IQ imbalance

Definition

A direct conversion receiver uses two quadrature sinusoidal signals to perform the so-called quadrature down conversion. This process requires shifting the local oscillator (LO) signal by 90 degrees to produce a quadrature sinusoidal component. When mismatches exist between the gain and phase of the two sinusoidal signals and/or along the two branches of down-conversion mixers, amplifiers, and low-pass filters, the quadrature baseband signals will be corrupted.

Suppose the received passband signal is identical to the transmitted signal and is given by

y ( t ) = R e { x ( t ) e j 2 π f c t } = x I ( t ) cos ⁡ ( 2 π f c t ) − x Q ( t ) sin ⁡ ( 2 π f c t ) , where    x ( t ) = x I ( t ) + j x Q ( t )  is the transmitted baseband signal

Assume that the gain error is 20 log ⁡ [ ( 1 + ϵ A ) / ( 1 − ϵ A ) ] dB and the phase error is ε θ degrees. Then we can model such imbalance using mismatched local oscillator output signals.

2 ( 1 + ϵ A ) cos ⁡ ( 2 π f c t − ε θ / 2 ) ,

− 2 ( 1 − ϵ A ) sin ⁡ ( 2 π f c t + ε θ / 2 ) .

Multiplying the passband signal by the two LO signals and passing through a pair of low-pass filters, one obtains the demodulated baseband signals as

x ~ I ( t ) = ( 1 + ε A ) [ x I ( t ) cos ⁡ ( ε θ / 2 ) − x Q ( t ) sin ⁡ ( ε θ / 2 ) ] ,

x ~ Q ( t ) = ( 1 − ε A ) [ x Q ( t ) cos ⁡ ( ε θ / 2 ) − x I ( t ) sin ⁡ ( ε θ / 2 ) ] .

The above equations clearly indicate that IQ imbalance causes interference between the I and Q baseband signals.

To analyze IQ imbalance in the frequency domain, above equation can be rewritten as

x ( t ) = x ~ I ( t ) + j x ~ Q ( t ) = [ cos ⁡ ( ε θ / 2 ) + j ε A sin ⁡ ( ε θ / 2 ) ] x ( t ) + [ ε A cos ⁡ ( ε θ / 2 ) − j sin ⁡ ( ε θ / 2 ) ] x ∗ ( t ) = η α x ( t ) + η β x ∗ ( t )

where ( . ) ∗ denotes the complex conjugate. In an OFDM system, the baseband signal consists of several subcarriers. Complex-conjugating the baseband signal of the kth subcarrier carrying data X k is identical to carrying X k ∗ on the (-k)th subcarrier:

( ( X k , I + j X k , Q ) e j 2 π k f S t ) ∗ = ( X k , I − j X k , Q ) e − j 2 π k f S t = X k ∗ e j 2 π ( − k ) f S t

where f S is the subcarrier spacing.

Equivalently, the received baseband OFDM signal under the IQ imbalance effect is given by

X k = η α X k + η β X − k ∗

In conclusion, besides a complex gain imposed on the current sub carrier data Xk, IQ imbalance also introduces ICI from the mirror subcarrier. The ICI term makes OFDM receivers very sensitive to the IQ imbalance effect. To solve this problem, the designer can request a stringent specification of the matching of the two branches in the frond-end or compensate for the imbalance in the baseband receiver.

Simulation

IQ imbalance can be simulated by computing the gain and phase imbalance and applying them to the baseband signal by means of several real multipliers and adders.

Synchronization Errors

The time-domain baseband Signals with IQ imbalance can be represented by

z i , n = η α z ( t ) + η β z ∗ ( t ) | t = i ( N + N | g ) T s + N g T s + n T s

Note that η α and η β can be assumed to be time-invariant and frequency-invariant, meaning that they are constant over several sub carriers and symbols. With this property, multiple OFDM subcarriers and symbols can be used to jointly estimate η α and η β to increase the accuracy. Transforming to the frequency domain, we have the frequency-domain OFDM signals under the influence of IQ imbalance given by

z i , k = η α H i , k X i , k + η β H i , − k ∗ X i , − k ∗ + V i , k

Note that the second term represents interference coming from the mirrored subcarrier X i , − k

IQ imbalance estimation in MIMO-OFDM systems

In MIMO-OFDM systems, each RF channel has its own down-converting circuit. Therefore, the IQ imbalance for each RF channel is independent of those for the other RF channels. Considering a 2 x 2 MIMO system as an example, the received frequency-domain signal is given by

{ Z i , k ( 0 ) = η α ( 0 ) ( H i , k ( 0 , 0 ) X i , k ( 0 ) + H i , k ( 0 , 1 ) X i , k ( 1 ) ) + η β ( 0 ) ( H i , − k ( 0 , 0 ) X i , − k ( 0 ) + H i , − k ( 0 , 1 ) X i , − k ( 1 ) ) ∗ + V i , k ( 0 ) Z i , k ( 1 ) = η α ( 1 ) ( H i , k ( 1 , 0 ) X i , k ( 0 ) + H i , k ( 1 , 1 ) X i , k ( 1 ) ) + η β ( 1 ) ( H i , − k ( 1 , 0 ) X i , − k ( 0 ) + H i , − k ( 1 , 1 ) X i , − k ( 1 ) ) ∗ + V i , k ( 1 )

where η α ( q ) and η β ( q ) are the IQ imbalance coefficients of the qth receive RF channel.

Estimation of η α ( q ) and η β ( q ) is the same for each RF channel. Therefore, we take the first RF channel as an example. The received signals at the pilot subcarriers of the first RF channel are stacked into a vector z i , α ( q ) ,

z i , α ( 0 ) = [ z i , α 0 ( 0 ) z i , α 1 ( 0 ) ⋮ z i , α J − 1 ( 0 ) ] = A i , α ( 0 ) [ η α ( 0 ) η β ( 0 ) ] + v i , α ( 0 )

where A i , α ( 0 ) is the J × 2 matrix

A i , α ( 0 ) = [ ( H i , α 0 ( 0 , 0 ) X i , α 0 ( 0 ) + H i , α 0 ( 0 , 1 ) X i , α 0 ( 1 ) ) ( H i , α J − 1 ( 0 , 0 ) X i , α J − 1 ( 0 ) + H i , α J − 1 ( 0 , 1 ) X i , α J − 1 ( 1 ) ) ∗ ( H i , α 1 ( 0 , 0 ) X i , α 1 ( 0 ) + H i , α 1 ( 0 , 1 ) X i , α 1 ( 1 ) ) ( H i , α J − 2 ( 0 , 0 ) X i , α J − 2 ( 0 ) + H i , α J − 2 ( 0 , 1 ) X i , α J − 2 ( 1 ) ) ∗ ⋮ ⋮ ( H i , α J − 1 ( 0 , 0 ) X i , α J − 1 ( 0 ) + H i , α J − 1 ( 0 , 1 ) X i , α J − 1 ( 1 ) ) ( H i , α 0 ( 0 , 0 ) X i , α 0 ( 0 ) + H i , α 0 ( 0 , 1 ) X i , α 0 ( 1 ) ) ∗ ]

Clearly, the above formula is similar to that of the SISO case and can be solved using the LS method. Moreover, the estimation complexity can be reduced by using fewer pilot subcarriers in the estimation.

IQ imbalance compensation

The IQ imbalance can be compensated in either the time domain or the frequency domain. In the time domain, the compensated signal Zm in the current mth sample point is given by

z ¯ m = η ^ α ∗ z m − η ^ β z m ∗ | η ^ α | 2 − | η ^ β | 2 = η ^ α ∗ | η ^ α | 2 − | η ^ β ∗ | 2 ( z m − η ^ β η ^ α ∗ z m ∗ )

We can see that, by using the ratio η ^ β η ^ α ∗ to mitigate the IQ imbalance, there is a loss factor η ^ α ∗ / ( | η ^ α | 2 − | η ^ β | 2 ) . When the noise is added before the IQ imbalance, the SNR remains the same, because both noise and signal suffer this loss. However, if the noise is added after IQ imbalance, the effective SNR degrades. In this case, η α and η β , respectively, should be computed.

Compared with the time-domain approach, compensating in the frequency domain is more complicated because the mirrored subcarrier is needed. The frequency domain compensated signal at the ith symbol and the kth subcarrier

Z ¯ i , k = η ^ α ∗ Z i , k − η ^ β Z i , k ∗ | η ^ α | 2 − | η ^ β | 2

Nevertheless, in reality, the time-domain compensation is less preferred because it introduces larger latency between IQ imbalance estimation and compensation.

IQ imbalance estimation

Frequency-domain OFDM signals under the influence of IQ imbalance is given by

z i , k = η α H i , k X i , k + η β H i , − k ∗ X i , − k ∗ + V i , k

the IQ imbalance coefficients η α and η β are mixed with the channel frequency responses, making both the IQ imbalance estimation and channel estimation difficult.

In the first half of the training sequence, only sub carriers ranging from 1 to N/2 - 1 transmit pilot symbols; the remaining subcarriers are not used. In the second half, the sub carriers from -1 to -N/2 are used for pilot transmission. Such a training scheme easily decouples the IQ imbalance and the channel frequency response. Assuming the value of the pilot symbols is + 1, the received signals at subcarriers from 1 to N/2 - 1 are given by

Z i , k = η α H i , k + V i , k         k = 1 , ⋯ , N / 2 − 1 ,

while the received signals at the mirrored sub carriers take the form

Z i , − k = η β H i , k ∗ + V i , − k         k = 1 , ⋯ , N / 2 − 1

From the two sets of received signals, the ratio η β η α ∗ can be easily estimated by Z i , − k / Z i , k ∗ . The second half of the training sequence can be used in a similar way. Furthermore, the accuracy of this ratio estimation can be improved by averaging over several training symbols and several sub carriers. Although the IQ imbalance estimation using this training symbol is simple, this method suffers from low spectrum efficiency, as quite a few OFDM symbols must be reserved for training. Note that, when the thermal noise is added before the IQ imbalance, the ratio η β η α ∗ is sufficient to compensate the IQ imbalance. However, when the noise is added after the IQ imbalance, compensation using only η β η α ∗ can degrade the ensuing demodulation performance.