SANDIA REPORT SAND2014-2252 Unlimited Release Printed March 2014
Mitigating I/Q Imbalance in RangeDoppler Images Armin W. Doerry
Prepared by Sandia National Laboratories Albuquerque, New Mexico 87185 and Livermore, California 94550 Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-AC04-94AL85000. Approved for public release; further dissemination unlimited.
Issued by Sandia National Laboratories, operated for the United States Department of Energy by Sandia Corporation. NOTICE: This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government, nor any agency thereof, nor any of their employees, nor any of their contractors, subcontractors, or their employees, make any warranty, express or implied, or assume any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represent that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government, any agency thereof, or any of their contractors or subcontractors. The views and opinions expressed herein do not necessarily state or reflect those of the United States Government, any agency thereof, or any of their contractors. Printed in the United States of America. This report has been reproduced directly from the best available copy. Available to DOE and DOE contractors from U.S. Department of Energy Office of Scientific and Technical Information P.O. Box 62 Oak Ridge, TN 37831 Telephone: Facsimile: E-Mail: Online ordering:
(865) 576-8401 (865) 576-5728
[email protected] http://www.osti.gov/bridge
Available to the public from U.S. Department of Commerce National Technical Information Service 5285 Port Royal Rd. Springfield, VA 22161 Telephone: Facsimile: E-Mail: Online order:
(800) 553-6847 (703) 605-6900
[email protected] http://www.ntis.gov/help/ordermethods.asp?loc=7-4-0#online
-2-
SAND2014-2252 Unlimited Release Printed March 2014
Mitigating I/Q Imbalance in RangeDoppler Images Armin Doerry ISR Mission Engineering Sandia National Laboratories PO Box 5800 Albuquerque, NM 87185-0519
Abstract It has become the norm for modern high-performance radar systems to sample and digitize the raw radar returns for subsequent processing by Digital Signal Processing (DSP) techniques. Furthermore, the data is often rendered to distinguish between positive and negative frequencies necessitating complex data values, with real and imaginary constituents typically termed In-phase (I) and Quadrature (Q) elements respectively. However, the reality of non-ideal component and circuit behavior will allow deleterious imbalances to manifest between I and Q data elements. A number of techniques may be employed to mitigate the effects of I/Q imbalance. These are presented, as well as recommendations for new designs, as well as retrofits to existing hardware.
-3-
Acknowledgements This report is the result of an unfunded Research and Development effort. Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000.
-4-
Contents Foreword ............................................................................................................................. 6 Classification .................................................................................................................. 6 1 Introduction ................................................................................................................. 7 2 Discussion – I/Q Imbalance......................................................................................... 9 2.1 What Is It? ............................................................................................................ 9 2.2 How Does It Manifest in the Range-Doppler Image? ........................................ 11 2.3 What Causes It? .................................................................................................. 13 2.3.1 Analog Down-Conversion to Quadrature Channels ................................... 13 2.3.2 Digital Down-Conversion to Quadrature Channels .................................... 14 3 Mitigation Techniques ............................................................................................... 21 3.1 Get Rid of It........................................................................................................ 21 3.2 Move the Effects ................................................................................................ 22 3.3 Smear the Effects ............................................................................................... 24 3.3.1 Range Smearing .......................................................................................... 24 3.3.2 Azimuth Smearing ...................................................................................... 27 3.3.3 Combined Range and Azimuth Smearing................................................... 33 3.4 Image Post-Processing ....................................................................................... 38 3.4.1 Image Ghost Subtraction............................................................................. 38 3.4.2 I/Q Imbalance Spur Apodization ................................................................ 38 4 Recommendations for Radar Systems ....................................................................... 39 4.1 New Designs ...................................................................................................... 39 4.2 Retrofit to Existing Designs ............................................................................... 39 5 Conclusions ............................................................................................................... 41 References ......................................................................................................................... 43 Distribution ....................................................................................................................... 44
-5-
Foreword This report details the results of an academic study. It does not presently exemplify any operational systems with respect to modes, methodologies, or techniques.
Classification The specific mathematics and algorithms presented herein do not bear any release restrictions or distribution limitations. This distribution limitations of this report are in accordance with the classification guidance detailed in the memorandum “Classification Guidance Recommendations for Sandia Radar Testbed Research and Development”, DRAFT memorandum from Brett Remund (Deputy Director, RF Remote Sensing Systems, Electronic Systems Center) to Randy Bell (US Department of Energy, NA-22), February 23, 2004. Sandia has adopted this guidance where otherwise none has been given. This report formalizes preexisting informal notes and other documentation on the subject matter herein.
-6-
1
Introduction
It has become the norm for modern high-performance radar systems to sample and digitize the raw radar returns for subsequent processing by Digital Signal Processing (DSP) techniques. Furthermore, the data is often rendered to distinguish between positive and negative frequencies necessitating complex data values, with real and imaginary constituents typically termed In-phase (I) and Quadrature (Q) elements respectively. Such techniques are common to 1. Synthetic Aperture Radar (SAR), and Inverse-SAR (ISAR), 2. Moving Target Indicator (MTI) radar, including Ground-MTI (GMTI) and similar radar modes, and 3. Wide-Area Search (WAS) modes, including Maritime-WAS (MWAS) and GMTI-WAS (GWAS) modes. Signal path characteristics such as gain and phase to the point where I and Q signal elements become digital data requires a very high degree of balance in order to maintain signal integrity. Failure to do so will allow artifacts to be generated and degrade the radar’s performance. Achieving an acceptable level of I/Q balance for some applications is problematic, requiring careful attention to system design and implementation. Doerry, et al. 1, suggest that for Dismount-MTI (DMTI) radar the suggested target RCS detection threshold of −10 dBsm would be exceeded about once per square mile if false alarm artifacts (like those due to I/Q imbalance) were higher than −55 dBc. An artifact limit of −55 dBc is a pretty harsh requirement for I/Q balance. That is not to say that worse performance would render a radar useless. However, it is undeniable that it would impact the goodness of performance. An oft-touted mitigation strategy is to collect digital data samples at an Intermediate Frequency (IF) and creates I/Q components via Digital Signal Processing (DSP) techniques. Ostensibly, I/Q balance should then be perfect. But, as we shall show, this is somewhat optimistic (to be polite). Where it makes a difference, we shall herein presume a Linear-FM (LFM) chirp waveform with “stretch processing,” as proposed by Caputi. 2 This is where the received echo analog signal is de-ramped, or de-chirped prior to digitization; a popular technique that allows wide-bandwidth echoes to be sampled at a lower data rate.
-7-
“Intellectuals solve problems, geniuses prevent them.” -- Albert Einstein
-8-
2 Discussion – I/Q Imbalance 2.1 What Is It? Consider a complex signal, with characteristics of variations in magnitude and phase. We identify this signal as x(t ) = input signal (complex).
(1)
This signal can be decomposed into real and imaginary elements, also known as In-phase (I) and Quadrature (Q) elements respectively. That is x(t ) = α (t ) + jβ (t ) = input signal (complex).
(2)
It is customary and routine to decompose a signal into these elements for signal processing. Nevertheless, a key point is that element signals α (t ) and β (t ) themselves are real-valued, and are created from x(t ) , or at least that is the intent. To investigate further, we recognize that we may also represent x(t ) as
x ( t ) = A ( t ) exp ( jB ( t ) ) ,
(3)
where we identify the real-valued elements as A(t ) = magnitude function, and B(t ) = phase function,
(4)
and identify based on the Euler formula
α ( t ) = A ( t ) cos ( B ( t ) ) , and
β ( t ) = A ( t ) sin ( B ( t ) ) .
(5)
So, the desired signal x(t ) can be written as
= x ( t ) A ( t ) cos ( B ( t ) ) + jA ( t ) sin ( B ( t ) ) .
(6)
In fact, for real systems, due to the inescapable reality of non-ideal components, what are typically really created are perturbed components of the signal, both in amplitude and phase, or delay. That is, we really end up with signals more accurately described by
-9-
t ϕ a t A ( t ) cos ( B ( t ) ) → 1 + A t − a cos B t − a − a 2 2 2 2
, and
−1
t t ϕ a A ( t ) sin ( B ( t ) ) → 1 + A t + a sin B t + a + a , 2 2 2 2
(7)
where a = amplitude perturbation factor, ta = delay perturbation factor, and
φa = phase offset perturbation factor.
(8)
When we think we are representing x(t ) , we are actually representing a different signal a ta ta ϕ a 1 + A t − cos B t − − 2 2 2 2 y (t ) = −1 + j 1 + a A t + ta sin B t + ta + ϕa 2 2 2 2
.
(9)
We note that the ratio of real to quadrature component amplitude is Re {max y ( t )}
2
a = 1 + ≈ (1 + a ) . Im {max y ( t )} 2
(10)
We also note that the phase difference between real and quadrature components is
∠ Re { y ( t )} − ∠ Im { y ( t )} = −ta B′ ( t ) − ϕa +
π . 2
(11)
To be sure, for small perturbations, y (t ) looks very much like the desired signal x(t ) , but it in fact is slightly different. Consequently, I/Q imbalance is perturbation of the relationship between Real and Imaginary components of the signal.
- 10 -
2.2 How Does It Manifest in the Range-Doppler Image? We can assess the effects of the perturbation by decomposing the components of y (t ) as follows. y ( t ) ≈ y ( t ) ta ==00 + a a
∂y ( t )
+ ta
∂a
∂y ( t )
+ ϕa
∂t
∂y ( t ) ∂ϕ
a 0= a 0 = a 0 =
.
(12)
a a t 0 0 ϕa == t a 0= ta 0 = a = ϕ a 0= ϕa 0 = ϕa 0
Of course, this allows us to separate the original signal from the error manifestations as y (t ) ≈ x (t ) + a
∂y ( t ) ∂a
+ ta
∂y ( t ) ∂t
+ ϕa
∂y ( t ) ∂ϕ
a 0= a 0 = a 0 = a a t 0 t a 0= ta 0 = = a ϕ a 0= ϕa 0 = ϕa 0 =
.
(13)
By performing the partial derivatives, this can be expanded to aA ( t ) 1 + jϕ a A ( t ) y (t ) ≈ x (t ) + exp j ( − B ( t ) ) , 2 −ta A′ ( t ) + jt A t B′ t a ( ) ( )
(14)
where A′(t ) = time derivative of A(t ) , and B′(t ) = time derivative of B(t ) .
(15)
This can be simplified to y ( t ) ≈ x ( t ) + Cimbalance x* ( t )
(16)
where x* (t ) = complex conjugate of x(t ) , and a + j (φa + ta B′(t )) ta A′(t ) Cimbalance = − . 2 2 A(t )
- 11 -
(17)
0
-10
50
-20 100 -30 150 -40 200 -50
250 50
100
150
200
250
-60
Figure 1. Example with I/Q Imbalance resulting in −26 dBc artifacts. The colorbar denotes dBc.
We observe the following •
Perturbations in amplitude, phase, and/or delay will cause the addition of an error signal that is the complex conjugate of the input signal.
•
If the input signal is a complex sinusoid, then A′(t ) = 0 . In SAR images, point targets will exhibit A′(t ) = 0 .
•
The error signal will exhibit opposite phase.
•
The error signal will also exhibit opposite phase rate.
•
In a range-Doppler image, this will manifest as both an opposite range offset, and an opposite Doppler offset. That is, this will manifest as a ‘ghost’ image reflected through the zero-range and zero-Doppler point, the actual DC point in the image.
We illustrate the effects of I/Q imbalance in Figure 1. The actual point targets are the brighter points in the bottom half of the image. For this example a = 0.1 , resulting in I/Q imbalance ghosts at –26 dBc. In this case the DC point is at the center of the image. Note the ghost targets rotated 180 degrees about the DC point.
- 12 -
2.3 What Causes It? We discuss below the typical root causes of I/Q imbalance. 2.3.1 Analog Down-Conversion to Quadrature Channels We identify a real-valued IF signal as = xIF ( t ) x ( t ) e j 2π f IF t + x* ( t ) e − j 2π f IF t .
(18)
where
f IF = the IF frequency on which the desired signal is modulated.
(19)
We desire to extract or isolate the complex baseband signal x ( t ) . We accomplish this as follows. First we down-convert the IF signal to a baseband signal created and identified as f IF t xbaseband = x ( t ) + x* ( t ) e − j 2π 2 f IF t . ( t ) xIF ( t ) e− j 2π=
(20)
Note that we may use the Euler formula and identify elemental signals
Re { xbaseband ( t )} = xIF ( t ) cos ( 2π f IF t ) , and
Im { xbaseband ( t )} = − xIF ( t ) sin ( 2π f IF t ) .
(21)
Analog lowpass filtering and subsequent sampling with Analog-to-Digital Converter (ADC) components of the down-converted signal yields x ADC ( n ) = x ( n f s ) ,
(22)
f s = ADC sampling frequency,
(23)
where
and the ADC samples are at uniformly spaced sample times of
t = n fs ,
(24)
n = sample index value.
(25)
where
- 13 -
The architecture to accomplish this is illustrated in Figure 2. Note that we require two ADC components to implement this architecture. Clearly, if the two channels were not identical to each other in all but the IF demodulation employed, or the demodulation were not exactly π 2 radians apart in phase, then an imbalance would be generated in the ADC samples. Since we are talking about analog signal paths, imbalance to some degree is virtually guaranteed. Furthermore, balance levels required by modern high-performance radar systems are virtually impossible to achieve without further mitigation techniques. 2.3.2 Digital Down-Conversion to Quadrature Channels We begin with a single ADC sampling the IF signal directly. It is particularly convenient for the input signal to be at an IF center frequency of f s 4 , that is, we will presume
f IF = f s 4 .
(26)
Consequently, we identify the IF signal as xIF ( t ) x ( t ) e =
j 2π ( f s 4 ) t
+ x* ( t ) e
− j 2π ( f s 4 ) t
.
(27)
Recall that this signal is real-valued. Furthermore, x(t ) is band-limited to something less than f s 2 , and often to less than f s 4 . The ADC then generates the data samples described by x ADC ( n ) = xIF ( n f s ) .
(28)
Note that the sampled signal x ADC ( n ) is also real-valued. The digital down-conversion is implemented as the complex multiplication
xdem ( n ) = e
− j (π 2 ) n
x ADC ( n ) .
(29)
Combining the previous results yields the signal of interest, namely x= x ( n f s ) + x* ( n f s ) e − jπ n . dem ( n )
(30)
This normally undergoes additional low-pass filtering to remove the e − jπ n modulated term, and typically some data decimation as well. We model the filtered demodulated signal as xdemfil ( n ) = x ( n f s ) .
(31)
- 14 -
Lowpass Filter
X xIF ( t )
ADC
cos ( 2π f IF t )
fs
− sin ( 2π f IF t )
Σ
Lowpass Filter
ADC
Σ
x ( n fs )
Σ
x ( n fs )
X j
Figure 2. Analog down-conversion architecture.
Lowpass Filter
X xIF ( t )
ADC
fs
cos ( 2π f IF t ) − sin ( 2π f IF t )
Σ
Lowpass Filter
X j
Figure 3. Digital down-conversion architecture. This is usually implemented with
f IF = f s 4 .
As stated, for digital down-conversion it is quite typical to build in some data decimation, usually by a factor of 4, so that the new data rate is in fact equal to the original IF frequency, which we recall as f IF = f s 4 . However, for the purposes of this report we will ignore any decimation. It is this term that is of interest to us for further processing. Note that inadequate filtering may also ultimately cause unacceptable I/Q imbalance all by itself. We will ignore this, and assume that the filter was designed and operates correctly. Consider the output of the digital down-conversion step xdem (n ) . We observe that the signal can be described with the following expansion. - 15 -
x ADC ( n ) n = 0, 4,8,... n − jx ADC ( n ) n = 1,5,9,... . xdem ( n ) = ( − j ) xADC ( n ) = − x ADC ( n ) n = 2, 6,10,... jx ADC ( n ) n = 3, 7,11,...
(32)
Note that even-valued values of index n yield purely real values for xdem (n ) , and odd-
valued values for index n yield purely imaginary values for xdem (n ) . That is
xIF ( n f s ) n = 0, 4,8,... Re { xdem ( n )} = 2, 6,10,... − xIF ( n f s ) n = 0 n = odd
(33)
− xIF ( n f s ) n = 1,5,9,... = xIF ( n f s ) n 3, 7,11,... . Im { xdem ( n )} = n = even 0
(34)
and
Combining some things, we identify explicitly the digital demodulation sequences as
π cos ( 2π f IF= t ) cos n= 1, 0, −1, 0,1,... , and 2 π sin ( 2π = f IF t ) sin = n 0,1, 0, −1, 0,... 2
(35)
The architecture to accomplish this is illustrated in Figure 3. Note that we require but a single ADC component to implement this architecture. The fact that this final demodulation can be accomplished by multiplications limited to 1, −1, and 0, is particularly attractive, and illustrates the utility of this digital downconversion scheme, in particular the relationship of IF frequency to ADC sampling frequency. 2.3.2.1
Digital Down-Conversion with Clock Interference
Now consider that the ADC samples are taken at nonuniform-spaced sample times where the sample times are modulated at a rate of f s 2 . The model for sample times becomes
f n = t n f s − cos 2π s ( n f s ) ( ta 2= ) n f s − ( −1) ( ta 2 ) . 2
- 16 -
(36)
The ADC samples then become the time-shifted (time-modulated) signal values
(
)
x ADC = ( n ) xIF n f s − ( −1) ( ta 2 ) . n
(37)
The output of the digital down-conversion step xdem (n ) then exhibits I/Q components of
(
)
x n f − ( −1)n ( t 2 ) s a IF n Re { xdem ( n )} = − xIF n f s − ( −1) ( ta 2 ) 0
(
n =0, 4,8,...
)
n = 2, 6,10,...
(38)
n = odd
and
(
)
− x n f − ( −1)n ( t 2 ) n =1,5,9,... s a IF n Im { x= ) n 3, 7,11,... . xIF n f s − ( −1) ( ta 2= dem ( n )} 0 n = even
(
)
(39)
Of course, this simplifies to the expressions for I and Q components of
xIF ( n f s − ( ta 2 ) ) n= 0, 4,8,... Re { xdem ( n )} = 2, 6,10,... − xIF ( n f s − ( ta 2 ) ) n = n = odd 0
(40)
− xIF ( n f s + ( ta 2 ) ) n = 1,5,9,... Im { xdem ( n )} = n= 3, 7,11,... . xIF ( n f s + ( ta 2 ) ) n = even 0
(41)
and
Clearly, all real terms are advanced by (ta 2 ) , and all imaginary terms are retarded by (ta 2) . That is
{
}
Re {= xdemfil ( n )} Re x ( n f s − ( ta 2 ) ) and
- 17 -
(42)
{
}
Im {= xdemfil ( n )} Im x ( n f s + ( ta 2 ) ) .
(43)
Putting it all together yields
{
}
{
}
xdemfil ( n= ) Re x ( n f s − ( ta 2 ) ) + j Im x ( n f s + ( ta 2 ) ) .
(44)
Hence, ADC clock modulation resulting from some interference-source yields an I/Q imbalance. From the earlier analysis, we can describe the output signal as xdemfil ( n ) ≈ x ( n f s ) +
A′ ( n f s ) * ta jB′ ( n f s ) − x ( n fs ) . 2 A ( n f s )
(45)
That is, repeating from above, xdemfil ( n ) ≈ x ( n f s ) + Cimbalance x* ( n f s ) ,
(46)
Point Target Response Consider the case where the desired signal is described by the rotating phasor x ( t ) = A0 e j 2π f0t ,
(47)
A0 = constant amplitude, and f 0 = constant frequency.
(48)
where
This model is consistent with a point reflector radar echo after dechirping the echo response from an LFM chirp waveform. From this model, we note that A′(t ) = 0 , and B′(t ) = 2πf 0 .
(49)
This allows the demodulated and filtered output samples to be described as
xdemfil ( n ) ≈ x ( n f s ) +
ta [ j 2π f0 ] x* ( n f s ) . 2
(50)
Interestingly, the amount of imbalance energy depends on the signal frequency f 0 . In a range-Doppler image, this corresponds to the range offset of the target reflector.
- 18 -
2.3.2.2
Digital Down-Conversion with Ping-Pong ADCs
The foregoing analysis suggests that whenever digital down-conversion is used, any modulation applied to the ADC in any manner has the potential of manifesting an I/Q imbalance in the data. This is because of the feature that alternating raw ADC samples become I and Q data components respectively. In the previous section, we showed how Electromagnetic Interference (EMI) modulating the ADC strobe signal can cause I/Q imbalance. We put forward now that even in the absence of EMI, we may still have sample-to-sample modulation of the data. Modern high-speed ADC components, of the sort that are normally attractive to radar designers, often achieve their high speed with an internal architecture that multiplexes two or more internal ADC functional blocks. One such example is the National ADC08D1520 A/D Converter 3, when operating in Dual Edge Sampling (DES) mode. † In principle, this works great. However, also in principle, this relies on the application not being sensitive to subtle differences in the internal ADC functional blocks. In practice, we observe that 1. Real ADC functional blocks are not perfect, and exhibit some degree of nonlinearity. 2. Real ADC functional blocks are not identically imperfect. They exhibit different nonlinearities from each other. The nonlinearity of an ADC conversion is often embodied in its Integrated Non-Linearity (INL) specification. There is also often a Differential Non-Linearity (DNL) specification as well. Nonlinearity affects gain. Different nonlinearities in ADC functional blocks affect gain differently in each functional block. Alternating between ADC functional blocks means alternating between signal gain characteristics. This in turn yields an amplitude modulation that is at a rate of f s 2 . This results in different gains for I values than for Q values, and hence an I/Q gain imbalance. Recall that an I/Q imbalance due to gain differences is not dependent on range offset in the image. However, it will be dependent on actual signal levels at the ADC. 2.3.2.3
Some Additional Comments
A detailed analysis of the internal operation of any particular ADC may reveal other possible EMI-based pathologies. While INL has been examined, we suggest that other EMI entry points into the ADC should not be neglected, such as the power and ground connections.
†
The National ADC08D1520 A/D Converter is a dual ADC device. In DES mode, the pair operates in a ping-pong fashion. However, even an individual ADC of the pair is itself a ping-ponged set of ADCs. Consequently, even if operating only half of the ADC08D1520, we are not able to escape ping-pong operation. The same is true for the ADC08D1500 part.
- 19 -
“Quality means doing it right when no one is looking.” -- Henry Ford
- 20 -
3 Mitigation Techniques Mitigation techniques for unwanted spurious energy are generally implemented with the following order of precedence. 1. Get rid of the spurious energy, 2. Move the spurious energy to unwanted signal space, 3. Smear the spurious energy to reduce its coherence and peak value, and 4. Reduce manifestation directly via image post-processing. We examine these in turn. First, however, we note that mitigating the effects of I/Q imbalance is very related to mitigating the effects of other sources of spurious energy in a radar receiver, such as harmonic spurs due to ADC INL.
3.1 Get Rid of It We recall the circuit design axiom “The circuit you have isn’t the circuit you think you have.” In particular for analog down-conversion architectures, where I and Q analog channels are created, if the imbalance is stable, then calibration might be employed to achieve improved balance. To achieve the required signal fidelity, the calibration might be for amplitude, phase, and/or delay, and furthermore may need to be a function of signal frequency. In practice, for some applications (like DMTI, etc.) it is quite difficult to achieve satisfactory performance. In digital down-conversion architectures, I/Q imbalance in some cases can be a result of inadequate isolation of the susceptible ADC connections (e.g. clock, power, ground, etc.) from other inadvertent signal emitters. This is an EMI problem. Consequently, the preferred solution is EMI mitigation to effectively eliminate the coupling mechanism. This involves one or more of the following. 1. Reducing or eliminating EMI emissions at their source. 2. Decreasing the EMI coupling mechanism. 3. Reducing or eliminating the EMI susceptibility of the ADC clock and other ADC connections in the radar receiver. This is aided by good mixed-signal design practices as reported by Dudley. 4
- 21 -
3.2 Move the Effects In a typical SAR image, the scene of interest is only a portion of the overall Doppler bandwidth, and sometimes a very small portion. Consequently, if the imbalance energy can be moved to a Doppler region that will be cropped away, then it will not appear in the image. A technique for doing this is presented in a patent by Doerry and Tise. 5 We briefly repeat a description of this technique here. Recall that the IF input signal is = xIF ( t ) x ( t ) e j 2π f IF t + x* ( t ) e − j 2π f IF t .
(51)
Now let us create a new IF signal that incorporates a particular pulse-dependent rolling phase shift and is described as = xIF , ps ( t , m ) x ( t ) e j 2π f IF t e
+ x* ( t ) e − j 2π f IF t e
j 2π (1 4 ) m
− j 2π (1 4 ) m
,
(52)
where m = pulse index number.
(53)
Note that e j 2π (1 4 )m represents a π 4 phase accumulation with each increment in pulse index. We now define an equivalent input signal with the phase shift as x ps ( t , m ) = x ( t ) e
j 2π (1 4 ) m
,
(54)
We can now write the new IF signal as = xIF , ps ( t , m ) x ps ( t , m ) e j 2π f IF t + x ps* ( t , m ) e − j 2π f IF t .
(55)
For convenience, we will assume digital down-conversion. Consequently, the demodulated and filtered signal may be expanded to
xdemfil , ps ( n, m ) ≈ x ps ( n f s , m ) + Cimbalance x ps* ( n f s , m ) ≈ x ( n fs ) e
j 2π (1 4 ) m
+ Cimbalance x* ( n f s ) e
− j 2π (1 4 ) m
.
(56)
Finally, we remove the previously inserted pulse-dependent phase shift by forming xdemfil ( n, m ) = e
− j 2π (1 4 ) m
xdemfil , ps ( n, m ) ,
(57)
which can be expanded and simplified to
xdemfil= ( n, m ) x ( n f s ) + Cimbalance x* ( n f s ) e− jπ m .
- 22 -
(58)
0
-10
50
-20 100 -30 150 -40 200 -50
250 50
100
150
200
250
-60
Figure 4. Example of Figure 1 but with rolling phase shift prior to ADC. For convenience, we have allowed the imbalance to be constant for all targets. Note that the anomalous responses have been shifted to the edge of the image, but the true signals are properly located.
We note the following. •
The output contains the signal x(n f s ) as desired. This is good.
•
The conjugate term x* (n f s ) is modulated by e − jπ m , which shifts the ghost echoes by half the image extent in azimuth (Doppler).
•
It is important that the rolling phase shift be applied before the I/Q imbalance occurs, and then removed afterwards. That is, for digital down-conversion, this is normally applied before the ADC, and removed from the data after the ADC.
•
For a range-Doppler image, a tacit requirement is that the radar PRF be at least twice the Doppler bandwidth of the in-beam clutter. That is the ‘shifted to’ region has to be at least as large as the ‘shifted from’ region in Doppler. The ‘shifted to’ region is then cropped away, thereby removing the I/Q imbalance energy.
•
For GMTI and other techniques that use the larger Doppler space, including the exo-clutter region, this technique is not appropriate. That is, generally the ‘shifted to’ region is not cropped away.
This effect of this technique is illustrated in Figure 4, with shifted imbalance artifacts.
- 23 -
3.3 Smear the Effects If the offending I/Q imbalance energy cannot be suppressed or moved to a harmless location in the range-Doppler space, then the next best mitigation strategy normally is to smear it by reducing its coherence, thereby reducing its obviousness in the range-Doppler map. This is simply diluting the imbalance energy. The smearing can be applied in either or both of the range and Doppler directions. 3.3.1 Range Smearing A convenient means of smearing the I/Q imbalance energy in range is to impart a quadratic phase error in range to the imbalance energy. The effects of a quadratic phase error on impulse response is discussed in an earlier report by Doerry. 6 Of course, since a quadratic phase error is simply a LFM chirp, one way to impart a quadratic phase error onto the imbalance energy is to leave a residual chirp on the data after the normal dechirping operation. This is discussed in a patent by Dubbert & Tise. 7 Here we briefly discuss the details of why a residual chirp helps with smearing I/Q imbalance energy. Recall that the IF input signal is = xIF ( t ) x ( t ) e j 2π f IF t + x* ( t ) e − j 2π f IF t .
(59)
Now let us create a new IF signal that incorporates a residual chirp, due to perhaps not completely de-chirping the radar echo energy. We describe this residual chirp effect as xIF ,rc ( t ) x ( t ) e j 2π f IF t e =
j 2π ( γ rc 2 )t 2
+ x* ( t ) e − j 2π f IF t e
− j 2π ( γ rc 2 )t 2
,
(60)
where
γ rc = residual chirp rate.
(61)
We now define an equivalent input signal with the phase shift as xrc ( t ) = x ( t ) e
j 2π ( γ rc 2 )t 2
,
(62)
We can now write the new IF signal as = xIF ,rc ( t ) xrc ( t ) e j 2π f IF t + xrc* ( t ) e − j 2π f IF t .
(63)
Using perfect digital down-conversion, this is then the signal that is sampled by the ADC, and then gets digitally down-converted and filtered to become
xdemfil ,rc ( n ) ≈ xrc ( n f s ) + Cimbalance xrc* ( n f s ) .
- 24 -
(64)
This can be expanded again to include the original signal as x ( n f ) e j 2π (γ rc 2)( n f s ) s xdemfil ,rc ( n ) ≈ − j 2π ( γ rc * +C imbalance x ( n f s ) e 2
2 )( n
. 2 fs )
(65)
Finally, we remove the previously inserted sample-dependent phase shift by forming xdemfil ( n ) = e
− j 2π ( γ rc 2 )( n f s )
2
xdemfil ,rc ( n ) ,
(66)
which can be expanded and simplified to xdemfil = ( n ) x ( n f s ) + Cimbalance x* ( n f s ) e
− j 2π ( γ rc )( n f s )
2
.
(67)
We note the following. • •
•
The output contains the signal x(n f s ) as desired. This is good. The conjugate term x* (n f s ) is modulated by e − j 2π (γ rc )(n f s ) , which is a residual chirp at twice the chirp rate of the original residual chirp on the signal prior to the ADC. 2
It is important that the residual chirp be applied before the I/Q imbalance occurs, and then removed afterwards. That is, for digital down-conversion, this is normally applied before the ADC, and removed from the data after the ADC.
The relevant parameter of the residual chirp is its time-bandwidth product. We define
TB = the residual chirp time-bandwidth product at the ADC input.
(68)
The time-bandwidth product of the imbalance energy in the final image is twice this, and is also the amount of smearing as measured at the −3 dB points of the smeared response in terms of resolution units. Consequently, the expected attenuation factor of the imbalance energy is
Limb ≈ 2 TB ,
(69)
where a loss is indicated by Limb > 1 . Since the smearing is not entirely ‘flat’, the amount of actual loss is typically about a dB less than this. This will be dependent on sidelobe filtering, i.e. the window function employed. Consequently, a better prediction for the case of using a −35 dB Taylor window with nbar = 4 is
Limb ≈ 1.6 TB ,
(70)
- 25 -
0
-10
50
-20 100 -30 150 -40 200 -50
250 50
100
150
200
250
-60
Figure 5. Example of Figure 1 but with a residual chirp applied prior to ADC. For convenience, we have allowed the imbalance to be constant for all targets. Note that the anomalous responses have been smeared in range and reduced in peak value by a measured 12 dB.
This effect of this technique is illustrated in Figure 5. In this example, the sampled data retained a residual chirp with a time-bandwidth product of 10. This means that the I/Q imbalance of Figure 1 is smeared in the final image with a residual chirp with timebandwidth product of 20. Theoretically this should reduce peak value of the imbalance energy by approximately 12 dB. In this example, we in fact measured a peak value for I/Q imbalance of −38 dBc, for a reduction of 12 dB. Fortunately the math works. Without any residual chirp, normal dechirping imposes a tight relationship between range swath and signal bandwidth. That is, the scene range dimension is constrained by the signal passband imposed by the various signal filters. However, allowing a residual chirp on the signal causes an increase in the required signal bandwidth for a particular scene range dimension. Alternatively, a particular filter will reduce the scene dimension if a residual chirp is employed, or at least the scene dimension for unaffected regions. The affected regions at near and far ranges will exhibit a loss in energy and fidelity. This is an ‘edge effect’ at near and far ranges. For this reason, residual chirps should be limited so that the smearing is small compared to the overall desired scene range dimension. For example, with perhaps 3000 range resolution cells, smearing across 100 range resolution cells represents only a 3.3% reduction in full fidelity data.
- 26 -
This of course applies to the residual chirp as manifested in the data prior to the removal of the eventual residual chirp from the desired signal. It does not apply to the residual chirp on the imbalance energy in the final focused image. 3.3.2 Azimuth Smearing As with the residual chirp for range smearing, accomplishing smearing in azimuth requires applying and removing a phase modulation in the raw data in azimuth, that is, on a pulse-to-pulse basis. We examine this here. Recall once again that the IF input signal is = xIF ( t ) x ( t ) e j 2π f IF t + x* ( t ) e − j 2π f IF t .
(71)
Now let us create a new IF signal that incorporates a particular pulse-dependent generalized phase shift and is described as = xIF ,rp ( t , m ) x ( t ) e j 2π f IF t e
jϕ ( m )
+ x* ( t ) e − j 2π f IF t e
− jϕ ( m )
,
(72)
where φ (m ) = generalized pulse-dependent phase value.
(73)
The nature of φ (m ) may be a specific function, or perhaps selections from a set of random values. For smearing, we will require φ (m ) to be nonlinear with m. We now define an equivalent input signal with the phase shift as xrp ( t , m ) = x ( t ) e
jϕ ( m )
,
(74)
We can now write the new IF signal as = xIF ,rp ( t , m ) xrp ( t , m ) e j 2π f IF t + xrp* ( t , m ) e − j 2π f IF t .
(75)
Using digital down-conversion, this is then the signal that is sampled by the ADC, and then gets down-converted and filtered to become
xdemfil ,rp ( n, m ) ≈ xrp ( n f s , m ) + Cimbalance xrp* ( n f s , m ) .
(76)
This can be expanded again to include the original signal as xdemfil ,rp ( n, m ) ≈ x ( n f s ) e
jϕ ( m )
+ Cimbalance x* ( n f s ) e
− jϕ ( m )
.
Finally, we remove the previously inserted pulse-dependent phase shift by forming
- 27 -
(77)
xdemfil ( n, m ) = e
− jϕ ( m )
xdemfil ,rp ( n, m ) ,
(78)
which can be expanded and simplified to xdemfil= ( n, m ) x ( n f s ) + Cimbalance x* ( n f s ) e
− j 2ϕ ( m )
.
(79)
We note the following. •
The output contains the signal x(n f s ) as desired. This is good.
•
The conjugate term x* (n f s ) is modulated by e − j 2φ (m ) , which is twice the phase as what was applied.
•
With proper modulation of phase shifts, the imbalance energy will smear in azimuth, perhaps even across the entire Doppler space. A proper phase shift
modulation is one for which e − j 2φ (m ) will still cause smearing, that is, for example, does not degenerate to a constant. This is addressed more completely below. •
It is important that the phase shift be applied before the I/Q imbalance occurs, and then removed afterwards. That is, for digital down-conversion, this is normally applied before the ADC, and removed from the data after the ADC.
Subsequent Azimuth Preprocessing As with range smearing, any phase modulation in azimuth (i.e. pulse to pulse) will spread the Doppler spectrum. This implies that it should ideally be removed before any subsequent Doppler filtering, so as to not impact image fidelity at the azimuth edges. For some modes, like SAR, Doppler filtering is often implemented in two main stages. The first is often in high-speed logic with the purpose of data reduction, often termed azimuth preprocessing, for example via presumming and/or azimuth prefiltering. The second is in the range-Doppler processing normally implemented in the subsequent general purpose processors. Depending on where in the processing chain the pulse-dependent phase modulation is removed, this will impart constraints onto the nature of the pulse-dependent modulation, in particular the Doppler spreading allowed by the modulation. We now specifically consider presumming in SAR. If the pulse-dependent phase modulation can be removed before the presummer, then no constraints are imposed on the phase modulation. Pulse-to-pulse phase variations are unconstrained. If the pulse-dependent phase modulation is removed after the presummer, then the modulation bandwidth needs to be constrained to be less than or equal to the presummer
- 28 -
passband. This implies a high degree of correlation for pulse-to-pulse phase variations during a presum interval. One way to achieve this is to only allow modulation phase changes to occur only at presum boundaries, but keep modulation phase constant during the presum interval. This applies only to the modulation meant to deal with spurious energy, and not the normal modulation that might be a part of motion compensation. Of the two, applying the widest possible bandwidth producing modulation and removing it prior to the presummer will generally spread the I/Q imbalance energy the most, thereby minimizing its peak manifestation in the SAR image. In GMTI there is no presumming, so no azimuth preprocessing is generally employed. In the subsequent analysis we will ignore any azimuth preprocessing, allowing that modulation can always be applied on a presum interval basis. 3.3.2.1
Azimuth Chirps
As with the residual chirp technique, we can spread imbalance energy in azimuth by applying an azimuth chirp prior to the ADC and removing it afterwards. This is implemented via a quadratic phase modulation. The effect of quadratic phase modulation is illustrated in Figure 6. In this example, a quadratic phase function with time-bandwidth product of 20 was applied and then removed from the data. The smearing across the Doppler space for this example should theoretically reduce the peak imbalance energy by approximately 15 dB. In this example, we in fact measured a peak value for I/Q imbalance of −41 dBc, for a reduction of 15 dB. Larger time-bandwidth products would improve this, although at some point aliasing would cause constructive interference leading to peak imbalance energy levels even larger yet than theoretical. 3.3.2.2
Pulse Random Phase Coding
We now address a random phase error of some type applied on a pulse to pulse basis. Any random phase modulation applied will be ‘doubled’ in phase value, which except in pathological cases retains its randomness. Consequently, the imbalance energy is affected by a random phase modulation. The Doppler processing implements a filter bank. However, filtering a random signal yields only another random signal. Uncorrelated phase modulation will yield a ‘white’ spectrum in the image. Central Limit Theorem arguments indicate that in the final image, the phase modulation will yield White Gaussian Noise, manifested as a streak across the Doppler spectrum with complex Gaussian characteristics. The magnitude would of course be Rayleigh distributed. Essentially, the random phase modulation will smear the imbalance energy into a noisy streak across the Doppler space. The noisy streak will have magnitude peaks substantially higher than the magnitude average, consistent with the Rayleigh distribution. Statistically, a uniform smearing across Doppler will yield a reduction in imbalance energy of - 29 -
0
-10
50
-20 100 -30 150 -40 200 -50
250 50
100
150
200
250
-60
Figure 6. Example of Figure 1 but with quadratic pulse-dependent phase shift prior to ADC. For convenience, we have allowed the imbalance to be constant for all targets. Note that the anomalous responses have been smeared substantially in azimuth, but the true signals are properly located.
Limb ≈
M , aw
(80)
where
M = number of pulses, aw = the impulse response broadening factor due to window function.
(81)
However, this is for a uniform distribution of energy across the Doppler spectrum. The random nature of the noisy streak will cause individual pixels to depart from this, and allow substantially higher peak values. The peak value is again a statistical value, based on the Rayleigh distribution. Typical peak to mean values encountered are in the range of 8-10 dB. Greater values are more likely encountered with larger Doppler dimensions, due to more opportunities. It’s a statistical thing. The effect of random phase modulation is illustrated in Figure 7. In this example, random values from a continuum of values in the range [0,2π ) were applied and then removed from the data. A uniform smearing across the Doppler space for this example would theoretically reduce the peak imbalance energy by 23 dB (assuming the −35 dB - 30 -
0
-10
50
-20 100 -30 150 -40 200 -50
250 50
100
150
200
250
-60
Figure 7. Example of Figure 1 but with pulse-dependent random phase shift prior to ADC. For convenience, we have allowed the imbalance to be constant for all targets. Note that the anomalous responses have been smeared across the extent of the image, but the true signals are properly located.
Taylor window with nbar = 4). Allowing for an 8 dB peak to average discount, we would expect perhaps a 15 dB reduction in peak I/Q imbalance energy. In this example, we in fact measured a peak value for I/Q imbalance of −41 dBc, for a reduction of 15 dB. We do note in the image that the smearing is not absolutely uniform, as expected. 3.3.2.3
Pulse Quantized Random Phase Coding
We recall that the requirement on random phase modulation is that e − j 2φ (m ) is still a random phase shift, or at least not constant. In the example above, φ (m ) was a random value in the continuous interval [0,2π ) . While sufficient, this is not absolutely necessary. In general, φ (m ) may be quantized. We define the quantization process as a rounding operation, where round( z , ζ ) = the nearest integer multiple of ζ to the value z.
We now discuss some specific examples.
- 31 -
(82)
Random 0/π Phase Modulation In this example, we limit
ϕ ( m ) = round ( z , π ) , where z ∈ [ 0, 2π ) .
(83)
This means we limit values for φ (m ) to 0 and π. Of course, this means that e − j 2φ (m ) = 1 , which means that the imbalance energy remains unaffected. Consequently, random 0/π modulation will not work. Random π/2 Multiples Phase Modulation In this example, we limit
ϕ ( m ) = round ( z , (π 2 ) ) , where z ∈ [ 0, 2π ) .
(84)
This means we limit values for φ (m ) to integer multiples of π 2 . This also means that the 2φ (m ) phase modulation applied to the I/Q imbalance energy in the final rangeDoppler map is ultimately modulated by a random 0/π modulation. This is sufficient to spread the imbalance energy in Doppler. This effect of this quantized phase modulation is illustrated in Figure 8. In this example, random integer multiples of π 2 phase in the interval [0,2π ) were applied and then removed from the data. As before, a uniform smearing across the Doppler space would theoretically reduce the peak imbalance energy by 23 dB (assuming the −35 dB Taylor window with nbar = 4). Allowing for an 8 dB peak to average ratio would yield a peak imbalance reduction of 15 dB. In this example, we in fact again measured a peak value for I/Q imbalance of −41 dBc, for a reduction of 15 dB. This is the same performance as an unquantized random phase modulation. Random 0/(π/2) Phase Modulation In this example, we limit
ϕ ( m ) = round ( z , (π 2 ) ) , where z ∈ [ 0, π ) .
(85)
This means we limit values for φ (m ) to only 0 and π 2 . This also means that the 2φ (m ) phase modulation applied to the I/Q imbalance energy in the final range-Doppler map is ultimately modulated by a random 0/π modulation. As before, this is sufficient to spread the imbalance energy in Doppler. The effect of this quantized phase modulation is illustrated in Figure 9.
- 32 -
0
-10
50
-20 100 -30 150 -40 200 -50
250 50
100
150
200
250
-60
Figure 8. Example of Figure 1 but with pulse-dependent quantized random phase shift prior to ADC. For convenience, we have allowed the imbalance to be constant for all targets. Note that the anomalous responses have been smeared across the extent of the image, but the true signals are properly located.
In this example, random integer multiples of π 2 phase in the interval [0, π ) were applied and then removed from the data. This again yields the same performance as the previous unquantized random phase modulation example. 3.3.3 Combined Range and Azimuth Smearing In previous sections we analyzed range smearing via a residual chirp, and azimuth smearing via quadratic and random phase coding. In this section we combine range and azimuth smearing, and simultaneously do both. All the previously developed constraints still apply. 3.3.3.1
Range Residual Chirp and Azimuth Chirp
The effect of combining range smearing with a residual chirp and azimuth smearing with chirp is illustrated in Figure 10. In this example, the sampled data retained a residual chirp with a time-bandwidth product of 10, and azimuth chirp with a time-bandwidth product of 20 were applied and then removed from the data.
- 33 -
0
-10
50
-20 100 -30 150 -40 200 -50
250 50
100
150
200
250
-60
Figure 9. Example of Figure 1 but with pulse-dependent random 0/(π/2) phase shift prior to ADC. For convenience, we have allowed the imbalance to be constant for all targets. Note that the anomalous responses have been smeared across the extent of the image, but the true signals are properly located.
Theoretically, a chirped smearing across a range band and Doppler band would reduce the peak imbalance energy from a single target by a combined 27 dB (assuming the −35 dB Taylor window with nbar = 4). However, overlapping smears from multiple smeared imbalance energies yield opportunities for constructive interference. Two equal targets constructively interfering would yield an expected extra 6 dB of peak response over a single target. Three equal targets constructively interfering would yield an extra 9 dB over a single target, but the odds of this are reduced. So, although a chirped smearing across a range band and Doppler band would reduce the peak imbalance energy by a combined 27 dB, making an allowance for perhaps three of the more closely spaced targets’ imbalance energy to constructively interfere would yield a net expected reduction of more like 18 dB. In this example, we in fact measured a peak value for I/Q imbalance of −45 dBc, for a reduction of 19 dB.
- 34 -
-10
50
-20 100 -30 150 -40 200 -50
250 50
100
150
200
250
-60
Figure 10. Example of Figure 1 but with pulse-dependent quadratic phase shift along with a residual chirp, both prior to ADC. For convenience, we have allowed the imbalance to be constant for all targets. Note that the anomalous responses have been smeared across some amount of range as well as some amount of azimuth extent of the image, but the true signals are properly located
Reducing the number of point targets to one would cause us to expect the ideal 27 dB reduction in peak imbalance energy, and simulation of this yields a peak imbalance energy of −53 dBc, for a reduction of 27 dB, as predicted. We also note that a larger Doppler space and larger azimuth chirp time-bandwidth product would allow more azimuth smearing with a corresponding decrease in peak imbalance energy. For example, quadrupling the data set from 256 pulses to 1024, and quadrupling the azimuth chirp time-bandwidth product to 80 yields a measured peak imbalance energy of −59 dBc, for an improvement of 33 dB, albeit for a single target. 3.3.3.2
Range Residual Chirp and Azimuth Random Phase
The effect of combining range smearing with a residual chirp and azimuth smearing with random phase modulation is illustrated in Figure 11. In this example, the sampled data retained a residual chirp with a time-bandwidth product of 10, and random integer multiples of π 2 phase in the interval [0,2π ) were applied and then removed from the data.
- 35 -
-10
50
-20 100 -30 150 -40 200 -50
250 50
100
150
200
250
-60
Figure 11. Example of Figure 1 but with pulse-dependent quantized random phase shift along with a residual chirp, both prior to ADC. For convenience, we have allowed the imbalance to be constant for all targets. Note that the anomalous responses have been smeared across some amount of range as well as the azimuth extent of the image, but the true signals are properly located.
Theoretically, a uniform smearing across a range band and Doppler space would reduce the peak imbalance energy by a combined 36 dB (assuming the −35 dB Taylor window with nbar = 4). However, the uniformity of the energy spreading would be impacted by nonuniform smearing by the range chirp, the random phase noise, and by interference. The likelihood of constructive interference is reduced by the random nature of the phase modulation. If we allow 1 dB for the chirp, 8 dB peak to average ratio due to the random phase modulation, and perhaps a further 6 dB for a ‘lucky’ interference situation, then the uniform smearing performance would be degraded by a combined 15 dB, for a net expected peak imbalance energy reduction of 21 dB. In this example, we in fact again measured a peak value for I/Q imbalance of −48 dBc, for a reduction of 22 dB. Also as with the azimuth chirp, reducing the number of point targets to one would cause us to expect azimuth smearing to yield 23 dB attenuation discounted to perhaps 15 dB assuming 8 dB peak to average ratio due to the random phase modulation, in addition to 12 dB for the spreading by the range chirp, yielding a combined expected only 27 dB reduction in peak imbalance energy. In fact, simulation yields a measured peak imbalance energy of −54 dBc, for a reduction of 28 dB. - 36 -
We also note that a larger Doppler space would allow more azimuth smearing with a corresponding decrease in peak imbalance energy. For example, quadrupling the data set from 256 pulses to 1024 yields a measured peak imbalance energy of −59 dBc, for an improvement of 33 dB. Some Final Comments This report concerns itself only with I/Q imbalance mitigation. The mitigation techniques discussed herein may be useful for other problematic spurious energies. These include additive interference, ADC biases, and harmonic distortion due to ADC nonlinearities. However, the various techniques presented have different degrees of effectiveness against these other problems. Spreading due to range and azimuth chirps is expected to generally work well. Random phase coding with a continuum of phase values is expected to also generally work well. However, some of the quantized random phase coding may have limited effectiveness against other problems. Random 0/(π/2) phase coding may be unable to adequately smear DC biases. Random π/2 multiples will smear DC biases, but will be unable to smear fourth-order harmonics, say from ADC nonlinearities. In fact, random multiples of 2π K phase modulation will generally be unable to smear Kth order harmonics. The bottom line is that I/Q imbalance mitigation is very related to mitigation techniques for other radar non-ideal behaviors, and perhaps ought to be considered in the larger framework of data fidelity in general.
- 37 -
3.4 Image Post-Processing Here we concern ourselves with operations after the range-Doppler image has been formed, and still contains manifestations of I/Q imbalance in the original data. 3.4.1 Image Ghost Subtraction The locations of I/Q imbalance artifacts are deterministic. That is, we know that the energy is symmetrical across the DC point in the range-Doppler image. This suggests that the imbalance energy might be subtracted from the image by subtracting a rotated and scaled version of the original image from the original image itself. We note, however, that care must be taken to rotate the subtracted image in the rangeDoppler space of the image, or otherwise account for any range-Doppler warping to effect geometric corrections due to wavefront curvature. We also note that this subtraction technique works best if no other I/Q imbalance mitigation techniques are employed, say, to smear the I/Q imbalance energy. 3.4.2 I/Q Imbalance Spur Apodization A recent report by Doerry and Bickel 8 presents a technique for cropping or apodizing spurious energy due to I/Q imbalance among other sources. This technique passes the signal through two parallel channels, each with different pulse-to-pulse phase modulations applied, and then compares the two resulting range-Doppler images. As previously shown, I/Q imbalance spurs are displaced by such modulations. Any energy not common to both channels, i.e. collocated in both images, is deemed a spurious response and discarded or otherwise attenuated in some fashion.
- 38 -
4 Recommendations for Radar Systems Mitigating the effects of I/Q imbalance in the radar systems can be divided into two principal realms. 1. Mitigation techniques for new radar designs. This allows us to touch hardware, firmware, and software. 2. Mitigation techniques to be retrofit into existing designs. We address this by limiting our attention to software solutions only.
4.1 New Designs For new designs we recommend the following strategy. •
EMI should be minimized with superb mixed signal board design.
•
An ADC should be selected with excellent INL characteristics, especially if internal multiplexing is employed.
•
Range smearing should be employed using a residual chirp with time-bandwidth product limited to a few percent of the range extent of the range swath.
•
Azimuth smearing should be employed using a true pulse-to-pulse modulation that is removed before any subsequent azimuth filtering. Azimuth smearing should be over an extent that is at least half the Doppler space before any presumming takes place. Modulation should be either a quadratic phase, or a true random phase with at best very fine quantization.
4.2 Retrofit to Existing Designs For retrofitting existing designs we recommend the following strategy. •
Range smearing should be employed using a residual chirp with time-bandwidth product limited to a few percent of the range extent of the range swath.
•
Azimuth smearing should be employed using a modulation that is constant during a presum interval, and changes only at interval boundaries. It is removed in software after any presumming. Azimuth smearing should be over an extent that is at least half the Doppler space after any presumming takes place. Modulation should be either a quadratic phase, or a true random phase with at best very fine quantization.
- 39 -
“Quality is never an accident. It is always the result of intelligent effort.” -- John Ruskin
- 40 -
5 Conclusions We reiterate the following points. •
It is difficult to match real components in analog down-conversion techniques such the I and Q channels are sufficiently balanced.
•
In spite of IF sampling and digital down-conversion, I/Q imbalance can still manifest in the sampled data.
•
EMI can cause I/Q imbalance.
•
ADC ping-pong operation can also cause I/Q imbalance, due to INL differences.
•
Pulse-to-pulse phase modulation can move the imbalance energy to otherwise cropped Doppler space for some range-Doppler (e.g. SAR) images. This is not viable for GMTI.
•
Combinations of residual chirps in range, and pulse to pulse phase modulation can substantially smear the imbalance energy by several tens of dB.
•
New radar designs have a number of options to prevent unacceptable I/Q imbalance. Reasonable retrofits to existing radar designs are also possible.
•
When all else fails, it is sometimes possible to use image processing to substantially diminish the manifestation of I/Q imbalance in the range-Doppler images themselves.
- 41 -
“It is the quality of our work which will please God and not the quantity.” -- Mahatma Gandhi
- 42 -
References 1
A. W. Doerry, D. L. Bickel, A. M. Raynal, “Some comments on performance requirements for DMTI radar,” Proceedings of the Radar Sensor Technology XVIII conference, SPIE 2014 Defense & Security Symposium, Vol. 9077, Baltimore, MD, 5 - 7 May 2014.
2
William J. Caputi, Jr., “Stretch: A Time-Transformation Technique,” IEEE Transactions on Aerospace and Electronic Systems, Vol. AES-7, No. 2, pp 269-278, March 1971.
3
Data Sheet, ADC08D1520 Low Power, 8-Bit, Dual 1.5 GSPS or Single 3.0 GSPS A/D Converter, National Semiconductor Corp., April 7, 2008.
4
Peter Dudley, “Design Guidelines for SAR Digital Receiver/Exciter Boards”, Sandia Report SAND2009-7276, Unlimited Release, November 2009.
5
Armin W. Doerry, Bertice L. Tise, “Correction of I/Q channel errors without calibration,” US Patent 6,469,661, October 22, 2002.
6
Armin W. Doerry, “SAR Impulse Response with Residual Chirps”, Sandia Report SAND2009-2957, Unlimited Release, June 2009.
7
Dale Dubbert, Bert Tise, “Radar echo processing with partitioned de-ramp,” US Patent 8,400,349, March 19, 2013.
8
Armin W. Doerry, Douglas L. Bickel, “Apodization of Spurs in Radar Receivers Using Multi-Channel Processing,” Sandia Report SAND2014-1678, Unlimited Release, March 2014.
- 43 -
Distribution Unlimited Release 1 1 1
MS 0519 MS 0519 MS 0519
J. A. Ruffner A. W. Doerry L. Klein
5349 5349 5349
1
MS 0532
J. J. Hudgens
5240
1
MS 0899
Technical Library
9536
- 44 -
(electronic copy)
- 45 -
