The third-order intercept point (TOI) is a property of the device transfer function O (see diagram). This transfer function relates the output signal voltage level to the input signal voltage level. We assume a “linear” device having a transfer function whose small signal form may be expressed in terms of a power series containing only odd terms, making the transfer function an odd function of input signal voltage, i.e., O = −O. Where the signals passing through the actual device are modulated sinusoidal voltage waveforms (e.g., RF amplifier), device nonlinearities can be expressed in terms of how they affect individual sinusoidal signal components. For example, say the input voltage signal is the sine wave
and the device transfer function produces an output of the form
where G is the amplifier gain and D3 is cubic distortion. We may substitute the first equation into the second and, using the trigonometric identity
we obtain the device output voltage waveform as
The output waveform contains the original waveform, cos(ωt), plus a new harmonic term, cos(3ωt), the third-order. The coefficient of the cos(ωt) harmonic has two terms, one that varies linearly with V and one that varies with the cube of V. In fact, the coefficient of cos(ωt) has nearly the same form as the transfer function, except for the factor ¾ on the cubic term. In other words, as signal level V is increased, the level of the cos(ωt) term in the output eventually levels off, similar to how the transfer function levels off. Of course, the coefficients of the higher-order harmonics will increase (with increasing V) as the coefficient of the cos(ωt) term levels off (the power has to go somewhere).
If we now restrict our attention to that portion of the cos(ωt) coefficient which varies linearly with V, and then ask ourselves, at what input voltage level, V, will the coefficients of the first and third order terms have equal magnitudes (i.e., where the magnitudes intersect), we find that this happens when
which is the Third-Order Intercept Point (TOI). So, we see that the TOI input power level is simply 4/3 times the ratio of the gain and the cubic distortion term in the device transfer function. The smaller the cubic term is in relation to the gain, the more linear the device is and the higher the TOI is, which clearly makes sense. The TOI, being related to the magnitude squared of the input voltage waveform, is a power quantity, typically measured in milliwatts (mW). The TOI is always beyond operational power levels because the output power saturates before reaching this level.
The TOI is closely related to the amplifier's "1 dB compression point," which is defined as that point at which the total coefficient of the cos(ωt) term is 1 dB below the linear portion of that coefficient. We can relate the 1 dB compression point to the TOI as follows. Since 1 dB = 20 log10 1.122, we may say, in a voltage sense, that the 1dB compression point occurs when
In a power sense (V2 is a power quantity), a factor of 0.10875 corresponds to −9.636 dB, so by this approximate analysis, the 1 dB compression point occurs roughly 9.6 dB below the TOI.
Recall: decibel figure = 10 dB × log10(power ratio) = 20 dB × log10(voltage ratio).
Read more about this topic: Third-order Intercept Point
Other articles related to "theory":
... Minimax (sometimes minmax) is a decision rule used in decision theory, game theory, statistics and philosophy for minimizing the possible loss for a worst case (maximum ... Originally formulated for two-player zero-sum game theory, covering both the cases where players take alternate moves and those where they make simultaneous moves, it has also been extended to more ...
... Rushton's book Race, Evolution, and Behavior (1995) uses r/K selection theory to explain how East Asians consistently average high, blacks low, and whites in the middle on an evolutionary scale of ... He first published this theory in 1984 ... He theorizes that r/K selection theory explains these differences ...
... The most widely held theory is put forth by Marc Bloch ... This Germanic origin theory was also shared by William Stubbs in the nineteenth century ... Another theory was put forward by Archibald R ...
... Zermelo set theory, as set out in an important paper in 1908 by Ernst Zermelo, is the ancestor of modern set theory ...
... It is useful to know if a statement or theory is falsifiable, if for no other reason than that it provides us with an understanding of the ways in which one might assess the theory ... One might at the least be saved from attempting to falsify a non-falsifiable theory, or come to see an unfalsifiable theory as unsupportable ... Popper claimed that, if a theory is falsifiable, then it is scientific ...
Famous quotes containing the word theory:
“It is not enough for theory to describe and analyse, it must itself be an event in the universe it describes. In order to do this theory must partake of and become the acceleration of this logic. It must tear itself from all referents and take pride only in the future. Theory must operate on time at the cost of a deliberate distortion of present reality.”
—Jean Baudrillard (b. 1929)
“The struggle for existence holds as much in the intellectual as in the physical world. A theory is a species of thinking, and its right to exist is coextensive with its power of resisting extinction by its rivals.”
—Thomas Henry Huxley (182595)
“The theory [before the twentieth century] ... was that all the jobs in the world belonged by right to men, and that only men were by nature entitled to wages. If a woman earned money, outside domestic service, it was because some misfortune had deprived her of masculine protection.”
—Rheta Childe Dorr (18661948)