imbalance is a performance-limiting issue in the design of direct conversion receivers, also known as zero intermediate frequency (IF) or homodyne receivers. Such a design translates the received radio frequency (RF, or pass-band) signal directly from the carrier frequency () to baseband using only one mixing stage. The traditional heterodyne receiver structure needs an IF stage between the RF and baseband signals. The direct conversion receiver structure does not have an IF stage and does not need an image rejection filter. Due to the lower component count, it is easier to integrate. However, a direct-conversion RF front-end suffers from two major drawbacks: one is imbalance and the other is DC offset. When designing a homodyne receiver, control of imbalance is necessary to limit signal demodulation error.
imbalances occur due to mismatches between the parallel sections of the receiver chain dealing with the in-phase () and quadrature () signal paths. The local oscillator (LO) generates a sine wave and a copy of that sine wave that is delayed by . When the direct LO output is mixed with the original signal, this produces the signal, whereas when the delayed LO output is mixed with the original signal, that produces the signal. In the analog domain, the delay is never exactly . Similarly, the analogue gain is never perfectly matched for each of the signal paths.
A direct-conversion receiver uses two quadrature sinusoidal signals to perform the so-called quadrature down-conversion. This process requires shifting the LO signal by to produce a quadrature sinusoidal component, and a matched pair of mixers converting the same input signal with the two versions of the LO. Mismatches between the two LO signals and/or along the two branches of down-conversion mixers, and any following amplifiers, and low-pass filters, cause the quadrature baseband signals to be corrupted, either due to amplitude or phase differences. Suppose the received pass-band signal is identical to the transmitted signal and is given by:
is the transmitted base-band signal. Assume that the gain error is
dB and the phase error is
degrees. Then we can model such imbalance using mismatched local oscillator output signals:
Multiplying the pass-band
signal by the two LO
signals and passing through a pair of low-pass filters, one obtains the demodulated base-band signals as:
The above equations clearly indicate that
imbalance causes interference between the
base-band signals. To analyze
imbalance in the frequency domain, the above equation can be rewritten as:
denotes the complex conjugate of
. In an OFDM
system, the base-band signal consists of several sub-carriers. Complex-conjugating the base-band signal of the kth sub-carrier carrying data
is identical to carrying
on the (-k)th sub-carrier:
is the sub-carrier spacing. Equivalently, the received base-band OFDM
signal under the
imbalance effect is given by:
In conclusion, besides a complex gain imposed on the current sub-carrier data
imbalance also introduces Inter Carrier Interference (ICI) from the adjacent carrier or sub-carrier. The ICI term makes OFDM
receivers very sensitive to
imbalances. 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 base-band receiver. On the other hand, can be used a digital Odd-Order I/Q-demodulator with only the one input,
but such design has a bandwidth limitation.
imbalance can be simulated by computing the gain and phase imbalance and applying them to the base-band signal by means of several real multipliers and adders.
The time domain base-band Signals with imbalance can be represented by :
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 sub-carriers
and symbols can be used to jointly estimate
to increase the accuracy. Transforming to the frequency domain
, we have the frequency domain OFDM
signals under the influence of
imbalance given by:
Note that the second term represents interference coming from the mirrored sub-carrier
imbalance estimation in MIMO-OFDM systems
In MIMO-OFDM systems, each RF channel has its own down-converting circuit. Therefore, the imbalance for each RF channel is independent of those for the other RF channels. Considering a MIMO system as an example, the received frequency domain signal is given by:
imbalance coefficients of the qth receive RF
channel. Estimation of
is the same for each RF
channel. Therefore, we take the first RF
channel as an example. The received signals at the pilot sub-carriers
of the first RF
channel are stacked into a vector
, where is the matrix defined by:
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 sub-carriers
in the estimation.
The imbalance can be compensated in either the time domain or the frequency domain. In the time domain, the compensated signal in the current mth sample point is given by:
We can see that, by using the ratio
to mitigate the
imbalance, there is a loss factor
. When the noise is added before the
imbalance, the SNR
remains the same, because both noise and signal suffer this loss. However, if the noise is added after
imbalance, the effective SNR
degrades. In this case,
, respectively, should be computed.
Compared with the time domain
approach, compensating in the frequency domain
is more complicated because the mirrored sub-carrier
is needed. The frequency domain
compensated signal at the ith symbol and the kth sub-carrier:
Nevertheless, in reality, the time domain compensation is less preferred because it introduces larger latency between
imbalance estimation and compensation.
Frequency domain OFDM signals under the influence of imbalance is given by:
are mixed with the channel frequency responses
, making both the
imbalance estimation and channel estimation difficult. In the first half of the training sequence, only sub-carriers
to N/2 - 1 transmit pilot symbols; the remaining sub-carriers 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
imbalance and the channel frequency response
. Assuming the value of the pilot symbols is + 1, the received signals at sub-carriers
from 1 to N/2 - 1 are given by
, while the received signals at the mirrored sub-carriers
take the form
From the two sets of received signals, the ratio can be easily estimated by . 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 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 imbalance, the ratio is sufficient to compensate the imbalance. However, when the noise is added after the imbalance, compensation using only can degrade the ensuing demodulation performance.
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