DE10234166B4 - A method of estimating the I / Q imbalance of an OFDM signal - Google Patents

Abstract

Method for estimating the I / Q imbalance of an OFDM signal (r (t)) consisting of several sub-signals (Kmod al, k), which are transmitted via one of several subcarriers with different subcarrier frequencies (ωk) via a transmission channel (9), the one unknown transfer function (H (ω)) has been transferred, with the following process steps: Forming a cost function (L) depending on the values (Hk) of the transfer function (H (ω)) of the transmission channel (9) at the subcarrier frequencies (ωk ) and depending on a parameter (ΔQ) that describes the I / Q imbalance, - partial differentiation of the cost function (L) according to the values (Hk) of the transfer function (H (ω)) of the transmission channel (9) at the subcarrier frequencies (ωk) and according to the parameter (ΔQ), which describes the I / Q imbalance, - Forming a system of equations from the assumption that all partial differentials of the cost function (L) are zero and therefore a minimum of the cost function (L) is present, and - releasing the slip system, where the system of equations is non-linear and the solution of the system of equations is iterative, where the ...

Description

The invention relates to a method for estimating the I / Q imbalance of an OFDM signal.

For data transmission, the so-called OFDM (Orthogonal Frequency Division Multiplex) method is used in modern telecommunication systems. In this principle, the transmission takes place via a plurality of orthogonal subcarriers. This OFDM method is described, for example, in more detail in Hermann Rohling, Thomas May, Karsten Brüninghaus and Rainer Grünheld, Broadband OFDM Radio Transmission for Multimedia Applications, PROCEEDINGS OF THE IEEE, Vol. 87, No. 10, October 1999, p 1778 ff.

In the OFDM method, to increase the data rate, each subcarrier is usually complex, i. H. modulated with an inphase (I) component and a quadrature phase (Q) component. In the DVB-T (Digital Video Broadcasting) system, for example, 1705 or 6817 subcarriers are modulated with the modulation type 64 QAM, that is, with 8 combinable values of the I component and the Q component, which results in 64 subcarriers in the subcarrier Constellation diagram leads. Due to linear or non-linear distortions, for example in the filters of the final stage of the transmitter, there is the so-called I / Q imbalance. The Q component is no longer completely orthogonal to the I component due to a phase shift. In addition, the I-component and the Q-component may be subjected to different gains. This manifests itself in a crosstalk of a particular subcarrier to another subcarrier in the mirror image channel of the baseband, i. H. the subcarrier with the channel number + k overrides the subcarrier with the channel number -k. The I / Q imbalance should therefore be determined on the receiver side for measurement purposes.

The technical background is still on the WO 00/77961 A1 which proposes a correlation method for estimating symbol timings and frequency synchronization in an OFDM signal. However, a method for estimating the I / Q imbalance is not apparent from this document. The present invention presupposes that the symbol times and the subcarrier frequencies as well as possibly a time offset are already estimated with sufficient accuracy before the inventive I / Q imbalance estimation is used.

The document Chen, P. Kobayashi, H .: Maximum Likelihood Channel Estimation and Signal Detection for OFDM Systems, IEEE International Conference on Communications, 28 April-2 May 2002, Vol. 3, pp. 1640-1645 discloses a method of estimating the Parameters of a transmission channel and the transmitted symbols in an OFDM transmission system. For this purpose, a cost function is set up as a function of the transfer function of the channel and the transmitted symbols, a system of equations is set up by partial differentiation and equal zeroing, and the equation system is solved iteratively.

The document Rezeanu, S.-C .; Zimmer, R. E .; Wickert, M.A .: Joint Maximum-Likelihood Parameter Estimation for Burst DS Spread Spectrum Transmission, IEEE Transactions to Communications, vol. 45, no. 2, February 1997, pages 227-238 shows a method by which a plurality of frequency, phase and time offset indicative parameters in a burst DS spread spectrum signal can be determined using the maximum likelihood algorithm ,

The document "A Novel Low Complexity Technique to Reduce Nonlinear Distortion Effects in OFDM Systems," D. Dardari, V. Tralli, A. Vaccari, 1998, IEEE International Symposium on Human Resources, Indoor and Mobile Radio Communications, 8-11 September 1998, vol 2, pp. 795-800 discloses a method for reducing nonlinear distortions in OFDM systems. It uses for this purpose virtual carriers which are modulated by means of known symbols in order to obtain a transmission signal of low envelope variation.

Martin Buchholz, Andreas Schuchert, Ralph Hasholzner, 2000, IEEE International Caracas Conference on Devices, Circuits and Systems, 15-17 March 2000, p. T65 / 1-T65 / 6 discloses an OFDM system employing multi-carrier modulation. The I / Q imbalance is examined.

The invention has for its object to provide an optimal method for estimating the I / Q imbalance, which requires a small implementation effort or a low computation time.

The method is solved by the features of claim 1.

According to the invention, a cost function is formed as a function of the transfer function of the transmission channel at the subcarrier frequencies and in dependence on a parameter which describes the I / Q imbalance. After a partial differentiation according to the transfer function of the transmission channel at the subcarrier frequencies and after the parameters describing the I / Q imbalance, a system of equations is set up. The equation system is nonlinear, so that the solution is iterative. In this case, the parameter describing the I / Q imbalance is set to zero as the starting value and first the values of the transfer function of the transmission channel at the subcarrier frequencies are calculated. Subsequently, the parameter describing the I / Q imbalance is calculated as a function of the values of the transmission function of the transmission channel at the subcarrier frequencies. Further iteration steps increase the accuracy.

The subclaims relate to advantageous developments of the invention.

The resulting equation system is not linear, so that the solution is preferably iterative. In this case, the parameter describing the I / Q imbalance is set to zero as the starting value and first the values of the transfer function of the transmission channel at the subcarrier frequencies are calculated. Subsequently, the parameter describing the I / Q imbalance is calculated as a function of the values of the transmission function of the transmission channel at the subcarrier frequencies. By further iteration steps, the accuracy can be increased.

The invention will be described in more detail by way of example with reference to the drawing. In the drawing show:

1 a block diagram for explaining the invention underlying signal processing;

2 the constellation diagram of the received OFDM symbols for a first example with ΔQ = 0.1; and

3 the constellation diagram for the received OFDM symbols for a second example with ΔQ = 0.1 · e ^{j · 45} .

1 shows a block diagram of the signal processing of the method according to the invention underlying model.

multzliziert. Die Normierungskonstante K_mod ist erforderlich, da unterschiedliche Modulationsalphabete für die einzelnen Subträger vorkommen können. k steht dabei für den Kanal- bzw. Subträger-Index. l steht für den Zeitindex des OFDM-Symbols. Es sind N + 1 Kanäle bzw. Subträger vorhanden. Die kleinste Subträgerfrequenz ist ω_-N/2 die größte Subträger-Frequenz ist ω_N/2. Die Multiplikation erfolgt in entsprechenden Multiplizierern 2_-N/2 bis 2_N/2. In einem Addierer 3 werden die Signale der Kanäle 3_-N/2 bis 3_N/2 zu dem gemeinsamen Sendesignal s₁(i) addiert.In a transmitting device l, the error-free, ie I / Q imbalance-free transmission signal s ₁ (t) is generated. For this purpose, the transmitted OFDM symbols a _{l, k} , which are multiplied by a normalization constant K _mod , so that the subcarrier power for each subcarrier is equal to one, with zuororigen Subtragern

multzliziert. The normalization constant K _mod is necessary since different modulation alphabets can occur for the individual subcarriers. k stands for the channel or subcarrier index. l stands for the time index of the OFDM symbol. There are N + 1 channels or subcarriers. The smallest subcarrier frequency is ω _{-N / 2,} the largest subcarrier frequency is ω _{N / 2} . The multiplication takes place in corresponding multipliers 2 _{-N / 2} to 2 _{N / 2} . In an adder 3 the signals of the channels 3 _{-N / 2} to 3 _{N / 2 are added} to the common transmission signal s ₁ (i).

In the I / Q imbalance unit 5 the generation of the I / Q imbalance is modeled. In fact, this I / Q imbalance is caused, for example, in the filters of the transmitter by linear or non-linear distortions. In the in 1 the model shown, the real part of the transmission signal s ₁ (t), ie the I-component, unchanged to the adder 6 transmitted or in the schematic representation of 1 in a multiplier 7 multiplied by I: = 1, where the added imaginary part is 0. The imaginary part of the transmission signal s ₁ (t), ie the Q-component, is in a multiplier 8th but multiplied by a constant value Q: = 1 + ΔQ, where the quadrature error ΔQ is a complex number. The output of the multiplier 8th becomes the adder 6 fed. Through this model of the I / Q imbalance unit 5 Any I / Q imbalance can be modeled.

The transmission signal s ₂ (t) subject to the I / O imbalance subsequently becomes the transmission channel 9 fed. The transmission channel 9 has an impulse response h (t) or a Fourier-transformed transfer function H (ω) represented by the block 10 , This produces the signal s ₃ (t). In a subsequent adder 11 a noise signal n (t) is added. This at the end of the transmission channel 9 Resulting, noisy, distorted and the I / Q imbalance subject receive signal r (t) is a receiving unit 12 fed.

für den k-ten Kanal für jeden der insgesamt N + 1 Empfangskanäle 14_–N/2 bis 14_N/2 multipliziert und in Integrierern 15_–N/2 bis 15_N/2 über den nutzbaren Teil T des OFDM-Symbols integriert. In den Integrierern wird für den k-ten Kanal 14_k folglich berechnet:

wodurch sich das an dem Abtastzeitpunkt l·T₅ empfangene OFDM-Symbol des k-ten Kanals 14_k ergibt. Dabei bedeutet T₅ die Symbolperiode. In multipliers 13 _{-N / 2} to 13 _{N / 2} , the received signal r (t) becomes the subcarrier

for the k-th channel for each of the total N + 1 receive channels 14 _{-N / 2} to 14 _{N / 2} multiplied and integrated in integrators 15 _{-N / 2} to 15 _{N / 2} over the usable part T of the OFDM symbol. In the integrators, _{k k is} therefore calculated for the k th channel as follows:

whereby the OFDM symbol of the k-th channel 14 _k received at the sampling time l · T ₅ results. Where T _{5 is} the symbol period.

a_l,k

gesendetes OFDM-Symbol

EVM

EVM (Error Vector Magnitude)

EVM_k

EVM vom Kanal k

l

Symbol-Index l = [l, nof_Symbols]

nof_Symbols

Symbolanzahl

H_k

Kanalfrequenzgang vom Kanal k, d. h. der Wert der Übertragungsfunktion H(ω) des Übertragungskanals 9 an der Subträgerfrequenz ω = ω_k des k-ten Kanals bzw. des k-ten Subträgers

h(t)

Impulsantwort des Übertragungskanals 9

k

Kanal-Index, k = [–N/2, N/2]

K_mod

Normierungskonstante, damit die Subträger-Leistung gleich Eins ist

n(t)

AWGN(Additive White Gaussian Noise)-Rauschen

r_l,k

empfangenes OFDM-Symbol

r(t)

Empfangssignal

s(t)

Sendesignal

T

Dauer des Nutzbereichs (useful Part) des OFDM-Symbols

T_s

Symbolperiode

The following is a mathematical description of this model and a mathematical description of the method according to the invention. The following symbols are used:

a _{l, k}

sent OFDM symbol

EVM

EVM (Error Vector Magnitude)

EVM _k

EVM from channel k

l

Symbol index l = [l, nof_symbols]

nof_Symbols

Symbol number

H _k

Channel frequency response from channel k, ie the value of the transfer function H (ω) of the transmission channel 9 at the subcarrier frequency ω = ω _{k of} the k-th channel or of the k-th subcarrier

h (t)

Impulse response of the transmission channel 9

k

Channel index, k = [-N / 2, N / 2]

K _mod

Normalization constant, so that the subcarrier power is equal to one

n (t)

AWGN (Additive White Gaussian Noise) noise

r _{l, k}

received OFDM symbol

r (t)

receive signal

s (t)

send signal

T

Useful part of the OFDM symbol

T _s

symbol period

The following symbol alphabets are possible, for example:

The factor K _mod has the purpose of determining the average subcarrier power in each modulation K 2 / mod · E {| a _{l, k} | ² } = 1 (2) to normalize to one. This results in the modulation-dependent factor K _mod

multipliziert. Based on 1 the baseband model of an OFDM transmission has been described considering the I / Q imbalances. Since the synchronization has already been performed in the T / Q imbalance estimation, an ideal synchronization is assumed in the model. The ideal transmission signal is designated s ₁ (t). The real part of s ₁ (t) is with I = 1 (4) and the imaginary part with

multiplied.

• 1. Die Definition I = 1 ist sinnvoll, weil auch der Kanal h(t) geschätzt wird. Folglich ist in h(t) bereits die Verstärkung (Gain) und auch die Phase enthalten. Würde I allgemein definiert werden, entstünde ein überbestimmtes Gleichungssystem.
• 2. Der Parameter Q ist im allgemeinen komplex. Im fehlerfreien Fall ist Q = 1. Der Betrag von Q gibt die Verstärkung (Gain) und die Phase von Q gibt den Phasenversatz der Quadraturzweiges (Imaginärteil von s₁(t)) gegenüber der Sollphase an.
• 3. Wie die nachfolgenden Herleitungen zeigen werden, empfiehlt sich die Definition von ΔQ in Gleichung (5). Hierbei handelt es sich nicht um eine Einschränkung, sondern lediglich um eine Definition.
• 4. Weiterhin hat ΔQ auch eine anschauliche Bedeutung: Es wird gezeigt, daß in jedem Punkt des Konstellations (Constellation)-Diagramms nochmals das Konstellations (Constellation)-Diagramm multipliziert mit ΔQ auftritt (siehe 2 und 3).

These definitions are chosen for the following reasons:

• 1. The definition I = 1 makes sense, because also the channel h (t) is estimated. Consequently, h (t) already contains the gain and also the phase. If I were defined in a general way, an overdetermined system of equations would result.
• 2. The parameter Q is generally complex. In the error-free case, Q = 1. The magnitude of Q gives the gain and the phase of Q indicates the phase offset of the quadrature branch (imaginary part of s ₁ (t)) from the target phase.
• 3. As the following derivations show, the definition of ΔQ in equation (5) is recommended. This is not a limitation, it's just a definition.
• 4. Furthermore, ΔQ also has a descriptive meaning: It is shown that in each point of the constellation (Constellation) diagram again the constellation (Constellation) diagram multiplied by ΔQ occurs (see 2 and 3 ).

beschrieben. Das Sendesignal nach den I/Q-Imbalances ergibt sich durch s₂(t) = I·Re{s₁(t)) + jQ·Im{s₁(t)} Corresponding 1 the error-free transmission signal is through

described. The transmission signal after the I / Q imbalances results from s ₂ (t) = I * Re {s ₁ (t)) + jQ * In {s ₁ (t)}

By substituting equations (4) and (5) one obtains s ₂ (t) = Re {s ₁ (t)} + j (1 + ΔQ) In {s ₁ (t)} = = s ₁ (t) + jΔQ · Im {s ₁ (t)}

By inserting In {s ₁ (t)} = 1 / 2j · [s ₁ (t) - s ^* ₁ (t)] surrendered s ₂ (t) = s ₁ (t) + ΔQ / 2 * [s ₁ (t) -s ^* ₁ (t)] = = (1 + ΔQ / 2) * s ₁ (t) -ΔQ / 2 s ^* ₁ (t)

After inserting equation (6) one obtains

The output signal of the channel results in the interesting ISI (Inter Symbol Interference) free range

The substitution k → -k in the second expression yields

Due to the orthogonality of the subcarriers, r _{1, k} can be derived directly from this representation r _{l, k} = (1 + ΔQ / 2) * K _mod * a _{l, k} * H _k -ΔQ / 2 * K _mod * a ^* _{1, -k} * H _k + n _{1, k} (7) be determined. The above point 4 can be confirmed from equation (7): In addition to the desired symbols a _{l, k} , if there is an I / Q imbalance ΔQ, the symbols a _{l, -k of} the opposite subcarrier are discussed. Consequently, in each Constellation point (point in the constellation diagram), another Constellation diagram multiplied by ΔQ is created.

ergibt sich die kompaktere Darstellung gemäß r_l,k = K_mod·a_l,k·H_k + ΔQ·K_mod·aa_l,k·H_k + n_l,k (9) With definition of the symbol sequence

results in the more compact representation according to r _{l, k} = K _mod · a _{l, k} · H _k · K + .DELTA.Q _mod · aa _{l, k} · H _k n + _{l, k} (9)

For the estimation of the I / Q imbalance, see k ε k _pilots , k _Data = [-N / 2, N / 2] used both the pilot and the data subcarriers. Note that the factor K _mod is different for the pilot and data subcarriers (subcarriers) unless BPSK is used, as shown by equation (3).

It can be seen from equation (9) that an optimal estimation is possible only if H _{k is} collectively estimated according to the invention as ΔQ. In principle, of course, the already existing estimates Ĥ _k used, but what a suboptimal estimate ΔQ ^ would lead. The shared estimation according to the invention pays off especially with large I / Q imbalances ΔQ.

In the maximum likelihood estimation used here, a cost function, in the exemplary embodiment, the so-called log-likelihood function, L (.DELTA.Q ~, H ~ _k) = Σ l kΣ | r _{l, k} - K · a _{mod l, k} · H _k ~ - ~ .DELTA.Q · K · _mod aa _{l, k} · H ~ _k | ² (10) be minimized. The derivative to H _K yields

After solving this equation one obtains

Sample: Substituting Equation (9) results in a trouble-free case

Correspondingly, the derivative leads to ΔQ

It will be appreciated that by substituting equation (11) into equation (12), an equation is produced whose numerator polynomial is the highest power ΔQ ^ ³ and its denominator polynomial is the highest power ΔQ ^ ⁴ has. Consequently, it is ΔQ ^ not calculable analytically. For this reason, the equation system is solved iteratively. It will be shown below that the numerical complexity is low and the method always converges.

The resolution of equation (12) after ΔQ ^ results

Sample: Substituting Equation (9) results in a trouble-free case

• 1. Initialisiere ΔQ ^ = 0
• 2. Berechne für alle K = [–N/2, N/2] jeweils den Kanalschätzwert H ^_K nach Gleichung (11)
• 3. Berechne ΔQ ^ nach Gleichung (13)
• 4. Falls eine weitere Iteration notwendig ist, springe erneut zu 2.

The iterative solution is obtained by the following procedure:

• 1. Initialise ΔQ ^ = 0
• 2. Calculate the channel estimate for each K = [-N / 2, N / 2] H ^ _K according to equation (11)
• 3. Calculate ΔQ ^ according to equation (13)
• 4. If another iteration is necessary, jump to 2 again.

The numerical effort of the iterative solution is very small if all partial sums in equation (11) and equation (13) are calculated in advance:

The decomposition of equation (11) into partial sums yields

Accordingly, the decomposition of equation (13) into partial sums yields:

und der Quadraturfehler

berechnet. Arg ist die Argument- oder Arcus-Tangens-2-Funktion, die den Winkel φ der komplexen Zahl 1 + ΔQ ^ = A·e^jφ ergibt.Usually, the modulator gain balance

and the quadrature error

calculated. Arg is the argument or arc tangent 2 function, which is the angle φ of the complex number 1 + ΔQ ^ = A · e ^jφ results.

First determine the range in which Modulator_Gain_Balance and Quadrature_Mismatch may lie. The criterion considered is the noise-free case. In the following table, an example range is given in which there are no symbol error decisions and thus the I / Q imbalance is estimated exactly. In the listed areas, a sufficiently large reserve was ensured. modulation Modulator_Gain_Balance Quadratur_Mismatch BPSK [0.55; 1,8] [-40 °; + 40 °] QPSK [0.55; 1,8] [-40 °; + 40 °] 16 QAM [0.76, 1.3] [-15 °, + 15 °] 64 QAM [0,9, 1,1] [-5 °; + 5 °]

The performance of the method according to the invention is illustrated below by means of a few examples.

• – In den Beispielen werden zwecks Übersichtlichkeit nur 4 Subträger (Subcarrier) verwendet.
• – Die Übertragungsfunktion der 4 Subcarrier ist: H_k = [1 1 1 1]. H_k kann beliebig sein. H_k wurde jedoch ideal gewählt, weil dann das Constellation-Diagramm anschaulich ist.
• – Es wurde die Modulation 64 QAM verwendet.
• – Die Anzahl der Symbole wurde willkürlich zu
• nof_Symbols = 400 Symbole gewählt. Die Anzahl der Iterationen wurde zu nof_Iterations = 30 Iterationen gewählt. was immer ausreichend ist.
• – Das Signal/Rausch-Verhältnis wurde zu S/N = ∞ gewählt. In diesem Fall müssen die Schätzwerte ideal sein.

The simulations were performed under the following conditions:

• - In the examples, only 4 subcarriers are used for clarity.
• - The transfer function of the 4 subcarriers is: H _k = [1 1 1 1]. H _k can be arbitrary. H _k, however, was ideal, chosen because the constellation diagram then is vividly.
• - Modulation 64 QAM was used.
• - The number of symbols became arbitrary
• nof_Symbols = 400 symbols selected. The number of iterations was chosen to be nof_Iterations = 30 iterations. whatever is sufficient.
• The signal-to-noise ratio was chosen to be S / N = ∞. In this case, the estimates must be ideal.

Example 1: ΔQ = 0.1

ΔQ was set to ΔQ = 0.1 in this example. The table below shows the EVM error and the EVM error EVM_diff difference from the previous iteration for each iteration. In the EVM calculation, the reference signal is subject to I / Q imbalance, i. H. the EVM error is zero if the I / Q imbalance is ideally estimated. Thus, the EVM error indicates which additional EVM error is due to the non-ideal I / Q estimate.

EVM is in the table the residual error without I / Q imbalance. dQ_est is the estimate for ΔQ. H_k_Used_est are the estimated values of the transfer function. dQ_est EVM [%] EVM diff [%] 1st iteration 0.0486 * exp (j * -0.1149 °) 3.4954 - 2nd iteration 0.07298 * exp (j * -0.06041 °) 1.7962 -1.699 3rd iteration 0.08564 * exp (j * -0.03211 °) .9435 -0.8526 4th iteration 0.09233 * exp (j * -0.01716 °) .5011 -0.4424 5th iteration 0.09589 * exp (j · -0.009195 °) .2677 -0.2334 6th iteration 0.09779 * exp (j · -0.004935 °) .1434 -0.1243 7. iteration 0.09881 * exp (j * -0,002,651 °) .0770 -006646 8. iteration 0.09936 * exp (j · -0.001425 °) 0.0413 -0.03563 9th iteration 0.09966 * exp (j * -0.0007657 °) 0.0222 -0.01913 10th iteration 0.09982 * exp (j * -0.0004116 °) 0.0119 -0.01028 11th iteration 0.0999 * exp (j * -0.0002213 °) 0.0064 -0.005522 12th iteration 0.09995 * exp (j · -0.000119 °) 0.0035 -0.002968 13th iteration 0.09997 * exp (j * -6,396e-005 °) 0.0019 -0.001596 14. iteration 0.09998 * exp (j * -3,439e-005 °) 0.0010 -0.0008578 15th iteration 0.09999 * exp (j * -1,849e-005 °) 0.0005 -0.0004612 16th iteration 0.1 · exp (j · -9,941e-006 °) 0.0003 -0.000248 17th iteration 0.1 · exp (j · -5,345e-006 °) 0.0002 -0.0001333 18th iteration 0.1 · exp (j · -2,874e-006 °) 0.0001 -7,167e-005 19th iteration 0.1 · exp (j · -1,545e-006 °) 0.0000 3,854e-005 20th iteration 0.1 · exp (j · -8,306e-007 °) 0.0000 -2,072e-005 21st iteration 0.1 · exp (j · -4,466e-007 °) 0.0000 -1,114e-005 22nd iteration 0.1 · exp (j · -2,401e-007 °) 0.0000 -5,989e-006 23rd iteration 0.1 · exp (j · -1,291e-007 °) 0.0000 -3,22e-006 24th iteration 0.1 · exp (j · -6,941e-008 °) 0.0000 -1,731e-006 25th iteration 0.1 · exp (j · -3,732e-008 °) 0.0000 -9,308e-007 26th iteration 0.1 · exp (j · -2,006e-008 °) 0.0000 -5,004e-007 27th iteration 0.1 · exp (j · -1,079e-008 °) 0.0000 -2,691e-007 28th iteration 0.1 · exp (j · -5,8e-009 °) 0.0000 -1,447e-007 29th iteration 0.1 · exp (j · -3,118e-009 °) 0.0000 -7,777e-008 30th iteration 0.1 · exp (j · -1,676e-009 °) 0.0000 -4,182e-008

Result:

• dQ_est = 0,1 · exp (j · -1,676e-009 °) H_k_Used = 1 1 1 1 H_k_Used_est = 1.0000 + 0.0000i 1.0000 + 0.0000i 1.0000 + .0000i 1.0000 + 0.0000i

• 1. Mit zunehmender Iteration konvergiert ΔQ ^ gegen den zu schätzenden Wert ΔQ = 0,1.
• 2. Entsprechend konvergiert der EVM-Fehler mit zunehmender Iteration gegen Null.
• 3. Der EVM-Fehler reduziert sich von Iteration zu Iteration um den Faktor 2.
• 4. Bereits nach ca. 14 Iterationen ist der I/Q-Schätzfehler und damit auch der EVM-Fehler vernachlässigbar klein.
• 5. Die Kanalschätzwerte H ^_k werden ebenfalls (praktisch) fehlerfrei geschätzt (siehe H_k_Used_est).

From the result the following conclusions can be drawn:

• 1. converges with increasing iteration ΔQ ^ against the value to be estimated ΔQ = 0.1.
• 2. Accordingly, the EVM error converges to zero with increasing iteration.
• 3. The EVM error is reduced by a factor of 2 from iteration to iteration.
• 4. Already after approx. 14 iterations the I / Q estimation error and thus also the EVM error are negligibly small.
• 5. The channel estimates H ^ _k are also estimated to be (practically) error-free (see H_k_Used_est).

In 2 is the Constellation diagram of r _{l, k shown} for Example 1. It can be seen clearly that instead of the individual Constellation points at each state position in each case the multiplied by ΔQ = 0.1 Constellation diagram occurs. This effect is described by equation (7).

Example 2: ΔQ = 0.1-e ^{j45 *}

ΔQ was set to ΔQ = 0.1 · e ^{j45 *} in this example. In this further example, the following table also documents the convergence.

EVM is in the table again the residual error without I / Q imbalance dQ_est EVM [%] EVM diff [%] 1st iteration 0.04796 * exp (j * 43.03 °) 3.5427 - 2nd iteration 0.07247 * exp (j * 43.96 °) 1.8440 -1.699 3rd iteration 0.08531 * exp (j * 44.44 °) .9754 .8686 4th iteration 0.09213 * exp (j * 44.7 °) .5201 -0.455 5th iteration 0.09578 * exp (j * 44.84 °) .2785 -0.2416 6th iteration 0.09773 * exp (j * 44.91 °) .1495 -0.129 7. iteration 0.09878 * exp (j * 44.95 °) .0803 -0.06916 8. iteration 0.09934 * exp (j * 44.98 °) 0.0432 -0.03714 9th iteration 0.09965 * exp (j * 44.99 °) 0.0232 -0.01996 10th iteration 0.09981 * exp (j * 44.99 °) 0.0125 -0.01073 11th iteration 0.0999 · exp (j · 45 °) 0.0067 -0.005774 12th iteration 0.09995 · exp (j · 45 °) 0.0036 -0.003106 13th iteration 0.09997 · exp (j · 45 °) 0.0019 -0.001671 14. iteration 0.09998 · exp (j · 45 °) 0.0010 -0.0008992 15th iteration 0.09999 · exp (j · 45 °) 0.0006 -0.0004838 16th iteration 0.1 · exp (j · 45 °) 0.0003 -0.0002603 17th iteration 0.1 · exp (j · 45 °) 0.0002 -0.0001401 18th iteration 0.1 · exp (j · 45 °) 0.0001 -7,538e-005 19th iteration 0.1 · exp (j · 45 °) 0.0000 -4,056e-005 20th iteration 0.1 · exp (j · 45 °) 0.0000 -2,183e-005 21st iteration 0.1 · exp (j · 45 °) 0.0000 -1,174e-005 22nd iteration 0.1 · exp (j · 45 °) 0.0000 -6,319e-006 23rd iteration 0.1 · exp (j · 45 °) 0.0000 -3,4e-006 24th iteration 0.1 · exp (j · 45 °) 0.0000 -1,83e-006 25th iteration 0.1 · exp (j · 45 °) 0.0000 -9,845e-007 26th iteration 0.1 · exp (j · 45 °) 0.0000 -5,298e-007 27th iteration 0.1 · exp (j · 45 °) 0.0000 -2,851e-007 28th iteration 0.1 · exp (j · 45 °) 0.0000 -1,534e-007 29th iteration 0.1 · exp (j · 45 °) 0.0000 -8,253e-008 30th iteration 0.1 · exp (j · 45 °) 0.0000 -4,441e-008

Result:

• dQ_est = 0.1 · exp (j · 45 °) H_k_Used_est = 1.0000 + 0.0000i 1.0000 + 0.0000i 1.0000 + 0.0000i 1.0000 + 0.0000i

In 3 is the Constellation diagram of r _{l, k shown} for Example 2. One recognizes the rotation described by equation (7) by 45 degrees of the individual Constellations diagrams due to the multiplication by ΔQ = 0.1 · e ^j45 .

Claims (8)

Method for estimating the I / Q imbalance of an OFDM signal (r (t)) consisting of a plurality of sub-signals (K _mod a _{l, k} ), each one of a plurality of subcarriers

with different subcarrier frequency (ω _k ) over a transmission channel ( 9 ) having an unknown transfer function (H (ω)), comprising the following method steps: - forming a cost function (L) in dependence on the values (H _k ) of the transfer function (H (ω)) of the transfer channel ( 9 ) at the subcarrier frequencies (ω _k ) and in dependence on a parameter (ΔQ) describing the I / Q imbalance, - partial differentiation of the cost function (L) according to the values (H _k ) of the transfer function (H (ω)) the transmission channel ( 9 ) at the subcarrier frequencies (ω _k ) and after the parameter (ΔQ) describing the I / Q imbalance, - Forming a system of equations from the assumption that all partial differentials of the cost function (L) are zero and thus a minimum of the cost function (L) is present, and - solving the system of equations, wherein the system of equations is nonlinear and the solution of the system of equations is iterative, whereby the parameter (ΔQ) describing the I / Q imbalance is set to zero as the starting value of the iteration, the values ( H _k ) of the transfer function (H (ω)) of the transmission channel ( 9 ) at the subcarrier frequencies (ω _k ), the parameter (ΔQ) describing the I / Q imbalance being dependent on the values (H _k ) of the transfer function (H (ω)) of the transmission channel ( 9 ) at the subcarrier frequencies (ω _k ), the values (H _k ) of the transfer function (H (ω)) of the transmission channel ( 9 ) at the subcarrier frequencies (ω _k ) in dependence on the previously calculated parameter (ΔQ) of the I / Q imbalance, and wherein the last two steps are repeated alternately. A method according to claim 1, characterized in that in the transmission channel ( 9 ), I / Q imbalance-free transmit signal

where K _{mod is} a normalization constant a _{l, k is} the transmitted OFDM symbols and ω _{k is} the subcarrier frequencies. A method according to claim 1 or 2, characterized in that a log-likelihood function is used as the cost function. Method according to one of claims 1 to 3, characterized in that the cost function (L) the form L (.DELTA.Q ~, H ~ _k) = Σ l kΣ | r _{l, k} - K · a _{mod l, k} · H _k ~ - ~ .DELTA.Q · K · _mod aa _{l, k} · H ~ _k | ² has, where is true

and where r _{l, k are} the received OFDM symbols H ~ _k the values to be estimated of the transfer function of the transmission channel at the subcarrier frequencies ω _k and ΔQ ~ the parameter of the I / Q imbalance to be estimated. Method according to claim 4, characterized in that the values H ^ _k the transfer function of the transmission channel ( 9 ) at the subcarrier frequencies ω _k according to

be calculated. Method according to claim 4 or 5, characterized in that the parameter describing the I / Q imbalance ΔQ ^ according to

is calculated. A digital storage medium having electronically readable control signals capable of interacting with a programmable computer or digital signal processor such that the method of any one of claims 1 to 6 is practiced. Computer program with program code means for carrying out all steps according to one of claims 1 to 6, when the program is executed on a computer or a digital signal processor.

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