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Noise refers to any electrical output other than the desired signal. Some noise sources are fundamental, and for several reasons cannot be avoided. A few of these reasons are: Photons do not arrive at a constant rate. Atoms in the detector vibrate slightly. Electrons move randomly. Other noise sources arise externally and can be minimized, such as: Electrical interference Temperature fluctuation Vibration that causes electrical components to shift
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Photon noise or shot noise is unavoidable. It is due to the random arrival of photons at the detector. Photon noise ( A/ Hz ) for a photovoltaic detector (InSb) is given by: I photon = 2 e I detector (10.6)
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It becomes I photon = e I detector for a photoconductor (MCT). Although photon noise seems smaller in the case of a photoconductor detector than in a photovoltaic one, the sensitivity of photoconductors is, in general, relatively small. Because photon noise is unavoidable, it is the ultimate limiting factor of the system s performance (SNR). Whenever possible, all other sources of noise should be kept lower than the photon noise. Other sources of noise, such as the Johnson noise from the first amplifier stage feedback resistor, quantization noise, and detector dark noise contribute to the total noise.
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Johnson noise, roughly defined, is the random variation of voltage due to the thermal agitation of charge carriers in a resistor. Originally described and measured by J. B. Johnson, a physicist at Bell Laboratories who performed his experiments with a vacuum tube amplifier and thermocouple, the effect was first explained theoretically by H. Nyquist in 1928. The Johnson noise ( A/ Hz ) of the feedback resistor of the first stage of amplification is given by: I Johnson = 4kT Rf (10.7)
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where Rf = feedback resistor ( ) k = 1.38 10 23 J/K is the Boltzmann constant = noise factor for imperfect resistors (no units), estimated by the manufacturer to be 1.5 T = temperature of the resistor (K) In the SpectRx design, Johnson noise never dominates the total noise of the system. It increases the total system noise by less than 10 percent with the coldest object view temperature and by even less as the object view temperature is increased.
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The circuit model of Nyquist is adopted in order to understand the behavior of the resistor. Imagine a resistor R at temperature T connected in series with an identical resistor R at the same temperature, through a transmission line of length L. To avoid losses due to radiation from the wires connecting the resistors, one line may be enclosed by the other, as in a coaxial cable. Imagine the first resistor to be held at a finite temperature T, so that thermally agitated electrons within produce a fluctuating voltage signal, and then represent this signal with the source V, as shown in Fig. 10.8. By placing the second resistor at the distant end of the transmission line, a boundary condition has been achieved. The transmission
R R V
FIGURE 10.8
Thermally agitated electrons to produce voltage uctuation.
Ten
line between 0 and L ought to accommodate voltage waves V(x, t) obeying boundary conditions V (0, t ) = V (L , t ) where, generally, V (x, t) = V0e i( kx t ) The voltage waves propagate at the speed of light, c= |k | (10.10) (10.9) (10.8)
The boundary condition implies kL = 2 n, n Z (10.11)
or, assuming the transmission line to be very long (so that many modes n are permitted), L dk = 2 dn 1 1 dn = dk L 2 (10.12)
This is simply the density of modes the number of modes, or voltage states allowed per unit length in the line. Both right- and left-propagating voltage waves are admitted in the solution, so that both k and its associated frequency w range over positive and negative values. Propose that each mode of voltage oscillation is a mode in which the resistor on the right can absorb radiation from the (quantized) electromagnetic field. Since one imagines the resistor to be an idealized one-dimensional lumped element, propose also that it absorbs radiation in the single dimension of the transmission line. Then the power energy per unit time absorbed by the resistor is the energy of a certain mode multiplied by the number of quanta in that mode, integrated over all modes, multiplied by the rate at which the quanta are absorbed (c/L), Pabsorbed = c L
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