Usual and unusual concepts
2.7 POLYPHASE, OR BOTTLE-BRUSH, NOISE
Cosmic noise comes from outer space into the antenna. A greater amount of noise is generated by the first active component in the receiver. Formerly, these components were crystal diodes, but currently they are mostly transistors.
This noise is filtered by the waveguide between the antenna and the receiver and is equally distributed in the band. The noise power in a resistor is given by the Nyquist expression:
where k is Boltzmann’s constant, 1.380622 10 J/K [2, pp. 3-16];-23 T is the absolute temperature of the resistor in Kelvin;
B is the bandwidth in Hz.
The filtering at radio frequency gives a band of noise around a central radio frequency extending from half the bandwidth on one side to half the bandwidth on the other side. There is no carrier frequency. In the receiver, the noise is block converted to intermediate frequency, filtered again, and then detected. The bandwidth defining the noise power sent for detection is determined by this final filter, as is shown in Figure 2.18.
With polar detection zero frequency components convert to stationary vectors. The components with a frequency of up to B/2 convert to positive phase sequence components, and those with a notional “negative” frequency convert to negative phase sequence components. Frequency is defined in cycles per second so that negative frequencies do not exist, and the negative portion, shown by a dotted line in Figure 2.18, is said to be folded into the positive domain. A linear detector gives the voltage of the vector, and a square law detector gives the power of the vector.
Signals that undergo Cartesian detection are resolved into two orthogonal components that represent the polar vector.
These components vary with time if the polar vector rotates. When the noise is detected by a Cartesian or polar converter, the complex waveform is represented as in Figure 2.18. Remember, a constant polyphase voltage, as in electrical power systems, is represented by a helix. In contrast, the noise has the appearance of a shaggy, stochastic bottlebrush, as shown in Figure 2.19. This is a bivariate Gaussian distribution with zero mean, and the standard deviation corresponds to the root mean square noise phase voltage. If the noise passes through a Cartesian detector, then the noise is resolved into the in-phase component and the quadrature component as shown in Figure 2.20.
-3 -2 -1 0 1 2 3
200 400
600 800
1000 -3
-2 -1 1
2 3
Noise sample number or time Real, in phase, or I axis
Imaginary, quadrature, or Q axis
In-phase or I component
Polyphase noise
Quadrature or Q component
.
.
p(x, y) '
exp & x2 2F2xy 2BF2xy
exp & y2 2F2xy 2BF2xy
'
exp &x2%y2 2F2xy 2BF2xy Figure 2.19 Bottle brush or polyphase noise.
Figure 2.20 Bottle-brush, or polyphase, noise and its resolution into two single-phase components.
(2.20) Figure 2.21 shows the scatter diagram of the noise, or the tips of the bottle-brush, as seen from the end. In the x and y directions, the components are part of a bivariate Gaussian distribution in that the samples are bunched towards the origin. The noise samples are uniformly distributed in phase. This is shown in three dimensions in Figure 2.22. The amplitude output of a polar detector is the radius, r, of the noise sample in Figure 2.22. A representation of amplitude with time, as on an A-scope, is shown in Figure 2.23. This probability distribution of the noise around the origin is, in x and y, the product of the two Gaussian distributions
where Fxy is the root mean square (rms) value of the x or y component of the noise voltage. The radial measure for F is
-3 -2 -1 0 1 2 3
-3 -2 -1 1 2 3
Root mean square
p2 F
Median p(2 ln 2)F
Gaussian F
r2 ' x2 % y2
Height of hump '
exp & r2 2F2 2BF2
r dr
dr Normal or Gaussian
distribution in two dimensions
Rayleigh distribution
2
Figure 2.21 A scatter diagram showing the amplitudes and phases of the bottle-brush noise samples.
(2.21)
Figure 2.22 Bivariate Gaussian distribution of noise.
the sum of the equal x and y components or Fxy.
Substituting for x and y:
p(r) ' 2 B r dr × Height of hump
'
r exp & r2 2F2 F2
0 1 2 3 4
200 400 600 800 1000
Gaussian F Mean radial error Median
Root mean square
Sample number or range
p(r) ' r
F2 exp &r2%S2 2F2 I0 rS
F2 B
2Fxy ' 1.2533 Fxy
2Fxy ' 1.4142 Fxy 2 ln 2Fxy ' 1.1774 Fxy
(2.22)
Figure 2.23 Rayleigh noise, or the plot of the modulus of bottle-brush noise.
(2.23) The probability at the radius, r, is
This is the Rayleigh distribution and is the probability distribution of the detected noise, as shown on the right of Figure 2.22.
The following quantities are defined for the Rayleigh distribution:
• Fxy, the standard deviation of each of the two parts of the bivariate Gaussian distribution;
• The mean of the Rayleigh distribution , measured by a direct voltage voltmeter;
• The root mean square value of the Rayleigh distribution = , measured by a root mean square voltmeter;
• The median of the Rayleigh distribution (half the points are inside this radius, and half are outside).
Traditionally, the sum of a vector and noise is shown as a washing up mop, as in Figure 2.24. The individual signal plus noise samples are the vectors from the origin to each of the noise samples. The probability distribution for the distance of a point from the origin is a Ricean distribution [3, p. 325] and is given by
where r is the variable to a noise sample;
S is the value of the steady vector or signal;
F is the standard deviation of the bivariate Gaussian distribution;
I is a Bessel function of zero order and imaginary argument. 0
Origin Signal vector
S
Added bivariate Gaussian noise with standard deviation F
Vector to signal + noise
sample r S
r Rayleigh root mean square = /2 Fxy
Median Fxy
dr
0 0.1 0.2 0.3 0.4 0.5
2 4 6 8 10
r S=0
S=1 S=2 S=3 S=4 S=5S=5 S=6 S=7
Rectangular pulse ' 1 for &J/2 < t < J/2 ' 0 otherwise
Figure 2.24 Bivariate Gaussian noise added to a signal vector.
Figure 2.25 Ricean distribution for various values of S.
(2.24) Figure 2.25 shows the Ricean distribution for F = 1 for various values of S. When S = 0, this is a Rayleigh distribution, which is used to determine the probability of detection for echoes that do not fade.