Comparing this frequency-selective compensation equation with the frequency-independent one, it is found that the frequency-independent IQM scenario has constants
α
k ,β
k , andH
cm.k. Consequently, this compensation formulation degenerates to the time-domain expression when replacing constant SG, IG, and common filter responses into the frequency-selective IQM calibration scheme. In other words, this frequency-selective based signal expression can be regarded as a general form for the IQM problem modeling.3-4 IQM Parameter Estimations
3-4-1 Narrow-band adaptive estimation
The IQM calibration is achieved via the estimations of signal gain and mirror gain. Instead of directly estimation SG and IG, the estimation errors are computed first to update the SG and IG parameters that are used for the signal calibration. If the signal is transmitted via a physical channel, there is inevitably noise added to the
information signals, resulting in the error-estimation of essential parameters for the IQM calibration. Accordingly, the estimation error is defined for the SG and IG parameters as delta-SG (dSG) and delta-MG (dMG).
β
where the hat notation denotes a parameter that comes from estimations. The IQM-distorted signal compensated by the estimated SG and IG can be expressed as
2 ˆ2
Therefore, there is absolutely an error amount between
y and
cyˆ due to non-ideal
c signal compensation. In order to represent the compensated signal as a function of error terms, (3-17) is substituted intoyˆ , and we have
cK
. According to (3-10), we find out the property− ∗
= β
α 1 and αˆ=1−βˆ∗ (3-20)
This is further inferred that Δα =−Δ∗β from (3-17) and (3-20). As a result, (3-13) can be summarized as
K y y
y
y
ˆc = c−(Δ∗β⋅ c −Δβ⋅ c∗)⋅ ˆ (3-21)Prior to SG and MG estimation, the incoming signals should be left uncompensated.
This is equivalent to set parameters as
α
ˆ =1 and βˆ =0, i.e.β
=Δβ. Therefore, our problem becomes how to estimate the dMG parameter Δ . After the CFO βcompensation of
yˆ with
ce
−jω⋅t, we haveK e
y y
y
y
ˆ= −(Δ∗β⋅ −Δβ⋅ ∗⋅ −j2ω⋅t)⋅ ˆ (3-22)If the received signal belongs to preamble, it will be used for channel estimation.
Then (3-22) is transformed into frequency domain and divided by the pre-defined preamble. The estimated channel may be expressed as
* , , 2
* ˆ) ( ˆ)
1 ˆ (
ˆ k N
k N k k
k
k k
D K H
X H X
X K
H
=Y
= −Δβ⋅ ⋅ + Δβ ⋅ − ⋅ − ε ⋅ ⋅ − (3-23)with }
X
k∈{+1,−1 . This result is composed of two product terms. The first product term shows that the primitive channel response is multiplied by a complex gain, and the second product term can be regarded as interference since the channel response comes from the mirror part and the variables in the bracket are all non-ideal effects.On the other hand, the ratio X-k,N
/X
k may be positive or negative depending on the values defined in preamble. Therefore, this ratio decides if the second product term is added or deducted by the first product term, resulting in large value transitions in estimated channel responses on specific subcarrier indices. These transitions are due to IQM effect and can be seen in Fig. 3-4.0 10 20 30 40 50 60 0
0.5 1 1.5
Primitive and Mirror Channel Response
Subcarrier Index (k) abs(H k)
0 10 20 30 40 50 60
0 0.5 1 1.5
Subcarrier Index (k) abs(H -k)
(a)
0 10 20 30 40 50 60
0 0.5 1 1.5 2
Primitive Channel v.s. Estimated Channel under IQM
Subcarrier Index (k)
abs(H)
Primitive Channel Response Estimated Channel Hk with IQM
Transitions
(b)
Fig. 3-4. Channel response with IQM effect (a) primitive channel response and the mirror response (b) estimated channel response under gain error 1 dB and phase 10 degree.
If gain error or phase error becomes more severe, the amplitude transitions in the consecutive channel responses will get larger. Therefore, we can extract Δ from β
those transitions of estimated channel responses. From (3-23), we consider two
consecutive subcarriers with large transitions, i.e. X-k,N
/X
kand X
-k-1,N/X
k+1have
opposite signs with subcarrier indices k and k+1 respectively. Then the difference between the estimated channel responses becomes)
where the block size N is omitted for notation simplicity. We assume our operating environment has slow varying frequency response, so the consecutive channel subcarriers are approximately equal, i.e.
H
k −H
k+1≈0. So, the first product term incan be derived by replacing index k with -k in (3-23), then we have
ˆ)
substitute these two terms into (3-24), eliminate the product term associated with ) can update our parameters to improve the compensation accuracy.
⎪⎩
where
μ
is the adaptive step size. As βˆ approaches β, the large transitions in estimated channel response will become smaller and smoother. This in turn makes the estimation error Δβ smaller, and achieves the parameter adaptation. To sum up, we apply the parameters, αˆ and βˆ, to do signal compensation, and the rest transitions in estimated channel response are used to update αˆ and βˆ. So, the compensation accuracy is improved.To summarize the overall adaptation algorithm, we depict the operation loop in Fig. 3-5. Before MG and SG estimation, the received signal is left uncompensated of I/Q mismatch, so the compensation parameters are set to be αˆ =1 and βˆ =0 respectively as the initial values.
Moreover, the signal is compensated in the order of IQM followed by CFO. The estimation and compensation are done in frequency and time domain respectively, and αˆ and
β
ˆ are adaptively updated without any delayed-line filters.Fig. 3-5. Summary of the overall adaptation loop.
The SG and IG parameters in a wideband scenario are expressed by the base of frequency indices k. Again, both of SG and IG are regarded as the composition of estimation values (αˆk and βˆk) and estimation errors (Δαk and Δβk).
time
IQM Correction
Packet Detection (in preamble)
IQM Estimation (in long preamble) Parameter Update
Data Decoding
Fig. 3-6. Summary of the overall adaptation loop.
All of the incoming signals are compensated by the IQM correction building block. Although the IQM is not estimated before any packet detection, the data-path and compensation-flow go through the IQM block. Compensating the IQM or not depends on the values of the SG and IG parameters. In other words, the signals may be considered as uncompensated with the SG=1 and IG=0, even the compensation is applied to the received signals. The next step after the IQM-correction building block is the packet detection. This conventionally takes the preamble for packet recognition.
Then the channel response is estimated after the packet detection. The estimated channel response is used for the IQM parameter estimation at the same time. With this value, the IQM SG and IG are updated, and the later-on coming signals are compensated by the latest estimated parameters as they are received in a baseband receiver. This means the rest of incoming signals are compensated by IQM, and the normal decoding flow starts for the signal demodulation. This reference time-chart is illustrated in Fig. 3-6.