Reducing image spectral leakage due to I-Q imbalance

Abstract

Methods of reducing spectral leakage due to I-Q imbalance within a transmitter are disclosed. The method includes the transmitter receiving a data stream of coefficients of a multi-carrier modulation signal. The data stream is pre-processed by processing a set of the coefficients that correspond with selected notch frequencies to reduce the effects of I-Q imbalance of the transmitter, wherein at least a portion of the set of coefficients corresponds to non-symmetrical notch frequencies. The pre-processed data stream is multi-carrier modulated. The multi-carrier modulated pre-processed data stream is I-Q modulated before transmission.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows a prior art wireless network.

FIG. 2 shows one example of a multi-carrier modulation transmitter the can implement methods of pre-processing to reduce spectral leakage due to I-Q imbalance.

FIG. 3 shows a frequency spectrum corresponding to the coefficients of a multi-carrier modulated signal.

FIG. 4 is a flow chart of one example of a method of reducing spectral leakage due to I-Q imbalance within a transmitter.

FIG. 5 shows a transmit frequency spectrum depicting frequency notching in which a portion of the notched spectrum is symmetric about a center frequency and therefore not affected by I-Q imbalance, and another portion which is not symmetric about the center frequency is therefore affected by I-Q imbalance.

FIG. 6 shows another example of an embodiment of a transmitter that includes measurement of I-Q imbalance to be used by methods of reducing spectral leakage.

DETAILED DESCRIPTION

The invention includes transmitter preprocessing methods and transmitter systems that reduce spectral leakage. The transmitters provide for reduction of transmitter distortion (causing spectral leakage) that can reduce the effectiveness of frequency notching of the transmitter to avoid particular portion of frequency spectrum.

A primary source of spectral leakage of a wireless transmitter is from transmitter distortion caused by I-Q imbalance of an I-Q modulator used for direct frequency up-conversion within the transmitter. The use of direct frequency up-conversion radios is becoming increasingly popular because direct frequency up-conversion radios avoid the need for expensive intermediate frequency bandpass filters, and avoid the need for double up-conversion transmitter architectures. I-Q imbalance is a major source of distortion in direct conversion transmitters.

The notch depths achievable by baseband signal frequency notching is severely compromised without reducing or eliminating the effects of I-Q imbalance. Unmitigated I-Q imbalance can force the use of more frequency spectrum being wasted during transmission to successfully achieve the requirements of DAA.

FIG. 2 shows one example of an embodiment of a transmitter that can implement methods of reducing spectral leakage, and therefore, improve the effectiveness of baseband signal processing for avoiding/notching particular portions of frequency spectrum. The transmitter receives a data stream that includes data coefficients X_−N . . . X_N. The data stream coefficients are pre-processed by an I-Q pre-processor 210 that pre-distorts the data stream so that multi-carrier signals generated by the data stream coefficients have their I-Q imbalance suppressed at particular frequencies.

A multi-carrier modulator 220 receives the data coefficients and generates a multi-carrier signal from them. One embodiment includes 2N+1 carriers corresponding to the 2N+1 data stream coefficients. A common choice for the multi-carrier modulation is orthogonal frequency division multiplexing (OFDM) modulation.

The multi-carrier modulated signal is converted to an analog signal by a digital-to-analog converter (DAC) 230. The analog signal is I-Q modulated by an I-Q modulator 240. The I-Q modulator 240 used to frequency up-convert the analog signals before transmission includes an in-phase (I) chain and a quadrature-phase (Q) chain. Ideally, the Q chain up-converting signal (−sin(ωt)) is 90 degrees out of phase with the I chain up-converting signal (cos(ωt)). The two chains are amplified by gain elements g before being summed and transmitted. Ideally, the gain elements g of the two chains are identical.

FIG. 2 also shows another depiction of an I-Q modulator 240′ that includes I-Q gain and phase imbalances. An I-Q phase imbalance is shown by φ, and an I-Q gain imbalance is shown by γ (γ=1 implies no gain imbalance). One embodiment of the transmitter includes the φ and γ parameters being frequency dependent, and another embodiment includes the φ and γ parameters not being frequency dependent. The I-Q phase and I-Q gain imbalance are shown in the I-Q modulator. However, it is to be understood that the I-Q phase and I-Q gain imbalances can also be due to other components within the transmitter. For example, I and Q DACs and imperfect matching between the analog portions of the I and Q paths can also result in I-Q imbalance.

The transmitter of FIG. 2 includes pre-processing or pre-distorting of the data stream coefficients within the I-Q pre-processor which mitigates the I-Q gain and the I-Q phase imbalances of the transmitter. The result is that pre-processing of the data stream coefficients for avoiding specific portions of frequency spectrum is not hindered by I-Q imbalance to the extent it would be in the absence of the pre-processing. Several different methods of pre-processing can be implemented. But all of the methods can reduce the effects of I-Q imbalance on the selected notch frequencies of the specific portions of the frequency spectrum.

FIG. 3 shows one example of a frequency spectrum of a multi-carrier signal that is generated by the multi-carrier modulator 220. For this example, the multi-carrier signal has carriers at frequencies f_−N . . . f₀ . . . f_N, with corresponding complex coefficients (corresponding to each carrier's amplitude and phase) of X_−N . . . X₀ . . . X_N. Here, the center frequency f₀ corresponds to the local oscillator frequency of the I-Q modulator. The multi-carrier signal can be an OFDM signal. For OFDM signals, the output spectrum Y(f), which is the Fourier transform of y(t), has the property that Y_m=Y(f_m)=X_m. However, other types of multi-carrier signals can be used as well.

FIG. 4 is a flow chart of one example of a method of reducing spectral leakage due to I-Q imbalance within a transmitter. A first step 410 of the method includes the transmitter receiving a data stream of coefficients of a multi-carrier modulation signal. A second step 420 includes pre-processing the data stream by processing a set of the coefficients that correspond with selected notch frequencies to reduce the effects of I-Q imbalance of the transmitter, wherein at least a portion of the set of coefficients corresponds to non-symmetrical notch frequencies. A third step 430 includes multi-carrier modulating the pre-processed data stream. A fourth step 440 includes I-Q modulating the multi-carrier modulated pre-processed data stream.

The set of coefficients of the data stream includes coefficients that correspond to non-symmetrical notch frequencies. However, the set can additionally include coefficients that correspond to symmetrical notch frequencies as well.

FIG. 5 shows a transmit frequency spectrum depicting frequency notching in which a portion of the notched spectrum is symmetric about a center frequency, f₀, and another portion which is not symmetric about the center. As discussed in the next section, the I-Q imbalance will not affect symmetric notch frequencies because the coefficients will be symmetrically nulled. The notch frequencies include a portion 510 of the frequency spectrum that is symmetrical about center frequency f₀, and a portion 520 of the frequency spectrum that is not symmetrical to the center frequency f₀. Asymmetrical refers to notch frequencies f_m that do not have a corresponding notch frequency f_−m. Achieving the desired symmetrical and asymmetrical notches in the output spectrum can be achieved by nulling the corresponding data stream coefficients and further pre-processing only those corresponding to the asymmetric frequencies.

Mathematical Representations of the Effects of I-Q Imbalance

A transmission signal y(t) of the transmitter of FIG. 2 is ideally given as:

y(t)=g[X₁(t)cos(ωt)−X_Q(t)sin(ωt)].

However, the phase and gain imbalances can be included and depicted as:

y(t)=g[X₁(t)cos((ωt)−X_Q(t)γsin(ωt+φ)]

where γ-1 represent IQ gain imbalance (y=1 means no gain imbalance) and φ represents phase imbalance.

For ease of presentation, but without loss of generality, it is initially assumed that notching is only required near a single frequency f_m. It is to be understand that the index m can, in fact, be allowed to represent multiple frequencies. Letting X_m be the coefficient (component) of the notch portion of the signal corresponding to frequency f_m and X_−m, be the coefficient (component) corresponding to the mirror frequency f_−m. The output of the baseband processing at the notch frequency due to I-Q distortion can be represented by:

$Y_m=c_1X_m+c_2^{*}X_{-m}^{*}$ $Y_m=\left(\frac{1+\gamma ·\cos \left(\phi \right)}{2}+j\frac{\gamma \sin \left(\phi \right)}{2}\right)X_m+{\left(\frac{1-\gamma ·\cos \left(\phi \right)}{2}+j\frac{\gamma \sin \left(\phi \right)}{2}\right)}^{*}X_{-m}^{*}$

where * means the complex conjugate and Y_m is the output spectrum at f_m.

This mathematical representation can be derived by applying the I-Q imbalance to generate y(t) and observing its Fourier transform. That is, first passing x(t) (input data stream) through an I-Q modulator that includes the amplitude and phase imbalances. Next, y(t) is demodulated with an ideal I-Q demodulator. Finally, a discrete fourier transform is taken on the demodulated y(t). It can be observed from this representation that setting X_m to zero or another value that ignores the contribution of X_−m to the output spectrum Y_m will not produce a satisfactory notch. For example, zeroing just the X_m coefficient results in leakage due to I-Q imbalances:

$Y_m={\left(\frac{1-\gamma ·\cos \left(\phi \right)}{2}-j\frac{\gamma \sin \left(\phi \right)}{2}\right)}^{*}X_{-m}^{*}.$

One example of data stream pre-processing to reduce the effects of I-Q imbalance includes eliminating leakage by nulling both X_m and X_−m. The processing of selected coefficients includes pre-processing of −m and m coefficients of the data stream, effecting a notch at a transmit carrier frequency f_m. This approach works because leakage due to I-Q imbalance does not affect the output spectrum at frequencies corresponding to symmetrically nulled coefficients X_m and X_−m. However, by nulling both coefficients, the result is a notch in y(f) at f_−m as well as f_m. That is, this approach suffers the drawback that the amount of frequency spectrum which is notched is doubled.

Another example of data stream preprocessing includes selecting a value for at least one data stream coefficient X_m to effectively cancel Y_m. The leakage at the transmit frequency f_m is reduced by canceling the leakage at the transmit frequency f_m by proper selection of the value of X_m. For example, Y_m of the above equation can be zeroed by setting:

$X_m=-\left(\frac{c_2^{*}}{c_1}\right)X_{-m}^{*}=-\left(\frac{1-{\gamma }^2-\mathrm{j2\gamma }\sin \phi }{1+2\gamma \cos \phi +{\gamma }^2}\right)X_{-m}^{*}.$

By determining or estimating γ and φ, Y_m can be canceled by setting X_m to this value. Algorithms used to determine γ and φ are known in the art. The benefit of this approach is that X_−m can be used to transmit useful information while X_m is being used to cancel leakage from X_−m due to I-Q imbalance, thereby providing a notch in Y(f) at f_m.

One embodiment of the above-example includes selecting a value for at least one data stream coefficient X_m reducing leakage at transmit frequency f_m due to data stream coefficient X_−m. As stated, the leakage Y_m is caused by I-Q imbalance of the transmitter.

One embodiment includes γ and φ being independent of frequency, and another embodiment includes γ and φ being frequency dependent.

Setting X_m as described can be undesirable because complete notching of Y_m requires X_m to satisfy the equation. However, just as I-Q imbalance causes leakage of X_−m into Y_m, I-Q imbalance also causes leakage X_m into Y_−m. Therefore, proper selection of X_m according to the method just described, rather than just zeroing it, out has of the side effect of introducing additional noise elements into Y_−m, effectively lowering the SNR on the portion of the frequency spectrum that is not being notched.

Another example of data stream preprocessing includes choosing X_m to effect notching of Y_m just enough to meet any DAA requirements. This can provide the benefit of limiting the SNR degradation of Y_−m. The pre-processing of the data stream includes selecting a value for at least one data stream coefficient X_m reducing leakage at transmit frequency f_m due to data stream coefficient X_−m the leakage. This can be accomplished by setting:

$X_m=-\beta \left(\frac{1-{\gamma }^2-\mathrm{j2\gamma }\sin \phi }{1+2\gamma \cos \phi +{\gamma }^2}\right)X_{-m}^{*}$

where 0<β<1 is chosen as the minimum value such that the notching for Y_m meets the avoidance requirement.

FIG. 6 shows another example of an embodiment of a transmitter that includes measurement of I-Q imbalance to be used by methods of reducing spectral leakage. The I-Q imbalance is determined by an I-Q imbalance measurement block 650. The measurements are feedback to the I-Q pre-processing 610. The measurements can be of γ and φ, or of leakage constants c₁ and c₂ as given above, and will be described next.

The pre-processing of the data stream is based at least in part on the measured I-Q imbalance. The selected value of a data stream coefficient X_m can be based at least in part on measured I-Q imbalance. One embodiment includes measuring the I-Q imbalance at a single frequency. Another embodiment includes measuring the I-Q imbalance at each notch frequency f_m or it's corresponding mirror frequency f_−m.

Based on the above equations, it can be determined that the spectral leakage is linearly proportional to the data coefficients X_m and X_−m This suggests that rather than estimating γ and φ, the leakage can be estimated directly during a calibration mode, and then canceled out during normal operation using the leakage estimates. As noted above, the output of the baseband processing at the notch frequency due to I-Q distortion can be represented by:

Y _m =c ₁ X _m+(c₂X_−m)*

This embodiment can include a calibration mode in which c₁ and c₂ are determined. A first step of the calibration can include setting X_m=0, measuring Y_m^c1=c₁X_m, and setting the estimate {tilde over (c)}₁=Y_m^c1/X_m. A second step can include setting X_m=0, measuring Y_m^c2=(c₂X_−m)*, and setting the estimate {tilde over (c)}₂=(Y_m^c2)*/X_−m. During normal operation, a third step can be executed that includes setting

$X_m=-\frac{{\left({\tilde{c}}_2X_{-m}\right)}^{*}}{{\tilde{c}}_1}.$

Generally, this embodiment includes determining X_m by measuring spectral leakage at f_m due to X_−m when X_m is zeroed, and by measuring spectral leakage at f_m due to X_m when X_−m is zeroed.

In general, the choice of X_m=0 does not provide the best notch around frequency f_m (even without IQ imbalance). This is due to energy leakage from adjacent tones arising from the digital to analog conversion in the transmitter. To provide a better notch, the values of X_m maybe chosen to minimize some function (generally the choice will be to minimize some measure of the energy surrounding the notch frequencies). Methods for doing so have been discussed in the art. However, in the presence of IQ imbalance, such methods for choosing of X_m have to be modified.

To start, the previous equation representing the spectral leakage due to I-Q imbalance is re-written with the introduction of an operator IQ_m(, ). IQ_m represents the effect of I-Q imbalance on the indices m but can, in general, be any bi-linear operator which is surjective in the first argument:

Y _m =IQ _m(X_m,X_−m)≡c₁X_m+(c₂X_−m)*.

The subscript m is replaced with {right arrow over (m)}={m₁,m₂, . . . , m_n} for generalization so that multiple tones can be explicitly treated simultaneously. In this notation, let −{right arrow over (m)}={−m₁,−m₂, . . . , −m_n} be the mirror image of m with m=0 corresponding to frequency f₀, the middle of the spectrum. Now, the output equation can be replaced with:

Y _{{right arrow over (m)}} =IQ _{{right arrow over (m)}}(X_{{right arrow over (m)}},X_{−{right arrow over (m)}})≡c₁X_{{right arrow over (m)}}+(c₂X_{−{right arrow over (m)}})*.

Now, define the operator P which maps the X coefficients to some energy function in the notch frequencies at a subset of the frequencies indexed by {right arrow over (m)}. Also, define ⁻P as the operator mapping the X coefficients to the energy function in the subset of mirror frequencies indexed by −{right arrow over (m)}. Finally, define:

P_{{right arrow over (m)}} as the restriction of P to the range of {right arrow over (m)}

^−P _{{right arrow over (m)}} as the restriction of ⁻P to the range of −{right arrow over (m)}

g as the vector of coefficients X with values at indices {right arrow over (m)} replaced by 0.

Then the energy in the notch frequencies can be expressed using the last equation,

IQ _{{right arrow over (m)}}(P_{{right arrow over (m)}}X_{{right arrow over (m)}}+Pg, ⁻P_{{right arrow over (m)}}X_{{right arrow over (m)}}+⁻Pg)=(P_{{right arrow over (m)}}X_{{right arrow over (m)}}+Pg)+(c₂(⁻P_{{right arrow over (m)}}X_{{right arrow over (m)}}+^−Pg))*.

By minimizing this expression, the energy in the notch frequencies is minimized. This can be done by choosing X_{{right arrow over (m)}} to satisfy

c ₁ P _{{right arrow over (m)}} X _{{right arrow over (m)}}+(c₂⁻P_{{right arrow over (m)}}X_{{right arrow over (m)}})*=−(c₁Pg+(c₂⁻Pg)*).

This equation can be solved by making the real part of the left-hand side equal the real part of the right-hand side, and similarly for the imaginary parts. For a complex variable, z, we have z+z*=2 Re{z} and z−z=j2Im{z}. Therefore, defining a≡−2 Re{c₁Pg +(c₂⁻Pg)*} and b≡−j2Im{c₁⁻Pg+(c₂⁻Pg)*}, the last equation can be split into separate equations for the real and imaginary parts as

c ₁ P _{{right arrow over (m)}} X _{{right arrow over (m)}}+(c₂⁻P_{{right arrow over (m)}}X_{{right arrow over (m)}})*+(c₁P_{{right arrow over (m)}}X_{{right arrow over (m)}})*+c₂⁻P_{{right arrow over (m)}}X_{{right arrow over (m)}}=a

c ₁ P _{{right arrow over (m)}} X _{{right arrow over (m)}}+(c₂⁻P_{{right arrow over (m)}}X_{{right arrow over (m)}})*−(c₁P_{{right arrow over (m)}}X_{{right arrow over (m)}})*−c₂⁻P_{{right arrow over (m)}}X_{{right arrow over (m)}}=b.

Let C₁≡c₁P_{{right arrow over (m)}}+c₂⁻P_{{right arrow over (m)}} and C₂≡c₁P_{{right arrow over (m)}}−c₂⁻P_{{right arrow over (m)}}. Then the last two equations become

C ₁ X _{{right arrow over (m)}}+(C₁X_{{right arrow over (m)}}Y)*=a

C ₂ X _{{right arrow over (m)}}−(C₂X_{{right arrow over (m)}})*=b.

Defining the matrix

$C=\left[\begin{array}{cc}{C}_1& C_1^{*}\\ C_2& -C_2^{*}\end{array}\right]$

the last two equations can be written more compactly as

$C\left[\begin{array}{c}{X}_{\stackrel{\_}{m}}\\ X_{\stackrel{\_}{m}}^{*}\end{array}\right]=\left[\begin{array}{c}a\\ b\end{array}\right]$

One solution to this equation is given by the Moore-Penrose pseudo-inverse:

$\left[\begin{array}{c}{X}_{\stackrel{\_}{m}}\\ X_{\stackrel{\_}{m}}^{*}\end{array}\right]={\left(C^TC\right)}^{-1}C^T\left[\begin{array}{c}a\\ b\end{array}\right],$

where T indicates Hermitian transpose. Other solutions corresponding to other energy metrics are possible.

Although specific embodiments of the invention have been described and illustrated, the invention is not to be limited to the specific forms or arrangements of parts so described and illustrated. The invention is limited only by the appended claims.

Claims

1. A method of reducing spectral leakage due to I-Q imbalance within a transmitter, comprising: the transmitter receiving a data stream of coefficients of a multi-carrier modulation signal;pre-processing the data stream by processing a set of the coefficients that correspond with selected notch frequencies to reduce the effects of I-Q imbalance of the transmitter, wherein at least a portion of the set of coefficients corresponds to non-symmetrical notch frequencies;multi-carrier modulating the pre-processed data stream;I-Q modulating the multi-carrier modulated pre-processed data stream.

2. The method of claim 1, wherein the set of coefficients additionally corresponds to symmetrical notch frequencies.

3. The method of claim 1, further comprising: measuring I-Q imbalance of the transmitter; and whereinthe pre-processing of the data stream is based at least in part on the measured I-Q imbalance.

4. The method of claim 3, wherein measuring I-Q imbalance of the transmitter comprises measuring the I-Q imbalance at a single frequency.

5. The method of claim 2, further comprising pre-processing data stream coefficients corresponding to symmetric notch frequencies to affect a notch at a transmit carrier frequency fm.

6. The method of claim 1, wherein pre-processing of the data stream comprises: selecting a value for at least one data stream coefficient Xm reducing leakage at transmit frequency fm due to data stream coefficient X−m, the leakage being caused by I-Q imbalance of the transmitter.

7. The method of claim 6, wherein the measured I-Q imbalance is frequency dependent.

8. The method of claim 6, wherein the measured I-Q imbalance is frequency independent.

9. The method of claim 6, wherein selecting the value of the data stream coefficient Xm is based at least in part on measured I-Q imbalance.

10. The method of claim 6, wherein the value for Xm is determined by measuring spectral leakage at fm due to X−m when Xm is zeroed, and by measuring spectral leakage at fm due to Xm when X−m is zeroed.

11. The method of claim 6, wherein reducing the leakage at the transmit frequency fm comprises canceling the leakage at the transmit frequency fm by proper selection of the value of Xm.

12. A transmitter that includes means for reducing I-Q imbalance of the transmitter, comprising: the transmitter receiving a data stream of coefficients of a multi-carrier modulation signal;means for pre-processing the data stream by processing a set of the coefficients that correspond with selected notch frequencies to reduce the effects of I-Q imbalance of the transmitter, wherein at least a portion of the set of coefficients corresponds to non-symmetrical notch frequencies;a multi-carrier modulator for multi-carrier modulating the pre-processed data stream;an I-Q modulator for I-Q modulating the multi-carrier modulated pre-processed data stream.

13. The transmitter of claim 12, wherein the set of coefficients additionally corresponds to symmetrical notch frequencies.

14. The transmitter of claim 12, further comprising: means for measuring I-Q imbalance of the transmitter; and whereinthe pre-processing of the data stream is based at least in part on the measured I-Q imbalance.

15. The transmitter of claim 14, wherein measuring I-Q imbalance of the transmitter comprises measuring the I-Q imbalance at a single frequency.

16. The transmitter of claim 13, further comprising pre-processing data stream coefficients corresponding to symmetric notch frequencies to affect a notch at a transmit carrier frequency fm.

17. The transmitter of claim 12, wherein pre-processing of the data stream comprises: selecting a value for at least one data stream coefficient Xm reducing leakage at transmit frequency fm due to data stream coefficient X−m the leakage being caused by I-Q imbalance of the transmitter.

18. The transmitter of claim 17, wherein selecting the value of the data stream coefficient Xm is based at least in part on measured I-Q imbalance.

19. The transmitter of claim 17, wherein the value for Xm is determined by measuring spectral leakage at fm due to X−m when Xm is zeroed, and by measuring spectral leakage at fm due to Xm when X−m is zeroed.

20. The transmitter of claim 17, wherein reducing the leakage at the transmit frequency fm comprises canceling the leakage at the transmit frequency fm by proper selection of the value of Xm.