Training Sequence Design

Theoretically, the optimal training sequence is the one that minimizes MSE

{ }

ˆ 2

E⎡⎢⎣ f f− ⎤⎥⎦=tr ϒ ϒC^{v H} . As seen in (5.29), however, C is a function of _v hR,₊

( )

n and

( )

,

hR₋ n , and therefore the optimal training sequence differs from a transceiver to another. Here, the simplified measure

{ }

^H

{ (

^H

)

¹

}

tr ϒϒ =tr Φ Φ ⁻ (5.43) is adopted for the search of the optimum training sequence. The measure is optimal if

( )

^{, 0, , 1} For this case, it was derived in [80] that the minimum MSE in (5.43) is achieved provided that

4 3 order to have the best performance, and that complicates the design very significantly. In the fol-lowing, a simpler method is proposed.

Consider a periodic training sequence that consists of P+ periods with K samples in 1

each period, i.e., ^{s n}

( ) (

^{=s n+K}

)

^{, n= −K,…, 0,…,N−1, where N =KP, and} Ng = . De-K

Fur-thermore, the condition _{1 1 4 3}

f

H

K L ₊

= ⋅

Φ Φ I , called Condition A, splits into the following three sub-conditions:

Condition A.3 : ₁

f

T

N = L

S 1 0 . (5.54) In [81], methods were given to design sequences that satisfy Conditions (A.1) and (A.2), whereas (A.3) just says that the designed sequence has a zero mean. As an example, using the fre-quency-domain nulling (FDN) method in [81], the sequence (K =64)

( )

⁶³

( )

²⁶⁴

5.3 Performance Analysis

In this section, the mean and variance of the calibrated IRRT

( )

f , IRRR

( )

f , ε_T and ε_R are analyzed, and numerical results in Section V confirms the accuracy of the analysis. To begin with, define During the internal loop-back, the signal-to-noise ratio (SNR) is usually very high (SNR 1), and, hence, it is reasonable to assume that ˆ_{1, 1,}

+ ≈ +

f f . Recall that f is the desired channel re-_1,+

sponse from transmitter to receiver, as is given in (5.25). The approximation is accurate for the SNRs of interest as to be discussed in Section 5.4. Using this approximation, the estimated cali-bration filters ^{w nˆ}

( )

^{and ρˆ n}

( )

in (5.30) and (5.32) are rewritten as

in-verse filters f1,⁻₊¹

( )

n and

(

^f1,*+

( )

^{n e−j2πμn}

)

^{−1 ; that is} w^{ˆ =⎡}⎣^wˆ

^{( ) ( )}

^{0 ,wˆ 1 ,}…^{,w Lˆ}

⁽

⁻¹

⁾

^⎤⎦^T , and

( ) ( ) ( )

ˆ =⎡⎣ρˆ 0 ,ρˆ 1 ,…,ρˆ L−1 ⎤⎦^T

ρ , where L is the filter length that can be selected as long as one wishes for the desirable accuracy. Using this modeling, ^{w nˆ}

( )

^{and ρˆ n}

( )

can be rearranged in the following vector-matrix form

( )

( ) ( ) ( ) ( ) ( ) ( )

impairments in real systems (see Section 5.4), and (5.74) is good because ideally ^{w n}

( )

^is

sought to make h_T,₋

( )

n +w n

( )

⊗h_T,₊

( )

n =0 in our method. Next, using the approximations in

Vw f is an exponentially distributed random variable with the probability density function (pdf) given by

( ( )

²

) _{( ) ( )}

^{2 ( ) ( )1 02 ( )2}

( ) ( )

Consequently, it can be shown that

{ ( ) } ^{( )}

( )

Similarly, it can be shown that

{ } { }

2 6

VAR _T VAR _R 10 log10e

π

ε = ε = ⎜^⎛⎜ ^⎞⎟⎟

⎝ ⎠ .

Some observations are in order based on our simplified analysis. First, it is observed that the estimates ˆ_1,

f , + ˆ_1, f , − ˆ_2,

f , − b^ˆ₁, and d^ˆ₀ in (5.60)-(5.64) are all zero mean with different variances, and so are the estimated calibration parameters wˆ , ρ and ˆ b^ˆ in (5.67), (5.68) and (5.86).

Second, for the calibration performance indices IRRT

( )

f , IRRR

( )

f , ε_T and ε_R, they have different means as functions of operating SNRs in (5.78), (5.84), (5.91) and (5.94) but with the same variances equal to a constant

2 6 2

10 log10e 5.57

⎛ π ⎞

⎜ ⎟ =

⎜ ⎟

⎝ ⎠ independent of operating SNRs. This is an interesting analysis result which indeed agrees well with the simulation results as seen in next section.

Table 5.1: The RF impairments

In this section, the performance of the proposed method is evaluated through analysis and com-puter simulations. Table 5.1 summarizes the transmitter and receiver RF impairments. In all the results, 1 /T_s =20MHz,

( )

² 0²

∑

= , and the optimal training employs the se-quence in (5.55) with K =64, P=3, and ^{μ =23 3 64}

(

^⋅

)

. Simulation results are obtained with 10 realizations 6

Figures 5.2 and 5.3 investigate the effects of L on the performance of the calibrated _f

( )

E⎡⎣IRR_T f ⎤⎦ and ^E⎡⎣IRRR

( )

f ⎤⎦ by computer simulations, respectively. Recall that L is the ^f length of the filters f_i,_±

( )

n , 1, 2i= . As can be seen, the calibration performances are insensitive to the values of L as long as it is selected larger than 6 in this case; similarly results are ob-_f served for the dc-offset calibration. In practice, since L may not be exactly known in advance, _f

Figure 5.2: Performance of the calibrated ^E

[

IRRT

]

with different L ’s under optimal training. _f

Figure 5.3: Performance of the calibrated ^E

[

IRRR

]

with different L ’s under optimal training. _f

Figure 5.4: Performance of the calibrated ^E

[

IRRT

]

with different training designs.

it is advisable to use a sufficiently large L to avoid performance degradation. In the rest of this _f section, 7L_f = is used.

In Figures 5.4 and 5.5, the calibrated ^E⎡⎣IRRT

( )

f ⎤⎦ and ^E⎡⎣IRRR

( )

f ⎤⎦ are investigated with three periodic training designs (K =64, P=3): the optimal training, Training A and Training B.

Training A uses the sequence in (5.55) but with ^{S k}

( )

^{=ejφk ∀k and μ =23 3 64}

(

^⋅

)

^{, and}

Training B uses the optimal sequence in (5.55) but with μ =8 64. Here, for comparison purpose, Training A is selected to violate Conditions A.2 and A.3 and Training B to violate Condition B.

As is shown, the optimal training indeed gives a superior performance than the other two. Com-pared to the case of no calibration, the proposed method provides around 20-35 dB performance improvement over the whole frequency band. Shown in the figures also include the simulation

Figure 5.5: Performance of the calibrated ^E

[

IRRR

]

with different training designs.

results which show perfect match to those with analysis.

Figure 5.6 shows the empirical cumulative distribution functions (CDFs) of the cali-brated IRRT

( )

f and IRRR

( )

f at f=4 MHz, which are constructed with 10 realizations. ⁶ The empirical and analytical means and standard deviations are given in Table 2 where it shows very good agreement between simulation and analysis. The smallest IRR and _T IRR observed _R are both 37 dB which are around 16 dB and 20 dB better than the cases of no calibration, respec-tively. Figure 5.7 shows the empirical CDFs of the calibrated ε_T and ε_R. The largest ε_T and εR observed are -26 dB and -24 dB, respectively. The empirical and analytical means and stan-dard deviations of the calibrated ε_T and ε_R are also shown in Table 5.2. Again, it shows very good match between simulation and analysis. Furthermore, Table 5.2 confirms that the variances of the calibrated IRR , _T IRR , _R ε_T, and ε_R are same and independent of the operating SNRs.

Figure 5.6: Empirical cumulative distribution functions of the calibrated IRR and _T IRR con-_R structed with 10 realizations. ⁶

Figure 5.7: Empirical cumulative distribution functions of the calibrated ε_T and ε_R constructed with 10 realizations. ⁶

Table 5.2: Example mean and standard deviation of the calibrated IRR , _T IRR , _R ε_T, and ε_R. Mean (dB) Standard deviation (dB)

Parameters SNR(dB)

Simulation Analysis Simulation Analysis

35 50.8 50.9 5.59 5.57 45 60.8 60.9 5.57 5.57

IRRT

(

^4MHz

)

55 70.8 70.9 5.56 5.57 35 50.8 50.8 5.53 5.57

45 60.8 60.8 5.58 5.57

IRRR

(

^4MHz

)

55 70.8 70.8 5.58 5.57 35 -40 -40.3 5.57 5.57

45 -50 -50.3 5.57 5.57

εT

55 -60 -60.3 5.56 5.57 35 -38 -38 5.6 5.57 45 -48 -48 5.57 5.57

εR

55 -58 -58 5.59 5.57

Figures 8 and 9 show a sample received signal constellation and bit error rate performance re-spectively for an un-coded 64-QAM OFDM (orthogonal frequency-division multiplexing) system with and without calibration. A one-tap equalizer is employed at the receiver for the simulated OFDM system that uses 64-point FFT (fast Fourier transform) with 52 subcarriers carrying data.

As are shown, the adverse effects due to radio impairments are removed almost completely by the proposed calibration.

Figure 5.8: Sample signal constellation with and without calibrations (σ₀² = ). 0

Figure 5.9: Bit error rate performance with and without calibration (64QAM).

5.5 Summary

In Chapter 5, a digital self-calibration method is proposed for the direct-conversion radio trans-ceiver to calibrate its own transmitter and retrans-ceiver radio impairments, including fre-quency-independent I-Q imbalance, frequency-dependent I-Q imbalance, and dc offset. By in-troducing a shift between transmit and receive frequencies, the radio impairments appearing at the transmitter and receiver can be calibrated simultaneously without a dedicated analog circuitry in the feedback loop. The calibration parameters are estimated based on the non-linear least-squares principle, and the calibration performance is analyzed that agrees very well with the simulations. In addition, the issue of optimal training design is investigated; sufficient conditions for optimal training are provided, and an example of optimal training is given for the periodic training structure. Analytical and simulation results confirm the effectiveness of the proposed method.

Chapter 6 Conclusions

In this dissertation, the radio impairment estimation and compensation techniques are investi-gated for the wideband MIMO communication systems with direct-conversion radio architecture.

A complete set of radio impairments is taken into consideration, including frequency-independent and dependent I-Q imbalances, dc offset and frequency offset. Both estimation/compensation and self-calibration techniques are studied in this dissertation.

The estimation/compensation technique is to remove the impairments from the received sig-nal during communication at the receiving side. In Chapter 3 and Chapter 4, the estimation and cancellation techniques for MIMO systems are investigated, with Chapter 3 focusing on receiver radio impairments for any types of MIMO systems and extended to cascaded transmitter and re-ceiver radio impairments for MIMO-OFDM systems in Chapter 4. First, a two-stage cancellation architecture is developed, which enables to explicitly cancel the radio impairments without in-creasing the dimension of signal detection. The two-stage architecture generalizes the cancella-tion architectures for various types of system configuracancella-tions such as wireless peer-to-peer com-munication, downlink and uplink of a mobile cellular system. Moreover, it is general to accom-modate different forms of MIMO operation including spatial multiplexing, STBC (space-time

block coded) and transmit beam forming, with any number of transmit and receive antennas.

Second, several methods of estimation of radio parameters are proposed. The optimum method is the joint least squares estimation of all radio parameters, which is shown to be unbiased and ap-proaches to CRLB for the SNRs of interest, but with the highest complexity. Several other meth-ods are then developed aiming to reduce the estimation complexity, including the special phase-rotated periodic training design, simplified frequency and dc offset estimators and low-complexity iterative estimation aided by periodic training. Simulation results show that the un-coded BER performance degradation is negligible when using the reduced-complexity designs, and the proposed techniques outperform the existing ones in error-rate performance and/or the number of training symbols required.

On the other hand, the self-calibration is a technique to remove the transceiver’s own radio impairments before communication. In Chapter 5, we propose to calibrate its own transmitter and receiver radio impairments, including frequency-independent I-Q imbalance, fquency-dependent I-Q imbalance, and dc offset. By introducing a shift between transmit and re-ceive frequencies, the radio impairments appearing at the transmitter and rere-ceiver can be cali-brated simultaneously without a dedicated analog circuitry in the feedback loop. Based on the time-domain approach, the proposed method is applicable to all types of communication systems.

In addition, the issue of optimal training design is investigated; sufficient conditions for optimal training are provided with an example of optimal periodic training design. Analytical and simula-tion results confirm the effectiveness of the proposed self-calibrasimula-tion technique.

Some possible extensions and future research topics are addressed in the following. Both types of techniques can be further extended to combine with PA linearization and other synchro-nization issues such as signal detection, timing and clock offset estimation. The estimation meth-ods can be investigated in the pilot-based systems without initial acquisition or training, and non-data-aided estimation technique can also be explored. Training sequence design to optimize the performance of the estimation/compensation technique might be another issue.

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