The Response to Arbitrarily Bandlimited Gaussian Noise of the Complex Stretch Processor Using a Conventional Range-Sidelobe-Reduction Window

This paper derives a mathematical description of the complex stretch processor’s response to bandlimited Gaussian noise having arbitrary center frequency and bandwidth. The description of the complex stretch processor’s random output comprises highly accurate closed-form approximations for the probability density function and the autocorrelation function. The solution supports the complex stretch processor’s usage of any conventional rangesidelobe-reduction window. The paper then identifies two practical applications of the derived description. Digital-simulation results for the two identified applications, assuming the complex stretch processor uses the rectangular, Hamming, Blackman, or Kaiser window, verify the derivation’s correctness through favorable comparison to the theoretically predicted behavior.


Introduction
Stretch processing [1]- [6] in radar uses relatively narrowband techniques to process wideband pulses with linear frequency modulation (LFM). Basic stretch processing [2] [3] (i.e., with no range-sidelobe-reduction window) yields the same fine range resolution and the same relatively high range-sidelobe levels produced by matched filtering. To reduce the range-sidelobe levels produced by basic stretch processing of LFM pulses, a practical stretch processor may apply a only the case of broadband noise (e.g., receiver thermal noise). References [7] and [8] respectively characterized the response to bandlimited Gaussian noise (BLGN) having arbitrary center frequency and bandwidth of the complex stretch processor having no range-sidelobe-reduction window and the complex stretch processor employing a Hamming or Hann window. This paper extends the work in [7] [8] to characterize the output noise's probability density function (PDF) and autocorrelation function when the complex stretch processor uses any conventional multiplicative window to reduce the range-sidelobe levels. The output noise's PDF and autocorrelation function provide sufficient information for high-fidelity simulation of the complex stretch processor's output noise via standard techniques. Since the complex stretch processor is a linear system, a radar modeler may simply add the simulated noise to the complex stretch processor's simulated response to targets and clutter.
The derivation assumes the BLGN has arbitrary center frequency and bandwidth. Therefore, the results can describe the output noise due to input receiver thermal noise, broadband-noise jamming, spot-noise jamming, or even spectrally offset narrowband interference. The paper specifies a mathematical form for the window which can exactly represent the commonly used rectangular, Hamming, Hann, and Blackman windows and can closely approximate all other conventional windows.
Section 2 firstly specifies a simplified functional model of a radar employing a complex stretch processor with a range-sidelobe-reduction window. Section 2 then describes the processor's response to target-return signals. Section 3 derives a mathematical description, comprising the PDF and the autocorrelation function, of the complex stretch processor's theoretical response to arbitrarily bandlimited Gaussian noise. Section 4 presents simulation results which verify the derived expressions for two practical applications. Section 5 summarizes the technical approach, presents key findings, and suggests additional research.

Review of Complex Stretch Processing
This section reviews the fundamental operations of a radar using complex stretch processing. Figure 1 shows a simplified block diagram of the basic functional elements of a monostatic, pulsed radar employing complex stretch processing. This section's discussion uses the mathematical notation shown in Figure 1 which pictorially represents the complex stretch processor's stimulation by a target-return signal. For analytical convenience we assume the complex stretch processor comprises exclusively continuous-time (CT) subsystems.

Transmitted Signal
The radar's transmitter sends a single pulse, Journal of Signal and Information Processing  , where T A is the pulse amplitude in volts, RF f is the center radio frequency (RF) in hertz, t is time in seconds, p τ is the pulse duration in seconds, and is the instantaneous phase deviation in radians, to the transmit antenna.
The transmit antenna radiates the pulse to a stationary point target at a slant range R meters from the radar. In Equation (1) is the dimensionless unit-pulse function, and where, for an up-chirped LFM pulse with sweep bandwidth B hertz, is the transmitted pulse's instantaneous frequency deviation in hertz. We substitute Equation (4)

Received Signal
The stationary point target instantaneously reradiates the incident pulse, so the receive antenna produces the voltage signal In Equation ( is the round-trip propagation delay, and c is the speed of light. The radar-range equation [9] determines the dimensionless ratio R T A A .

Quadrature Demodulator's Output
Using reference frequency RF f , the receive system's quadrature demodulator Journal of Signal and Information Processing indicates the operation of an ideal lowpass filter having a dimensionless passband gain of unity and a cutoff frequency between 2 B . Thus, the quadrature demodulator's output has units of volts.

Complex Multiplier's Output
Assuming the stretch processor considers target slant ranges from min R to max R , the slant ranges on this interval correspond to round-trip propagation delays To support processing on slant ranges from min R to max R , the complex multiplier of Figure 1 multiplies is a complex heterodyne signal and ( ) w t is a sidelobe-reduction window. In and ( ) ( ) Note that Equation (15) is the instantaneous phase deviation corresponding to the instantaneous frequency deviation In Equation (11) Mathematically, For any conventional window, the peak magnitude of Equation ( and ( ) ( )

Complex Stretch Processor's Theoretical Response to BLGN
This section mathematically characterizes the complex stretch processor's theoretical response to BLGN having arbitrary bandwidth and center frequency. We firstly describe the BLGN. We then determine the PDF and autocorrelation function of the receive system's response to the BLGN. Specifically, we show the complex stretch processor's output is complex, zero mean, and Gaussian with independent real and imaginary parts. We then derive the autocorrelation function of the complex stretch processor's output. From the autocorrelation function, we find the variance to complete the PDF's description. This section's discussion uses the mathematical notation shown in Figure 2 which pictorially Journal of Signal and Information Processing represents the complex stretch processor's stimulation by arbitrarily bandlimited Gaussian noise.

BLGN Description
The BLGN at the complex stretch processor's input is a real random-voltage signal having mathematical form where y f is the BLGN's center RF. As given in [11], as depicted in Figure 3.

Quadrature Demodulator's Output
The quadrature demodulator applies the mathematical action of Equation (8)

Fourier Transform's Output
The Fourier transform of the complex multiplier's output Since the real and imaginary parts of ( ) 1 CM

Y f are uncorrelated and Gaussian
RVs, the RVs are also independent. Since the mean, correlation, and variance of its real and imaginary parts completely specify the complex RV's PDF (i.e., the joint PDF of the RV's real and imaginary parts [13]). The RV has mean where ( ) E Z denotes the expected value of the generally complex RV Z. Thus, the mean of both the real and imaginary parts of Since the real and imaginary parts are independent and zero mean, their correlation is zero.
We find the variance of the RV's real and imaginary parts by finding the autocorrelation function of ( ) CM Y f , setting both frequency arguments equal to 1 f , and dividing the result by two.
The autocorrelation function of ( ) Since ( ) QD y t is WSS, its autocorrelation function is the inverse Fourier transform of its PSD, so QD QD In a practical stretch processor, the heterodyne signal's time-bandwidth product M M B T very greatly exceeds unity, so [7] ( ) Substituting Equation (51) into Equation (50) gives (after simplification) All conventional windows have energy spectral densities concentrated around 0 f = [14], so for some positive integer W N . Therefore, we can make the further approxima- For practical stretch processors, the sweep bandwidth M B very greatly ex-Journal of Signal and Information Processing Equation (55) and Equation (56) permit the further approximation From Equation (57) we immediately obtain Substituting Equation (41), Equation (58), and Equation (59) For values of 1 f outside the frequency interval the two Π functions in the integrand of Equation (61) Now, we respectively define ( ) and ( ) , u f f f . Assuming values of ( ) In Equation (67) Analysis of Equation (70)

Simulation Results
To demonstrate the correctness and utility of Equation (70), we simulate a radar having the parameter values listed in Table 1. With these parameters a 1-kHz frequency separation in the Fourier transform's output maps to a 1.5-m slantrange separation.
To achieve various compromises between Rayleigh range resolution [2] and peak sidelobe levels [14], the radar can use the CT rectangular, Hamming, Blackman, and Kaiser windows, mathematically described by [15] ( ) and respectively. In Equation (74) to specify a Kaiser window having a temporally broader characteristic than the Hamming and Blackman windows, as shown in Figure 4. Table 2 shows the key performance characteristics corresponding to these four windows, assuming the returned pulse is temporally centered in each window.    10 10 as the final expression for the output's theoretical autocorrelation function. In agreement with [3] [4], for any frequency considered by the complex stretch processor, the output noise will have a variance of ( ) ( ) The numerically approximated correlation coefficient for the simulated output's real and imaginary components is −0.00073. Since this value is practically zero, the real and imaginary components are practically uncorrelated. Since the real and imaginary components are also Gaussian, they are practically independent, as previously stated. The complex correlation coefficient [16] ( quantitatively characterizes the correlation between samples of ( ) We evaluate Equation ( Table 2 and W N + = terms, indicating that, even after significant simplification, the closed-form expression for Equation (100) will certainly be mathematically unwieldy. Therefore, we make no attempt to obtain a closedform solution without the double summation. Figure 12 shows excellent agree-

Conclusion
This paper presented a detailed mathematical development which characterized the response to arbitrarily bandlimited Gaussian noise of a complex stretch processor using a conventional range-sidelobe-reduction window. The paper specified the complex stretch processor's functional structure and the input BLGN's mathematical description. The subsequent development then propagated the BLGN through the complex stretch processor's functional components, characterizing the noise at the key components' outputs.
The effort produced four significant findings. Firstly, the final output is complex, zero-mean, Gaussian noise with equal variance in its independent real and imaginary components. Secondly, the output noise's autocorrelation function has a highly accurate closed-form approximation readily determined from the radar's and input BLGN's parameters. Thirdly, the output noise is generally not WSS (whereas it is for the case of a matched filter), which may complicate highfidelity modeling. Fourthly, we may determine the correlation between the noise components of any two output range samples by evaluating the complex correlation coefficient using the derived autocorrelation function.
The windows considered in this effort were all conventional (i.e., real, symmetric, and lowpass). The described approach also applies to complex and/or asymmetric windows so long as those functions have energy spectral densities concentrated around 0 Hz f = . In addition the approach readily extends to unconventional windows having energy spectral densities concentrated around a nonzero frequency.