For example, in Chap. 6 it will be seenthat detection of a constant-amplitude target signal in complex Gaussian noisewith a probability of detection of 0.9 and a probability of false alarm of 10-8requires a single-sample SNR of 14.2 dB (about 26.3 on a linear scale).

The same probabilities can be obtained byintegrating the magnitude of 10 samples each having an individual SNR of only5.8 dB (3.8 on a linear scale).

The reduction of 8.4 dB (a factor of26.3/3.8 = 6.9) in the required single-sample SNR when 10 samples arenoncoherently integrated is the implied noncoherent integration gain.

Noncoherent integration is much moredifficult to analyze than coherent integration, typically requiring derivationof the probability density functions of the noise-only and signal-plus-noisecases in order to determine the effect on detection and parameter estimation. Thus, noncoherent integration is lessefficient than coherent integration.

This should not be surprising, since notall of the signal information is used.

1.4.4带宽扩展

1.4.4 Bandwidth Expansion

The scaling property of Fourier transformsstates that if x(t) has Fourier transform X(Ω) = F{x(t)}, then Equation (1.34) states that if the signal xis compressed in the time domain by the factor α > 1, its Fourier transformis stretched (and scaled) in the frequency domain by the same factor (Papoulis,1987).

When α < 1, Eq. (1.34) shows thatstretching in the time domain results in compression in the frequency domain.

This reciprocal spreading behavior isillustrated in Fig. 1.17. Figure 1.17. 傅立叶变换时频域特性表现相反的描述。（a）正弦波脉冲及其傅立叶变换的主要频域部分。（b）时域窄脉冲具有更宽的频谱特性。Illustration of reciprocal spreading property of Fourier transforms.(a) A sinusoidal pulse and the main portion of its Fourier transform. (b) Anarrower pulse has a wider transform. See text for details.

Part (a) shows a sinusoidal pulse with afrequency of 10 MHz and a duration of 1 μs and its Fourier transform, which isa sinc function centered on 10 MHz and with a Rayleigh mainlobe width of 1 MHz,the reciprocal of the 1 μs pulse duration.

In part (b) the pulse has the samefrequency but only one-quarter the duration.

Its spectrum is still a sinc centered at 10MHz, but the Rayleigh width is now four times larger at 4 MHz.

The spectrum amplitude is also reduced by afactor of four.

——本文译自Mark A. Richards所著的《Fundamentals of Radar Signal Processing（Second edition）》