Carrier Triple Beat Test


Triple beat test is employed to mimic the cross-modulation distortion scenario and validate the performance in 3G and 4G/LTE devices. In Triple Beat test two tones f_1 and f_2 at Tx frequency and another tone f_3 at Rx frequency produces, intermod components at f_3 \pm (f_1 - f_2). If the two Tx tones are very close, then intermod components falls in Rx band. The highest power intermod component produced by triple beat test is 6dB higher than 3rd order intermod component produced by two tone test. This test is also called Carrier triple beat test (not composite triple beat test).

Consider a nonlinear amplifier whose input-output characteristic is represented by a 3rd order polynomial given by

(1)   \begin{equation*} v_o(t) = \alpha_o + \underbrace{\alpha_{\tiny 1} v_{i}(t)}_{{\huge \textcircled{\normalsize I}}}  + \underbrace{\alpha_{\tiny 2} v_{i}^{\tiny 2}(t)}_{{\huge \textcircled{\normalsize II}}}   + \underbrace{\alpha_3 v_{i}^3(t)}_{{\huge \textcircled{\normalsize III}}}  \end{equation*}

where, v_o(t) is the output voltage of the amplifier
and v_i(t) is the input voltage applied to the amplifier

Consider three carrier frequencies, f1, f2 and f3. Any combination involving one sum and one difference will produce a triple beat component near the original third carrier frequency. If the carriers are close to each other, the inter-modulation product at frequencies (f_1 - f_2 - f_3) and (f_1 - f_2 + f_3) fall near the original carrier frequency f_3.

(2)   \begin{equation*} v_{i} = A_{i,1} \cos(\varphi_{1}) + A_{i,2} \cos(\varphi_{2}) + A_{i,3} \cos(\varphi_{3}) \end{equation*}

where,
\varphi_{\tiny 1}=2\pi f_{\tiny 1}t+\phi_{\tiny 1},
\varphi_{\tiny 2}=2\pi f_{\tiny 2}t+\phi_{\tiny 2} and
\varphi_{\tiny 3}=2\pi f_{\tiny 3}t+\phi_{\tiny 3}.

Substituting Eq.(2) in Eq.(1), and grouping the same order terms we get

(3)   \begin{eqnarray*}  v_o(t) &=& \left. \alpha_o + {1\over 2}\left(A_{i,1}^2 +  A_{i,2}^2 +  A_{i,3}^2\right) \right\} \text{DC} \\ & &  \left. \begin{array}{l} \quad +\left(\alpha_1 +{3\over 4}\alpha_3 A_{i,1}^2 +{3\over 2}\alpha_3 (A_{i,2}^2+A_{i,3}^2)\right)A_{i,1}\cos\varphi_1 \\ \quad +\left(\alpha_1 +{3\over 4}\alpha_3 A_{i,2}^2 +{3\over 2}\alpha_3 (A_{i,1}^2+A_{i,3}^2)\right)A_{i,2}\cos\varphi_2 \\ \quad +\left(\alpha_1 +{3\over 4}\alpha_3 A_{i,3}^2 +{3\over 2}\alpha_3 (A_{i,1}^2+A_{i,2}^2)\right)A_{i,3}\cos\varphi_3  \end{array} \right\} \text{fundamental}\\ & & \left. \begin{array}{l} \quad +{1\over 2} \alpha_2 A_{i,1}^2\cos 2\varphi_1 + {1\over 2} \alpha_2 A_{i,2}^2\cos 2\varphi_2 + {1\over 2} \alpha_2 A_{i,2}^2\cos 2\varphi_2  \end{array} \right\} \text{2nd harmonic}\\ & & \left. \begin{array}{l} \quad + {1\over 4} \alpha_3 A_{i,1}^3\cos 3\varphi_1 + {1\over 4} \alpha_3 A_{i,2}^3\cos 3\varphi_2 + {1\over 4} \alpha_3 A_{i,3}^2\cos 2\varphi_3  \end{array} \right\} \text{3rd harmonic}\\ & & \left. \begin{array}{l} \quad +\alpha_2 A_{i,1}A_{i,2}\cos(\varphi_1-\varphi_2) + \alpha_2 A_{i,1}A_{i,3}\cos(\varphi_1-\varphi_3) \\ \quad + \alpha_2 A_{i,2}A_{i,3}\cos(\varphi_2-\varphi_3)+ \alpha_2 A_{i,1}A_{i,2}\cos(\varphi_1+\varphi_2) \\ \quad + \alpha_2 A_{i,1}A_{i,3}\cos(\varphi_1+\varphi_3) + \alpha_2 A_{i,2}A_{i,3}\cos(\varphi_2+\varphi_3) \\ \end{array} \right\} \text{2nd order IMD}\\ & & \left. \begin{array}{l} \quad + {3\over 4}\alpha_3 A_{i,1}^2 A_{i,2} \cos(2\varphi_1-\varphi_2) + {3\over 4}\alpha_3 A_{i,1}^2 A_{i,3} \cos(2\varphi_1-\varphi_3) \\  \quad + {3\over 4}\alpha_3 A_{i,2}^2 A_{i,3} \cos(2\varphi_2-\varphi_3) + {3\over 4}\alpha_3 A_{i,1} A_{i,2}^2 \cos(\varphi_1-2\varphi_2) \\ \quad + {3\over 4}\alpha_3 A_{i,1} A_{i,3}^2 \cos(\varphi_1-2\varphi_3) + {3\over 4}\alpha_3 A_{i,2} A_{i,3}^2 \cos(\varphi_2-2\varphi_3)\\            \quad + {3\over 4}\alpha_3 A_{i,1}^2 A_{i,2} \cos(2\varphi_1+\varphi_2) + {3\over 4}\alpha_3 A_{i,1}^2 A_{i,3} \cos(2\varphi_1+\varphi_3) \\ \quad + {3\over 4}\alpha_3 A_{i,2}^2 A_{i,3} \cos(2\varphi_2+\varphi_3) + {3\over 4}\alpha_3 A_{i,1} A_{i,2}^2 \cos(\varphi_1+2\varphi_2) \\ \quad + {3\over 4}\alpha_3 A_{i,1} A_{i,3}^2 \cos(\varphi_1+2\varphi_3) + {3\over 4}\alpha_3 A_{i,2} A_{i,3}^2 \cos(\varphi_2+2\varphi_3)\\  \quad + {6\over 4}\alpha_3 A_{i,1} A_{i,2} A_{i,3}  \underbrace{\left[ \begin{array}{l} \cos(\varphi_1 - \varphi_2 -\varphi_3) + \cos(\varphi_1 - \varphi_2 +\varphi_3) \\  + \cos(\varphi_1 + \varphi_2 -\varphi_3) + \cos(\varphi_1 + \varphi_2 + \varphi_3) \end{array} \right]}_{\text{triple beat}}\\ \end{array} \right\} \text{3rd order IMD} \end{eqnarray*}

Figure 1 shows the triple beat component produced due to beating two tones (f1 and f2) along with f3, and falling in the vicinity of thrid carrier tone(f3).

Triple beat test - RF carriers/tones and intermod components

Figure 1. Triple beat test – RF carriers/tones and intermod components

Here carrier triple beat ratio is defined as ratio of fundamental component power to triple beat 3rd order intermod component power.

    \[TB ratio = { v_{o,1} \over v_{o,tb}}\]

From Eq.(3),

(4)   \begin{equation*} TB~Ratio = {v_{o,1} \over v_{o,tb,f_3-(f_1-f_2)}} = \frac{\left(\alpha_1 +{3\over 4}\alpha_3 A_{i,1}^2 +{3\over 2}\alpha_3 (A_{i,2}^2+A_{i,3}^2)\right)A_{i,1}}{{6\over 4}\alpha_3 A_{i,1} A_{i,2} A_{i,3}}\\ \end{equation*}

At small signal levels, \alpha_1 \gg {3\over 4}\alpha_3 A_{i,1}^2 +{3\over 2}\alpha_3 (A_{i,2}^2+A_{i,3}^2). So

(5)   \begin{equation*} TB~Ratio  \approx \frac{\alpha_1}{{6\over 4}\alpha_3 A_{i,2} A_{i,3}} = {1\over 2}{4\over 3}{\alpha_1 \over \alpha_3} {1 \over A_{i,2} A_{i,3}} \end{equation*}

In terms of power

(6)   \begin{equation*} \boxed{\Delta P_{TB} = P_{o,1} - P_{o,tb,f_3-(f_1-f_2)} = 2IIP3 - (P_{i,2}+P_{i,3}) - 6} \end{equation*}

where, IIP3 is 3rd order intercept point from two tone test

The highest power intermod component produced by triple beat test is 6dB higher than 3rd order intermod component produced by two tone test.

For further reading on the test setup, refer to the articles or application notes here [1], [2] and [3]

References

[1] “K&L Microwave 3G, LTE/4G filter solutions for Intermodulation and Triple Beat testing.” [Online; Accessed Aug 2014]. Available: http://www.klmicrowave.com/catalog/KLMicrowaveIMDWebinar.wmv
[2] “Triple Beat and IMD for RF Switches, National Instruments .” [Online; Accessed Aug 2014]. Available: http://www.ni.com/white-paper/14506/en/
[3] “SKY13473–569LF: 0.4 to 2.7 GHz SP10T LTE Transmit/Receive Switch with MIPI RFFE Interface.” [Online; Accessed March 2015]. Available: http://www.skyworksinc.com/uploads/documents/SKY13473_569LF_202148F.pdf

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