IQ Rotation and Correction
In many multi-channel RF systems like the AD-FMCOMMS5™, or even on the AD-FMCOMMS2™ or AD-FMCOMMS3™, it is necessary to measure or correct for the phase difference between two complex (I/Q) RF signals.
From a pure mathematical description, a single sine wave has no phase, a phase can only develop between two different sine waves. The added complexity is that we do not have a single real signal (sine wave), we have a complex signal:
Rotating this signal by 180° or pi radians, becomes
and this is the result we want to be able to find, and eventually correct for.
(If you are not sure why the math works this way, check out this explanation).
I/Q correction - is phase shifting Q with respect to a fixed I, and making the magnitudes of I and Q equal. In the AD9361™/AD9364™ this is done inside the device automatically, and is know as Quadrature correction.
The first stage we will examine is measuring phase differences.
In the classical sense - when measuring phase synchronization of different signals, only the phase is important. No restriction on the amplitudes is imposed. Thus, the phase synchronization of coupled systems is defined as the appearance of relations between their phases, while the amplitudes can remain non-correlated. Many of the techniques below will measure phase/frequency/amplitude differences, and others will simply measure phase only. Depending on the application, and signal, different schemes may provide better results.
Although Simulink™ may have a block to directly measure phase difference between two complex input signals, most other environments do not, and we will need to get a better understanding how to do this.
The exact method of measuring a phase difference between two RF signals depends on the signal, and application in question:
- Is there noise in the signals, is the noise correlated, or is it random Gaussian noise in both?
- What does the signal look like?
- Is it a wide-band signal (like QPSK, QAM, LTE, etc), or is it a single-tone, constant-frequency signal?
- Is the frequency of the signal changing (like a chirp?)
- Is the amplitude of the two signals close to being the same? or are they very different?
- Does the application need the average phase lag across the entire frame/dataset, or does it require tracking of how the phase changes throughout time?
Depending on your answer to these questions, you could consider the following techniques:
- zero crossings - while this can be done with the I/Q separately, it's computationally intensive to look at 2 different I, and 2 different Q signals, and then average/combine the difference. This also has issues if there are small amounts of DC offset, or amplitude differences between the signals . The other options below do not have these same drawbacks.
- Cross Correlation : This is implemented in the iio oscilloscope application, by using the code found in David E. Narváez post about his thesis project, and in his Github repository. This measures similarity (phase and magnitude) of two series (in this case either real or complex signals) as a function of the lag of one relative to the other. The downside of this is its accuracy is based on the sample time relative to the signal being analyzed. In this case, both amplitude and phase/frequency are examined.
- single point analysis : this uses the above picture, and looks at the phase between 2 points. Since we are not really interested in a point in time (since this isn't how complex modulation works), this isn't applicable, or implemented. The below methods treat things with multiple captured samples, and are used, but are mostly based on this Law of cosines developed by Euclid of Alexandria in the 3rd century BC). The only issue is that the law of cosines, always provides the inside, or obtuse angle, not the reflex one, which may be the one we are interested in.
- Inner Product - although this works better for pure sinusoids, it can be used for arbitrary signals, but what you get is the angle between the two vectors (including all the noise), independent of amplitude. This uses the single point analysis described above, and uses it across the entire sampled data
where
denotes the complex conjugate
- Frequency Domain: This works well for signals which are dominated by a constant frequency. Here you convert your signals into the frequency domain (using fftw). Then, find the bin that corresponds to the frequency that you care about and get the angle between the two signals. For example, if we knew that the 18th bin was the frequency of interest, it's just simply :
You can use the Law of cosines to re-write this in the vector method (using dot products, and therefore multiplication of the complex conjugate as:
When you want to use the atan vs acos depends -- if the angles are close to 90º, use acos (tan of 90º is ∞). The downside is that cos/acos does have symmetry -- there is no way to tell the difference between 45º and -45º, or 90º and -90º so some extra testing of the real/imaginary signs may be necessary. If the angles are closer to 0º, use atan (the resolution of atan at 0º is higher than acos, so it will provide a more accurate number). Care also must be taken to ensure we are measuring the reflex angle when necessary. This is all implemented in the iio oscilloscope application, just choose 2 complex inputs for the frequency method, and the phase difference at different bins (the markers) will show up.
- Hilbert transform. This can be done, but I have not done it yet, so until I do, you can look it up on Google
Understanding what sort of issues we are trying to solve -- it's nice to understand the relationship between phase angle, and time lag/lead.
So for a 1MHz baseband sine wave, that is off by .5°, that is a
Like all electromagnetic waves, radio waves travel at the speed of light (~0.299792458 meter / ns). In this case 0.5°, means we could be ~41.64cm off any relative distance measurement (if that is what you are doing).
This is why we need to be sure we don't think about baseband.
For a 1MHz baseband sine wave, that we are modulating to 2.4GHz (2.401MHz wave), it's
This turns into a relative distance measurement of ~0.173 mm.
Needless to say - we want to be as precise as possible, and make sure we reference things to the air/RF interface, not the baseband signals.
When a single Continuous Wave (CW) RF tone is mixed to baseband, it creates two signals (in-phase and quadrature) that should be orthogonal to each other with the same amplitude. Without loss of generality, we normalize the magnitude and the phase, then the two signals can be expressed as:
and
As mentioned above, we want to rotate these signals around the same point, which is the same as providing a constant phase shift.
and
We can use the trigonometric sum/difference identities to re-write that as:
and
and substitute our original uncorrected values, (and swap the order of Q'), so we can see the matrix a little easier...
and
translating that into a 2 x 4 matrix multiplier is quite easy.
And while the coefficients are much different than adjusting Q with respect to I, the same hardware (the _2-by-2 multiplier)) can be used.
This is all well and good - but since we are dealing with fixed point math inside of an FPGA, we need to be careful of overflow. If we determine that we want to rotate something, psi is 45°, (or pi/4 radians),
since
,
,
If you are just investigating continuous tones, this would be fine. However, since simple CW tones don't really represent complex (I/Q) radio signals like QPSK, it's possible that the I/Q may be close to the max of the resolution, for the AD9361™ that would be 12 bits, or 1.11 bits (1 sign bit, 11 magnitude bits) result. For example: If I and Q are both at +1638 (about 80% of full scale), the results of the corrected I' and Q' would be I' = 0 and Q' = 2317. Q' will overflow an 1.11 bit representation.
There are two solutions:
- make the data path wider (which is what is done in the ADI designs, where we allow 16-bit results). This has a side effect of changing the scale of the device. Results are no longer just 12 bits, but could be half a bit higher (as mentioned above, we could change the scale by a factor of sqrt(2) or 1.41, which could add almost 1/2 a bit of magnitude).
- scaling down. While reducing the amplitude (which could throw away existing resolution) is always an issue, this does have the benefit of not changing the scale from ±1 (full scale will always be the same, no matter the rotation).
To scale, it's just a matter of dividing by the gain, 
It's important to take the absolute value, to ensure that 
(which would happen otherwise at +135 degrees (+3/4π radians) or +315 (-45) degrees (-π/4 radians)). This translates into:
and
OR
and
Which makes the matrix:
The above can be used directly, depending on what sort of functions your math library has available (sin, cos, tan) and the amount of rotation that you will allow (the above is for ±180 degrees). You can re-write in terms of other trig functions, but care must be taken to ensure that denominator never approaches zero.
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