Page 66 - Fundamentals of Radar Signal Processing

complex Gaussian noise with a probability of detection of 0.9 and a probability

–8
of false alarm of 10 requires a single-sample SNR of 14.2 dB (about 26.3 on a
linear scale). The same probabilities can be obtained by integrating the
magnitude of 10 samples each having an individual SNR of only 5.8 dB (3.8 on
a linear scale). The reduction of 8.4 dB (a factor of 26.3/3.8 = 6.9) in the
required single-sample SNR when 10 samples are noncoherently integrated is

the implied noncoherent integration gain.
Noncoherent integration is much more difficult to analyze than coherent
integration, typically requiring derivation of the probability density functions of
the noise-only and signal-plus-noise cases in order to determine the effect on
detection and parameter estimation.
Chapter 6 will show that in many useful cases, the noncoherent integration
α
gain is approximately N , where α ranges from about 0.7 or 0.8 for small N to
about 0.5 ( ) for large N, rather than in direct proportion to N. Thus,
noncoherent integration is less efficient than coherent integration. This should
not be surprising, since not all of the signal information is used.

1.4.4 Bandwidth Expansion
The scaling property of Fourier transforms states that if x(t) has Fourier

transform X(Ω) = F{x(t)}, then

(1.34)

Equation (1.34) states that if the signal x is compressed in the time domain by
the factor α > 1, its Fourier transform is stretched (and scaled) in the frequency
domain by the same factor (Papoulis, 1987). When α < 1, Eq. (1.34) shows that
stretching in the time domain results in compression in the frequency domain.
This reciprocal spreading behavior is illustrated in Fig. 1.17. Part (a) shows a

sinusoidal pulse with a frequency of 10 MHz and a duration of 1 μs and its
Fourier transform, which is a 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 same frequency but only one-quarter the duration. Its
spectrum is still a sinc centered at 10 MHz, but the Rayleigh width is now four

times larger at 4 MHz. The spectrum amplitude is also reduced by a factor of
four. This effect can also be viewed in the opposite direction: if the signal gets
wider in the frequency domain, it must get narrower in the time domain.