                     k 
                   
                 
               
               + 
               
                 W 
                 k 
               
             
           
         
       
       
         
           
             
               Z 
               ⁡ 
               
                 [ 
                 
                   - 
                   k 
                 
                 ] 
               
             
             = 
             
               
                 
                   
                     A 
                     2 
                   
                   ⁢ 
                   
                     hK 
                     1 
                   
                 
                 
                   ︸ 
                   
                     ICl 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     from 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     bin 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     k 
                   
                 
               
               + 
               
                 
                   
                     B 
                     2 
                   
                   ⁢ 
                   
                     h 
                     * 
                   
                   ⁢ 
                   
                     K 
                     2 
                   
                 
                 
                   ︷ 
                   
                     signal 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     term 
                   
                 
               
               + 
               
                 W 
                 
                   - 
                   k 
                 
               
             
           
         
       
     
     It is easy to show that each expression is composed of an expected signal term and an ICI term from the mirror frequency. We shall now show that the ICI terms are much smaller than the signal terms and can thus be neglected. 
     The ICI induced at bin k is due to the fact that the image signal that results from I/Q imbalance is produced at a digital frequency of −k/N−φ′ (where φ′=(φ−½ε)) which is not on the FFT grid. The further away this frequency is from the FFT grid of 1/N the larger the ICI. Since k is restricted to be in the interval {[146,186},[217,249]} the image is produced far away from the desired signal and the ICI noise it produces at frequency k/N is very small. To see this considers the ratio between the signal and ICI terms at bin k. We denote this ratio as the SNR between the desired and ICI terms and it is given by: 
     
       
         
           
             
               SNR 
               k 
             
             = 
             
               
                 10 
                 ⁢ 
                 
                   
                     log 
                     10 
                   
                   ( 
                   
                     
                       
                          
                         
                           hNK 
                           1 
                         
                          
                       
                       2 
                     
                     
                       
                          
                         
                           
                             h 
                             * 
                           
                           ⁢ 
                           
                             ⅇ 
                             
                               - 
                               
                                 j 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     2 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       π 
                                       ⁡ 
                                       
                                         ( 
                                         
                                           N 
                                           - 
                                           1 
                                         
                                         ) 
                                       
                                     
                                     ⁢ 
                                     
                                       ( 
                                       
                                         
                                           k 
                                           N 
                                         
                                         + 
                                         
                                           φ 
                                           ′ 
                                         
                                       
                                       ) 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                           ⁢ 
                           
                             
                               sin 
                               ⁡ 
                               
                                 ( 
                                 
                                   2 
                                   ⁢ 
                                   
                                     πφ 
                                     ′ 
                                   
                                   ⁢ 
                                   N 
                                 
                                 ) 
                               
                             
                             
                               sin 
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     
                                       2 
                                       ⁢ 
                                       π 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       k 
                                     
                                     N 
                                   
                                   + 
                                   
                                     2 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     π 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       φ 
                                       ′ 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                           ⁢ 
                           
                             K 
                             2 
                           
                         
                          
                       
                       2 
                     
                   
                   ) 
                 
               
               = 
               
                 
                   10 
                   ⁢ 
                   
                     
                       log 
                       10 
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           
                             N 
                             2 
                           
                           ⁢ 
                           
                             
                                
                               
                                 K 
                                 1 
                               
                                
                             
                             2 
                           
                         
                         
                           
                              
                             
                               K 
                               2 
                             
                              
                           
                           2 
                         
                       
                       ) 
                     
                   
                 
                 - 
                 
                   10 
                   ⁢ 
                   
                     
                       log 
                       10 
                     
                     ( 
                     
                       
                         [ 
                         
                           
                             sin 
                             ⁡ 
                             
                               ( 
                               
                                 2 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 π 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   φ 
                                   ′ 
                                 
                                 ⁢ 
                                 N 
                               
                               ) 
                             
                           
                           
                             sin 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   
                                     2 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     π 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     k 
                                   
                                   N 
                                 
                                 + 
                                 
                                   2 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   π 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     φ 
                                     ′ 
                                   
                                 
                               
                               ) 
                             
                           
                         
                         ] 
                       
                       2 
                     
                     ) 
                   
                 
               
             
           
         
       
     
     The worst case SNR is found by minimizing the above expression with respect to {g, θ, k, φ′}. It is easy to show that minimizing the above expression is separable and thus minimization is achieved by
         minimizing the first term with respect to g and θ under the constraint that gε[0.5,2] (max 3[dB] amplitude imbalance) and θε[−10°,10°]   maximizing the second term under the constraint that φ′ε[−200e−6*1.5e9/50e6:−200e−6*1.5e9/50e6] and kε[146:186, 217:249]
 
The above minimizations were performed numerically using a Matlab simulation.  FIG. 12  a depicts the first term as a function of g and θ
       

     It is easy to see (analytically as well) that the minimum is at the edges of the argument interval namely for g=0.5,2 and Teta=±10° and thus 
     
       
         
           
             57.4216 
             = 
             
               
                 min 
                 
                   g 
                   , 
                   θ 
                 
               
               ⁢ 
               
                 
                   { 
                   
                     10 
                     ⁢ 
                     
                       
                         log 
                         10 
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             
                               N 
                               2 
                             
                             ⁢ 
                             
                               
                                  
                                 
                                   
                                     K 
                                     1 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       g 
                                       , 
                                       θ 
                                     
                                     ) 
                                   
                                 
                                  
                               
                               2 
                             
                           
                           
                             
                                
                               
                                 
                                   K 
                                   2 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     g 
                                     , 
                                     θ 
                                   
                                   ) 
                                 
                               
                                
                             
                             2 
                           
                         
                         ) 
                       
                     
                   
                   } 
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 st 
                 ⁢ 
                 
                   
                     
                       
                         g 
                         ∈ 
                         
                           [ 
                           
                             0.5 
                             , 
                             2 
                           
                           ] 
                         
                       
                     
                   
                   
                     
                       
                         θ 
                         ∈ 
                         
                           [ 
                           
                             
                               - 
                               
                                 10 
                                 ∘ 
                               
                             
                             , 
                             
                               10 
                               ∘ 
                             
                           
                           ] 
                         
                       
                     
                   
                 
               
             
           
         
       
     
       FIG. 12   b  depicts the second term as a function of φ and k 
     From  FIG. 12   b  it is easy to see that the second term is maximized for k=249, for such a k the second term is depicted in  FIG. 13 . 
     Maximum is achieved for Df=±245 Khz and thus 
             17.04   =       max     g   ,   θ       ⁢       {     10   ⁢       log   10     (       [       sin   ⁡     (     2   ⁢           ⁢   π   ⁢           ⁢   φ   ⁢           ⁢   N     )         sin   ⁡     (         2   ⁢   π   ⁢           ⁢   k     N     +     2   ⁢   π   ⁢           ⁢   φ       )         ]     2     )       }     ⁢           ⁢   st   ⁢           φ   ∈     [         -   Δ     ⁢           ⁢   f     ,     Δ   ⁢           ⁢   f       ]                 k   ∈       [     146   ,   186     ]     ⋃     [     217   ,   249     ]                         
Thus the worst case SNR induced by the ICI term is 40.3816[dB]
 
     
       
         
           
             
               40.3816 
               ⁡ 
               
                 [ 
                 dB 
                 ] 
               
             
             = 
             
               
                 min 
                 ⁡ 
                 
                   ( 
                   
                     SNR 
                     k 
                   
                   ) 
                 
               
               = 
               
                 
                   min 
                   
                     φ 
                     , 
                     k 
                     , 
                     g 
                     , 
                     θ 
                   
                 
                 ⁢ 
                 
                   ( 
                   
                     
                       10 
                       ⁢ 
                       
                         
                           log 
                           10 
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               
                                 N 
                                 2 
                               
                               ⁢ 
                               
                                 
                                    
                                   
                                     
                                       K 
                                       1 
                                     
                                     ⁡ 
                                     
                                       ( 
                                       
                                         g 
                                         , 
                                         θ 
                                       
                                       ) 
                                     
                                   
                                    
                                 
                                 2 
                               
                             
                             
                               
                                  
                                 
                                   
                                     K 
                                     2 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       g 
                                       , 
                                       θ 
                                     
                                     ) 
                                   
                                 
                                  
                               
                               2 
                             
                           
                           ) 
                         
                       
                     
                     - 
                     
                       10 
                       ⁢ 
                       
                         
                           log 
                           10 
                         
                         ( 
                         
                           
                             [ 
                             
                               
                                 sin 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     2 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     π 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     φ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     N 
                                   
                                   ) 
                                 
                               
                               
                                 sin 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       
                                         2 
                                         ⁢ 
                                         π 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         k 
                                       
                                       N 
                                     
                                     + 
                                     
                                       2 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       π 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       φ 
                                     
                                   
                                   ) 
                                 
                               
                             
                             ] 
                           
                           2 
                         
                         ) 
                       
                     
                   
                   ) 
                 
               
             
           
         
       
     
     Thus the ICI term is at the worst case 40[dB] below the signal term and so can be neglected. A similar analysis can be performed for the negative bins. The FFT outputs at bins k and −k+i after neglecting the ICI terms is given by: 
     
       
         
           
             
               Z 
               ⁡ 
               
                 [ 
                 k 
                 ] 
               
             
             = 
             
               
                 hAK 
                 1 
               
               + 
               
                 W 
                 k 
               
             
           
         
       
       
         
           
             
               Z 
               ⁡ 
               
                 [ 
                 
                   
                     - 
                     k 
                   
                   + 
                   i 
                 
                 ] 
               
             
             = 
             
               
                 
                   h 
                   * 
                 
                 ⁢ 
                 
                   B 
                   1 
                 
                 ⁢ 
                 
                   K 
                   2 
                 
               
               + 
               
                 W 
                 
                   
                     - 
                     k 
                   
                   + 
                   i 
                 
               
             
           
         
       
       
         
           
             A 
             = 
             
               
                 ⅇ 
                 
                   j 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   π 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ɛ 
                     ⁡ 
                     
                       ( 
                       
                         N 
                         - 
                         1 
                       
                       ) 
                     
                   
                 
               
               ⁢ 
               
                 
                   sin 
                   ⁡ 
                   
                     ( 
                     
                       π 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       ɛ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       N 
                     
                     ) 
                   
                 
                 
                   sin 
                   ⁡ 
                   
                     ( 
                     
                       π 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       ɛ 
                     
                     ) 
                   
                 
               
             
           
         
       
       
         
           
             
               B 
               i 
             
             = 
             
               
                 ⅇ 
                 
                   
                     - 
                     j 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   2 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     π 
                     ⁡ 
                     
                       ( 
                       
                         φ 
                         - 
                         
                           
                             1 
                             2 
                           
                           ⁢ 
                           ɛ 
                         
                         + 
                         
                           i 
                           
                             2 
                             ⁢ 
                             N 
                           
                         
                       
                       ) 
                     
                   
                   ⁢ 
                   
                     ( 
                     
                       N 
                       - 
                       1 
                     
                     ) 
                   
                 
               
               ⁡ 
               
                 ( 
                 
                   
                     
                       
                         ( 
                         
                           - 
                           1 
                         
                         ) 
                       
                       i 
                     
                     ⁢ 
                     
                       sin 
                       ⁡ 
                       
                         ( 
                         
                           2 
                           ⁢ 
                           π 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             N 
                             ⁡ 
                             
                               [ 
                               
                                 φ 
                                 - 
                                 
                                   
                                     1 
                                     2 
                                   
                                   ⁢ 
                                   ɛ 
                                 
                               
                               ] 
                             
                           
                         
                         ) 
                       
                     
                   
                   
                     sin 
                     ⁡ 
                     
                       ( 
                       
                         2 
                         ⁢ 
                         
                           π 
                           ⁡ 
                           
                             [ 
                             
                               φ 
                               - 
                               
                                 
                                   1 
                                   2 
                                 
                                 ⁢ 
                                 ɛ 
                               
                               + 
                               
                                 i 
                                 
                                   2 
                                   ⁢ 
                                   N 
                                 
                               
                             
                             ] 
                           
                         
                       
                       ) 
                     
                   
                 
                 ) 
               
             
           
         
       
     
     Since we cannot estimate the channel response h we cannot solve a linear LS problem for ge −jθ , instead we first solve a LS problem for the estimation of hK* 2  from the two negative bins −k+i1 and −k+i2 
     
       
         
           
             
               [ 
               
                 
                   
                     
                       
                         
                           Z 
                           ^ 
                         
                         * 
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             - 
                             k 
                           
                           + 
                           
                             i 
                             1 
                           
                         
                         ] 
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           Z 
                           ^ 
                         
                         * 
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             - 
                             k 
                           
                           + 
                           
                             i 
                             2 
                           
                         
                         ] 
                       
                     
                   
                 
               
               ] 
             
             = 
             
               
                 
                   [ 
                   
                     
                       
                         
                           B 
                           1 
                           * 
                         
                       
                     
                     
                       
                         
                           B 
                           2 
                           * 
                         
                       
                     
                   
                   ] 
                 
                 ⁢ 
                 
                   ( 
                   
                     hK 
                     2 
                     * 
                   
                   ) 
                 
               
               + 
               
                 [ 
                 
                   
                     
                       
                         W 
                         
                           k 
                           + 
                           
                             i 
                             1 
                           
                         
                         * 
                       
                     
                   
                   
                     
                       
                         W 
                         
                           k 
                           + 
                           
                             i 
                             2 
                           
                         
                         * 
                       
                     
                   
                 
                 ] 
               
             
           
         
       
       
         
           
             
               
                 ( 
                 
                   hK 
                   2 
                   * 
                 
                 ) 
               
               LS 
             
             = 
             
               
                 
                   
                     B 
                     1 
                   
                   ⁢ 
                   
                     
                       
                         Z 
                         ^ 
                       
                       * 
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           - 
                           k 
                         
                         + 
                         
                           i 
                           1 
                         
                       
                       ] 
                     
                   
                 
                 + 
                 
                   
                     B 
                     2 
                   
                   ⁢ 
                   
                     
                       
                         Z 
                         ^ 
                       
                       * 
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           - 
                           k 
                         
                         + 
                         
                           i 
                           2 
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   
                      
                     
                       B 
                       1 
                     
                      
                   
                   2 
                 
                 + 
                 
                   
                      
                     
                       B 
                       2 
                     
                      
                   
                   2 
                 
               
             
           
         
       
     
     Thus we can estimate 
               K   1       K   2   *           
without knowledge of the channel h by
 
     
       
         
           
             
               C 
               ≡ 
               
                 
                   
                     K 
                     _ 
                   
                   1 
                 
                 
                   K 
                   2 
                   * 
                 
               
             
             = 
             
               
                 
                   ( 
                   
                     hK 
                     1 
                   
                   ) 
                 
                 
                   
                     ( 
                     
                       hK 
                       2 
                       * 
                     
                     ) 
                   
                   LS 
                 
               
               = 
               
                 
                   
                     
                       Z 
                       ⁡ 
                       
                         [ 
                         k 
                         ] 
                       
                     
                     A 
                   
                   
                     
                       
                         
                           B 
                           1 
                         
                         ⁢ 
                         
                           
                             
                               Z 
                               ^ 
                             
                             * 
                           
                           ⁡ 
                           
                             [ 
                             
                               
                                 - 
                                 k 
                               
                               + 
                               
                                 i 
                                 1 
                               
                             
                             ] 
                           
                         
                       
                       + 
                       
                         
                           B 
                           2 
                         
                         ⁢ 
                         
                           
                             
                               Z 
                               ^ 
                             
                             * 
                           
                           ⁡ 
                           
                             [ 
                             
                               
                                 - 
                                 k 
                               
                               + 
                               
                                 i 
                                 2 
                               
                             
                             ] 
                           
                         
                       
                     
                     
                       
                         
                            
                           
                             B 
                             1 
                           
                            
                         
                         2 
                       
                       + 
                       
                         
                            
                           
                             B 
                             2 
                           
                            
                         
                         2 
                       
                     
                   
                 
                 = 
                 
                   
                     
                       ( 
                       
                         
                           
                              
                             
                               B 
                               1 
                             
                              
                           
                           2 
                         
                         + 
                         
                           
                              
                             
                               B 
                               2 
                             
                              
                           
                           2 
                         
                       
                       ) 
                     
                     · 
                     
                       Z 
                       ⁡ 
                       
                         [ 
                         k 
                         ] 
                       
                     
                   
                   
                     A 
                     · 
                     
                       ( 
                       
                         
                           
                             B 
                             1 
                           
                           ⁢ 
                           
                             
                               
                                 Z 
                                 ^ 
                               
                               * 
                             
                             ⁡ 
                             
                               [ 
                               
                                 
                                   - 
                                   k 
                                 
                                 + 
                                 
                                   i 
                                   1 
                                 
                               
                               ] 
                             
                           
                         
                         + 
                         
                           
                             B 
                             2 
                           
                           ⁢ 
                           
                             
                               
                                 Z 
                                 ^ 
                               
                               * 
                             
                             ⁡ 
                             
                               [ 
                               
                                 
                                   - 
                                   k 
                                 
                                 + 
                                 
                                   i 
                                   2 
                                 
                               
                               ] 
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
             
           
         
       
     
     Since probe2 is composed of two tones one at k1 and the other at k2 we can average the result from these two tones and thus 
               C   _     ≡         1   2     ⁢         (              B   1          2     +            B   2          2       )     ·     Z   ⁡     [     k   1     ]           A   ·     (         B   1     ⁢         Z   ^     *     ⁡     [       -     k   1       +     i   1       ]         +       B   2     ⁢         Z   ^     *     ⁡     [       -     k   1       +     i   2       ]           )           +       1   2     ⁢         (              B   1          2     +            B   2          2       )     ·     Z   ⁡     [     k   2     ]           A   ·     (         B   1     ⁢         Z   ^     *     ⁡     [       -     k   2       +     i   1       ]         +       B   2     ⁢         Z   ^     *     ⁡     [       -     k   2       +     i   2       ]           )                   
It is easy to see that
 
     
       
         
           
             
               g 
               ⁢ 
               
                   
               
               ⁢ 
               
                 ⅇ 
                 
                   
                     - 
                     j 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   θ 
                 
               
             
             = 
             
               
                 
                   
                     K 
                     1 
                   
                   / 
                   
                     K 
                     2 
                     * 
                   
                 
                 - 
                 1 
               
               
                 
                   
                     K 
                     1 
                   
                   / 
                   
                     K 
                     2 
                     * 
                   
                 
                 + 
                 1 
               
             
           
         
       
     
     And thus its estimate is given by 
     
       
         
           
             
               
                 g 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   ⅇ 
                   
                     - 
                     jθ 
                   
                 
               
               _ 
             
             = 
             
               
                 
                   C 
                   _ 
                 
                 - 
                 1 
               
               
                 
                   C 
                   _ 
                 
                 + 
                 1 
               
             
           
         
       
     
     The I/Q compensation is then easily computed by 
               ξ   ^     =     1     real   ⁢     {     g   ⁢           ⁢     ⅇ       -   j     ⁢           ⁢   θ         }                       ρ   ^     =     -       imag   ⁢     {       g   ⁢           ⁢     ⅇ       -   j     ⁢           ⁢   θ         _     }         real   ⁢     {     g   ⁢           ⁢     ⅇ       -   j     ⁢           ⁢   θ         }                 
FFT Processing of Probe2 (Multi OFDM Symbol)
 
     When looking at multiple OFDM symbols we need to take into account the phase error induced by the accumulation of the residual frequency error ε. It is easy to show that the phase of the m′th OFDM symbol relative to the first one is given by 
     
       
         
           
             
               Z 
               ⁡ 
               
                 [ 
                 
                   k 
                   . 
                   m 
                 
                 ] 
               
             
             = 
             
               { 
               
                 
                   
                     
                       k 
                       ≥ 
                       0 
                     
                   
                   
                     
                       
                         ⅇ 
                         
                           j 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           2 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             πɛ 
                             ⁡ 
                             
                               ( 
                               
                                 N 
                                 + 
                                 
                                   N 
                                   CP 
                                 
                               
                               ) 
                             
                           
                           ⁢ 
                           m 
                         
                       
                       ⁢ 
                       
                         Z 
                         ⁡ 
                         
                           [ 
                           
                             k 
                             , 
                             0 
                           
                           ] 
                         
                       
                     
                   
                 
                 
                   
                     
                       k 
                       &lt; 
                       0 
                     
                   
                   
                     
                       
                         ⅇ 
                         
                           j 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           2 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             π 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   2 
                                   ⁢ 
                                   φ 
                                 
                                 - 
                                 ɛ 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             ( 
                             
                               N 
                               + 
                               
                                 N 
                                 CP 
                               
                             
                             ) 
                           
                           ⁢ 
                           m 
                         
                       
                       ⁢ 
                       
                         Z 
                         ⁡ 
                         
                           [ 
                           
                             k 
                             , 
                             0 
                           
                           ] 
                         
                       
                     
                   
                 
               
             
           
         
       
     
     Note that the above does not take into account the sampling frequency error its effect is assumed to be small and was neglected throughout the analysis. 
     The phase accumulated from the starting time of carrier frequency compensation to the start of the first FFT window should be accounted for. Since our algorithm computed the ratio between Zk and conj(Z−k) any constant phase term will not cancel out but on the contrary double itself. 
     
       
         
         
             
             
         
       
     
     Thus the FFT output at bins +k, −k+i for the m′th OFDM symbol is given by 
               Z   ⁡     [   k   ]       =         D     ︸   #1       ⁢           ⁢       ⅇ       -   j     ⁢           ⁢   2   ⁢           ⁢   π   ⁢           ⁢   Δ   ⁢           ⁢     n   ⁡     (     φ   -   ɛ     )             ︷   #2       ⁢       ⅇ     j   ⁢           ⁢   2   ⁢           ⁢   π   ⁢           ⁢     ɛ   ⁡     (     N   +     N   CP       )       ⁢   in         ︸   #3       ⁢   h   ⁢           ⁢     AK   i       +     W   k                       Z   ⁡     [       -   k     +   i     ]       =           D   *       ︸   #1       ⁢       ⅇ       -   j     ⁢           ⁢   2   ⁢           ⁢   π   ⁢           ⁢   Δ   ⁢           ⁢     n   ⁡     (     φ   -   ɛ     )             ︷   #2       ⁢       ⅇ     j   ⁢           ⁢   2   ⁢           ⁢     π   ⁡     (       2   ⁢   φ     -   ɛ     )       ⁢     (     N   +     N   CP       )     ⁢   m         ︸   #3       ⁢     h   *     ⁢     B   i     ⁢     K   2       +     W       -   k     +   1           ⁢                         #1   -     due   ⁢           ⁢   to   ⁢           ⁢   Δ   ⁢           ⁢   n   ⁢           ⁢   samples   ⁢           ⁢   between   ⁢           ⁢   start       ⁢                       of   ⁢           ⁢   compensation   ⁢           ⁢   to   ⁢           ⁢   FFT   ⁢           ⁢   will   ⁢           ⁢   not   ⁢           ⁢   cancell   ⁢           ⁢   out                 #2   -     due   ⁢           ⁢   to   ⁢           ⁢   Δ   ⁢           ⁢   n   ⁢           ⁢   samples   ⁢           ⁢   between   ⁢           ⁢   start   ⁢           ⁢   of       ⁢                       compensation   ⁢           ⁢   to   ⁢           ⁢   FFT   ⁢           ⁢   start   ⁢           ⁢   will   ⁢           ⁢   not   ⁢           ⁢   cancell   ⁢           ⁢   out               #3   =     Phase   ⁢           ⁢   due   ⁢           ⁢   to   ⁢           ⁢   residual   ⁢           ⁢   carrier   ⁢           ⁢   freq   ⁢           ⁢   error           
Residual Carrier Frequency Estimation
 
     To use the information from all L OFDM symbols we need to compensate for the residual frequency offset ε and then compute the average of the compensated signals form each bin to reduce the AWGN variance. Since ε can be large enough such that phase wrapping can occur several times during the L OFDM symbols we propose the following estimator which is immune to phase wrapping (as long as no more than one wrap occurs between two consecutive samples which is the case here). 
     Residual Frequency Estimation 
     The residual frequency error estimate may be computed by 
     
       
         
           
             
               ɛ 
               ^ 
             
             = 
             
               
                 angle 
                 ⁡ 
                 
                   ( 
                   
                     
                       ∑ 
                       
                         m 
                         = 
                         0 
                       
                       
                         L 
                         - 
                         2 
                       
                     
                     ⁢ 
                     
                       
                         Z 
                         ⁡ 
                         
                           [ 
                           
                             
                               k 
                               i 
                             
                             , 
                             m 
                           
                           ] 
                         
                       
                       ⁢ 
                       
                         
                           Z 
                           * 
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               k 
                               i 
                             
                             , 
                             
                               m 
                               + 
                               1 
                             
                           
                           ] 
                         
                       
                     
                   
                   ) 
                 
               
               
                 2 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   π 
                   ⁡ 
                   
                     ( 
                     
                       N 
                       + 
                       
                         N 
                         CP 
                       
                     
                     ) 
                   
                 
               
             
           
         
       
       
         
           
             
               Where 
               ⁢ 
               
                   
               
               ⁢ 
               
                 k 
                 i 
               
             
             = 
             
               { 
               
                 
                   
                     
                       k 
                       1 
                     
                   
                   
                     
                       
                         SNR 
                         
                           k 
                           1 
                         
                       
                       &gt; 
                       
                         SNR 
                         
                           k 
                           2 
                         
                       
                     
                   
                 
                 
                   
                     
                       k 
                       2 
                     
                   
                   
                     
                       
                         SNR 
                         
                           k 
                           2 
                         
                       
                       &gt; 
                       
                         SNR 
                         
                           k 
                           1 
                         
                       
                     
                   
                 
               
             
           
         
       
     
     The residual frequency error compensation and averaging is given by Residual Frequency Compensation and time averaging 
     
       
         
           
             
               
                 Z 
                 _ 
               
               
                 k 
                 1 
               
             
             = 
             
               
                 ∑ 
                 
                   m 
                   = 
                   0 
                 
                 
                   L 
                   - 
                   1 
                 
               
               ⁢ 
               
                 
                   Z 
                   ⁡ 
                   
                     [ 
                     
                       
                         k 
                         1 
                       
                       , 
                       m 
                     
                     ] 
                   
                 
                 ⁢ 
                 
                   ⅇ 
                   
                     
                       - 
                       j 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     2 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     π 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         ɛ 
                         ^ 
                       
                       ⁡ 
                       
                         ( 
                         
                           N 
                           + 
                           
                             N 
                             CP 
                           
                         
                         ) 
                       
                     
                     ⁢ 
                     m 
                   
                 
               
             
           
         
       
       
         
           
             
               
                 Z 
                 _ 
               
               
                 
                   - 
                   
                     k 
                     1 
                   
                 
                 + 
                 
                   i 
                   1 
                 
               
             
             = 
             
               
                 ∑ 
                 
                   m 
                   = 
                   0 
                 
                 
                   L 
                   - 
                   1 
                 
               
               ⁢ 
               
                 
                   Z 
                   ⁡ 
                   
                     [ 
                     
                       
                         
                           - 
                           
                             k 
                             1 
                           
                         
                         + 
                         
                           i 
                           1 
                         
                       
                       , 
                       m 
                     
                     ] 
                   
                 
                 ⁢ 
                 
                   ⅇ 
                   
                     
                       - 
                       j 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     2 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       π 
                       ⁡ 
                       
                         ( 
                         
                           
                             2 
                             ⁢ 
                             
                               φ 
                               ^ 
                             
                           
                           + 
                           
                             ɛ 
                             ^ 
                           
                         
                         ) 
                       
                     
                     ⁢ 
                     
                       ( 
                       
                         N 
                         + 
                         
                           N 
                           CP 
                         
                       
                       ) 
                     
                     ⁢ 
                     m 
                   
                 
               
             
           
         
       
       
         
           
             
               
                 Z 
                 _ 
               
               
                 
                   - 
                   
                     k 
                     1 
                   
                 
                 + 
                 
                   i 
                   2 
                 
               
             
             = 
             
               
                 ∑ 
                 
                   m 
                   = 
                   0 
                 
                 
                   L 
                   - 
                   1 
                 
               
               ⁢ 
               
                 
                   Z 
                   ⁡ 
                   
                     [ 
                     
                       
                         
                           - 
                           
                             k 
                             1 
                           
                         
                         + 
                         
                           i 
                           2 
                         
                       
                       , 
                       m 
                     
                     ] 
                   
                 
                 ⁢ 
                 
                   ⅇ 
                   
                     
                       - 
                       j 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     2 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       π 
                       ⁡ 
                       
                         ( 
                         
                           
                             2 
                             ⁢ 
                             
                               φ 
                               ^ 
                             
                           
                           + 
                           
                             ɛ 
                             ^ 
                           
                         
                         ) 
                       
                     
                     ⁢ 
                     
                       ( 
                       
                         N 
                         + 
                         
                           N 
                           CP 
                         
                       
                       ) 
                     
                     ⁢ 
                     m 
                   
                 
               
             
           
         
       
       
         
           
             
               
                 Z 
                 _ 
               
               
                 k 
                 2 
               
             
             = 
             
               
                 ∑ 
                 
                   m 
                   = 
                   0 
                 
                 
                   L 
                   - 
                   1 
                 
               
               ⁢ 
               
                 
                   Z 
                   ⁡ 
                   
                     [ 
                     
                       
                         k 
                         2 
                       
                       , 
                       m 
                     
                     ] 
                   
                 
                 ⁢ 
                 
                   ⅇ 
                   
                     
                       - 
                       j 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     2 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     π 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         ɛ 
                         ^ 
                       
                       ⁡ 
                       
                         ( 
                         
                           N 
                           + 
                           
                             N 
                             CP 
                           
                         
                         ) 
                       
                     
                     ⁢ 
                     m 
                   
                 
               
             
           
         
       
       
         
           
             
               
                 Z 
                 _ 
               
               
                 
                   - 
                   
                     k 
                     2 
                   
                 
                 + 
                 
                   i 
                   1 
                 
               
             
             = 
             
               
                 ∑ 
                 
                   m 
                   = 
                   0 
                 
                 
                   L 
                   - 
                   1 
                 
               
               ⁢ 
               
                 
                   Z 
                   ⁡ 
                   
                     [ 
                     
                       
                         
                           - 
                           
                             k 
                             2 
                           
                         
                         + 
                         
                           i 
                           1 
                         
                       
                       , 
                       m 
                     
                     ] 
                   
                 
                 ⁢ 
                 
                   ⅇ 
                   
                     
                       - 
                       j 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     2 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       π 
                       ⁡ 
                       
                         ( 
                         
                           
                             2 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               φ 
                               ^ 
                             
                           
                           + 
                           
                             ɛ 
                             ^ 
                           
                         
                         ) 
                       
                     
                     ⁢ 
                     
                       ( 
                       
                         N 
                         + 
                         
                           N 
                           CP 
                         
                       
                       ) 
                     
                     ⁢ 
                     m 
                   
                 
               
             
           
         
       
       
         
           
             
               
                 Z 
                 _ 
               
               
                 
                   - 
                   
                     k 
                     2 
                   
                 
                 + 
                 
                   i 
                   2 
                 
               
             
             = 
             
               
                 ∑ 
                 
                   m 
                   = 
                   0 
                 
                 
                   L 
                   - 
                   1 
                 
               
               ⁢ 
               
                 
                   Z 
                   ⁡ 
                   
                     [ 
                     
                       
                         
                           - 
                           
                             k 
                             2 
                           
                         
                         + 
                         
                           i 
                           2 
                         
                       
                       , 
                       m 
                     
                     ] 
                   
                 
                 ⁢ 
                 
                   ⅇ 
                   
                     
                       - 
                       j 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     2 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       π 
                       ⁡ 
                       
                         ( 
                         
                           
                             2 
                             ⁢ 
                             
                               φ 
                               ^ 
                             
                           
                           + 
                           
                             ɛ 
                             ^ 
                           
                         
                         ) 
                       
                     
                     ⁢ 
                     
                       ( 
                       
                         N 
                         + 
                         
                           N 
                           CP 
                         
                       
                       ) 
                     
                     ⁢ 
                     m 
                   
                 
               
             
           
         
       
     
     The phasor ge −jθ  can then be estimated using the same estimator derived above, namely 
     
       
         
           
             
               C 
               i 
             
             = 
             
               
                 ⅇ 
                 
                   j 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   4 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     π 
                     ⁡ 
                     
                       ( 
                       
                         φ 
                         - 
                         ɛ 
                       
                       ) 
                     
                   
                   ⁢ 
                   
                     ( 
                     
                       Δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       n 
                     
                     ) 
                   
                 
               
               ⁢ 
               
                 
                   
                     ( 
                     
                       
                         
                            
                           
                             B 
                             1 
                           
                            
                         
                         2 
                       
                       + 
                       
                         
                            
                           
                             B 
                             2 
                           
                            
                         
                         2 
                       
                     
                     ) 
                   
                   · 
                   
                     
                       Z 
                       _ 
                     
                     
                       k 
                       1 
                     
                   
                 
                 
                   
                     
                       
                         B 
                         1 
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             Z 
                             _ 
                           
                           
                             
                               - 
                               
                                 k 
                                 1 
                               
                             
                             + 
                             
                               i 
                               1 
                             
                           
                         
                         ) 
                       
                     
                     * 
                   
                   + 
                   
                     
                       
                         B 
                         2 
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             Z 
                             _ 
                           
                           
                             
                               - 
                               
                                 k 
                                 1 
                               
                             
                             + 
                             
                               i 
                               2 
                             
                           
                         
                         ) 
                       
                     
                     * 
                   
                 
               
             
           
         
       
     
     Where the phase term e j4π(φ-ε)(Δn)  compensates for the initial phase error accumulated from the start time of frequency compensation till the start of the first FFT window. Simplification of the Coefficients Bi and A 
     For pragmatic implementation we need to simplify the expressions of Bi and A, simplification can be obtained by introducing some approximations. Let&#39;s look at 
     
       
         
           
             A 
             = 
             
               
                 ⅇ 
                 
                   j 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   π 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ɛ 
                     ⁡ 
                     
                       ( 
                       
                         N 
                         - 
                         1 
                       
                       ) 
                     
                   
                 
               
               ⁢ 
               
                 
                   sin 
                   ⁡ 
                   
                     ( 
                     
                       π 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       ɛ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       N 
                     
                     ) 
                   
                 
                 
                   sin 
                   ⁡ 
                   
                     ( 
                     
                       π 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       ɛ 
                     
                     ) 
                   
                 
               
             
           
         
       
     
     The residual frequency error is typically smaller then 10 khz (7 ppm) for such an error 
               ⅇ     j   ⁢           ⁢   π   ⁢           ⁢     ɛ   ⁡     (     N   -   1     )           =       0.9872   +     0.1595   ⁢   i       ≅   1                     sin   ⁡     (     πɛ   ⁢           ⁢   N     )         sin   ⁡     (   πɛ   )         =       254.8975   ≅   256     =   N           
Thus we make the following approximation
 
             A   =         ⅇ     j   ⁢           ⁢   π   ⁢           ⁢     ɛ   ⁡     (     N   -   1     )           ⁢       sin   ⁡     (     π   ⁢           ⁢   ɛ   ⁢           ⁢   N     )         sin   ⁡     (   πɛ   )           ≅   N           
As for Bi
 
     
       
         
           
             
               
                 
                   
                     B 
                     i 
                   
                   = 
                     
                   ⁢ 
                   
                     
                       ⅇ 
                       
                         
                           - 
                           j 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         2 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           π 
                           ⁡ 
                           
                             ( 
                             
                               φ 
                               - 
                               
                                 
                                   1 
                                   2 
                                 
                                 ⁢ 
                                 ɛ 
                               
                               + 
                               
                                 i 
                                 
                                   2 
                                   ⁢ 
                                   N 
                                 
                               
                             
                             ) 
                           
                         
                         ⁢ 
                         
                           ( 
                           
                             N 
                             - 
                             1 
                           
                           ) 
                         
                       
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           
                             
                               ( 
                               
                                 - 
                                 1 
                               
                               ) 
                             
                             i 
                           
                           ⁢ 
                           
                             sin 
                             ⁡ 
                             
                               ( 
                               
                                 2 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 π 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   N 
                                   ⁡ 
                                   
                                     [ 
                                     
                                       φ 
                                       - 
                                       
                                         
                                           1 
                                           2 
                                         
                                         ⁢ 
                                         ɛ 
                                       
                                     
                                     ] 
                                   
                                 
                               
                               ) 
                             
                           
                         
                         
                           sin 
                           ⁡ 
                           
                             ( 
                             
                               2 
                               ⁢ 
                               
                                 π 
                                 ⁡ 
                                 
                                   [ 
                                   
                                     φ 
                                     - 
                                     
                                       
                                         1 
                                         2 
                                       
                                       ⁢ 
                                       ɛ 
                                     
                                     + 
                                     
                                       i 
                                       
                                         2 
                                         ⁢ 
                                         N 
                                       
                                     
                                   
                                   ] 
                                 
                               
                             
                             ) 
                           
                         
                       
                       ) 
                     
                   
                 
               
             
             
               
                 
                   ≅ 
                     
                   ⁢ 
                   
                     
                       ⅇ 
                       
                         
                           - 
                           j 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         2 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           π 
                           ⁡ 
                           
                             ( 
                             
                               φ 
                               - 
                               
                                 
                                   1 
                                   2 
                                 
                                 ⁢ 
                                 ɛ 
                               
                             
                             ) 
                           
                         
                         ⁢ 
                         
                           ( 
                           N 
                           ) 
                         
                       
                     
                     ⁢ 
                     
                       
                         ⅇ 
                         
                           
                             - 
                             j 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           π 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           i 
                         
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             
                               
                                 ( 
                                 
                                   - 
                                   1 
                                 
                                 ) 
                               
                               i 
                             
                             ⁢ 
                             
                               sin 
                               ⁡ 
                               
                                 ( 
                                 
                                   2 
                                   ⁢ 
                                   π 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     N 
                                     ⁡ 
                                     
                                       [ 
                                       
                                         φ 
                                         - 
                                         
                                           
                                             1 
                                             2 
                                           
                                           ⁢ 
                                           ɛ 
                                         
                                       
                                       ] 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                           
                             sin 
                             ⁡ 
                             
                               ( 
                               
                                 2 
                                 ⁢ 
                                 
                                   π 
                                   ⁡ 
                                   
                                     [ 
                                     
                                       φ 
                                       - 
                                       
                                         
                                           1 
                                           2 
                                         
                                         ⁢ 
                                         ɛ 
                                       
                                       + 
                                       
                                         i 
                                         
                                           2 
                                           ⁢ 
                                           N 
                                         
                                       
                                     
                                     ] 
                                   
                                 
                               
                               ) 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
             
             
               
                 
                   ≅ 
                     
                   ⁢ 
                   
                     
                       
                         
                           ⅇ 
                           
                             
                               - 
                               j2π 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               N 
                               ⁡ 
                               
                                 ( 
                                 
                                   φ 
                                   - 
                                   ɛ 
                                 
                                 ) 
                               
                             
                           
                         
                         ⁡ 
                         
                           ( 
                           
                             - 
                             1 
                           
                           ) 
                         
                       
                       
                         2 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         i 
                       
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           sin 
                           ⁡ 
                           
                             ( 
                             
                               2 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               π 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 N 
                                 ⁡ 
                                 
                                   [ 
                                   
                                     φ 
                                     - 
                                     ɛ 
                                   
                                   ] 
                                 
                               
                             
                             ) 
                           
                         
                         
                           sin 
                           ⁡ 
                           
                             ( 
                             
                               2 
                               ⁢ 
                               
                                 π 
                                 ⁡ 
                                 
                                   [ 
                                   
                                     φ 
                                     - 
                                     ɛ 
                                     + 
                                     
                                       i 
                                       
                                         2 
                                         ⁢ 
                                         N 
                                       
                                     
                                   
                                   ] 
                                 
                               
                             
                             ) 
                           
                         
                       
                       ) 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       ⅇ 
                       
                         
                           - 
                           j 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         2 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         π 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           N 
                           ⁡ 
                           
                             ( 
                             
                               φ 
                               - 
                               ɛ 
                             
                             ) 
                           
                         
                       
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           sin 
                           ⁡ 
                           
                             ( 
                             
                               2 
                               ⁢ 
                               π 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 N 
                                 ⁡ 
                                 
                                   [ 
                                   
                                     φ 
                                     - 
                                     ɛ 
                                   
                                   ] 
                                 
                               
                             
                             ) 
                           
                         
                         
                           sin 
                           ⁡ 
                           
                             ( 
                             
                               2 
                               ⁢ 
                               
                                 π 
                                 ⁡ 
                                 
                                   [ 
                                   
                                     φ 
                                     - 
                                     ɛ 
                                     + 
                                     
                                       i 
                                       
                                         2 
                                         ⁢ 
                                         N 
                                       
                                     
                                   
                                   ] 
                                 
                               
                             
                             ) 
                           
                         
                       
                       ) 
                     
                   
                 
               
             
           
         
       
       
         
           
             
               
                 Where 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 φ 
               
               - 
               ɛ 
             
             = 
             
               CFO 
               
                 2 
                 ⁢ 
                 π 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     B 
                     i 
                   
                   ≅ 
                     
                   ⁢ 
                   
                     
                       ⅇ 
                       
                         
                           - 
                           j2π 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           N 
                           ⁡ 
                           
                             ( 
                             
                               φ 
                               - 
                               ɛ 
                             
                             ) 
                           
                         
                       
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           sin 
                           ⁡ 
                           
                             ( 
                             
                               2 
                               ⁢ 
                               π 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 N 
                                 ⁡ 
                                 
                                   [ 
                                   
                                     φ 
                                     - 
                                     ɛ 
                                   
                                   ] 
                                 
                               
                             
                             ) 
                           
                         
                         
                           sin 
                           ⁡ 
                           
                             ( 
                             
                               2 
                               ⁢ 
                               
                                 π 
                                 ⁡ 
                                 
                                   [ 
                                   
                                     φ 
                                     - 
                                     ɛ 
                                     + 
                                     
                                       i 
                                       
                                         2 
                                         ⁢ 
                                         N 
                                       
                                     
                                   
                                   ] 
                                 
                               
                             
                             ) 
                           
                         
                       
                       ) 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       
                         sin 
                         ⁡ 
                         
                           ( 
                           
                             2 
                             · 
                             N 
                             · 
                             CFO 
                           
                           ) 
                         
                       
                       + 
                       
                         j 
                         ⁡ 
                         
                           [ 
                           
                             
                               cos 
                               ⁡ 
                               
                                 ( 
                                 
                                   2 
                                   · 
                                   N 
                                   · 
                                   CFO 
                                 
                                 ) 
                               
                             
                             - 
                             1 
                           
                           ] 
                         
                       
                     
                     
                       2 
                       ⁢ 
                       
                         sin 
                         ⁡ 
                         
                           ( 
                           
                             2 
                             ⁢ 
                             
                               π 
                               ⁡ 
                               
                                 [ 
                                 
                                   φ 
                                   - 
                                   ɛ 
                                   + 
                                   
                                     i 
                                     
                                       2 
                                       ⁢ 
                                       N 
                                     
                                   
                                 
                                 ] 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
             
           
         
       
     
     Since the frequency shift along with the residual frequency error is smaller than (200+7)ppm and −3≦i≦3 follows that the argument of the sin( ) in the denominator is small 
     
       
         
           
             
               
                 2 
                 ⁢ 
                 
                   π 
                   ⁡ 
                   
                     [ 
                     
                       φ 
                       - 
                       ɛ 
                       + 
                       
                         i 
                         
                           2 
                           ⁢ 
                           N 
                         
                       
                     
                     ] 
                   
                 
               
               &lt; 
               
                 
                   207 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   ⅇ 
                 
                 - 
                 
                   6 
                   * 
                   1.5 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅇ9 
                     / 
                     50 
                   
                   ⁢ 
                   ⅇ6 
                 
                 + 
                 
                   3 
                   / 
                   512 
                 
               
             
             = 
             0.0121 
           
         
       
     
     For such a small angle a simple linear approximation has very little error 
     
       
         
           
             
               sin 
               ⁡ 
               
                 ( 
                 0.0121 
                 ) 
               
             
             ≅ 
             
               0.0121 
               ⁢ 
               
                 
 
               
               ⇓ 
               
                 
 
               
               ⁢ 
               
                 sin 
                 ⁡ 
                 
                   ( 
                   
                     2 
                     ⁢ 
                     
                       π 
                       ⁡ 
                       
                         [ 
                         
                           φ 
                           - 
                           ɛ 
                           + 
                           
                             i 
                             
                               2 
                               ⁢ 
                               N 
                             
                           
                         
                         ] 
                       
                     
                   
                   ) 
                 
               
             
             ≅ 
             
               2 
               ⁢ 
               
                 π 
                 ⁡ 
                 
                   [ 
                   
                     φ 
                     - 
                     ɛ 
                     + 
                     
                       i 
                       
                         2 
                         ⁢ 
                         N 
                       
                     
                   
                   ] 
                 
               
             
           
         
       
     
     Thus follows that 
               B   i     ≅         sin   ⁡     (     2   ·   N   ·   CFO     )       +     j   ⁡     [       cos   ⁡     (     2   ·   N   ·   CFO     )       -   1     ]           2   ⁢     sin   ⁡     (     2   ⁢     π   ⁡     [     φ   -   ɛ   +     i     2   ⁢   N         ]         )           ≅       1   2     ⁢         sin   ⁢     (     2   ·   N   ·   CFO     )       +     j   ⁡     [       cos   ⁡     (     2   ·   N   ·   CFO     )       -   1     ]           CFO   +       i   ⁢           ⁢   π     N                 
And so the simplified coefficients are given by
 
 A=N  
 
     
       
         
           
             
               B 
               1 
             
             = 
             
               
                 
                   1 
                   2 
                 
                 ⁢ 
                 
                   ( 
                   
                     
                       sin 
                       ⁡ 
                       
                         ( 
                         
                           2 
                           · 
                           CFO 
                           · 
                           N 
                         
                         ) 
                       
                     
                     
                       CFO 
                       + 
                       
                         
                           π 
                           · 
                           
                             i 
                             1 
                           
                         
                         N 
                       
                     
                   
                   ) 
                 
               
               + 
               
                 
                   j 
                   2 
                 
                 ⁢ 
                 
                   ( 
                   
                     
                       
                         cos 
                         ⁡ 
                         
                           ( 
                           
                             2 
                             · 
                             CFO 
                             · 
                             N 
                           
                           ) 
                         
                       
                       - 
                       1 
                     
                     
                       CFO 
                       + 
                       
                         
                           π 
                           · 
                           
                             i 
                             1 
                           
                         
                         N 
                       
                     
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               B 
               2 
             
             = 
             
               
                 
                   1 
                   2 
                 
                 ⁢ 
                 
                   ( 
                   
                     
                       sin 
                       ⁡ 
                       
                         ( 
                         
                           2 
                           · 
                           CFO 
                           · 
                           N 
                         
                         ) 
                       
                     
                     
                       CFO 
                       + 
                       
                         
                           π 
                           · 
                           
                             i 
                             2 
                           
                         
                         N 
                       
                     
                   
                   ) 
                 
               
               + 
               
                 
                   j 
                   2 
                 
                 ⁢ 
                 
                   ( 
                   
                     
                       
                         cos 
                         ⁡ 
                         
                           ( 
                           
                             2 
                             · 
                             CFO 
                             · 
                             N 
                           
                           ) 
                         
                       
                       - 
                       1 
                     
                     
                       CFO 
                       + 
                       
                         
                           π 
                           · 
                           
                             i 
                             2 
                           
                         
                         N 
                       
                     
                   
                   ) 
                 
               
             
           
         
       
     
     APPENDIX B 
     Exemplary Measurements 
     Ideal Channel No AWGN with 200 ppm carrier &amp; sampling frequency offset The following plots summarize simulation results for a 3[dB] amplitude imbalance, 10° phase imbalance, 200 ppm frequency offset, ideal channel and no AWGN. Before RX I/Q compensation routine is invoked the receiver SNR is around 10.1 [dB] as can be seen in the following  FIG. 14 . 
       FIG. 14  shows a Slicer input Ideal Channel, no AWGN before cancellation 
     After processing of the first probe2, the SNR is around 33[dB]. In  FIG. 16 , one can compare the image signal magnitude before and after the first iteration. 
     After processing of the second Probe II, the SNR is around 39.6[dB]. In  FIG. 17 , one can compare the image signal magnitude before and after the second iteration. The image signal is no longer visible after the second iteration. 
     After processing of the third Probe II, the SNR is around 40.6[dB], as shown in  FIG. 18 . 
       FIG. 19  depicts the slicer SNR after processing of the fourth Probe II transmission. SNR is shown to be around 40.9[dB]. 
       FIG. 20  depicts the slicing SNR without I/Q imbalance, the SNR is around 41.3 [dB]. Thus comparing this SNR to that of the SNR obtained after four probe II transmission we can conclude that the residual I/Q imbalance degrades performance by around 0.4[dB] relative to a noise floor of 41.3 [dB]. 
     The I/Q imbalance parameter estimation after each one of the four iterations is summarized in the following table 
                                                     Iteration Number   Gain   Theta [°]   Slicer SNR [dB]                                                            True Value   0.70795   10   41.3           #0   1   0   10.1           #1   0.71345   11.4402   33           #2   0.71103   10.3227   39.6           #3   0.70989   10.1742   40.6           #4   0.70944   10.1358   40.9                        
Channel MoCA10408, SNR AWGN 15[dB]
 
     The following plots summarize simulation results for a 3[dB] amplitude imbalance, 10° phase imbalance, 200 ppm frequency offset, MoCA10408 channel and 15[dB] AWGN SNR. 
     Before RX I/Q compensation routine is invoked the receiver SNR is around 5.1 [dB] as can be seen in  FIG. 21 . 
     After processing of the first Probe II, the SNR is around 9.5[dB]. In  FIG. 22 , one can compare the image signal magnitude before and after the first iteration. 
       FIG. 23  shows slicer input MoCA10408 channel 15[dB] AWGN SNR after first iteration. 
     After processing of the second Probe II, the SNR is around 11 [dB], as shown in  FIG. 24 . 
       FIG. 25  shows slicer input MoCA10408 channel 15[dB] AWGN SNR after second iteration. 
     After processing of the third Probe II, the SNR is around 11.6[dB], as shown in  FIG. 26 . 
       FIG. 27  shows slicer input MoCA10408 channel 15[dB] AWGN SNR after second iteration. 
     The SNR when no I/Q imbalance is present at the receiver is around 11.3[dB], as shown in  FIG. 28 . Thus, the residual I/Q imbalance is well below the noise floor of our demodulator and the estimation and compensation algorithm is robust even under harsh channel conditions. 
     Decoupling of TX and RX I/Q Imbalance 
     The intentional frequency shift specified by MoCA results in the decoupling of the TX and RX imbalance parameters, to show that our algorithm can estimate the RX parameters in the presence of X imbalance we show simulation results for the following scenario
         TX amplitude imbalance 1[dB]   TX phase imbalance 2°   RX amplitude imbalance 3 [dB]   RX phase imbalance 10°   Frequency Offset 20 ppm   Channel=Ideal, no AWGN
 
Before transmission of Probe2 the SNR was around 11.3 [dB], as shown in  FIG. 29 .
       

     After three probe2 transmissions the SNR was around 21.2[dB], as shown in  FIG. 30 . 
       FIG. 31  shows a slicer input Ideal Channel no AWGN, under RX and TX I/Q imbalance after third iteration. 
     The estimated RX I/Q imbalance parameters after 3 iterations were 
     
       
         
           
               
               
               
             
               
                   
               
               
                 Parameter 
                 True 
                 Estimated 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
            
               
                 G 
                 0.7071 
                 0.70785 
               
               
                 Θ 
                 10.0 
                 10.2014 
               
               
                   
               
            
           
         
       
     
     Thus parameters were correctly estimated, for comparison the SNR in a scenario where only TX imbalance is present is around 21.2[dB]. 
       FIG. 15  shows a: Slicer input Ideal Channel no AWGN under TX I/Q imbalance only. 
     Thus the proposed algorithm is robust in the presence of TX I/Q imbalance. 
     APPENDIX C 
     Fixed Point Pseudo Code 
     The following flow Pseudo code gives a fixed point implementation of the above algorithm. Note that complex variables have the letter “c” prepended. 
     Function 1: Probe2Processing 
     
       
         
           
               
               
               
             
               
                   
               
             
            
               
                   
                 function [rho, theta, Scale_Q] = Probe2Processing (CFO,  
                   
               
               
                   
                 cIQparameters_log,i1,i2, N_delta) 
                   
               
               
                   
                 if (CFO==0) 
                   
               
               
                   
                  CFO=−1; 
                   
               
               
                   
                 end 
                   
               
               
                   
                 % Residual Frequency Estimation 
                   
               
               
                   
                 [cphasor_p,cphasor_m] = Residual_Frequency_Estimation 
                   
               
               
                   
                 (cIQparameters_log,CFO); 
                   
               
               
                   
                 % Residual Frequency Correction 
                   
               
               
                   
                 cZ = Residual_frequency_Correction(cIQparameters_log, 
                   
               
               
                   
                 cphasor_p,cphasor_m); 
                   
               
               
                   
                 % Coefficient Computation 
                   
               
               
                   
                 [cfB1 ,cfB2,Scale_ratio] = Coeff_Computation(CFO, i1 ,i2); 
                   
               
               
                   
                 % Phasor Estimation 
                   
               
               
                   
                 [cg_exp_mTeta_M_16,Scale_g,scale_inv] = Phasor  
                   
               
               
                   
                 Estimation (cZ,cfB1,cfB2,CFO,N_delta,Scale_ratio); 
                   
               
               
                   
                 % IQ Coeff Computation 
                   
               
               
                   
                 [theta,rho, Scale_Q] = Compensation_Params_Estimation 
                   
               
               
                   
                 (cg_exp_mTeta_M_16, scale_inv,Scale_g); 
               
               
                   
               
            
           
         
       
     
                     TABLE C-1                  Probe2Processing Variable Definition Table                             Name   Size   Comments   I/O               CFO   32 bit   Carrier Frequency Offset   I       cIQpara-   16 bit × 6 ×    Recorded FFT outputs   I       meters_log   Nsym Complex               I1   8 bit   Negative Bin offset 1   I       I2   8 bit   Negative Bin offset 2   I       N_delta   16 bit   NCO reset time offset   I       cphasor_p   16 bit Complex   Frequency compensation   NA       cphasor_m   16 bit Complex   Frequency compensation   NA       cZ   16 bit × 6 Complex   Rotated FFT output   NA       cfB 1   16 bit Complex   Estimation Coefficient   NA       cfB2   16 bit Complex   Estimation Coefficient   NA       Scale_ratio   16 bit   Scaling factor   NA       Cg_exp_   16 bit Complex   Estimated I/Q Phasor   NA       mTeta_M_16                   Theta   16 bit   I/Q Compensation   O       Rho   16 bit   I/Q Compensation   O                    
Function 2: Residual_Frequency_Estimation
 
                                            function [cphasor_p,cphasor_m] = Residual_Frequency_Estimation               (cIQparameters_log,CFO)               %TBD Select 1 or 2 according to SNR               cfphasor_64 =0;               for i=0:Nsym-2                cfphasor_64 = Cmplx_Add_64_32(cfphasor_64, Cmplx_Mult_                16_16(cIQparameters_log(i,1),... conj(cIQparameters_log(i+1,1));                end               %Level control to 16 bit signed               Nphasor_bits = Nfft_out-1;               csphasor =Scale_Complex_64(cfphasor_64, Nphasor_bits);               %Get phasor angle and magnitude               [angle_Ef rPhasor] = cordic_SW( csphasor,1);               %Generate ‘Exp_Vec_p’                                     fScale = 26981;   %const 16 bit - round(gcordic)*2{circumflex over ( )}(Nfft_out-1))                                 csphasor_div = Cmplx_real_div_32_16 (csphasor&lt;&lt;(Nfft_out-2)),                rPhasor);                %32bit complex / l6bit real division               % csphasor_div &lt;+−2{circumflex over ( )}15               cphasor_p_32 = (Cmplx_real_mult_16_16(fScale,csphasor_div) )&gt;&gt;                (Nfft_out-2); %Scale back to 16bit, known               cphasor_p = Cmplx_Saturate(cphasor_p_32, Nphasor_bits);               %Compute angle for IQ image rotation (Coridic Preparations)               angle_m = 2*CFO*(Nfft+ LCP) - angle_Ef; %1rad = 2{circumflex over ( )}(Fr_bits-1)               %Generate ‘Exp_Vec_m’               [cphasor_m tmp] = cordic_SW( angle_m,0); %gives -phasor(angle)                    
Function 3: Scale_Complex — 64
 
     function [csphasor]=Scale_Complex — 64(cfphasor — 64, Nphasor_bits) 
     Ceil_Log 2_Abs_Real_Cfphasor — 64=ceil_log 2(abs(real(cfphasor — 64))); 
     Ceil_Log 2_Abs_Imag_Cfphasor — 64=ceil_log 2(abs(imag(cfphasor — 64))); 
     Scale=Nphasor_bits-max(Ceil_Log 2_Abs_Real_Cfphasor — 64, Ceil_Log 2_Abs_Imag_Cfphasor — 64); 
     if (Scale&gt;=0) 
     csphasor — 32=(cfphasor — 64&lt;&lt;Scale); 
     else 
     csphasor — 32=(cfphasor — 64&gt;&gt;(−Scale)); 
     end 
     csphasor=Cmplx_Saturate(csphasor — 32, Nphasor_bits); 
     Function 4: ceil_log 2 
                                                        function [i] = ceil_log2 (X)                   i=0;                   while(X!=0)                    X=X&gt;&gt;1;                    i=i+1;                   end                        
Function: 5: Sign
 
                                                        function [Y] = Sign (X)                   y=1;                   if (X&lt;O)                    y=−1;                   end                        
Function 6: Cmplx_Saturate
 
     
       
         
           
               
               
               
               
             
               
                   
                   
               
             
            
               
                   
                   
                 function [X_16] = Cmplx_Saturate(X_32,Nbits) 
                   
               
               
                   
                   
                 Sign_Real_X_32 = sign(real(X_32)); 
                   
               
               
                   
                   
                 Sign_Imag_X_32 = sign(imag(X_32)); 
                   
               
               
                   
                   
                 Abs_Real_X_32 = abs(real(X_32)); 
                   
               
               
                   
                   
                 Abs_Imag_X_32 = abs(imag(X_32)); 
                   
               
               
                   
                   
                 if (Abs_Real_X_32 &gt;= (1&lt;&lt;Nbits) ) 
                   
               
               
                   
                   
                  if(Sign_Real_X_32==1) 
                   
               
               
                   
                   
                   X_32.r= (1&lt;&lt;Nbits)-1; 
                   
               
               
                   
                   
                  else 
                   
               
               
                   
                   
                 X_32.r=-( (1&lt;&lt; Nbits)-1 ); 
                   
               
               
                   
                   
                  end 
                   
               
               
                   
                   
                 end 
                   
               
               
                   
                   
                 if (Abs_Imag_X_32 &gt;= (1&lt;&lt;Nbits) ) 
                   
               
               
                   
                   
                  if(Sign_Image_X_32==1) 
                   
               
               
                   
                   
                   X_32.i=(1&lt;&lt; Nbits)-1; 
                   
               
               
                   
                   
                  else 
                   
               
               
                   
                   
                 X_32.1=-( (1&lt;&lt; Nbits)-1 ); 
                   
               
               
                   
                   
                  end 
                   
               
               
                   
                   
                 end 
                   
               
               
                   
                   
                 X_16 = X_32; %casting to 16bit 
               
               
                   
                   
               
            
           
         
       
     
                     TABLE C-2                  Residual_Frequency_Estimation, Scale_       Complex_64 Variable Definition Table                             Name   Size   Comments   I/O               cfphasor64   64 bit complex   Phasor Acc   NA       Nphasor_bits   8 bit   Target Num of phasor bits   NA       csphasor_32   32 bit complex   Tmp Variable   NA       csphasor   16 bit complex   Scaled phasor for freq rot    NA       rPhasor   16 bit   Magnitude csphasor   NA       csphasor_div   16 bit complex   Normalized phasor   NA       cphasor_p   16 bit complex   Frequency Compensation   O       angle_Ef   32 bit   Angle of csphasor   NA       angle_m   32 bit   Angle for phasor_m rotation   NA       cphasor_m   16 bit complex   Frequency Compensation   O                    
Function 7: Cordic_SW
 
     
       
         
           
               
               
               
             
               
                   
               
             
            
               
                   
                 function [z,x] = cordic_SW(data_in ,mode) 
                   
               
               
                   
                 %Inits &amp; constants 
                   
               
            
           
           
               
               
               
               
            
               
                   
                 QUARTER=102944; 
                 %round( pi/2*2{circumflex over ( )}(Fr_bits-1)); 
                   
               
               
                   
                 Niter=13; 
                 %Do not Change without Zak&#39;s permission! 
                   
               
            
           
           
               
               
               
            
               
                   
                 Ntan=16; 
                   
               
               
                   
                 x_tmp(0)=0;x_tmp(1)=0; 
                   
               
               
                   
                 Atan_Table =[51472,30386,16055,8150,4091,2047,1024,512, 
                   
               
               
                   
                 256,128,64,32,16]; 
                   
               
               
                   
                 if(mode==0) 
                   
               
               
                   
                  %map phase into [-pi/2 pi/2]interval and init Cordic 
                   
               
               
                   
                  [Angle x] = Cordic_Pre_Process(data_in,0); 
                   
               
               
                   
                  %Initial direction of rotation 
                   
               
               
                   
                  if(Angle==0) 
                   
               
               
                   
                   Sgn =1; 
                   
               
               
                   
                  else 
                   
               
               
                   
                   Sgn = -sign(Angle); 
                   
               
               
                   
                  end 
                   
               
               
                   
                 else 
                   
               
               
                   
                  %init Cordic 
                   
               
               
                   
                  Angle=0; 
                   
               
               
                   
                  %map phasor into [-pi/2 pi/2]interval and init Cordic 
                   
               
               
                   
                  [data in Angle_Offset] = Cordic_Pre_Process(data_in,1); 
                   
               
               
                   
                  x(0) = real(data_in); 
                   
               
               
                   
                  x(1) = imag(data_in); 
                   
               
               
                   
                  if(x(1) ==0) 
                   
               
               
                   
                   Sgn = 1; 
                   
               
               
                   
                  else 
                   
               
               
                   
                   Sgn = sign(x(1)); 
                   
               
               
                   
                  end 
                   
               
               
                   
                 end 
               
               
                   
               
            
           
         
       
     
                     TABLE C-3                  SW CORDIC variable definition                             Name   Size   Comments   I/O               x_tmp   32 bit × 2 array   Local memory (no more than 17    NA               bits used)           Atan_Table   16 bit × 13    ATan Table (const)   NA           array               data_in   32 bit complex   CORDIC input   I       Angle   32 bit   Phasor Angle   NA       Angle_offset   32 bit   Phasor Angle offset   NA       X   32 bit × 2 array   CORDIC out 1   O       z   32 bit × 2 array   CORDIC out 2   O       mode   Boolean (1 bit)   CORDIC mode select   I                    
Function 8: Cordic_Pre_Process
 
     
       
         
           
               
               
               
             
               
                   
               
             
            
               
                   
                 function [data_in, X] = Cordic_Pre_Process(data_in,mode) 
                   
               
               
                   
                 %******************************************************************************** 
                   
               
               
                   
                 % Pre Cordic Processing Bring angle to [-pi/2,pi/2] and find Quadrate 
                   
               
               
                   
                 %******************************************************************************** 
                   
               
               
                   
                 if(mode==0) 
                   
               
               
                   
                  Npi = 0; 
                   
               
               
                   
                  Sgn=0; 
                   
               
               
                   
                  while (data_in &lt; -QUARTER) 
                   
               
               
                   
                   data_in = data_in + (QUARTER&lt;&lt;1); 
                   
               
               
                   
                   Npi = 1-Npi; 
                   
               
               
                   
                  end 
                   
               
               
                   
                  while (data_in &gt; QUARTER) 
                   
               
               
                   
                   data_in =data_in − (QUARTER&lt;&lt;l); 
                   
               
               
                   
                   Npi=1-Npi; 
                   
               
               
                   
                  end 
                   
               
            
           
           
               
               
               
               
            
               
                   
                  X(0) =19898;  
                 % (1/GainCordic)*2{circumflex over ( )}(Ntan-1) 
                   
               
            
           
           
               
               
               
            
               
                   
                  X(1) = 0; 
                   
               
               
                   
                  if(Npi) 
                   
               
               
                   
                   X=-X; 
                   
               
               
                   
                  end 
                   
               
               
                   
                 else 
                   
               
               
                   
                  X=0; 
                   
               
               
                   
                  Sgn_real = sign(real(data_in)); 
                   
               
               
                   
                  Sgn_imag = sign(imag(data_in)); 
                   
               
               
                   
                  if ( Sgn_real ==−1 ) 
                   
               
               
                   
                   data_in = -data_in; 
                   
               
               
                   
                   if(Sgn_imag == 1) 
                   
               
               
                   
                    X = QUA RTER&lt;&lt;1; 
                   
               
               
                   
                   else 
                   
               
               
                   
                    X = -(QUARTER&lt;&lt; 1); 
                   
               
               
                   
                   end 
                   
               
               
                   
                  end 
                   
               
               
                   
                 end 
               
               
                   
               
            
           
         
       
     
                     TABLE C-4                  Cordic_SW, Cordic_Pre_Process variable definition                             Name   Size   Comments   I/O               data_in   32 bit complex   CORDIC input   I/O       X   16 bit × 2 array   CORDIC Phasor   O                    
Function 9: Residual_frequency_Compensation
 
     
       
         
           
               
             
               
                   
               
             
            
               
                 function [cZ_16] = Residual_frequency_Correction(cIQparameters_log,cphasor_p,cphasor_m) 
               
               
                 cdphasor_p = cphasor_p; 
               
               
                 cdphasor_m = cphasor_m; 
               
               
                 cZ_ACC_64 = zeros(1,6);%64bit Acc array (16bit phasor * 16bit FFT output + log2(40)bit for acc &gt;32 
               
               
                   !!!!) 
               
               
                 for i=1:Nsym-1 
               
               
                   cZ_ACC_64(1) = Cmplx_Add_64_32(cZ_ACC_64(1), Cmplx_Mult_16_16(cIQparameters_log(i+1,1) 
               
               
                   ,cphasor_p); 
               
               
                   cZ_ACC_64(2) = Cmplx_Add_64_32(cZ_ACC_64(2), Cmplx_Mult_16_16(cIQparameters_log(i+1,2) 
               
               
                   ,cphasor_p); 
               
               
                   cZ_ACC_64(3) = Cmplx_Add_64_32(cZ_ACC_64(3), Cmplx_Mult_16_16(cIQparameters_log(i+1,3) 
               
               
                   ,cphasor_m); 
               
               
                   cZ_ACC_64(4) = Cmplx_Add_64_32(cZ_ACC_64(4), Cmplx_Mult_16_16(cIQparameters_log(i+1,4) 
               
               
                   ,cphasor_m); 
               
               
                   cZ_ACC_64(5) = Cmplx_Add_64_32(cZ_ACC_64(5), Cmplx_Mult_16_16(cIQparameters_log(i+1,5) 
               
               
                   ,cphasor_m); 
               
               
                   cZ_ACC_64(6) = Cmplx_Add_64_32(cZ_ACC_64(6), Cmplx_Mult_16_16(cIQparameters_log(i+1,6) 
               
               
                   ,cphasor_m); 
               
               
                  cphasor_p = Cmplx_Saturate((Cmplx_Mult_16_16 (cphasor_p,cdphasor_p))&gt;&gt;(Ntan-1), Ntan-1); 
               
               
                  cphasor_m = Cmplx_Saturate( (Cmplx_Mult_16_16 (cphasor_m.cdphasor_m))&gt;&gt;(Ntan-1), Ntan-1); 
               
               
                 end 
               
               
                 %Scale Back to fit 32bit 
               
               
                 for i=1:6 
               
               
                 cZ_ACC_32(i) = cZ_ACC_64(i)&gt;&gt;(Nfft_out); 
               
               
                 end 
               
               
                 % Level control vector so that max fits in 16bit 
               
               
                 Max_Z=0; 
               
               
                 for i=1:6 
               
               
                   if (abs(real(cZ_ACC_32(i)))&gt;Max_Z) 
               
               
                    Max_Z = abs(real(Z_ACC_32(i))); 
               
               
                   end 
               
               
                   if (abs(imag(cZ_ACC_32(i)))&gt;Max_Z) 
               
               
                    Max_Z = abs(imag(Z_ACC_32(i))); 
               
               
                   end 
               
               
                 end 
               
               
                 %Scale Back to 16 bits 
               
               
                 Scale_Z = (Nfft_out-1) − ceil_log2(Max_Z); 
               
               
                 for i=1:6 
               
               
                 cZ_16(i) = Cmplx _Saturate (cZ_ACC_32(i)&gt;&gt;(Scale_Z), Nfft_out- l); 
               
               
                 end 
               
               
                   
               
            
           
         
       
     
                     TABLE C-5                  Residual_frequency_Compensation variable                                     Name   Size   Comments   I/O                   cdphasor_p   16 bit Complex       NA           cdphasor_m   16 bit Complex       NA           cphasor_p   16 bit Complex       I           cphasor_m   16 bit Complex       I           cZ_ACC_64   64 bit Complex × 6 array       NA           cZ_32   32 bit Complex × 6 array       NA           cZ_16   16 bit Complex × 6 array       O           Max_32   32 bit       NA           ScaleZ   16 bit       NA                    
Function: 10: Coeff Computation
 
     
       
         
           
               
             
               
                   
               
             
            
               
                 function [cfB1,cfB2,Scale_ratio] = Coeff_Computation(CFO,i1,i2) 
               
            
           
           
               
               
            
               
                 Log2Nfft = 8; 
                 %log2(Nfft) = log2(256)=8; 
               
               
                 PI_over_FFT = 804; 
                 % round(pi/Nfft*2{circumflex over ( )}(Fr_bits-1)) 
               
            
           
           
               
            
               
                 Nfft=256; 
               
               
                 N1=2; 
               
               
                 [cphasor11 tmp] = cordic_SW( (CFO&lt;&lt;(1+Log2Nfft)),0); 
               
               
                 cphasor11 = Switch_real_imag(cphasor11); 
               
               
                 cfs11 = Cmplx_Add_16_16 (phasor11, -sqrt(−1)*(1&lt;&lt;(Ntan-1))); 
               
               
                 fs21 = CFO+PI_over_FFT *i1; 
               
               
                 fs22 = CFO +PI_over_FFT *i2; 
               
               
                 if(fs21==0) 
               
               
                   cfB1_32 = 1&lt;&lt;( Nfft_out+ Log2Nfft ); 
               
               
                 else 
               
               
                   cfB1_32 = Cmplx_real_div_32_16 ((cfs11&lt;&lt;(Nfft_out-N1)),fs21); 
               
               
                 end 
               
               
                 if(fs22==0) 
               
               
                   cfB2_32 = 1&lt;&lt;( Nfft_out+ Log2Nfft ); 
               
               
                 else 
               
               
                   cfB2_32 = Cmplx_real_div_32_16 ((cfs11&lt;&lt;(Nfft_out-N1)),fs22); 
               
               
                 end 
               
               
                 %Scale to fB1,fB2 to 15bit signed 
               
               
                 Max =0; 
               
               
                 Max_abs_real_cfB1 = abs(real(cfB1_32)); 
               
               
                 Max_abs_imag_cfB1= abs(imag(cfB1_32)); 
               
               
                 Max_abs_real_cfB2 = abs(real(cfB2_32)); 
               
               
                 Max_abs_imag_cfB2= abs(imag(cfB2_32)); 
               
               
                 if(Max_abs_real_cfB1 &gt;Max) 
               
               
                   Max = Max_abs_real_cfB1; 
               
               
                 end 
               
               
                 if(abs Max_abs_imag_cfB1&gt;Max) 
               
               
                   Max = Max_abs_imag_cfB1; 
               
               
                 end 
               
               
                 if(Max_abs_real_cfB2 &gt;Max) 
               
               
                   Max = Max_abs_real_cfB2 
               
               
                 end 
               
               
                 if(Max_abs_imag_cfB2&gt;Max) 
               
               
                   Max Max_abs_imag_cfB2; 
               
               
                 end 
               
               
                  Scale_ratio = 15-(ceil_log2(Max)+1); 
               
               
                 If (Scale&gt;=0) 
               
               
                   cfB1 = Cmplx_Saturate( (cfB1_32&lt;&lt;Scale_ratio), Nfft_out-2); 
               
               
                   cfB2 = Cmplx_Saturate ((cfB2_32&lt;&lt;Scale_ratio),Nfft_out-2); 
               
               
                 else 
               
               
                   cfB1 = Cmplx_Saturate ((cfB1_32&gt;&gt;-Scale_ratio), Nfft_out-2); 
               
               
                   cfB2 = Cmplx_Saturate ((cfB2_32&gt;&gt;-Scale_ratio), Nfft_out-2); 
               
               
                 end 
               
               
                   
               
            
           
         
       
     
                     TABLE C-6                  Coeff Computation Variable Definition                             Name   Size   Comments   I/O               cphasor11   16 bit complex   CORDIC out   NA       cfs11   32 bit complex   Coeff Numerator   NA       fs21   16 bit   Coeff Denominator   NA       fs22   16 bit   Coeff Denominator   NA       cfB1_32   32 bit complex   Unsealed Coeff 1   NA       cfB2_32   32 bit complex   Unsealed Coeff 2   NA       cfB1   16 bit complex   Coeff 1   O       cfB2   16 bit complex   Coeff 2   O       Scale_ratio   16 bit   Coeff Scale factor   O                    
Function 11: Phasor Estimation Variable Definition
 
     
       
         
           
               
             
               
                   
               
             
            
               
                 function [cg_exp_mTeta_M_16,Scale_g,scale_inv] = 
               
               
                  Phasor_Estimation(cZ,cfB1,cfB2,CFO,N_delta,Scale_ratio) 
               
               
                 N2=2; 
               
               
                 %**************************************************************************** 
               
            
           
           
               
               
            
               
                 %  
                 Fix Point Computation of 
               
               
                 % 
                   
               
            
           
           
               
               
               
               
               
            
               
                 %  
                 Z1*( |fB1|{circumflex over ( )}2+|fB2|{circumflex over ( )}2) 
                   
                 Z1*( |fB1|{circumflex over ( )}2+|fB2|{circumflex over ( )}2)*(B1′*Z3 +B2*Z4′)′ 
                   
               
            
           
           
               
               
               
            
               
                 %  
                 ------------------------------ = ------------------------------------------- 
                   
               
            
           
           
               
               
               
               
               
            
               
                 %  
                 B1′*Z3 +B2*Z4′ 
                   
                 |B1′*Z3|{circumflex over ( )}2 +|B2*Z4′|{circumflex over ( )}2 
                   
               
            
           
           
               
            
               
                 %***************************************************************************** 
               
               
                 sumfB1SfB2S = (Real_Add_30_30_T(MAG_2_16(cfB1), MAG_2_16(cfB2)))&gt;&gt;15 
               
               
                 cNUMERATOR_32(0) = Cmplx_Real_Mul_16_16(cZ(0),sumfB1SfB2S); 
               
               
                 cNUMERATOR_32(1) = Cmplx_Real_Mul_16_16(cZ(1),sumfB1SfB2S); 
               
               
                 cDENOM_32(0) =Cmplx_Add_32_32(Cmplx_Mul_16_16(conj(cZ(2)),(cfB1)), 
               
               
                   Cmplx_Mul_16_16(conj(cZ(3)),(cfB2))); 
               
               
                 cDENOM_32(1) = Cmplx_Add_32_32(Cmplx_Mul_16_16(conj(cZ(4)),(cfB1)), 
               
               
                   Cmplx_Mul_16_16(conj(cZ(5)),(cfB2))); 
               
               
                 cMul_NUM_cDENOM_64(0) = Cmplx_Mul_32_32(cNUMERATOR_32(0),conj(cDENOM_32(0))); 
               
               
                 cMul_NUM_cDENOM_64(1) = Cmplx_Mul_32_32(cNUMERATOR_32(1),conj(cDENOM_32(1))); 
               
               
                 DENOM_2_64(0) = MAG_2_32(cDENOM_32(0)); 
               
               
                 DENOM_2_64(1) = MAG_2_32(cDENOM_32(1)); 
               
               
                 %Level Control for division scale denominator down to 32bit 
               
               
                 if(DENOM_2_64(0)&gt;DENOM_2_64(1) ) 
               
               
                  Scale = ceil_log2(DENOM_2_64(0))); 
               
               
                 else 
               
               
                  Scale = ceil_log2(DENOM_2_64(1))); 
               
               
                 end 
               
               
                 Scale = 31-Scale; 
               
               
                 If(Scale&gt;=0) 
               
               
                 DENOM_2_32(0) = Cmplx_Saturate (DENOM_2_64(0)&lt;&lt;(Scale),31); 
               
               
                 DENOM_2_32(1) = Cmplx_Saturate (DENOM_2_64(I)&lt;&lt; (Scale),31); 
               
               
                 else 
               
               
                 DENOM_2_32(0) = Cmplx_Saturate( DENOM_2_64(0)&gt;&gt;(-Scale),31); 
               
               
                 DENOM_2_32(1) = Cmplx_Saturate (DENOM_2_64(1)&gt;&gt;(-Scale),31); 
               
               
                 end 
               
               
                 %complex/real division 64bit/32bit gives 32bit result 
               
               
                 cfCM(0) = Cmplx_real_div_64_32 (cMul_NUM_cDENOM_64(0)&gt;&gt;N2 ,DENOM_2_32(0)); 
               
               
                 cfCM(1) = Cmplx_real_div_64_32 (cMul_NUM_cDENOM_64(1)&gt;&gt;N2 ,DENOM_2_32(1)); 
               
               
                 cfCM_avg = Cmplx_Add_32_32 (cfCM(0),cfCM(1)); 
               
               
                 % Rotation of FCM _avg needed only if HW can not Insure Delatn=0 
               
               
                 [cphasor_dn tmp] = cordic_SW( (CF0&lt;&lt;1)*N_delta, 0); 
               
               
                 cfCM_avg_rot_64 = Cmplx_Mul_32_16(cfCM_avg,cphasor_dn); 
               
               
                 cfCM_avg_rot_32 = fCM_avg_rot_64&gt;&gt;(Ntan); 
               
               
                 Scale_fCM = Ntan-1-N1+Scale_ratio-Scale-6-N2; 
               
               
                 fCM_one_level_32 = 1&lt;&lt;(Scale_fCM); 
               
               
                 cfp_m_32 = Cmplx_Real_Add_32_32( cfCM_avg_rot_32, -fCM_one_level_32); 
               
               
                 cfp_p_32 = Cmplx_Real_Add_32_32( cfCM_avg_rot_32, fCM_one_level_32); 
               
               
                 cNUMERATORf_64 = Cmplx_Mul_32_32(cfp_m_32,conj(cfp_p_32)); 
               
               
                 DENOMf_64 = MAG_2_32 (cfp_p_32); 
               
               
                 %Level Control for division scale denominator down to 32bit signed 
               
               
                 Abs_real_Numerator64 = abs(real(cNUMERATORf_64); 
               
               
                 Abs_imag_Numerator64 = abs(imag(cNUMERATORf_64); 
               
               
                 if(Abs_real_Numerator64) &gt; Abs_imag_Numerator64) 
               
               
                  Scale_fn =30 − ceil_log2(Abs_real_Numerator64); 
               
               
                 else 
               
               
                  Scale_fn =30 − ceil_log2(Abs_imag_Numerator64); 
               
               
                 end 
               
               
                 if(Scale_fn&gt;=0) 
               
               
                 cNUMERATORf _32= cNUMERATORf_64&lt;&lt;(Scale_fn); 
               
               
                 else 
               
               
                 cNUMERATORf_32= cNUMERATORf_64&gt;&gt;(-Scale_fn); 
               
               
                 end 
               
               
                 %Level Control for division scale denominator down to 16bit unsigned 
               
               
                 Scale_fd =16 − ceil_log2(DENOMf_64); 
               
               
                 if(Scale_fd &gt;=0) 
               
               
                 DENOMf_scaled_16= Cmplx_Saturate (DENOMf_64&lt;&lt;(Scale_fd),16); 
               
               
                 else 
               
               
                 DENOMf_scaled_16= Cmplx_Saturate (DENOMf_64&gt;&gt;(-Scale_fd),16); 
               
               
                 end 
               
               
                 % Division 32bit complex by 16bit real 
               
               
                 cg_exp_mTeta_M_16 = Cmplx_real_div_32_16 (cNUMERATORf_32,DENOMf_scaled_16); 
               
               
                 Scale_g = Scale_fd-Scale_fn; 
               
               
                 scale_inv = 14-Scale_fd+Scale_fn; 
               
               
                   
               
            
           
         
       
     
                     TABLE C-7                  Phasor Estimation Variable Definition                             Name   Size   Comments   I/O               sumfB1SfB2S   16 bit               cNUMERATOR_32   32 bit × 2 complex array               cDENOM_32   32 bit × 2 complex array               cMul_NUM_cDENOM_64   64 bit × 2 complex array               DENOM_2_64   64 bit × 2 array               DENOM_2_32   32 bit × 2 array               cfCM   32 bit × 2 complex array               cfCM_avg   32 bit complex               cphasor_dn   32 bit complex               cfs11   16 bit complex               fs21   16 bit               fs22   16 bit               cfCM_avg_rot_64   64 bit complex               cfCM_avg_rot_32   32 bit complex               cfCM_one_level_32   32 bit               cfp_m_32   32 bit complex               cfp_p_32   32 bit complex               cNUMERATORf_64   64 bit complex               DENOMf_64   64 bit               Abs_real_Numerator64   64 bit               Abs_imag_Numerator64   64 bit               cNUMERATORf_32   32 bit complex               DENOMf_scaled_16   16 bit complex               cg_exp_mTeta_M_16   16 bit complex       O       Scale_g   16 bit       O       scale_inv   16 bit       O       Scale_fd   16 bit               Scale_fn   16 bit               Scale_fCM   16 bit               Scale   16 bit                    
Function 12: Compensation_Params_Estimation
 
     
       
         
           
               
             
               
                   
               
             
            
               
                 function [theta,rho, Scale_Q] = Compensation_Params_Estimation(cg_exp_mTeta_M_16, 
               
               
                    scale_inv,Scale_g) 
               
               
                 Mue_bits=4; 
               
               
                 if(ProbeII_Num == 1) 
               
               
                 Mue =12; 
               
               
                 elseif(ProbeII_Num == 2) 
               
               
                 Mue = 8; 
               
               
                 else 
               
               
                 Mue = 4; 
               
               
                 End 
               
               
                 if(ProbeII_Num&gt;=1) 
               
               
                  % First Order Loop 
               
               
                  Mue_1m = (16-Mue); 
               
               
                  cg_ACC_32 = cg_exp_mteta &lt;&lt;(-Scale_g-Mue_bits); 
               
               
                  cg_Delta_32 = (Cmplx_Mul_16_16 (cg_exp_mteta *cg_exp_mTeta_M_16))&gt;&gt;4; 
               
               
                  cg_ACC_32 = Cmplx_Add_32_32 (Cmplx_Mul_32_16(cg_ACC_32,Mue_Fix_1m), 
               
               
                     Cmplx_Mul_32_16(cg_Delta_32,Mue_Fix ))&gt;&gt;(-Scale_g); 
               
               
                  cg_exp_mteta = g_ACC_32; 
               
               
                  % I/Q Correction Params Calculation 
               
               
                  [theta,rho, Scale_Q] = Compute_Fix_Point_IQ_Coeffs(cg_exp_mteta,scale_inv,Scale_g,0); 
               
               
                 else 
               
               
                   g_exp_mteta = cg_exp_mTeta_M_16; 
               
               
                   % I/Q Correction Params Calculation 
               
               
                  [theta,rho, Scale_Q] = Compute_Fix_Point_IQ_Coeffs(g_exp_mTeta_M_16,scale_inv, Scale_g,1); 
               
               
                 end 
               
               
                 ProbeII_Num = ProbeII_Num+1; 
               
               
                   
               
            
           
         
       
     
                     TABLE C-8                  Compensation_Params_Estimation variable definition                                     Name   Size   Comments   I/O                   ProbeII_Num   16 bit   Reset Value is 0   NA           Mue   16 bit   Loop Gain   NA           Mue_1m   16 bit   Complementary Loop   NA                   Gain               cg_ACC_32   32 bit complex                   cg_Delta_32   32 bit complex                   cg_exp_mteta   16 bit complex                   theta   16 bit   Reset Value 2048   O           rho   16 bit   Reset Value 0   O           Scale_Q   1 bit   Reset Value 1   O                    
Function 13: Compute_Fix_Point_IQ_Coeffs
 
     
       
         
           
               
             
               
                   
               
             
            
               
                 function [theta,rho, Scale_Q] = Compute_Fix_Point_IQ_Coeffs(cg_exp_mTeta_M_16,scale_inv, scale_g 
               
               
                  ,FirstTime) 
               
               
                 if(FirstTime) 
               
               
                  scale_delta=0; 
               
               
                  if (scale_inv&gt;31) 
               
               
                   scale_delta = scale_inv-31; 
               
               
                   scale_inv=31; 
               
               
                  end 
               
               
                  scale_inv_log = scale_inv; 
               
               
                  scale_delta_log = scale_delta; 
               
               
                  scale_g_log = scale_g 
               
               
                 else 
               
               
                  scale_inv = scale_inv_log; 
               
               
                  scale_delta= scale_delta_log; 
               
               
                  scale_g = scale_g_log; 
               
               
                 end 
               
               
                 real_g_exp_mTeta_M_16 = real(cg_exp_mTeta_M_16); 
               
               
                 imag_g_exp_mTeta_M_16 = imag(cg_exp_mTeta_M_16); 
               
               
                 if( real_g_exp_mTeta_M_16 &gt; 1&lt;&lt; (-scale_g_log) ) 
               
               
                  Inv_real_g_exp_mTeta_M_16 = Cmplx_real_div_32_16 (1 &lt;&lt;(scale_inv),real_g_exp_mTeta_M_16)); 
               
               
                  Scale_theta = -14+Teta_bits-1+scale_delta; 
               
               
                  If(Scale_theta&gt;=0) 
               
               
                 theta = Inv_real_g_exp_mTeta_M_ 16&lt;&lt;(Scale_theta); 
               
               
                  else 
               
               
                 theta = Inv_real_g_exp_mTeta_M_16&gt;&gt;(-Scale_theta); 
               
               
                  end 
               
               
                  Rho_32 = Cmplx_Mul_ 16_16 (-imag_g_exp_mTeta_M_16,Inv_real_g_exp_mTeta_M_16); 
               
               
                  Scale_rho = -Rho_bits+1+scale_inv; 
               
               
                  If(Scale_rho&gt;=0) 
               
               
                 rho = Rho_32&lt;&lt;(Scale_rho); 
               
               
                  else 
               
               
                 rho = Rho_32&gt;&gt;(-Scale_rho); 
               
               
                  end 
               
               
                  Scale_Q = 1; 
               
               
                 else 
               
               
                  Scale_theta = scale_g_log +Teta_bits-1; 
               
               
                  Scale_ rho = scale_g_log + Rho_bits −1; 
               
               
                  If(Scale_theta&gt;=0) 
               
               
                 theta = real_g_exp_mTeta_M_16&lt;&lt;(Scale_theta); 
               
               
                  else 
               
               
                 theta = real_g_exp_mTeta_M_16&gt;&gt;(-Scale_theta); 
               
               
                  end 
               
               
                  If(Scale_rho&gt;=0) 
               
               
                 rho = (-imag_g_exp_mTeta_M_16)&lt;&lt;(Scale_rho); 
               
               
                  else 
               
               
                 rho = (-imag_g_exp_mTeta_M_16)&gt;&gt;(-Scale_rho); 
               
               
                  end 
               
               
                  Scale_Q = 0; 
               
               
                 end 
               
               
                   
               
            
           
         
       
     
                     TABLE C-9                  Compute_Fix_Point_IQ_Coeffs variable definition                             Name   Size   Comments   I/O               scale_inv_log   16 bit               scale_delta_log   16 bit               scale_inv   16 bit               scale_delta   16 bit               real_g_exp_mTeta_M_16   16 bit               imag_g_exp_mTeta_M_16   16 bit               Inv_real_g_exp_mTeta_M_16   16 bit               Scale_theta   16 bit               Scale_rho   16 bit               Rho_32   32 bit               theta   16 bit       O       rho   16 bit       O       Scale_Q    1 bit       O                    
Complex Math Operation Definitions
 
[C — 32_r,C — 32_i]=Cmplx_Add — 16 — 16(A — 16_r,A — 16_i,B — 16_r,B — 16_i);
         C — 32_r=A — 16_r+B — 16_r   C — 32_i=A — 16_i+B — 16_i       

     [C — 32_r,C — 32_i]=Cmplx_Add — 32 — 32(A — 32 r,A — 32_i,B — 32_r,B — 32_i);
         C — 32_r=A — 32_r+B — 32_r   C — 32_i=A — 32_i+B — 32_i       

     [R — 32]=Real_Add — 30 — 30_T(A — 30,B — 30);
         R — 32=A — 30+B — 30       

     [C — 64_r,C — 64_i]=Cmplx_Add — 64 — 32(A — 64 r,A — 64 i,B — 32 r,B — 32_i);
         C — 64_r=A — 64_r+B — 32_r   C — 64_i=A — 64_i+B — 32_i       

     [C — 32_r,C — 32_i]=Cmplx_Real_Add — 32 — 16 (A — 32_r,A — 32_i,B — 16);
         C — 32_r=A — 32_r+B — 16   C — 32_i=A — 32_i       

     [C — 32_r,C — 32_i]=Cmplx_imag_Add — 32 — 16 (A — 32_r,A — 32_i,B — 16);
         C — 32_r=A — 32_r   C — 32_i=A — 32_i+B — 16       

     [C — 32_r]=MAG — 2 — 16(A — 16_r,A — 16_i)
         C — 32_r=A — 16_r*A — 16_r+A — 16_i*A — 16_i       

     [C — 64_r]=MAG — 2 — 32(A — 32_r,A — 32_i)
         C — 64_r=A — 32_r*A — 32_r+A — 32 — 1*A — 32 i       

     [C — 32_r,C — 32_i]=Cmplx_Mult — 16 — 16(A — 16_r,A — 16_i,B — 16_r,B — 16_i)
         C — 32_r=A — 16_r*B — 16_r−A — 16_i*B — 16_i   C — 32_i=A — 16_r*B — 16_i+A — 16_i*B — 16_r       

     [C — 64_r,C — 64_i]=Cmplx_Mult — 32 — 32(A — 32_r,A — 32_i,B — 32_r,B — 32_i)
         C — 64_r=A — 32_r*B — 32_r−A — 32_i*B — 32_i   C — 64_i=A — 32_r*B — 32_i+A — 32_i*B — 32_r       

     [C — 32_r,C — 32_i]=Cmplx_real_Mult — 16 — 16(A — 16_r,A — 16_i,B — 16_r)
         C — 32_r=A — 16_r*B — 16_r   C — 32_i=A — 16_i*B — 16_r       

     [C — 64_r,C — 64_i]=Cmplx_Mul — 32 — 16(A — 32_r,A — 32_i,B — 16_r)
         C — 64_r=A — 32_r*B — 16_r   C — 64_i=A — 32_i*B — 16_r       

     [C — 16_r,C — 16_i]=Cmplx_real_div — 32 — 16(A — 32_r,A — 32_i,B — 16_r)
         C — 32_r=A — 32_r/B — 16_r   C — 32_i=A — 32_i/B — 16_r       

     [C — 32_r,C — 32_i]=Cmplx_real_div — 64 — 32(A — 64_r,A — 64_i,B — 32_r)
         C — 32_r=A — 64_r/B — 32_r   C — 32_i=A — 64_i/B — 32_r       

     [C — 16_r,C — 16_i]=Switch_real_imag(A — 16_r,A — 16_i);
         C — 32_r=A — 16_i       

     C — 32_i=A — 16_r 
     APPENDIX D 
     Exemplary Parameters of a HW-SW Interface 
     The following table summarizes the information exchanged between the HW and SW during probe2 reception. Output refers to Output from the HW and Input refers to Input to the HW. 
     
       
         
           
               
               
               
               
               
             
               
                   
               
               
                   
                   
                   
                 Input 
                   
               
               
                   
                   
                   
                 to/Output from 
                   
               
               
                 HW Name 
                 SW Name 
                 Size 
                 HW 
                 Remarks 
               
               
                   
               
             
            
               
                 Δn 
                 N_delta 
                 16 bits 
                 Output 
                 Number of samples between NCO 
               
               
                   
                   
                   
                   
                 reset and first sample of the FFT 
               
               
                   
                   
                   
                   
                 window. 
               
               
                 i1 
                 I1 
                 3 bit 
                 Output 
                 Offset index of first negative FFT 
               
               
                   
                   
                   
                   
                 bin from nominal 
               
               
                 i2 
                 I2 
                 4 bit 
                 Output 
                 Offset index of second negative FFT 
               
               
                   
                   
                   
                   
                 bin from nominal 
               
               
                 CFO 
                 CFO 
                 17 bit 
                 Output 
                 Carrier Frequency Offset (after fine 
               
               
                   
                   
                   
                   
                 frequency offset) 
               
               
                 Z[k1,:] 
                 cIQparameters_log(0,:) 
                 16×2×Nprobe2_symbols bit 
                 Output 
                 FFT output at bin k1 for symbols 
               
               
                   
                   
                   
                   
                 1,..., Nprobe2_symbols 
               
               
                 Z[k2,:] 
                 cIQparameters_log(1,:) 
                 16×2×Nprobe2_symbols bit 
                 Output 
                 FFT output at bin k2 for symbols 
               
               
                   
                   
                   
                   
                 1,..., Nprobe2_symbols 
               
               
                 Z[−k1+i1,:] 
                 cIQparameters_log(2,:) 
                 16×2×Nprobe2_symbols bit 
                 Output 
                 FFT output at bin −k1+i1 for 
               
               
                   
                   
                   
                   
                 symbols 1,..., Nprobe2_symbols 
               
               
                 Z[−k1+i2,:] 
                 cIQparameters_log(3,:) 
                 16×2×Nprobe2_symbols bit 
                 Output 
                 FFT output at bin −k1+i2 for 
               
               
                   
                   
                   
                   
                 symbols 1,..., Nprobe2_symbols 
               
               
                 Z[−k2+i1,:] 
                 cIQparameters_log(4,:) 
                 16×2×Nprobe2_symbols bit 
                 Output 
                 FFT output at bin −k2+i1 for 
               
               
                   
                   
                   
                   
                 symbols 1,..., Nprobe2_symbols 
               
               
                 Z[−k2+i2,:] 
                 cIQparameters_log(5,:) 
                 16×2×Nprobe2_symbols bit 
                 Output 
                 FFT output at bin −k2+i2 for 
               
               
                   
                   
                   
                   
                 symbols 1,..., Nprobe2_symbols 
               
               
                 ζ 
                 theta 
                 12 bit unsigned 
                 Input 
                 I/Q Compensation parameter 
               
               
                 ρ 
                 Rho 
                 12 bit 
                 Input 
                 I/Q Compensation parameter 
               
               
                 Scale_Q 
                 Scale_Q 
                 1 bit 
                 Input 
                 I/Q Compensation parameter 
               
               
                   
               
            
           
         
       
     
     APPENDIX E 
     Frequency Offset Introduction, CP Length, Number of Symbols 
                                        function [dF] = Frequency_Offset_Introduction(CFO)                             Freq_Th = 41;   %round((5e3/50e6)*2*pi*2{circumflex over ( )}16)                         if (abs(CFO)&lt; Freq_Th)            dF = sign(CFO)*(TBD RF Interface introduce);           else            dF=0;           end                    
Tone Selection
 
                                        function [SC] = ProbeII_Tone_Selection(Sigma_2_32)           SC_MIN(0) = 146;           SC_MAX(0) = 186;           SC_MIN(2) = 217;           SC_MAX(2) = 249;           SC_DEFAULT(0) = 176;           SC_DEFAULT(1) = 249;           NLog_In_Bits=6;           NLog_Out_ Bits=14;           SC = SC_DEFAULT;           a=−7871; % round(−1.921625277102556*2{circumflex over ( )}(NLog_Out_Bits-2) ) ;           b= 8071; % round( 1.970377382221271*2{circumflex over ( )}(NLog_Out_Bits-2) );           C =32656; %round(0.0375*log2(10)*2{circumflex over ( )}(NLog_Out_Bits-2 + NLog_In_Bits));           %Find Default NSR log2(Sigma_2)           for i=0:1            Scale_16(i) = ceil_log2( Sigma_2_32(SC_DEFAULT(i)) );            Frac_16(i) = Sigma_2_32(SC_DEFAULT(i))&gt;&gt; (Scale_16(i) -NLog_In_Bits );            NSR_Default_32(i) = Scale_16(i) &lt;&lt; (NLog_Out_Bits - 2 + NLog_In_Bits)+ ...                (Real_Mult_16_16(b,Frac_16(i))+a&lt;&lt;(NLog_In_Bits) );            NSR_Best_32(i) = NSR_Default_32(i);            for k=[SC_MIN(i) : SC_MAX(i)]             Scale_k_16 = ceil_log2( Sigma_2_32 (k) );             Frac_k_16 = Sigma_2_32 (k)&gt;&gt;(Scale_k_16 -NLog_In_Bits );             NSR_32 = Scale_k_16 &lt;&lt; (NLog_Out_Bits-2 + NLog_In_Bits)+ ...                (Real_Mult_16_16(b,Frac_k_16)+a&lt;&lt;(NLog_In_Bits) );             if(NSR_32&lt; Add_Real_32_32(NSR_Default_32 (i),...                - Real_Mult_16_16(C, Abs_ 16(Add_Real_16_16(k, -SC_DEFAULT(i)) )              if(NSR_32 &lt; NSR_Best_32(i))               SC(i) = k;               NSR_Best_32(i) = NSR_32;              end             end            end           end           Where Sigma_32 is a vector of the estimated noise variance of the various FFT tones.                    
CP &amp; Number of OFDM Symbol Selection
 
     
       
         
           
               
             
               
                   
               
             
            
               
                 function [NUM_OF_SYMS, CP_LENGTH] = Probe2_CP_L_Select(CFO) 
               
               
                 CP_Max = 126; 
               
               
                 CP_Min = 64; 
               
               
                 L_Max = 40; 
               
               
                 L_Min = 28; 
               
            
           
           
               
               
            
               
                 PI = 205887; 
                 %round(pi*2{circumflex over ( )}Freq_bits-1) 
               
               
                 2P = 411775; 
                 %round(2pi*2{circumflex over ( )}Freq_bits-1) 
               
               
                 Phase_Th = 41177; 
                 %round(2pi/10*2{circumflex over ( )}Freq_bits-1) 
               
            
           
           
               
            
               
                 dPhase_64= 0; 
               
               
                 error= 0; 
               
               
                 Min_Err = Inf; 
               
               
                 NUM_OF_SYMS = L_Min; 
               
               
                 CP_LENGTH =CP_Min; 
               
               
                 for CP = CP_Min: CP_Max 
               
               
                  for L= L_Min: L_Max 
               
               
                   dPhase_32 = L*(CP+Nfft)*(CFO); 
               
               
                   while (abs(dPhase_32) &gt; 2P) 
               
               
                    dPhase_32= dPhase_32- sign(dPhase_32)* 2P; 
               
               
                   end 
               
               
                   if (abs(dPhase_32 )&gt;PI) 
               
               
                    error_32 = abs(2PI - abs(dPhase_32)); 
               
               
                   else 
               
               
                    error_32 = abs(dPhase_32); 
               
               
                   end 
               
               
                   if( error_32 &lt; Phase_Th ) 
               
               
                    Indicator=1; 
               
               
                    if(error_32 &lt;Min_Err) 
               
               
                     Min_Err = error; 
               
               
                     NUM_OF_SYMS = L; 
               
               
                     CP_LENGTH =CP; 
               
               
                    end 
               
               
                   end 
               
               
                  end 
               
               
                 end