An Analytic Description of a 

Harmonic Decomposition 

Technique for Correcting Signal 

Errors Due to Wideband Radar 

Phase Detector 

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Army Research Laboratory 

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ARL-TR-2306 November 2000 

An Analytic Description of a 

Harmonic Decomposition 

Technique for Correcting 

Signal Errors Due to Wideband 

Radar Phase Detector 

Thomas J. Pizzillo and Jerry Silvious 

Sensors and Electron Devices Directorate 

Approved for public release; distribution unlimited. 
Abstract 

A signal processing technique is presented for correcting imbalances and 

distortions introduced to the signals by the phase detectors of a coherent wideband radar. The signal model and sources of signal errors are described, an analytic description of how the corrections are derived is provided, and a sample application is presented with the use of simulated 

data. This report has been prepared in anticipation of a subsequent report in which the performance of this signal processing technique will be 

compared with the performances of several other techniques developed for 

similar purposes with actual data measured in the field. 

ii 
Contents 

1 Introduction 1 

2 Derivation of Correction Terms 2 

3 Technique Application 5 

4 Conclusion 7 

Distribution 9 

Report Documentation Page 

Figures 

13 

1 Simplified block diagram of a monopulse radar . . . . . . . . . . 2 

2 Simplified block diagram of IF section of a monopulse radar . . 3 

3 Simplified block diagram of rf section of a monopulse radar . . 5 

4 Simulated I/Q data: Squares indicate measured data and circles 

indicate corrected data . . . . . . . . . . . . . . . . . . . . . . . . 6 

. . . 

111 

1. Introduction 

The U.S. Army Research Laboratory (ARL) has a variety of instrumentation radars for research related to sensor technologies, to include modeling, simulation, analysis, and signal processing of radar signatures. Before 

any of these research objectives are met, the data collected by the instrumentation radars must be calibrated to relate the returned signal to the 

transmitted signal. This calibration removes errors introduced by components of the radar and scales the data. In particular, ARL’s Millimeter-Wave 

Branch of the Radio Frequency (rf) and Electronics Division has several inverse synthetic aperture radars containing phase detectors that require error correction. As these new radars are developed and introduced, new calibration techniques have been developed so that three distinct techniques 

are now being used. Two of these techniques, the method of Wallace and 

Pizzillo’ and a phase modulation and demodulation technique,2 have been 

documented as ARL technical reports. The third, a harmonic decomposition technique, has only been documented as an algorithm in software. 

This report provides an analytic description of the third algorithm in anticipation of publishing a report detailing a comparison of the three techniques 

with the use of data collected by the state-of-the-art 35-GHz monopulse instrumentation radar. This instrument uses the phase modulation and demodulation technique, an architecture-based calibration, and the only one 

of the various radars that can collect data that may be corrected by all three 

techniques, thereby allowing a direct comparison of the efficiency and accuracy of each scheme. 

IH. Bruce Wallace and Thomas J. Pizzillo, A Teckniqwfor Calibrating tke Pkase Detector 

of a Wideband Radar Using an External Target, U.S. Army Research Laboratory, ARL-TR-1521 

(March 1998). 

‘Thomas J. Pizzillo and H. Bruce Wallace, A Teckniquefor Calibrating tke Phase Detector 

of Widebaud Radars Using a Pkase Modulation and Demodulation Schenle, U.S. Army Research 

Laboratory, ARL-TR-1567 (May 1998). 

1 
2. Derivation of Correction Terms 

Figure 1. Simplified 

block diagram of a 

monopulse radar. 

Figure 1 is a simplified block diagram of a polarimetric, two-coordinate, 

amplitude-comparison monopulse system operating at 94 GHz, with a 

640-MHz bandwidth. Either linearly or circularly polarized radiation is 

transmitted and both components are received, i.e., left-circular transmit 

and right- and left-circular receive or vertical transmit and vertical and 

horizontal receive. Transmit polarization may be changed to pulse to pulse 

or ramp to ramp. 

The rf local oscillator (LO) is frequency-stepped synchronously with the 

transmitter and maintains a constant 3-GHz offset. This source is mixed 

with the received signal to provide the 3-GHz intermediate frequency (IF) 

to the in-phase/quadrature (I/Q) detectors. The IF is mixed with the 3-GHz 

LO to provide the final direct current (DC) signal to the analog-to-digital 

converter (ADC). The 3-GHz LO is also the source of the injected test signal 

used for correcting the phase and gain imbalances introduced by the I/Q 

detectors. A simplified block diagram of the IF section is shown in figure 2. 

This test signal is 

S = Acos2~f, (1) 

where f represents the test signal’s constant frequency and A is the amplitude of the test signal. However, this only provides a single DC output 

s QWP Diplexer 

Feed 

horn 

Toggle 

w&h.% 

Injected 

test 

signal 

Toggle 

swtches 

;_L ‘L 

3-GHz LO 

2 
Figure 2. Simplified 

block diagram of IF 

section of a monopulse 

radar. 

3-GHz tzsignal 

3GHz LO +20dBm xv/ I 

- 

I 01 QI Q I QI QI Q 

value for a given frequency from the I/Q detectors, and correcting the output of the detectors over a range of frequencies is necessary. To accomplish 

this, one must phase-modulate the test signal to provide an input to the 

I/Q detectors as 

S(m) = Aces 

( 

2nf + 27~: 

1 

, (2) 

where A4 is an even integer representing the number of phase modulation steps. Shifting the phase creates a pseudo-time-sampled signal with 

frequency of 1 Hz and a period of l/M. This signal is injected at the IF 

section and then propagates through the phase detector, where a second 

signal, shifted 90” relative to the test signal, is generated. These signals are 

referred to as the I and Q channels of the detector: 

Q(m) = GAsin (27rE + 6) + V, : (3) 

where Vi and V, are DC-offsets and G and 6 represent the relative gain and 

phase imbalances between the outputs of I and Q channels, all introduced 

by the imperfect detector. The 2~f term has been eliminated because the 

detection is a mixing process with f as the baseband. At this juncture, these 

3 
errors could be corrected by the method described in Wallace and Pizzillo. 

However, in addition to the errors introduced by the phase detector, the 

components that compose the IF section cause nonlinear distortion of the 

signal so that the signals may be better modeled by a set of trigonometric 

polynomials, 

I(M) = 5 A,, cos (2m;) 

n=O 

Q(M) = 5 B, sin (2mr~) + 1% . (4) 

n=O 

Although the phase and gain imbalance of the phase detector creates a response at the image range bin, 3 this error is subsumed by the appropriate 

terms and coefficients of the polynomial and needs not be explicitly enumerated. Equation (4) represents a set of functions orthogonal on the complex plane over the range [-7r, rr]. Hence, the coefficients A, and B, may be 

determined via the complex fast Fourier transform (CFFT) of the measured 

I/Q detector outputs by 

J’[IW).Q(Af)l = n$o{ A n cos ( 2nrz .,> + zB, sm ’ ( 2nrE “) + V } exp ( 27;~~ “) 

= I (A-) + iQ (A-) : (5) 

where V = vi + V, and I(K) and Q(K) are the nil outputs of the CFFT that 

represent the harmonic coefficients of the transformed signal. That is, for 

each k, only the trigonometric component X: = n contributes 

I(K) = Ao+A:!cos 

Q (IT) = B1 sin . (6) 

The maximum number of contributing harmonics is determined by the 

Nyquist rate or M/2 distributed symmetrically about the fundamental: 

I (K) = A0 + A+ cos (-YT) + . . . + AA+ cos (YT) , and 

Q (K) = By sin (27rG) + . . . + By sin (277%) . 

By repeating this procedure for several amplitudes and storing the values 

in a lookup table, one may correct any measured amplitude on the complex 

plane defined by I and Q up to the maximum calibration amplitude. 

‘Merril Skolnik, Rndn~ Handbook, 2nd ed., McGraw-Hill, Inc. (1990), p 3.41. 

4 
3. Technique Application 

The correction noted in section 2 is applied in the following procedure: A 

constant amplitude signal is injected into the IF section of the radar via the 

toggle switches (see fig. 2). This provides a DC signal to the ADC. Data 

are collected at 16 phase angles, 22.5”m, where M = 0,l.. .15, controlled by 

the phase-shifter shown in figure 3 for both the I and Q channels. Six thousand four hundred samples are averaged for each phase setting to minimize noise. The data are converted to signed integer values by subtracting 

half the dynamic range of the ADC from each set, i.e., 2048. A complex 

value is generated from the I and Q data and then sorted from the minimum to maximum phase. This process is repeated for four attenuation levels: 98, 78, 58, and 39 percent of the A/D dynamic range, i.e., 4095. The 

final results are four sets of complex data of 16 values each from which the 

correction coefficients are derived. 

The correction coefficients are used to correct measured data in both phase 

and amplitude by subtracting the unwanted harmonic components from 

each measured data point, Inz and Qm. Measured amplitudes are corrected 

with the correction amplitudes above and below the measured value, with 

Figure 3. Simplified 

block diagram of rf 

section of a monopulse 

radar. 

- 

94-GHz transmit signal 

92-GHz FA LO 
Figure 4. Simulated I/Q 

data: Squares indicate 

measured data and 

circles indicate 

corrected data. 

an appropriate weighting factor. This weighting factor is calculated with 

NT ‘I = (Au - Am) (A, - 4) 

j/q = CA,, - Ad : 

(Au - -4) 

p;, + l;l,; = 1> . (8) 

where A,, is the correction amplitude greater than the measured value, 

Al is the correction amplitude less than the measured value, and A,, = 

Jm=th e amplitude of the measured I and Q data points. 

Consider the simulated data in figure 4 created for Ar = 16. The set of measured I/Q pairs is indicated with squares and falls between the calibration 

amplitudes, A, = 1500 and Al = 1200. The corrected I and Q values are 

determined by 

I, = I,, - Wu [AOll + Apst, cos (-S&J + . . . + Asu cos (84,)] 

- 14,; [Aor + A-81 cos (-80,) + . . . + Ag( cos (M,,)] . and 

Qc = Q7, - NT,, [A--721 sin (-78,) + . . . + ATE, sin (78,)] 

- 14) [A-T/ sin (-7&,) + . . . + AT/ sin (70,,)] . (9) 

where 1, and Qc are the corrected I and Q data points. 8, = arctan 2 ( > p 171 

is the measured relative phase between 1, and Q,, . A,, and A,[ are the nth 

harmonic coefficient for the upper and lower calibration bounds, respectively. These expressions are applied to each of the 16 I/Q pairs indicated 

with circles in figure 4. This example is for demonstration only and for clarifying the application of this technique. A subsequent report will compare 

the application of this technique to measured data, and its performance 

will be compared with two other techniques developed for the same data 

correction problem. 

6 
4. Conclusion 

An algorithm currently used by ARL (Millimeter-Wave Branch) for correcting the I and Q errors introduced by the phase detectors of a wideband 

radar has been presented. The signals with errors have been modeled as 

trigonometric polynomials and the correction described as a harmonic decomposition based upon the CFFT. An analysis of the performance of this 

technique is left for a subsequent publication in which the performance of 

the two other techniques (see sect. 1) will be compared along with the results of this harmonic decomposition as applied to actual measured data 

from a state-of-the-art 35-GHz monopulse instrumentation radar. 

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P. TITLE AND SUBTITLE An Analytic Description of a Harmonic 5. FUNDING NUMBERS 

Decomposition Technique for Correcting Signal Errors Due to 

Wideband Radar Phase Detector DA PR: AH16 

PE: 62120A 

6. AUTHOR(S) Thomas J. Pizzillo and Jerry Silvious 

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS 8. PERFORMING ORGANIZATION 

U.S. Army Research Laboratory REPORT NUMBER 

Attn: AMSRL- SE-RM email: pizzillo@arl.army.mil ARL-TR-2306 

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13. ABSTRACT (Maximum 200 words) 

A signal processing technique is presented for correcting imbalances and distortions introduced 

to the signals by the phase detectors of a coherent wideband radar. The signal model and sources 

of signal errors are described, an analytic description of how the corrections are derived is 

provided, and a sample application is presented with the use of simulated data. This report has 

been prepared in anticipation of a subsequent report in which the performance of this signal 

processing technique will be compared with the performances of several other techniques 

developed for similar purposes with actual data measured in the field. 

I~.SUBJECTTEFM Error correction, imbalance 15. NUMBER OF PAGES 

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