Basic problem statement
Imperfect analog front-ends of mobile devices cause transmit signals to be non-ideal. Typically, the signal processing for inphase and quadrature (IQ) components is likewise affected. On the one hand, the undesirable zero‐offsets of both digital‐to‐analog converters lead to the IQ offset, and on the other hand, the IQ imbalance originates from an imperfect up‐conversion of the baseband signal.
The uplink of the latest mobile communication system Long Term Evolution (LTE) specified by 3GPP deviates in some aspects from the classical orthogonal frequency-division multiplex (OFDM) scheme. In particular, the spectral shift by half a subcarrier spacing before up‐conversion causes the IQ offset to interfere on all subcarriers. Its compensation is required in:
- Modulation analysis for a correct assessment of the signal quality for a transmitter according to the 3GPP measurement specification TS 36.101. That type of signal analysis is carried out in mobile communication testers, which are a special kind of high‐precision receiver. For compliance tests, the IQ offset must be removed before computing the error vector magnitude, which reflects the remaining signal error of the handheld compared to the ideal signal.
- Base-station receivers for reliable data detection. According to the limits stated in TS 36.101, high values for this distortion have to be accepted at the receiver. An optional compensation can significantly reduce the symbol error rate.
An estimation of the IQ offset has to take into account the special structure of the physical layer of LTE, which is defined in the 3GPP physical layer specification
TS 36.211. The literature provides solutions only for classical OFDM, which cannot be applied to the mobile communications system under consideration.Measures for IQ transmitter impairments
The first impairment to be considered is the IQ imbalance
, which results from the up-conversion of the baseband signal to the radio-frequency band with oscillators working with the carrier frequency fc. The corresponding modulation is done with
in the inphase branch and with
in the quadrature branch. Without loss of generality, the inphase branch is considered to be ideal and is used as a reference. For the non-ideal transmitter, the complex-valued IQ imbalance
represents a non-vanishing error vector that comprises the gain imbalance A
and the quadrature error ϕ. The down-conversion of the radio-frequency signal at the receiver is assumed to be done using ideal oscillators. Because signal processing in mobile testers or base stations is more accurate, the imbalance is usually less dominant here compared to low-cost mobile devices. Consequently, the receiver applies
for the demodulation of the inphase and
for the quadrature component of the receive signal.
The IQ offset is also regarded as a transmitter impairment. The complex-valued parameter b describes a non-ideal digital-to-analog conversion. Its real and imaginary parts represent the undesirable zero-offsets in inphase as well as quadrature branches, respectively.
Complete transmission model
The complete discrete-time transmission model, already adapted to the physical layer of the LTE uplink, is shown in figure 1. The inner blocks in orange
frames model the impairments in the baseband and the outer blocks in violet frames reflect spectral shifts that are part of the physical layer specification for LTE.
Click on image to enlarge.
Fig. 1: LTE specific discrete-time channel model in the baseband, comprising the transmitter impairments IQ imbalance and offset as well as the physical part of the channel.
Besides the non-linear IQ imbalance ΔQ
and the IQ offset b
, we additionally consider a frequency selective channel with coefficients hm
that describes the cascade of transmit filter, physical channel and receive filter. All of these parameters are unknown to the receiver, and the receive values are observed in the presence of additive noise samples that are assumed to be uncorrelated.
In this specific multicarrier system, the spectral shift by half a subcarrier distance that is carried out before transmitting the OFDM signal s’i
must be properly removed after its reception, which results in r
. It is applied per OFDM symbol l
, which is also known as single-carrier frequency-division multiple-access (SC-FDMA) symbol in the uplink of LTE. Each OFDM symbol consists of N
Taking into account an inverse discrete Fourier transform (IDFT) at the transmitter and a discrete Fourier transform (DFT) at the receiver for the OFDM symbols, the time-domain channel model of figure 1 leads to its equivalent frequency-domain representation, which can be described by
Click on image to enlarge.
The variables with small letters, introduced in figure 1, are represented in the preceding equation with capital letters after the DFT as corresponding frequency-domain parameters. The subcarrier index for the multicarrier system is given by k
It can be observed that the IQ impairments cause a deterministic loss of orthogonality. The IQ imbalance ΔQ
leads to pairwise interfering carriers (k, N-1-k
) whose transmit values are received on subcarrier k
weighted with the channel coefficient H’k
. Likewise, the IQ offset b’
corrupts all subcarriers with the interference characteristic Ck
as depicted in figure 2.
Fig. 2: Characteristic function for the interference of the IQ offset.