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where I + J is the number of all responses of which I are supplied with signals.
3.4 Discussion of noise quantities
Most often noise quantities are used to describe the noise properties of two-ports. In cascading these, one or more of the two-ports may be only conditionally stable. This means that a source or load for one of the stages is active even if the cascaded circuit is stable. This situation is quite common at higher frequencies. Therefore in extending the gain and noise quantity definitions to include active sources and loads, first priority was given to ensure that the well-known and often used formulae could be used unchanged.
When extending to multi-ports this choice will in rare cases give some values for extended noise quantities which - considered alone - do not give a correct feeling for the noise properties of the device. This is due to the fact that the extended definitions in special cases may have noise powers going in opposite directions and thus being subtracted. It has been decided to accept this rarely occurring calamity in order to preserve the simplicity of the often used formulae.
The same problem occurs for the average noise quantities if either the source, the load or the gain of the device changes sign in the frequency band of interest. At one frequency the noise power flows in one direction and at another in the opposite. Thus the noise power integrated over the frequency band may be a small and rather meaningless figure.
One way to get around these problems is to introduce yet another noise quantity ]7j. As all definitions of noise quantities consider only one output port no ambiguity exists on the sign of the load at a given output frequency. By referring a noise temperature to the output terminal many problems are solved. The proposed output noise temperature is called the extended load operating noise temperature and is defined for a multi response transducer as
Definition 3.11 The extended load operating noise temperature of
a multi-response transducer is defined as
where N£ is the noise power density delivered to the equivalent noise free immittance of the load circuit at the output frequency and к =
1.3807 x 10-23 J K-1 is Boltzmann’s constant.
3.4. Discussion of noise quantities
If the extended load operating noise temperature should have an average equivalent and the load immittance should change sign in the frequency band of interest the same problem occurs once again. Two possibiHties exist. One is to integrate numerically in order to add all the power flow over the frequency range. The other is just to integrate, which in some rare instances gives meaningless results without further information. But is an average extended load operating noise temperature needed? Perhaps an international, European or American standard committee on noise might be a good idea. If so the number of recommended definitions might be reduced. The first step could be to abolish the average definitions.
 Engberg, J. fc. Larsen, Т.: ''Extended definitions for noise temperatures of linear noisy one- and two-ports", IEE Proc. Part, II, vol. 138, dd. 86 - 90, February 1991.
 Haus, H. A. к Adler, R. B.: “Circuit theory of linear noisy networks'', Technology Press and Wiley, 1959.
 “IRE Standards on Electron Tubes: Definitions of Terms, 1957, 57 IRE 7. S2,!, Proc. IRE, vol. 45, pp. 983 - 1010, July 1957.
 Haus, H. A. U Adler, R. B.: ‘‘An extension of the noise figure definition'5. Froc. IRE, vol. 45, pp. 690 - 691, May 1957.
 “IRE Standards on Electron Tubes: Definitions of Terms, 1962, 62 IRE 7. S2". Proc. IRE, vol. 50, pp. 434 - 435, March 1963.
 Friis, H. Т.: “Noise figures of radio receivers1’, Proc. IRE, vol. 32. pp. 419 - 422. July 1944.
 Larsen, Т.: '"Extended source and load operating noise temperatures of nonlinear systems”, IEE Proc. Part II, vol.139, pp. 121 - 124, April 1992.
Most amplifiers consist of cascaded two-ports. In order to characterize the noisy behaviour of a two-port several sorts of noise parameters have beer, developed. Noise parameters can also be developed for multi-ports, but would be more clumsy to use. Two-ports are conveniently characterized by their small-signal parameters like
H, Y or S parameters. These consist of 8 (4 complex) numbers, which are functions of frequency and bias point in particular and to a minor extent temperature, radioactive radiation etc. They are, however, used extensively in practical circuit design. By adding 4 more numbers ('2 real and 1 complex) the noise properties of a two-port can be included as well. Those 4 numbers are called noise parameters and
- in analogy to small-signal parameters - noise parameters exist in many forms.
In this chapter the noise parameters are derived, their use explained, and the conversion formulae between different types of noise parameters given.
4.1 Noise voltages and currents
In 1955 Rothe and Dahlke [I] introduced noise parameters related to the chain or ABCD small-signal parameters. These are based on noise voltages and currents. In a similar way noise power waves related to the scattering parameters have been used by several authors, e.g. [2,3,4,5,6,7,8,9], but they are still not used as extensively as the Rothe and Dahlke type of noise parameters.