A HILBERT TRANSFORM BASED RECEIVER POST PROCESSOR
Antenna Measurement Techniques Association Conference
October 7-11, 1991
Nearfield Systems Inc.
1330 E. 223rd St. #524
Carson, Ca. 90745
This paper describes a software based receiver post
processor that corrects circularity and gain error in coherent
receivers. The receiver post processor additionally provides range
gating capabilities, signal quality estimation, mixer non-linearity
detection and various display functions. This paper will concentrate
primarily on the identification of circularity errors by the receiver
This paper describes a software based receiver post
processor now incorporated in near-field measurement systems built
by Nearfield Systems Inc. (NSI). The post processor concept is
generic in the sense that it may be used with any amplitude/phase
measuring receiver providing capabilities not normally available.
A significant advantage of the post processor is that a new class
of frequency agile yet simple and low cost receivers can be used
to provide high quality near-field measurements. A version of
this receiver built by NSI currently provides 37,500 CW or multifrequency
measurement per second. This type of receiver however has large
frequency dependent circularity errors which must be corrected.
This is the primary function of the post-processor. The receiver
post-processor provides the following functions:
- Gain and circularity corrector -- Provides quadrature
unbalance and gain correction to a measurement set. A Hilbert
transform technique is used to identify the frequency dependent
gain and circularity errors within the receiver.
- Range gate -- Swept frequency RF measurement
may gated by pulse compression to be within a certain path length
- Signal quality monitor -- Provides a real time
estimate of the receiver noise power and signal to noise ratio.
- Non-linearity detection -- Detects the presence
of receiver compression and non-linearities by a spatial distortion
- Display module -- The receiver post processor
includes an extensive signal display capability.
L.O. GENERATION IN SUPERHETERODYNE RECEIVERS
Virtually all receivers used in antenna and RCS measurements
are based on a superheterodyne design. The superheterodyne receiver
generally includes a mixer followed by an IF amplifier. The IF
amplifier output is then down-converted to DC in a second mixer.
The first mixer is driven from a local oscillator (LO) which is
offset in frequency from the receive frequency by the IF frequency.
The superheterodyne design minimizes leakage, drift and 1/f noise
terms as compared to a homodyne receiver.
A significant problem in the design of superheterodyne
receiver and vector network analyzers is developing the coherently
related LO offset frequency. Four methods of LO frequency generation
are summarized below:
- Phase locked loop.
- Dual coherent synthesizers
- Phase modulated (serrodyne) systems
- Hilbert transform methods
(Method 1) The most common method of generating the
LO frequency offset is to use an indirect phase locked loop (PPL).
This method is used in phase coherent antenna receivers and vector
network analyzers built by Hewlett-Packard, Scientific-Atlanta
and others. Disadvantages of this approach include limited frequency
switching speed and general complexity.
(Method 2) The frequency agility of a receiver or
vector network analyzer can be greatly increased by using a directly
synthesized LO. Several vendors offer the option of using a pair
of coherently related direct digital frequency synthesizers to
provided both the source frequency and offset LO frequency. While
this approach can provide very fast frequency switching, it is
even more costly and complex than method 1.
(Method 3) Another method of LO generation using
direct frequency synthesis is much simpler than either method
1 or 2. The LO frequency offset can be directly synthesized by
the phase modulation of the transmitted signal. The phase modulation
is accomplished by using a broadband bi-phase or quadra-phase
modulator. Either a single or quadrature mixer is then used to
down-convert the RF signal (Slater, 1991). The primary disadvantage
of this technique is that relatively large frequency dependent
circularity errors within the QOSK modulator or quadrature mixer
will result in the generation of spurious LO sidebands. These
frequency dependent circularity errors however are stable with
time and can be readily suppressed if correctly identified. The
primary function of the receiver post processor is the precise
and quantitative identification of the circularity errors and
the associated suppression of these errors.
(Method 4) The receiver post processor creates a
truth or reference model of the circularity errors using direct
synthesis of the LO frequency offset through the use of a doppler
shift technique. A Hilbert transformer is then used to extract
an error free quadrature signal (Slater, 1991, Slater, 1985).
The doppler frequency shift is directly equivalent
to the IF frequency offset. A doppler frequency shift is produced
when the path-length dimensions as measured in wavelengths is
dynamically varied. This variation can be accomplished either
by a physical change in the interferometer path length difference
or by changing the RF test frequency.
Because of the unknown circularity errors, the quadrature
signal component is derived through the use of a Hilbert transformer
implemented numerically in a digital computer. The Hilbert transformer
is conceptually equivalent to a broadband 90 phase shifter (Slater,
1991, Slater, 1985). One requirement for the Hilbert transform
to be valid is the input signal not contained any negative frequencies.
The path length changes producing the doppler signal are used
to suppress negative frequency components. The I channel signal
and Hilbert transform are combined to form an analytic signal
which is then used to identify the circularity and gain errors
within any receiver configuration.
A phase modulated interferometer suitable for near-field antenna measurement
manufactured by NSI uses method 3 to provide a measurement speed of 37,500 CW
or multifrequency measurements per second over a continues frequency span of
2 to 26 GHz. A modification of method 4 is used by the post processor to calibrate
the circularity errors in the hardware.
RECEIVER POST PROCESSOR
The receiver post processor includes the following
major signal processing software modules:
- Error corrector - applies a frequency dependent
gain and circularity correction to a given RF measurement.
- Chirp / doppler processor - identifies gain,
circularity and linearity errors and provides a range gating function.
- Signal quality monitor - provides an estimate
of the receiver noise power and signal to noise ratio.
Figure 1 -- PHASE MODULATED INTERFEROMETER AND RECEIVER POST PROCESSOR
Figure 1 shows a block diagram of the NSI phase modulated
interferometer and major receiver post processor elements.
CIRCULARITY ERROR IDENTIFICATION
Error in the circularity or orthogonality of a quadrature
mixer or quadra-phase modulator cause AM to PM and PM to AM conversion
error within the receiver. The circularity error consists of differences
in the relative gains of the I and Q channels and lack of orthogonality
between the I and Q channels. The term circularity error came
about from Q vs. I parametric plats where a swept frequency measurement
may trace out an ellipse instead of a circle if a quadrature signal
was unbalanced. The impact of circularity errors to near-field
measurements include spurious side lobes and gain errors.
The circularity calibration is performed as follows:
- A swept frequency measurement set is acquired
and read into an input vector. If the receiver is limited to operation
at a single frequency, the path length is swept instead of frequency.
The circularity calibration is not sensitive to multipath, source
phase noise or similar terms as long as these terms do not induce
negative frequencies. Aliased multipath and excessive source phase
noise can be identified in the pulse compressor display.
- The measurement set is tapered with a Kaiser-Bessel
window. This window significantly reduces the level of range side lobes
after the signal is compressed. A pulse compression is performed
by a Fourier transform resulting in the complex receiver output
sorted in terms of differential path length. In the case of a
CW swept path measurement, the output is sorted in terms of spatial
frequency. This mode is similar to the time domain mode often
seen in network analyzers.
- The pulse compressed measurement is gated in
terms of path length or spatial frequency by applying a range
gating window function. This window is designed so as to eliminate
all energy at negative ranges. Because the range gate does not
cross through zero path length or spatial frequency offset, a
Hilbert transform was performed.
- Step three is repeated with a mirror image range
- The results of step 4 and 5 are inverse Fourier
transformed and normalized. The normalization removes the frequency
or position dependent amplitude distortion introduced in step 2.
Figure 2-- CIRULARITY IDENTIFICATION
- Because both I and Q channels measurements were
passed through both transforms, the derived analytic signals need
to be separated. The sum and difference of the gated positive
and complex conjugate negative frequency domain vectors is then
formed. The sum corresponds to an uncorrupted gated I derived
analytic signal. The difference corresponds to the gated Q derived
- The Q derived analytic signal is divided by the
I derived analytic signal. The result is the circularity error.
The circularity errors are saved in a table for use by the gain
and circularity compensator. The operation of the chirp / doppler
processor is better understood by following the signal flow. A
simulated receiver input consisting of a swept frequency measurement
from 4 to 12 GHz through a 10 foot path length differential including
a pair of antennas is used for the following example. The simulation
includes a 3.5 dB high Q channel gain with a +5 phase unbalanced
from 4 to 6 GHz, a 3.5 dB low Q channel gain with a -5 phase unbalanced
from 6 to 11 GHz and no error from 11 to 12 GHz.
Figure 3 shows the amplitude of the receiver as a
function of frequency. The amplitude ripple in the pattern is
equal to the peak to peak quadrature unbalanced or 7 dB. Note
that the Q unbalanced is specified relative to the I channel.
I channel errors are accounted for by the overall frequency dependent
Figure 3 -- SIMULATED RECEIVER INPUT SIGNAL
Figure 4 shows the Fourier transform of the input
signal. The Fourier domain corresponds to path length differential.
A receiver signal with no quadrature error will have a single
peak at a 10 foot distance. The peak at the image range (-10 feet)
is directly related to the quadrature unbalance. This peak can
be alternately considered to the result of a serrodyne LO signal
at the image frequency
Figure 4 - PULSE COMPRESSOR OUTPUT
Figure 5 shows the analytic signal derived from the
I channel. This signal corresponds to measurements made without
quadrature or circularity error. Notice that there is no amplitude
ripple. The generally ascending power is due to the inverse square
law model in the simulator which transitions into a near-field
region at the right side.
Figure 5 - ANALYTIC SIGNAL DERIVED FROM THE I CHANNEL
Figure 6 shows the result of dividing the Q derived
analytic signal by the I derived analytic signal. This ratio is
frequency dependent and corresponds to the gain and phase unbalance
between the Q and I channels.
Figure 6 - DERIVED CIRCULARITY ERROR (GAIN)
The receiver post processor has been used with both
the HP-8510C network analyzer and with the NSI phase modulated
interferometer. The HP-8510C residual circularity errors were
extremely low, as expected.
Without compensation, circularity errors in the NSI
interferometer are quite large as is shown in figure 7. Figure
8 shows a near-field derived pattern of a standard gain horn without
circularity correction. Figure 9 shows the same pattern with circularity
correction enabled. This corrected pattern is virtually identical
to a pattern acquired with a HP-8510C receiver in the same test
Correct operation of the circularity calibration
can be verified by computing the residual circularity error of
the corrected signal and more importantly, by the ratio of the
negative to positive path-length amplitude in the pulse-compressor
display. This ratio is equivalent to the single-sideband-suppression
ratio. In the current version of the NSI interferometer, the sideband-suppression
ratio is typically better than 55 dB. This corresponds to a residual
rms gain and phase unbalance of 0.02 dB and 0.1 degree.
Figure 7 - PMI UNCORRECTED CIRCULIRITY ERROR
Figure 8 - HORN PATTERN WITH NO CIRCULARITY CORRECTION
This paper has described a receiver post-processor
which compensated for frequency dependent gain and circularity
errors as found in a new class of simple yet high performance
receivers. Additional capabilities include signal quality determination,
range gating and the detection of compression and non-linearities
in a coherent receiver.
Figure 9 - HORN PATTERN WITH CIRCULARITY CORRECTION ENABLED
1. Mensa, D., High Resolution Radar Imaging,
Artech House, Norwood, MA, 1981
Describes pulse compression and the related processing
2. Slater, D., Near-field Antenna Measurements,
Artech House, Norwood, MA, 1991
Book includes a section on receivers using direct
synthesis LO including serrodyne systems and Hilbert transform
3. Slater, D., Interverse Synthetic Aperture Imaging Radar,
AMTA Conference Proceedings, Melbourne, FL,
Paper describes a swept frequency pulse compression
receiver which derives phase information through the use of the
© Copyright 1996, Nearfield Systems Inc., All Rights