Page 143 - Fundamentals of Radar Signal Processing

–34
where h = 6.6254 × 10 J/s is Planck’s constant . If hF/kT 1, a series
approximation gives exp(hF/kT) ≈ 1 + hF/kT so that Eq. (2.76) reduces to the
white noise spectrum

(2.77)

Note that Eq. (2.77), when integrated over frequency, implies infinite
power in the white noise process. In reality, however, the noise is not white
[Eq. (2.76)] and, in any event, it is observed in any real system only over a
finite bandwidth. For frequencies below 100 GHz, the approximation of Eq.
(2.77) requires the equivalent noise temperature T’ (to be defined below) to be
larger than about 50 K, which is almost always the case. Consequently, thermal

noise has a white power spectrum. For many practical systems it is reasonable
to choose the temperature of the system to be the “standard” temperature T =
0
–21
290 K = 62.3°F so that kT ≈ 4 × 10 W/Hz.
0
In a coherent radar receiver, the noise present at the front end of the system
contributes noise to both the I and Q channels after the quadrature demodulation.
The I and Q channel noises are both zero-mean Gaussian random processes with

equal power. Since the total noise spectral density is kT W/Hz, the noise density
in each channel individually is kT/2 W/Hz. Furthermore, if the power spectrum
of the input noise is white, then the I and Q noise processes are uncorrelated and
their power spectra are also white. Since the I and Q noise processes are
Gaussian and uncorrelated, it follows that they are also independent (Papoulis
and Pillai, 2001). Finally, since the I and Q signals are independent zero-mean

Gaussian processes, it also follows that the magnitude of the complex signal I +
jQ is Rayleigh distributed, the magnitude-squared is exponentially distributed,
and the phase angle tan (Q/I) is uniformly distributed over (0, 2π].
–1
The bandwidths of the various components of a receiver vary, but the
narrowest bandwidth is generally approximately equal to the bandwidth of the
transmitted pulse. If the receiver contains any component of narrower bandwidth
signal, energy will be lost, reducing sensitivity. If the most narrowband

component has a bandwidth appreciably wider than the pulse bandwidth, the
signal will have to compete against more noise power than necessary, again
reducing sensitivity. Thus for the purpose of noise power calculation, the
frequency response of the receiver can be approximated as a bandpass filter
centered at the transmit frequency with a bandwidth equal to the waveform

bandwidth.
Real filters do not have perfectly rectangular passbands. For analyzing
noise power the noise-equivalent bandwidth β of a filter described by the
n
transfer function H(F) is used. Figure 2.24 illustrates the concept. The noise
equivalent bandwidth is the width an ideal rectangular filter with gain equal to
the peak gain of the actual filter must have so that the area under the two squared