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REFERENCES

1.Skolnik, M.I. 2008. Radar Handbook. 3rd edn. New York: McGraw-Hill, Inc.

2.Tzvetkov, A. 1971. Principles of Quantitative Ratings of Complex Radar System Efficiency. Moscow, Russia: Soviet Radio.

3.Hovanessian, S. 1984. Radar System Design and Analysis. Norwood, MA: Artech House, Inc.

4.Meyer, D. and H. Mayer. 1973. Radar Target Detection: Handbook of Theory and Practice. New York: Academic Press.

5.Drujinin, B. and D. Kontorov. 1976. Problems of Military Systems Engineering. Moscow, Russia: Military Press.

6.Gutkin, L.S., Pestryakov, V.V., and V.H. Tipugin. 1970. Radio Control. Moscow, Russia: Soviet Radio.

7.Skolnik, M.I. 2001. Introduction to Radar Systems. New York: McGraw-Hill, Inc.

8.Lacomme, P., Hardange, J.-P., Marchais, J.-C., and E. Normant. 2001. Air and Spaceborne Radar Systems: An Introduction. New York: William Andrew Publishing.

9.Tait, P. 2005. Introduction to Radar Target Recognition. Cornwall, U.K.: IEE Press.

10.Levanon, N. and E. Mozeson. 2004. Radar Signals. New York: IEEE Press, John Wiley & Sons, Inc.

11.Barton, D.K. 2005. Modern Radar System Analysis and Modeling. Canton, MA: Artech House, Inc.

12.Stimson, G.W. 1998. Introduction to Airborne Radar. 2nd edn. Raleigh, NC: SciTech Publishing, Inc.

13.Nitzberg, R. 1999. Radar Signal Processing and Adaptive Systems. Norwood, MA: Artech House, Inc.

14.Streetly, M. 2000. Radar and Electronic Warfare Systems. 11th edn. Surrey, U.K.: James Information Group.

15.Guerci, J.R. 2003. Space-Time Adaptive Processing for Radar. Norwood, MA: Artech House, Inc.

16.Barkat, M. 2005. Signal Detection and Estimation. 2nd Edn., Norwood, MA: Artech House, Inc.

17.DiFranco, J.V. and W.L. Rubin. 1980. Radar Detection. Norwood, MA: Artech House, Inc.

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19.Kay, S.M. 1993. Fundamentals of Statistical Signal Processing—Estimation Theory. Vol. I. Englewood Cliffs, NJ: Prentice Hall, Inc.

20.Kay, S.M. 1998. Fundamentals of Statistical Signal Processing—Detection Theory. Vol. II. Englewood Cliffs, NJ: Prentice Hall, Inc.

21.Knott, E.F., Shaeffer, J.F., and M.T. Tuley. 1993. Radar Cross Section. 2nd edn. Norwood, MA: Artech House, Inc.

22.Mahafza, B.R. 1998. Introduction to Radar Analysis. Boca Raton, FL: CRC Press, Taylor & Francis Group.

23.Mahafza, B.R. 2000. Radar System Analysis and Design Using MATLAB. Boca Raton, FL: CRC Press.

24.Nathanson, F.E. 1991. Radar Design Principles. 3rd edn. New York: McGraw Hill, Inc.

25.Peebles, Jr., P.Z. 1998. Radar Principles. New York: John Wiley & Sons, Inc.

26.Richards, M.A. 2005. Fundamentals of Radar Signal Processing. Englewood Cliffs, NJ: Prentice Hall, Inc.

27.Brookner, E. 1988. Aspects of Modern Radar. Boston, MA: Artech House, Inc.

28.Franceschetti, G. and R. Lanari. 1999. Synthetic Aperture Radar Processing. Boca Raton, FL: CRC Press.

29.Johnson, D.H. and D.E. Dudgeon. 1993. Array Signal Processing: Concepts and Techniques. Englewood Cliffs, NJ: Prentice Hall, Inc.

30.Klemm, R. 1998. Space-Time Adaptive Signal Processing: Principles and Applications. London, U.K.: INSPEC/IEEE.

31.Sullivan, R.J. 2000. Microwave Radar: Imaging and Advanced Concepts. Boston, MA: Artech House, Inc.

32.Tuzlukov, V.P. 2001. Signal Detection Theory. New York: Springer-Verlag.

33.Tuzlukov, V.P. 2002. Signal Processing Noise. Boca Raton, FL: CRC Press.

34.Tuzlukov, V.P. 2004. Signal and Image Processing in Navigational Systems. Boca Raton, FL: CRC Press.

35.Van Trees, H.L. 2002. Optimum Array Processing: Part IV of Detection, Estimation, and Modulation Theory. New York: John Wiley & Sons, Inc.

36.Tsui, J.B. 2004. Digital Techniques for Wideband Receivers. 2nd edn. Raleigh, NC: SciTech Publishing, Inc.

37.Bogler, P.L. 1990. Radar Principles with Applications to Tracking Systems. New York: John Wiley & Sons, Inc.

38.Cumming, I.G. and F.N. Wong. 2005. Digital Signal Processing of Synthetic Aperture Radar Data. Norwood, MA: Artech House, Inc.

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FIGURE 2.1  Illustration of the sampling process: (a) Analog waveform x(t) and (b) instantaneously sampled representation of x(t).

The instantaneously sampled target return signal xδ(t) has a mathematical form similar to that of the Fourier transform of a periodic signal. This is readily established by comparing (2.1) for xδ(t) with the Fourier transform of a periodic signal given by

where G(nfs) is the Fourier transform of x(t), evaluated at the frequency f = nfs. This correspondence suggests that we may determine the Fourier transform of the sampled target return signal xδ(t) by invoking the duality property of the Fourier transform [3], the essence of which is the following:

Representation of the continuous function x(t) in the form of the sequence {x(nTs)} is possible only under the well-known restrictions. One restriction in kind is a requirement of spectrum limitation of the sampled target return signal xδ(t). By Kotelnikov’s theorem concerning the signal space concept­ [4], the continuous target return signal x(t) possessing a limited spectrum is completely defined by countable set of samples spaced Ts ≤ 1/(2fmax) seconds apart, where fmax is the cutoff frequency of target return signal spectrum.

Under digital signal processing in CRSs, a stochastic process at the analog receiver output is an object to sample. This output process is a narrowband process at the condition fs / fc 1, where fs is the signal spectrum bandwidth and fc is the carrier frequency. This condition allows us to apply

an envelope procedure to represent the narrowband signal in the following form:

where

X(t) is the low-frequency signal (envelope)

φ(t) is the phase modulation law of the narrowband signal—slowly varied function in comparison with 2πfct

Since information about the target is extracted from the envelope X(t) and phase φ(t) of the target return signal and not from the carrier frequency fc, and, moreover, X(t) and φ(t) are the slowly varied functions in time, there is a need to convert the target return signal given by (2.4) in such a way that sampling intervals would be defined by the real signal spectrum bandwidth rather then the carrier frequency fc. In the case of the narrowband radio signal, we can write

phase-modulated narrowband signal spectrum. For example, in the case of a completely known deterministic signal, the spectral bandwidth of the envelope is defined from the following condition [6,7]:

where τ0 is the duration of the sampled signal. Consequently, a maximal sampling period of amplitude envelope is limited by the condition Ts ≤ τ0. In doing so, an initial phase of the envelope is known. By this reason, it becomes evident that there is no need to sample it. In the case of a wideband signal of the same duration τ0 (the case of chirp modulation), the spectrum bandwidth of the modulated signal is close to double the amount of frequency deviation [8]:

Consequently, for one-valued representation of the frequency-modulated signal with constant amplitude, there is a need to set the phase

Under the representation of complex envelope of the radio signal in the in-phase xI(t) and quadrature xQ(t) components, the maximal frequencies f1max and f2max are the same, that is,

Consequently, the sampling must be carried out simultaneously over every equal sampling interval:

In the case of chirp-modulated signal with random initial phase, we have

30 Signal Processing in Radar Systems

The number of quantization levels is determined by

where dr is the dynamic range of analog part of receiver. The number of bit site to represent the sampled target return signal is given by

where [z] is the nearest integer NLE z. To characterize the analog-to-digital converter, we often use the following formula defining a dynamic range in dB for quantized sequence of samples for one bit of conversion [9]:

Under detection of the target return signal, definition of its parameters, and noise compensation by digital signal processing subsystems in CRSs, there is a need to use a capacity digit, for example, Nb = 6 ÷ 8, analog-to-digital conversion of the sampled target return signal. Capacity digit quantization at a high sampling rate fs is a very difficult technical problem. Moreover, an increase in the sampling rate fs and capacity digit quantization leads to overdesign of digital signal processing subsystems in complex radar systems. Because of this, we use binary quantizers and binary detectors for target return signals, in addition to capacity digit quantization [10,11]. Binary detectors are very simple under realization by digital signal processing techniques.

2.1.3  Analog-to-Digital Conversion: Design Principles and Main Parameters

Manifold types of analog-to-digital conversion are employed by digital signal processing subsystems in CRSs, for instance, the analog-to-digital conversion of voltage/current, time intervals, phase, frequency, and angular displacement. The principal structure of majority of analog-to-digital converters is the same: There are sampling block, quantization block, and data encoding.

The main engineering data are as follows:

1.Time parameters defining a speed of operation (see Figure 2.4):

a.Sampling interval Ts

b.Time of conversion Tc, within the limits of which the target return signal is processed by the analog-to-digital converter

c.Time of conversion cycle Tcc, that is, the delay between the instant of appearance of the incoming target return signal at the input of the analog-to-digital converter and the instant of occurrence of output code generation

2.The number of code bits Nb of the target return signal

3.The element base of the analog-to-digital converter

Consider in detail the Q-factors or external parameters of analog-to-digital converter.

2.1.3.1  Sampling and Quantization Errors

There are two types of these errors: dynamical errors and statical errors. The dynamical errors are the discrete transform errors. The statical errors are the unit sample errors. The dynamical errors depend on the target return signal nature and time performance of the analog-to-digital

X

Ts

t

Tc

x´

t

FIGURE 2.4  Time parameters of ADC.

converter. The main constituent of this kind of error is an inaccuracy caused by variations in the target return signal parameters at the analog-to-digital converter input. For example, if variations in the amplitude of the target return signal within the limits of Tcc are less or comparable with the quantization step x, then

where Vxmax is the maximal speed of variations of the sampled and quantized target return signal. Taking into consideration that Tccmax = Ts , we can define the sampling interval Ts based on (2.28).

The unit sample error is caused, first, by quantization error with variance

and, second, by deviation of actual quantization performance from ideal one (see Figure 2.3). If this deviation at the i-th quantization step is equal to ξi and the input target return signal possesses a uniform distribution of instantaneous values within the limits of operating range, then the variance of statical quantization error can be determined by

2.1.3.2  Reliability

Reliability of the analog-to-digital converter is the ability to keep operation accuracy at given boundaries within the limits of a definite time interval under a given environment. As a rule, to evaluate the reliability of an analog-to-digital converter the probability of no-failure operation for time t is used, taking into account one of the following failure types:

•The outage—glitch

•Errors exceed a predefined value of accuracy—degradation failure

•Intermittent failure—malfunctions

An increase in the reliability of analog-to-digital converters is obtained owing to redundancy. Other parameters that pose a risk of limitation include power consumption, weight, and other

dimensions of analog-to-digital converters, cost under serial production, manufacturability, time that is required for design, and so on.

As a generalized Q-factor of analog-to-digital converter efficiency, we consider the following ratio:

is the data throughput of the analog-to-digital converter and Q is the size of enginery. At the digital signal processing of wideband radio signals in CRSs, rigid requirements in data throughput and reliability are applied to analog-to-digital converters. To design and produce such converters, the parallel Fourier transforms for target return signals are used.

2.2  DIGITAL GENERALIZED DETECTOR FOR COHERENT IMPULSE SIGNALS

2.2.1  Matched Filter

Recall briefly the main statements of classical detection theory. In accordance with a general theory of radar signal processing, the signal processing in time of stochastic process x(t) representing an additive mixture of the signal s(t) and stationary white Gaussian noise w(t) is reduced to calculation of the correlation integral, which in the case of a scalar real signal s(t, α) with known parameter α can be presented in the following form:

where s*(t, α) is the expected signal model generated by a local oscillator in the receiver or detector. If α is the delay of the expected signal with respect to the incoming initial input realization x(t), then the correlation integral is determined as