Apparatus and methods for compensating for signal imbalance in a receiver

Abstract

Apparatus, methods and systems for compensating for an I/Q imbalance 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 the frequency domain energy at negative frequencies is reduced; and modifying at least one of the components based on the value.

Description

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 MoCA™-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;

FIG. 12A 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; and

FIG. 15 shows another illustrative plot that summarizes simulation results.

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.

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 & 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).

$\begin{array}{cc}{\tilde{Y}}_{\mathrm{real}}=\{\begin{array}{cc}\mathrm{Bypass}==1& Y_{\mathrm{real}}\\ \mathrm{Bypass}==0& \{\begin{array}{cc}\mathrm{Scale\_Q}==0& \varsigma Y_{\mathrm{real}}\\ \mathrm{Scale\_Q}==1& Y_{\mathrm{real}}\end{array}\end{array}{\tilde{Y}}_{\mathrm{imag}}=\{\begin{array}{cc}\mathrm{Bypass}==1& Y_{\mathrm{imag}}\\ \mathrm{Bypass}==0& \{\begin{array}{cc}\mathrm{Scale\_Q}=0& {\tilde{Y}}_{\mathrm{imag}}=Y_{\mathrm{imag}}+\rho Y_{\mathrm{real}}\\ \mathrm{Scale\_Q}=1& \varsigma Y_{\mathrm{imag}}+\rho Y_{\mathrm{real}}\end{array}\end{array}& \mathrm{Equation}1\end{array}$

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 i₁ and i₂ are frequency bin indices computed as shown in Equation 2.

$\begin{array}{cc}{i}_2=-\mathrm{round}\left(\frac{N·\mathrm{CFO}}{\pi }\right)i_2=i_1+\mathrm{sign}\left(\mathrm{CFO}\right)·\mathrm{sign}\left(i_1-\frac{N·\mathrm{CFO}}{\pi }\right)& \mathrm{Equation}2\end{array}$ 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’=2¹⁴. The computation of i₁ and i₂ may be done via comparison to fixed thresholds. The value of the FFT grid in fixed point representation is given by Equation 3:

$\begin{array}{cc}{F}_k=\mathrm{round}\left(\frac{2\pi k}{N}·2^{\mathrm{Freq}\_\mathrm{bits}-1}\right),\mathrm{for}k=\left[-3,3\right],& \mathrm{Equation}3\end{array}$ in which Freq_bits may be set to 14 or any other suitable number. Indices i₁ and i₂ 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|	$\frac{F_0+F_1}{2}$	$\frac{F_1+F_2}{2}$	$\frac{F_2+F_3}{2}$	F1	F2	F3
Fixed point value	804	2413	4012	1608	3217	4825

Equation 4 sets forth a definition for the sign operation.

$\begin{array}{cc}\mathrm{sign}\left(x\right)=\{\begin{array}{cc}1& x\ge 0\\ -1& x<0\end{array}& \mathrm{Equation}4\end{array}$

In some embodiments, bin recording (at step 408, see FIG. 4) may involve 16 bit FFT outputs at bins k₁, −k₁+i₁, −k₁+i₂, k₂, −k₂+i₁, −k₂+i₂, 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: k1	Z[k1, m]	k1 ∈ [146, 186]
ADDR: k2	Z[k2, m]	k2 ∈ [217, 249]
ADDR: 256 −	Z[−k1 + i1, m],	−k1 + i1 = 256 − k1 + i1 ∈ [67, 113]
k1 + i1
ADDR: 256 −	Z[−k1 + i2, m],	−k1 + i2 = 256 − k1 + i2 ∈ [66, 112]
k1 + i2
ADDR: 256 −	Z[−k2 + i1, m],	−k2 + i1 = 256 − k2 + i1 ∈ [4, 36]
k2 + i1
ADDR: 256 −	Z[−k2 + i2, m]	−k2 + i2 = 256 − k2 + i2 ∈ [3, 37]
k2 + i2

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.

$\begin{array}{cc}\hat{\varepsilon }=\frac{\mathrm{angle}\left(\sum _{m=0}^{L-2}Z\left[k_i,m\right]Z^{*}\left[k_i,m+1\right]\right)}{2\pi \left(N+N_{\mathrm{CP}}\right)}\mathrm{Where}{k}_i=\{\begin{array}{cc}{k}_1& {\mathrm{SNR}}_{k_1}>{\mathrm{SNR}}_{k_2}\\ k_2& {\mathrm{SNR}}_{k_2}>{\mathrm{SNR}}_{k_1}\end{array}& \mathrm{Equation}5\end{array}$

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.

$\begin{array}{cc}{\stackrel{\_}{Z}}_{k_1}=\sum _{m=0}^{L-1}Z\left[k_1,m\right]e^{-\mathrm{j2\pi }\hat{\varepsilon }\left(N+N_{\mathrm{CP}}\right)m}{\stackrel{\_}{Z}}_{k_2}=\sum _{m=0}^{L-1}Z\left[k_2,m\right]e^{-\mathrm{j2\pi }\hat{\varepsilon }\left(N+N_{\mathrm{CP}}\right)m}{\stackrel{\_}{Z}}_{-k_1+i_1}=\sum _{m=0}^{L-1}Z\left[-k_1+i_1,m\right]e^{-j\left(2\mathrm{CFO}+2\pi \hat{\varepsilon }\right)\left(N+N_{\mathrm{CP}}\right)m}{\stackrel{\_}{Z}}_{-k_2+i_1}=\sum _{m=0}^{L-1}Z\left[-k_2+i_1,m\right]e^{-j\left(2\mathrm{CFO}+2\pi \hat{\varepsilon }\right)\left(N+N_{\mathrm{CP}}\right)m}{\stackrel{\_}{Z}}_{-k_1+i_2}=\sum _{m=0}^{L-1}Z\left[-k_1+i_2,m\right]e^{-j\left(2\mathrm{CFO}+2\pi \hat{\varepsilon }\right)\left(N+N_{\mathrm{CP}}\right)m}{\stackrel{\_}{Z}}_{-k_2+i_2}=\sum _{m=0}^{L-1}Z\left[-k_2+i_2,m\right]e^{-j\left(2\mathrm{CFO}+2\pi \hat{\varepsilon }\right)\left(N+N_{\mathrm{CP}}\right)m}& \mathrm{Equations}6\end{array}$

Equations 7 may be used to evaluate an I/Q imbalance phasor estimate, which may be computed using Equation 8.

$\begin{array}{cc}{B}_n=\{\begin{array}{cc}\mathrm{CFO}\ne 0& \frac{1}{2}\left(\frac{\sin \left(2·\mathrm{CFO}·N\right)}{\mathrm{CFO}+\frac{\pi ·i_n}{N}}\right)+\\ & \frac{j}{2}\left(\frac{\cos \left(2·\mathrm{CFO}·N\right)-1}{\mathrm{CFO}+\frac{\pi ·i_n}{N}}\right)\\ \mathrm{CFO}=0,i_n=0& N\\ \mathrm{CFO}=0,i_n\ne 0& 0\end{array}{C}_1=\frac{e^{j2\mathrm{CFO}\left(\Delta n\right)}}{N}\frac{\left({B_1}^2+{B_2}^2\right)·{\stackrel{\_}{Z}}_{k_1}}{{B_1\left({\stackrel{\_}{Z}}_{-k_1+i_1}\right)}^{*}+{B_2\left({\stackrel{\_}{Z}}_{-k_1+i_2}\right)}^{*}}{C}_2=\frac{e^{j2\mathrm{CFO}\left(\Delta n\right)}}{N}\frac{\left({B_1}^2+{B_2}^2\right)·{\stackrel{\_}{Z}}_{k_2}}{{B_1\left({\stackrel{\_}{Z}}_{-k_2+i_1}\right)}^{*}+{B_2\left({\stackrel{\_}{Z}}_{-k_2+i_2}\right)}^{*}}C=\frac{1}{2}\left(C_1+C_2\right)& \mathrm{Equations}7\\ \stackrel{\_}{g{e}^{-j0}}=\frac{C-1}{C-1}& \mathrm{Equation}8\end{array}$

I/Q imbalance compensation parameters ξ, ρ and Scale_Q (see, e.g., Equation 1) may then be computed in accordance with Equation 9.

$\begin{array}{cc}\{\begin{array}{cccc}\mathrm{real}\left\{\stackrel{\_}{g{e}^{-\mathrm{j\theta }}}\right\}\ge 1& \mathrm{ScaleQ}=1& \hat{\xi }=\frac{1}{\mathrm{real}\left\{\stackrel{\_}{g{e}^{-\mathrm{j\theta }}}\right\}},& \hat{\rho }=-\frac{\mathrm{imag}\left\{\stackrel{\_}{g{e}^{-\mathrm{j\theta }}}\right\}}{\mathrm{real}\left\{\stackrel{\_}{g{e}^{-\mathrm{j\theta }}}\right\}}\\ \mathrm{otherwise}& \mathrm{ScaleQ}=0& \hat{\xi }=\left\{\stackrel{\_}{g{e}^{-\mathrm{j\theta }}}\right\},& \hat{\rho }=-\mathrm{imag}\left\{\stackrel{\_}{g{e}^{-\mathrm{j\theta }}}\right\}\end{array}& \mathrm{Equation}9\end{array}$

Equation 9 avoids saturation at the receiver since ξ 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.

$\begin{array}{cc}\begin{array}{c}{\left(\stackrel{\_}{g{e}^{-\mathrm{j\theta }}}\right)}_i^{\mathrm{ACC}}=\left(1-\mu \right){\left(\stackrel{\_}{g{e}^{-\mathrm{j\theta }}}\right)}_{i-1}^{\mathrm{ACC}}+\\ {{\mu }_i\left(\stackrel{\_}{g{e}^{-\mathrm{j\theta }}}\right)}_{i-1}^{\mathrm{ACC}}·{\left(\stackrel{\_}{g{e}^{-\mathrm{j\theta }}}\right)}_i\\ {\left(\stackrel{\_}{g{e}^{-\mathrm{j\theta }}}\right)}_0^{\mathrm{ACC}}=1\end{array}& \mathrm{Equations}10\end{array}$

In Equations 10,

${\left(\stackrel{\_}{g{e}^{-\mathrm{j\theta }}}\right)}_i.$ 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_iε[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.

$\begin{array}{cc}{\mu }_i=\{\begin{array}{cc}1& i=1\\ 0.75& i=2\\ 0.5& i=3\\ 0.25& i=4\end{array}\hspace{1em}& \mathrm{Equation}11\end{array}$

Equations 12 set forth I/Q compensation parameters that may be used during the reception of the i'th probe2.

$\begin{array}{cc}\{\begin{array}{ccc}\mathrm{real}\left\{{\left(\stackrel{\_}{g{e}^{-\mathrm{j\theta }}}\right)}_{i-1}^{\mathrm{ACC}}\right\}\ge 1& \hat{\xi }=\frac{1}{\mathrm{real}\left\{{\left(\stackrel{\_}{g{e}^{-\mathrm{j\theta }}}\right)}_{i-1}^{\mathrm{ACC}}\right\}}& {\hat{\rho }}_i=-\frac{\mathrm{imag}\left\{{\left(\stackrel{\_}{g{e}^{-\mathrm{j\theta }}}\right)}_{i-1}^{\mathrm{ACC}}\right\}}{\mathrm{real}\left\{{\left(\stackrel{\_}{g{e}^{-\mathrm{j\theta }}}\right)}_{i-1}^{\mathrm{ACC}}\right\}}\\ \mathrm{otherwise}& \hat{\xi }=\mathrm{real}\left\{{\left(\stackrel{\_}{g{e}^{-\mathrm{j\theta }}}\right)}_{i-1}^{\mathrm{ACC}}\right\},& \widehat{{\rho }_i}=-\mathrm{imag}\left\{{\left(\stackrel{\_}{g{e}^{-\mathrm{j\theta }}}\right)}_{i-1}^{\mathrm{ACC}}\right\}\end{array}& \mathrm{Equation}s12\end{array}$

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(2πφn)−s_q[n] sin(2πφn)+w_i[n]z_q[n]=gs_i[n] sin(2πφn−θ)+gs_q[n] cos(2πφn−θ)+w_q[n]

Some algebra shows that the above can be expressed as

$z\left[n\right]=K_{1}s\left[n\right]e^{\mathrm{j2\pi \phi }n}+K_{2}s*\left[n\right]e^{-\mathrm{j2\pi \phi }n}+w\left[n\right]$ $K_1=\frac{1}{2}\left(1+g{e}^{-\mathrm{j\theta }}\right)$ $K_2=\frac{1}{2}\left(1-g{e}^{-\mathrm{j\theta }}\right)$

At the receiver I/Q compensation is performed the signal after I/Q compensation is given by:

$z^{\prime }\left[n\right]=\left\{\mathrm{real}\left(K_{1}s\left[n\right]e^{\mathrm{j2\pi \phi }n}+K_{2}s*\left[n\right]e^{-\mathrm{j2\pi \phi }n}+w\left[n\right]\right)\right\}+\mathrm{j\xi }\left\{\mathrm{imag}\left(K_{1}s\left[n\right]e^{\mathrm{j2\pi \phi }n}+K_{2}s*\left[n\right]e^{-\mathrm{j2\pi \phi }n}+w\left[n\right]\right)\right\}+j\rho \left\{\mathrm{real}\left(K_{1}s\left[n\right]e^{\mathrm{j2\pi \phi }n}+K_{2}s*\left[n\right]e^{-\mathrm{j2\pi \phi }n}+w\left[n\right]\right)\right\}$

Assuming transmission of a single frequency at frequency bin k, after some algebra the compensated signal is given by

$z^{\prime }\left[n\right]=\left\{h\cos \left(2\pi n\left(\frac{k}{N}+\phi \right)+\angle h\right)\right\}+\mathrm{j\xi }\left\{gh\left(2\pi n\left(\frac{k}{N}+\phi \right)+\angle h-\theta \right)\right\}+\mathrm{j\rho }\left\{h\cos \left(2\pi n\left(\frac{k}{N}+\phi \right)+\angle h\right)\right\}+\left[w_r\left[n\right]+j\left(\rho ·w_r\left[n\right]+\xi ·w_i\left[n\right]\right)\right]$

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:

$\begin{array}{c}\hat{Z}\left[k\right]=he^{\mathrm{j\angle }h}N\left[\frac{1}{2}\left(1+\xi g{e}^{-\mathrm{j\theta }}\right)+\frac{\rho }{2}{e}^{j\frac{\pi }{2}}\right]+\\ he^{-\mathrm{j\angle }h}{e}^{-j\left(2\pi \left(N-1\right)\left(\frac{k}{N}+\phi \right)\right)}\frac{\sin \left(2\pi \left(k+N\phi \right)\right)}{\sin \left(2\pi \left(\frac{k}{N}+\phi \right)\right)}\left[\frac{1}{2}\left(1-\xi g{e}^{\mathrm{j\theta }}\right)+\frac{\rho }{2}{e}^{j\frac{\pi }{2}}\right]\end{array}$ $\hat{Z}\left[-k\right]=he^{-\mathrm{j\angle }h}{e}^{-j\left(2\mathrm{\pi \phi }\left(N-1\right)\right)}\frac{\sin \left(2\pi \phi N\right)}{\sin \left(2\mathrm{\pi \phi }\right)}\left\{\frac{1}{2}\left(1-\xi g{e}^{\mathrm{j\theta }}\right)+\frac{\rho }{2}{e}^{j\frac{\pi }{2}}\right\}$

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

${\hat{Z}\left[-k\right]}^2={h}^2{\left(\frac{\sin \left(2\pi \phi N\right)}{\sin \left(2\mathrm{\pi \phi }\right)}\right)}^2\left\{\frac{1}{2}\left(1-\xi g{e}^{\mathrm{j\theta }}\right)+\frac{\rho }{2}{e}^{j\frac{\pi }{2}}\right\}\left\{\frac{1}{2}\left(1-\xi g{e}^{-\mathrm{j\theta }}\right)+\frac{\rho }{2}{e}^{-j\frac{\pi }{2}}\right\}=\frac{1}{4}{h}^2{\left(\frac{\sin \left(2\pi \phi N\right)}{\sin \left(2\mathrm{\pi \phi }\right)}\right)}^2\left\{1+{\xi }^2{g}^2+{\rho }^2-2\xi g\cos \theta -2\xi g\rho \sin \left(\theta \right)\right\}$

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

$\frac{\partial {\hat{Z}\left[-k\right]}^2}{\partial \xi }=\frac{\partial \left(1+{\xi }^2{g}^2+{\rho }^2-2\xi g\cos \theta -2\xi g\rho \sin \left(\theta \right)\right)}{\partial \xi }=0$ $\frac{\partial {\hat{Z}\left[-k\right]}^2}{\partial \rho }=\frac{\partial \left(1+{\xi }^2{g}^2+{\rho }^2-2\xi g\cos \theta -2\xi g\rho \sin \left(\theta \right)\right)}{\partial \rho }=0$ $2\xi g^2-2g\cos \theta -2\rho g\sin \left(\theta \right)=0$ $2\rho -2\xi g\sin \left(\theta \right)=0$

Solving the above yields

$\left[\begin{array}{cc}{g}^2& -g\sin \left(\theta \right)\\ -g\sin \left(\theta \right)& 1\end{array}\right]\left[\begin{array}{c}\xi \\ \rho \end{array}\right]=\left[\begin{array}{c}g\cos \theta \\ 0\end{array}\right]\left[\begin{array}{c}\xi \\ \rho \end{array}\right]=\frac{1}{{\left(g\cos \theta \right)}^2}\left[\begin{array}{c}g\cos \theta \\ g^2\sin \theta \cos \theta \end{array}\right]=\left[\frac{1}{\begin{array}{c}g\cos \theta \\ \tan \theta \end{array}}\right]$

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;

${\hat{h}}_k=\frac{A_k^{*}}{N{A_k}^2}\hat{Z}\left[k\right]=he^{\mathrm{j\angle }h}{K}_1+e_k$

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

$Z\left[k\right]=he^{\angle h}{K}_1{e}^{\mathrm{j\pi \varepsilon }\left(N-1\right)}\frac{\sin \left(\mathrm{\pi \varepsilon }N\right)}{\sin \left(\mathrm{\pi \varepsilon }\right)}+he^{-\angle h}{K}_2{e}^{-\mathrm{j2\pi }\left(N-1\right)\left(\frac{k}{N}+\phi -\frac{1}{2}\varepsilon \right)}\frac{\sin \left(2\pi N\left(\frac{k}{N}+\phi -\frac{1}{2}\varepsilon \right)\right)}{\sin \left(2\pi \left(\frac{k}{N}+\phi -\frac{1}{2}\varepsilon \right)\right)}+W_k$ $Z\left[-k\right]=he^{-\angle h}{K}_2{e}^{-\mathrm{j2\pi }\left(\phi -\frac{1}{2}\varepsilon \right)\left(N-1\right)}\left(\frac{\sin \left(2\pi N\left[\phi -\frac{1}{2}\varepsilon \right]\right)}{\sin \left(2\pi \left[\phi -\frac{1}{2}\varepsilon \right]\right)}\right)+he^{\angle h}{K}_1{e}^{-\mathrm{j2\pi }\left(\frac{k}{N}+\frac{1}{2}\varepsilon \right)\left(N-1\right)}\left(\frac{\sin \left(2\pi N\left[\frac{k}{N}+\frac{1}{2}\varepsilon \right]\right)}{\sin \left(2\pi \left[\frac{k}{N}+\frac{1}{2}\varepsilon \right]\right)}\right)+W_{-k}$ 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

$Z\left[-k+i\right]=he^{-\angle h}{K}_2{e}^{-\mathrm{j2\pi }\left(\phi -\frac{1}{2}\varepsilon +\frac{i}{2N}\right)\left(N-1\right)}\left(\frac{{\left(-1\right)}^i\sin \left(2\pi N\left[\phi -\frac{1}{2}\varepsilon \right]\right)}{\sin \left(2\pi \left[\phi -\frac{1}{2}\varepsilon +\frac{i}{2N}\right]\right)}\right)+he^{\angle h}{K}_1{e}^{-\mathrm{j\pi }\left(\frac{2k-i}{N}+\varepsilon \right)\left(N-1\right)}\left(\frac{\sin \left(\pi N\left[\frac{2k-i}{N}+\varepsilon \right]\right)}{\sin \left(\pi \left[\frac{2k-i}{N}+\varepsilon \right]\right)}\right)+W_{-k+i}$

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

$-\left(\frac{k+M}{N}+r\right),2\phi =\frac{M}{N}+r\mathrm{where}M=\lfloor 2\phi V\rfloor \mathrm{and}r<\frac{1}{N}.$

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:

$\mathrm{Loss}=10{\log }_{10}{N}^2-10{\log }_{10}\left(\sum _{i=-M-L}^{-M+L}{\left(\frac{\sin \left(\pi \left[M+\mathrm{Nr}\right]\right)}{\sin \left(\pi \left[\frac{M+i}{N}+r\right]\right)}\right)}^2\right)$

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] the following figure 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

$Z\left[k\right]=\stackrel{\stackrel{\mathrm{signal}\mathrm{term}}{︷}}{A_1{\mathrm{hK}}_1}+\underset{\underset{\mathrm{ICI}\mathrm{from}\mathrm{bin}-k}{︸}}{B_1{h}^{*}{K}_2}+W_k$ $Z\left[-k\right]=\underset{\underset{\mathrm{ICI}\mathrm{from}\mathrm{bin}k}{︸}}{A_2{\mathrm{hK}}_1}+\stackrel{\stackrel{\mathrm{signal}\mathrm{term}}{︷}}{B_2{h}^{*}{K}_2}+W_{-k}$

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

$-\frac{k}{N}-{\phi }^{\prime }\left(\mathrm{where}{\phi }^{\prime }=\left(\phi -\frac{1}{2}\varepsilon \right)\right)$ which is not on the FFT grid. The further away this frequency is from the FFT grid of

$\frac{l}{N}$ 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

$\frac{k}{N}$ 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:

$\begin{array}{c}{\mathrm{SNR}}_k=10{\log }_{10}\left(\frac{{{\mathrm{hNK}}_1}^2}{{h^{*}{e}^{-j\left(2\pi \left(N-1\right)\left(\frac{k}{N}+{\phi }^{\prime }\right)\right)}\frac{\sin \left(2{\mathrm{\pi \phi }}^{\prime }N\right)}{\sin \left(\frac{2\pi k}{N}+2\pi {\phi }^{\prime }\right)}{K}_2}^2}\right)\\ =10{\log }_{10}\left(\frac{N^2{K_1}^2}{{K_2}^2}\right)-10{\log }_{10}\left({\left[\frac{\sin \left(2{\mathrm{\pi \phi }}^{\prime }N\right)}{\sin \left(\frac{2\pi k}{N}+2\pi {\phi }^{\prime }\right)}\right]}^2\right)\end{array}$

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 g[0.5,2] (max 3[dB] amplitude imbalance) and θ[−10°,10°]
  • maximizing the second term under the constraint that φ′[−200e−6*1.5e9/50e6: −200e−6*1.5e9/50e6] and k[146:186, 217:249]

The above minimizations were performed numerically using a Matlab simulation. FIG. 12a 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

$57.4216={\min }_{g,\theta }\left\{10{\log }_{10}\left(\frac{N^2{K_1\left(g,\theta \right)}^2}{{K_2\left(g,\theta \right)}^2}\right)\right\}\mathrm{st}\begin{array}{c}g\in \left[0.5,2\right]\\ \theta \in \left[-10°,10°\right]\end{array}$

FIG. 12 b depicts the second term as a function of φ and k

From FIG. 12b 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

$17.04={\max }_{g,\theta }\left\{10{\log }_{10}\left({\left[\frac{\sin \left(2\mathrm{\pi \phi }N\right)}{\sin \left(\frac{2\pi k}{N}+2\pi \phi \right)}\right]}^2\right)\right\}$ $\mathrm{st}\begin{array}{c}\phi \in \left[-\Delta f,\Delta f\right]\\ k\in \left[146,186\right]\bigcup \left[217,249\right]\end{array}$

Thus the worst case SNR induced by the ICI term is 40.3816[dB]

$40.3816\left[\mathrm{dB}\right]=\min \left({\mathrm{SNR}}_k\right)={\min }_{\phi ,k,g,\theta }\left(10{\log }_{10}\left(\frac{N^2{K_1\left(g,\theta \right)}^2}{{K_2\left(g,\theta \right)}^2}\right)-10{\log }_{10}\left({\left[\frac{\sin \left(2\mathrm{\pi \phi }N\right)}{\sin \left(\frac{2\pi k}{N}+2\pi \phi \right)}\right]}^2\right)\right)$

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:

$Z\left[k\right]={\mathrm{hAK}}_1+W_k$ $Z\left[-k+i\right]=h^{*}{B}_i{K}_2+W_{-k+i}$ $A=e^{\mathrm{j\pi \varepsilon }\left(N-1\right)}\frac{\sin \left(\mathrm{\pi \varepsilon }N\right)}{\sin \left(\mathrm{\pi \varepsilon }\right)}$ $B_i=e^{-\mathrm{j2\pi }\left(\phi -\frac{1}{2}\varepsilon +\frac{i}{2N}\right)\left(N-1\right)}\left(\frac{{\left(-1\right)}^i\sin \left(2\pi N\left[\phi -\frac{1}{2}\varepsilon \right]\right)}{\sin \left(2\pi \left[\phi -\frac{1}{2}\varepsilon +\frac{i}{2N}\right]\right)}\right)$

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

$\left[\begin{array}{c}{\hat{Z}}^{*}\left[-k+i_1\right]\\ {\hat{Z}}^{*}\left[-k+i_2\right]\end{array}\right]=\left[\begin{array}{c}{B}_1^{*}\\ B_2^{*}\end{array}\right]\left({\mathrm{hK}}_2^{*}\right)+\left[\begin{array}{c}{W}_{k+i_1}^{*}\\ W_{k+i_2}^{*}\end{array}\right]$ ${\left({\mathrm{hK}}_2^{*}\right)}_{\mathrm{LS}}=\frac{B_1{\hat{Z}}^{*}\left[-k+i_1\right]+B_2{\hat{Z}}^{*}\left[-k+i_2\right]}{{B_1}^2+{B_2}^2}$

Thus we can estimate

$\frac{K_1}{K_2^{*}}$ without knowledge of the channel h by

$C\equiv \frac{{\stackrel{\_}{K}}_1}{K_2^{*}}=\frac{\left({\mathrm{hK}}_1\right)}{{\left({\mathrm{hK}}_2^{*}\right)}_{\mathrm{LS}}}=\frac{Z\left[k\right]/A}{B_1{\hat{Z}}^{*}\left[-k+i_1\right]+B_2{\hat{Z}}^{*}\left[-k+i_2\right]/{B_1}^2+{B_2}^2}=\frac{\left({B_1}^2+{B_2}^2\right)·Z\left[k\right]}{A·\left(B_1{\hat{Z}}^{*}\left[-k+i_1\right]+B_2{\hat{Z}}^{*}\left[-k+i_2\right]\right)}$

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

$\stackrel{\_}{C}\equiv \frac{1}{2}\frac{\left({B_1}^2+{B_2}^2\right)·Z\left[k_1\right]}{A·\left(B_1{\hat{Z}}^{*}\left[-k_1+i_1\right]+B_2{\hat{Z}}^{*}\left[-k_1+i_2\right]\right)}+\frac{1}{2}\frac{\left({B_1}^2+{B_2}^2\right)·Z\left[k_2\right]}{A·\left(B_1{\hat{Z}}^{*}\left[-k_2+i_1\right]+B_2{\hat{Z}}^{*}\left[-k_2+i_2\right]\right)}$

It is easy to see that

$g{e}^{-\mathrm{j\theta }}=\frac{K_1/K_2^{*}-1}{K_1/K_2^{*}+1}$

And thus its estimate is given by

$\stackrel{\_}{g{e}^{-\mathrm{j\theta }}}=\frac{\stackrel{\_}{C}-1}{\stackrel{\_}{C}+1}$

The I/Q compensation is then easily computed by

$\hat{\xi }=\frac{1}{\mathrm{real}\left\{\stackrel{\_}{g{e}^{-\mathrm{j\theta }}}\right\}}$ $\hat{\rho }=-\frac{\mathrm{imag}\left\{\stackrel{\_}{g{e}^{-\mathrm{j\theta }}}\right\}}{\mathrm{real}\left\{\stackrel{\_}{g{e}^{-\mathrm{j\theta }}}\right\}}$ 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

$Z\left[k·m\right]=\{\begin{array}{cc}k\ge 0& e^{\mathrm{j2\pi \varepsilon }\left(N+N_{\mathrm{CP}}\right)m}Z\left[k,0\right]\\ k<0& e^{\mathrm{j2\pi }\left(2\phi -\varepsilon \right)\left(N+N_{\mathrm{CP}}\right)m}Z\left[k,0\right]\end{array}\hspace{1em}$

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

$Z\left[k\right]=\underset{\underset{\mathrm{\#1}}{︸}}{D}\stackrel{\stackrel{\mathrm{\#2}}{︷}}{e^{-\mathrm{j2\pi \Delta }n\left(\phi -\varepsilon \right)}}\underset{\underset{\mathrm{\#3}}{︸}}{e^{\mathrm{j2\pi \varepsilon }\left(N+N_{\mathrm{CP}}\right)m}}{\mathrm{hAK}}_1+W_k$ $Z\left[-k+i\right]=\underset{\underset{\mathrm{\#1}}{︸}}{D^{*}}\stackrel{\stackrel{\mathrm{\#2}}{︷}}{e^{-\mathrm{j2\pi \Delta }n\left(\phi -\varepsilon \right)}}\underset{\underset{\mathrm{\#3}}{︸}}{e^{\mathrm{j2\pi }\left(2-\mathrm{\phi \varepsilon }\right)\left(N+N_{\mathrm{CP}}\right)m}}{h}^{*}{B}_i{K}_2+W_{-k+i}$ $\begin{array}{c}\mathrm{\#1}\text{-}\mathrm{due}\mathrm{to}\Delta n\mathrm{samples}\mathrm{between}\mathrm{start}\mathrm{of}\mathrm{compensation}\mathrm{to}\mathrm{FFT}\\ \mathrm{will}\mathrm{not}\mathrm{cancell}\mathrm{out}\\ \mathrm{\#2}\text{-}\mathrm{due}\mathrm{to}\Delta n\mathrm{samples}\mathrm{between}\mathrm{start}\mathrm{of}\mathrm{compensation}\mathrm{to}\mathrm{FFT}\\ \mathrm{start}\mathrm{will}\mathrm{not}\mathrm{cancell}\mathrm{out}\\ \mathrm{\#3}\text{-}\mathrm{Phase}\mathrm{due}\mathrm{to}\mathrm{residual}\mathrm{carrier}\mathrm{freq}\mathrm{error}\end{array}$ 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

$\hat{\varepsilon }=\frac{\mathrm{angle}\left(\sum _{m=o}^{L-2}Z\left[k_i,m\right]Z^{*}\left[k_i,m-1\right]\right)}{2\pi \left(N+N_{\mathrm{CP}}\right)}$ $\mathrm{Where}$ $k_i=\{\begin{array}{cc}{k}_1& {\mathrm{SNR}}_{k_1}>{\mathrm{SNR}}_{k_2}\\ k_2& {\mathrm{SNR}}_{k_2}>{\mathrm{SNR}}_{k_1}\end{array}$

The residual frequency error compensation and averaging is given by Residual Frequency Compensation and time averaging

${\stackrel{\_}{Z}}_{k_1}=\sum _{m=0}^{L-1}Z\left[k_1,m\right]e^{-\mathrm{j2\pi }\hat{\varepsilon }\left(N+N_{\mathrm{CP}}\right)m}$ ${\stackrel{\_}{Z}}_{k_2}=\sum _{m=0}^{L-1}Z\left[k_2,m\right]e^{-\mathrm{j2\pi }\hat{\varepsilon }\left(N+N_{\mathrm{CP}}\right)m}$ ${\stackrel{\_}{Z}}_{-k_1+i_1}=\sum _{m=0}^{L-1}Z\left[-k_1+i_1,m\right]e^{-\mathrm{j2\pi }\left(2\hat{\phi }+\hat{\varepsilon }\right)\left(N+N_{\mathrm{CP}}\right)m}$ ${\stackrel{\_}{Z}}_{-k_2+i_1}=\sum _{m=0}^{L-1}Z\left[-k_2+i_1,m\right]e^{-\mathrm{j2\pi }\left(2\hat{\phi }+\hat{\varepsilon }\right)\left(N+N_{\mathrm{CP}}\right)m}$ ${\stackrel{\_}{Z}}_{-k_1+i_2}=\sum _{m=0}^{L-1}Z\left[-k_1+i_2,m\right]e^{-\mathrm{j2\pi }\left(2\hat{\phi }+\hat{\varepsilon }\right)\left(N+N_{\mathrm{CP}}\right)m}$ ${\stackrel{\_}{Z}}_{-k_2+i_2}=\sum _{m=0}^{L-1}Z\left[-k_2+i_2,m\right]e^{-\mathrm{j2\pi }\left(2\hat{\phi }+\hat{\varepsilon }\right)\left(N+N_{\mathrm{CP}}\right)m}$

The phasor ge^−jθ can then be estimated using the same estimator derived above, namely

$C_i=e^{\mathrm{j4\pi }\left(\phi -\varepsilon \right)\left(\Delta n\right)}\frac{\left({B_1}^2+{B_2}^2\right)·{\stackrel{\_}{Z}}_{k_i}}{{B_1\left({\stackrel{\_}{Z}}_{-k_i+i_1}\right)}^{*}+{B_2\left({\stackrel{\_}{Z}}_{-k_i+i_2}\right)}^{*}}$

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's look at

$A=e^{\mathrm{j\pi \varepsilon }\left(N-1\right)}\frac{\sin \left(\mathrm{\pi \varepsilon }N\right)}{\sin \left(\pi \varepsilon \right)}$

The residual frequency error is typically smaller then 10 khz (7 ppm) for such an error

$e^{\mathrm{j\pi \varepsilon }\left(N-1\right)}=0.9872+0.1595i\cong 1$ $\frac{\sin \left(\mathrm{\pi \varepsilon }N\right)}{\sin \left(\pi \varepsilon \right)}=254.8975\cong 256=N$

Thus we make the following approximation

$A=e^{\mathrm{j\pi \varepsilon }\left(N-1\right)}\frac{\sin \left(\mathrm{\pi \varepsilon }N\right)}{\sin \left(\pi \varepsilon \right)}\cong N$

As for Bi

$\begin{array}{c}\begin{array}{c}{B}_i=e^{-\mathrm{j2\pi }\left(\phi -\frac{1}{2}\varepsilon +\frac{i}{2N}\right)\left(N-1\right)}\left(\frac{{\left(-1\right)}^i\sin \left(2\pi N\left[\phi -\frac{1}{2}\varepsilon \right]\right)}{\sin \left(2\pi \left[\phi -\frac{1}{2}\varepsilon +\frac{i}{2N}\right]\right)}\right)\\ \cong e^{-\mathrm{j2\pi }\left(\phi -\frac{1}{2}\varepsilon \right)\left(N\right)}{e}^{-\mathrm{j\pi i}}\left(\frac{{\left(-1\right)}^i\sin \left(2\pi N\left[\phi -\frac{1}{2}\varepsilon \right]\right)}{\sin \left(2\pi \left[\phi -\frac{1}{2}\varepsilon +\frac{i}{2N}\right]\right)}\right)\\ \cong {e^{-\mathrm{j2\pi }N\left(\phi -\varepsilon \right)}\left(-1\right)}^{2i}\left(\frac{\sin \left(2\pi N\left[\phi -\varepsilon \right]\right)}{\sin \left(2\pi \left[\phi -\varepsilon +\frac{i}{2N}\right]\right)}\right)\\ =e^{-\mathrm{j2\pi }N\left(\phi -\varepsilon \right)}\left(\frac{\sin \left(2\pi N\left[\phi -\varepsilon \right]\right)}{\sin \left(2\pi \left[\phi -\varepsilon +\frac{i}{2N}\right]\right)}\right)\end{array}\\ \mathrm{Where}\phi -\varepsilon =\frac{\mathrm{CFO}}{2\pi }\\ \begin{array}{c}{B}_i\cong e^{-\mathrm{j2\pi }N\left(\phi -\varepsilon \right)}\left(\frac{\sin \left(2\pi N\left[\phi -\varepsilon \right]\right)}{\sin \left(2\pi \left[\phi -\varepsilon +\frac{i}{2N}\right]\right)}\right)\\ =\frac{\sin \left(2·N·\mathrm{CFO}\right)+j\left[\cos \left(2·N·\mathrm{CFO}\right)-1\right]}{2\sin \left(2\pi \left[\phi -\varepsilon +\frac{i}{2N}\right]\right)}\end{array}\end{array}$

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

$2\pi \left[\phi -\varepsilon +\frac{i}{2N}\right]<207e-6*1.5e9\text{/}50e6+3\text{/}512=0.0121$

For such a small angle a simple linear approximation has very little error

$\begin{array}{c}\sin \left(0.0121\right)\cong 0.0121\\ \Downarrow \\ \sin \left(2\pi \left[\phi -\varepsilon +\frac{i}{2N}\right]\right)\cong 2\pi \left[\phi -\varepsilon +\frac{i}{2N}\right]\end{array}$

Thus follows that

$\begin{array}{c}{B}_i\cong \frac{\sin \left(2·N·\mathrm{CFO}\right)+j\left[\cos \left(2·N·\mathrm{CFO}\right)-1\right]}{2\sin \left(2\pi \left[\phi -\varepsilon +\frac{i}{2N}\right]\right)}\\ \cong \frac{1}{2}\frac{\sin \left(2·N·\mathrm{CFO}\right)+j\left[\cos \left(2·N·\mathrm{CFO}\right)-1\right]}{\mathrm{CFO}+\frac{\mathrm{i\pi }}{N}}\end{array}$

And so the simplified coefficients are given by

$A=N$ $B_1=\frac{1}{2}\left(\frac{\sin \left(2·\mathrm{CFO}·N\right)}{\mathrm{CFO}+\frac{\pi ·i_1}{N}}\right)+\frac{j}{2}\left(\frac{\sin \left(2·\mathrm{CFO}·N\right)-1}{\mathrm{CFO}+\frac{\pi ·i_1}{N}}\right)$ $B_2=\frac{1}{2}\left(\frac{\sin \left(2·\mathrm{CFO}·N\right)}{\mathrm{CFO}+\frac{\pi ·i_2}{N}}\right)+\frac{j}{2}\left(\frac{\sin \left(2·\mathrm{CFO}·N\right)-1}{\mathrm{CFO}+\frac{\pi ·i_2}{N}}\right)$

APPENDIX B

Exemplary Measurements

Ideal Channel No AWGN with 200 ppm carrier & 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. B-1.

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 the following FIG. B-2 one can compare the image signal magnitude before and after the first iteration.

FIG. B-2: Frequency Input Ideal Channel, No AWGN after First Iteration

After processing of the second Probe II, the SNR is around 39.6[dB]. In following FIG. B-3, one can compare the image signal magnitude before and after the second iteration. The image signal is no longer visible after the second iteration.

FIG. B-3: Frequency Plot Ideal Channel No AWGN after Two Iterations

After processing of the third Probe II, the SNR is around 40.6[dB].

FIG. B-4: Slicer Input Ideal Channel, No AWGN after Three Iterations

The following FIG. B-5 depicts the slicer SNR after processing of the fourth Probe II transmission SNR is shown to be around 40.9[dB]

FIG. B-5: Slicer Input Ideal Channel, No AWGN after Four Iterations

The following FIG. B-6 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 probeII 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].

FIG. B-6: Slicer Input Channel, No AWGN no I/Q Imbalance

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. B-7: Slicer Input MoCA10408 Channel, 15 [dB] AWGN SNR Before Cancellation

After processing of the first Probe II, the SNR is around 9.5 [dB]. In the following FIG. B-8 one can compare the image signal magnitude before and after the first iteration.

FIG. B-8: Frequency Plot MoCA10408 Channel 15 [dB] AWGN after First Iteration

FIG. B-9: Slicer Input MoCA10408 Channel 15 [dB] AWGN SNR after First Iteration.

After processing of the second Probe II, the SNR is around 11 [dB].

FIG. B-10: Frequency Plot MoCA10408 Channel 15 [dB] after Second Iteration

FIG. B-11: 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]

FIG. B-12: Frequency Plot MoCA10408 Channel 15 [dB] AWGN SNR after Third Iteration

FIG. B-13: 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] 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.

FIG. B-14: Slicer Input MoCA10408 Channel 15 [dB] AWGN SNR no I/Q Imbalance

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]

FIG. B-15: Slicer Input Ideal Channel No AWGN, Under RX and TX I/Q Imbalance

After three probe2 transmissions the SNR was around 21.2 [dB].

FIG. B-16: Frequency Plot Idea Channel No AWGN, Under RX and TX I/Q Imbalance after Third Iteration

FIG. B-17: 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
⊖	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
cIQparameters_log	16 bit × 6 × Nsym Complex	Recorded FFT outputs	I
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
cfB1	16 bit Complex	Estimation Coefficient	NA
cfB2	16 bit Complex	Estimation Coefficient	NA
Scale_ratio	16 bit	Scaling factor	NA
Cg_exp_mTeta_M_16	16 bit Complex	Estimated I/Q Phasor	NA
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<<(Nfft_out−2)), rPhasor); %32bit complex / 16bit real
division
% csphasor_div <+−2{circumflex over ( )}15
cphasor_p_32 = (Cmplx_real_mult_16_16(fScale,csphasor_div) )>>(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_Log2_Abs_Real_Cfphasor_64 =
ceil_log2(abs(real(cfphasor_64)));
Ceil_Log2_Abs_Imag_Cfphasor_64 =
ceil_log2(abs(imag(cfphasor_64)));
Scale=Nphasor_bits-max(Ceil_Log2_Abs_Real_Cfphasor_64,
Ceil_Log2_Abs_Imag_Cfphasor_64);
if (Scale>=0)
csphasor_32 = (cfphasor_64<< Scale);
else
csphasor_32 = (cfphasor_64>> (−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>>1;
i=i+1;
end

Function: 5: Sign

function [Y] = Sign (X)
y=1;
if (X<0)
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 >= (1<< Nbits) )
if(Sign_Real_X_32==1)
X_32.r= (1<< Nbits)−1;
else
X_32.r= −( (1<< Nbits)−1 );
end
end
if (Abs_Imag_X_32 >= (1<< Nbits) )
if(Sign_Image_X_32==1)
X_32.i= (1<< Nbits)−1;
else
X_32.i= −( (1<< 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 & constants
QUARTER= 102944; %round( pi/2*2{circumflex over ( )}(Fr_bits−1));
Niter=13;        %Do not Change without Zak'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	NA
		(no more than 17 bits used)
Atan_Table	16 bit × 13 array	ATan Table (const)	NA
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 < −QUARTER)
data_in = data_in + (QUARTER<<1);
Npi = 1−Npi;
end
while (data_in > QUARTER)
data_in = data_in − (QUARTER<<1);
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 = QUARTER<<1;
else
X = −(QUARTER<<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 >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))>>(Ntan−1), Ntan−1);
cphasor_m = Cmplx_Saturate( (Cmplx_Mult_16_16 (cphasor_m.cdphasor_m))>>(Ntan−1), Ntan−1);
end
%Scale Back to fit 32bit
for i=1:6
cZ_ACC_32(i) = cZ_ACC_64(i)>>(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)))>Max_Z)
Max_Z = abs(real(Z_ACC_32(i)));
end
if (abs(imag(cZ_ACC_32(i)))>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)>>(Scale_Z), Nfft_out−1);
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<<(1+Log2Nfft)),0);
cphasor11 = Switch_real_imag(cphasor11);
cfs11 = Cmplx_Add_16_16 (phasor11, −sqrt(−1)*(1<<(Ntan−1)));
fs21 = CFO+ PI_over_FFT *i1;
fs22 = CFO +PI_over_FFT *i2;
if(fs21==0)
cfB1_32 = 1<<( Nfft_out+ Log2Nfft );
else
cfB1_32 = Cmplx_real_div_32_16 ((cfs11<<(Nfft_out−N1)),fs21);
end
if(fs22==0)
cfB2_32 = 1<<( Nfft_out+ Log2Nfft );
else
cfB2_32 = Cmplx_real_div_32_16 ((cfs11<<(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 >Max)
Max = Max_abs_real_cfB1;
end
if(abs Max_abs_imag_cfB1>Max)
Max = Max_abs_imag_cfB1;
end
if(Max_abs_real_cfB2 >Max)
Max = Max_abs_real_cfB2
end
if(Max_abs_imag_cfB2>Max)
Max Max_abs_imag_cfB2;
end
Scale_ratio = 15−(ceil_log2(Max)+1);
If (Scale>=0)
cfB1 = Cmplx_Saturate( (cfB1_32<<Scale_ratio), Nfft_out−2);
cfB2 = Cmplx_Saturate ((cfB2_32<<Scale_ratio),Nfft_out−2);
else
cfB1 = Cmplx_Saturate ((cfB1_32>>−Scale_ratio), Nfft_out−2);
cfB2 = Cmplx_Saturate ((cfB2_32>>−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	Unscaled Coeff 1	NA
	cfB2_32	32 bit complex	Unscaled 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)))>>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)>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>=0)
DENOM_2_32(0) = Cmplx_Saturate (DENOM_2_64(0)<<(Scale),31);
DENOM_2_32(1) = Cmplx_Saturate (DENOM_2_64(1)<< (Scale),31);
else
DENOM_2_32(0) = Cmplx_Saturate( DENOM_2_64(0)>>(−Scale),31);
DENOM_2_32(1) = Cmplx_Saturate (DENOM_2_64(1)>>(−Scale),31);
end
%complex/real division 64bit/32bit gives 32bit result
cfCM(0) = Cmplx_real_div_64_32 (cMul_NUM_cDENOM_64(0)>>N2 ,DENOM_2_32(0));
cfCM(1) = Cmplx_real_div_64_32 (cMul_NUM_cDENOM_64(1)>>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( (CFO<<1)*N_delta, 0);
cfCM_avg_rot_64 = Cmplx_Mul_32_16(cfCM_avg,cphasor_dn);
cfCM_avg_rot_32 = fCM_avg_rot_64>>(Ntan);
Scale_fCM = Ntan−1−N1+Scale_ratio−Scale−6−N2;
fCM_one_level_32 = 1<<(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) > Abs_imag_Numerator64)
Scale_fn =30 − ceil_log2(Abs_real_Numerator64);
else
Scale_fn =30 − ceil_log2(Abs_imag_Numerator64);
end
if(Scale_fn>=0)
cNUMERATORf_32= cNUMERATORf_64<<(Scale_fn);
else
cNUMERATORf_32= cNUMERATORf_64>>(−Scale_fn);
end
%Level Control for division scale denominator down to 16bit unsigned
Scale_fd =16 − ceil_log2(DENOMf_64);
if(Scale_fd >=0)
DENOMf_scaled_16= Cmplx_Saturate (DENOMf_64<<(Scale_fd),16);
else
DENOMf_scaled_16= Cmplx_Saturate (DENOMf_64>>(−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>=1)
% First Order Loop
Mue_1m = (16−Mue);
cg_ACC_32 = cg_exp_mteta <<(−Scale_g−Mue_bits);
cg_Delta_32 = (Cmplx_Mul_16_16 (cg_exp_mteta *cg_exp_mTeta_M_16))>>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  ))>>(−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>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 > 1<< (−scale_g_log) )
Inv_real_g_exp_mTeta_M_16 = Cmplx_real_div_32_16
(1<<(scale_inv),real_g_exp_mTeta_M_16));
Scale_theta = −14+Teta_bits−1+scale_delta;
If(Scale_theta>=0)
theta = Inv_real_g_exp_mTeta_M_16<<(Scale_theta);
else
theta = Inv_real_g_exp_mTeta_M_16>>(−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>=0)
rho = Rho_32<<(Scale_rho);
else
rho = Rho_32>>(−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>=0)
theta = real_g_exp_mTeta_M_16<<(Scale_theta);
else
theta = real_g_exp_mTeta_M_16>>(−Scale_theta);
end
If(Scale_rho>=0)
rho = (−imag_g_exp_mTeta_M_16)<<(Scale_rho);
else
rho = (−imag_g_exp_mTeta_M_16)>>(−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_i*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)< 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))>> (Scale_16(i) −NLog_In_Bits );
NSR_Default_32(i) = Scale_16(i) << (NLog_Out_Bits − 2 + NLog_In_Bits)+ ...
(Real_Mult_16_16(b,Frac_16(i))+a<<(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)>>(Scale_k_16 −NLog_In_Bits );
NSR_32 = Scale_k_16 << (NLog_Out_Bits−2 + NLog_In_Bits)+ ...
(Real_Mult_16_16(b,Frac_k_16)+a<<(NLog_In_Bits) );
if(NSR_32< 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 < 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 & 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) > 2P)
dPhase_32= dPhase_32− sign(dPhase_32)* 2P;
end
if (abs(dPhase_32 )>PI)
error_32 = abs(2PI − abs(dPhase_32));
else
error_32 = abs(dPhase_32);
end
if( error_32 < Phase_Th )
Indicator=1;
if(error_32 <Min_Err)
Min_Err = error;
NUM_OF_SYMS = L;
CP_LENGTH =CP;
end
end
end
end

Claims

1. A system for compensating for an imbalance between a first signal and a second signal, the system comprising: a first module operative to record, based on the first and second signals, a first frequency domain parameter and a second frequency domain parameter corresponding to each of a first tone, a second tone and a carrier frequency; anda second module operative to compute at least one time domain compensation parameter based on the first and second frequency domain parameters, wherein:the carrier frequency is a receiver carrier frequency;the first and second signals are received from a transmitter operative to transmit the signals using a transmitter carrier frequency;the receiver and transmitter carrier frequencies differ by a carrier frequency offsetthe first and the second frequency domain parameters correspond to one of the first and the second tones: the first frequency domain parameter corresponds to a first bin in a discrete-valued frequency spectrum;the second frequency domain parameter corresponds to a second bin in the spectrum;the first bin is contiguous with the second bin;the first module is operative to select the second bin based on the location in the spectrum of the first bin and based on the ratio of the offset, in radians, to Pi radians.

2. The system of claim 1 further comprising a third module operative to output a compensated signal based on the time domain compensation parameter.

3. The system of claim 1 wherein the at least one time domain compensation parameter comprises three time domain compensation parameters.

4. The system of claim 1 wherein, when there is a residual carrier frequency offset, the second module is operative to: compute an estimate of the residual frequency offset; andcompute the time domain compensation parameter based on the estimate.

5. The system of claim 1 wherein, when the first module is operative to record the first and second frequency domain parameters for each of a plurality of symbols that are received serially in time, the second module is operative to compute a first average value for the first frequency domain parameter and a second average value for the second frequency domain parameter, the average values based, respectively, on the first and second frequency domain parameters corresponding to the symbols.

6. The system of claim 1 wherein: the first module is a hardware module; andthe second module is a software module.

7. The system of claim 2 wherein: the first module is a hardware module;the second module is a software module; andthe third module is a hardware module.

8. The system of claim 1 wherein: the first module is a hardware module; andthe second module is a hardware module.

9. The system of claim 2 wherein the third module is a hardware module.