Optimum Linear Filtering

Optimum Linear Filtering

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The majority of this chapter is devoted to the matched filter because of its importance in modern communication and radar systems. It is the optimum filter under a wide variety of criteria when the input noise is additive, white, and Gaussian. Even if this optimum filter cannot be exactly realized, it remains as the standard against which other realizations can be compared. The study of the matched filter here includes its derivation, its time- and frequency-domain characteristics, its synthesis, its sensitivity to waveform changes, its relationship to cross-correlation, and its characterization when the input noise spectrum is nonconstant. Also discussed is pulse compression, a topic of both theoretic and practical importance. For large time-bandwidth products, the pulse compression system is essentially a matched filter system, thus an optimum detection system. The two methods of pulse compression considered here use the linear FM signal and the Barker sequences. The compressed waveforms include undesirable sidelobes, and we discuss methods of reducing them by appropriate filtering. Again, these filters are not included in the usual treatments of electronic filtering.

Chapter Contents:

  • 7.1 Autocorrelation, Cross-Correlation, and Power Density Functions
  • 7.1.1 Periodic Function Analysis
  • 7.1.2 Aperiodic (Transient) Function Analysis
  • 7.1.3 Example of Aperiodic Function Calculation
  • 7.1.4 Random Function Analysis
  • 7.1.5 Properties of the Autocorrelation Function
  • 7.1.6 Cross-Correlation Function
  • 7.1.7 Relationship Between Convolution and Cross-Correlation
  • 7.1.8 Output Autocorrelation Function of a Linear System
  • 7.1.9 Output Power Density Spectrum of a Linear System
  • 7.2 Linear Mean-Square Estimation
  • 7.2.1 Wiener-Kolmogoroff Filtering
  • 7.2.2 Kalman Filtering
  • 7.2.3 Summary
  • 7.3 Matched Filtering
  • 7.3.1 Matched Filter Derivation
  • 7.3.2 Filtering the Rectangular Pulse
  • 7.3.3 Frequency-Domain Characterization
  • 7.3.4 Matched Filter Synthesis
  • 7.3.5 Synthesis of the Rectangular Impulse Response
  • 7.3.6 Synthesis of the Trapezoidal Impulse Response
  • 7.3.7 Relationship of Matched Filtering to Cross-Correlation
  • 7.3.8 Nonoptimal Conditions
  • 7.3.9 Nonwhite Input Noise Spectrum
  • 7.4 Pulse Compression Using Linear Frequency Modulation
  • 7.4.1 The Linear FM Signal
  • 7.4.2 The Linear FM Matched Filter
  • 7.4.3 Compression Mechanism
  • 7.4.4 Sidelobe Reduction
  • 7.4.5 The Tapped Delay Line
  • 7.4.6 Optimum Filter Realizations
  • 7.5 Phase-Coded Waveforms
  • 7.5.1 The Barker Codes
  • 7.5.2 Amplitude Spectra of Pulse Sequences
  • 7.5.3 Sidelobe Reduction
  • 7.6 Matched Filter Technology
  • 7.6.1 Optical Signal Processor
  • 7.6.2 Surface Acoustic Wave Device
  • 7.6.3 Digital Filter
  • 7.6.4 Charge Transfer Device
  • References
  • Problems

Inspec keywords: correlation methods; frequency-domain analysis; matched filters; pulse compression; AWGN; time-domain analysis

Other keywords: Gaussian noise; radar systems; white noise; Barker sequences; optimum detection system; modern communication; pulse compression; frequency-domain characteristics; matched filter; time-domain characteristics; cross-correlation; input noise spectrum; compressed waveforms; linear FM signal; optimum linear filtering; additive noise

Subjects: Mathematical analysis; Filters and other networks

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