Two-Tone vs. Single-Tone Measurement of 2nd-Order Non-linearity and IP2 Performance. Likewise for f4:

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1 CX7407 Two-Tone vs. Single-Tone Measurement of nd-order Non-linearity and IP Performance This paper covers the subject of how to correctly find IP from -tone and -tone tests, and then presents measurement results for the CX7407 GSM900 receive (RX) path, as used in the Advanced Mobile Phone Services (AMPS) band. Relation Between -Tone and -Tone Tests For nd- Order Non-linearity Second order non-linearity is a phenomenon that is important in Direct Conversion Receiver (DCR)-type receivers. Here we set out to show the test result of this kind of non-linearity, which can be used to predict the direct current (DC) offset, both with single tone or two tones. A two-tone condition is shown pictorially in Figure and Figure. Two input tones (f, f ) f f The relationship between f, f and f3, f4 is: f 3 = f f Likewise for f4: f + 4 = f f This shows that the unwanted output frequency components are mathematically related to the input tones. To have a better insight into the relationship between undesired components and the input terms, a more rigorous derivation is needed. Here is an attempt in showing the derivation with some simplifications. Using the Taylor series expansion the output of the gain stage can be modeled as: Vo(t) = k vi(t) + k vi (t) + k3 vi 3 (t) + k4 vi 4 (t) + k5 vi 5 (t) + () For a two-tone case then: Vi(t) = A Cos(ωt) + B Cos(ωt) () 073A _0690 Figure. Two-Tone Condition: Input Inserting () into () and using the well-known trigonometric equalities: f f f 3 f 4 f 4 = f + f f 3 = f + f 073A 3_0690 Figure. Two-Tone Condition: Output Application Note Conexant Proprietary 073A Preliminary Data Subject to Change July, 00

2 Two-Tone vs. Single-Tone Measurement CX7407 V o(t) = k A + k B + { k A k3 A3 + 3 k3 AB k5 A k5 A3 B k5 AB4 } Cos(ω t) + { k B k3 B3 + 3 k3 A B k5 B k5 A B k5 A4 B} Cos(ω t) + { k A + k4 A4 + 3 k4 A B } Cos(ω t) + { k B + k4 B4 + 3 k4 A B } Cos(ω t) + {k AB + 3 k4 A3 B + 3 k4 AB3 } Cos((ω + ω ) t ) + {k AB + 3 k4 A3 B + 3 k4 AB3 } Cos((ω - ω ) t ) + { 4 k3a k5a k5a3 B } Cos(3ω t) + { 4 k3b k5b k5a B 3 } Cos(3ω t) + { 3 4 k3 A B k5a4 B k5 A B 3 } Cos((ω ± ω ) t ) + { 3 4 k3 AB k5ab k5 A3 B } Cos((ω ± ω ) t ) + k4 A3 B Cos((3ω ± ω ) t ) + k4 AB3 Cos((ω ± 3ω ) t ) k4 A B Cos((ω + ω ) t ) + 8 k4a4 Cos(4ω t) + 8 k4b4 Cos(4ω t) (3) In this study the sum term is ignored since the baseband filters of the device will reject it. However the difference term is retained, since it can be used to evaluate the DC offset of the receiver. Note that in practice the input tones are chosen such that the difference can produce a tone near DC, that is, inside the receiver s baseband bandwidth. The pure DC terms: k A + k B The difference term is: k AB + k 4 A B + k 4 AB Cos (( ω ω ) t) Under a single tone condition, B is set to zero and the DC created is k A For a two-tone test and only nd order non-linearity, the higher order terms are ignored and coefficient of the output is then k AB. This coefficient is the peak amplitude of the difference frequency output. In calculating IP, we are concerned with power, so we want to know the RMS output; it is k AB. The ratio of the output of the -tone test to the output of the twotone test is then: k A R = (4) k AB The numerator is the DC output voltage from the -tone test, and the denominator is the ACRMS voltage from the -tone test. Remember that in the -tone test, the input amplitude is A, while in the -tone test the amplitudes are A and B. Assuming equal tones for the -tone test (B = A), then, or 3 db. Conexant Proprietary 073A Preliminary Data Subject to Change July, 00 R = This simply says that the root mean square (RMS) alternate current (AC) signal created from a two tone test is 3 db higher than the DC offset created from a single tone test, when the single tone test uses the same amplitude as one of the two tones. In other words, DC voltage (-tone) = AC RMS voltage (-tones) 3 db This is verified by measurement in the graph shown in Figure 3, where the receiver under test exhibits a significant amount of

3 CX7407 Two-Tone vs. Single-Tone Measurement Output DC and AC(rms) vs. Blocker Input Level Output DC (dbv) and AC (dbvrm s) Blocker Level (dbm ) Single tone Tw o tone 073A 4_0690 nd order non-linearity. In this test, the single tone was at 3 MHz offset, while the two tones were at 3 MHz and 3.06 MHz, so we are comparing a DC IM output in dbvdc to a 60 khz IM output in dbvrms. We expect the 60 khz output to be 3 db higher as derived above. But in the measured system, there is a lowpass response that rolls off by.4 db at 60 khz. Therefore, the 60 khz output should be.6 db higher than the DC. This is indeed the case over most of the tested range. IP Calculation From -Tone and -Tone Measurements In Figure 4, we plot the fundamental and nd order output powers vs. input power for a generic system that has some nd order non-linearity. In Figure 4, the gain is normalized to unity so IIP = OIP. In practice, we usually refer all quantities to the input. By convention, in any -tone test (including IP3 tests), the power levels plotted refer to one of the input tones and one of the output product tones. Therefore even though the system has an amount of power applied to it that is 3 db higher than the power of one tone alone, we only plot the power of one tone, not the sum of both. Likewise, we only plot the IM power at the difference frequency f - f, rather than the sum of powers at both f - f and f + f. From the slopes in Figure 4, we can see that IP can be calculated from one set of measurements as IP = (Pin) IM where all quantities are in dbm. Figure 3. Output DC and AC(rms) vs. Blocker Input Level This classical IP equation has a resemblance to the much more often used IP3 equation for a -tone test, which is: (IP3) = 3(Pin) IM3 P out IP P in IM Fundamental : slope nd order product : slope IP P in 073A 4_0690 Figure 4. Fundamental and nd Order Output Powers vs. Input Power The next question is how to correctly calculate IP when a -tone test is done and the IM product is DC. The answer must be the same as that found in the -tone test. First, we must choose the conventions for the -tone test. We calculate IP based on the power of the single applied tone 073A Conexant Proprietary 3 July, 00 Preliminary Data Subject to Change

4 Two-Tone vs. Single-Tone Measurement CX7407 (even though this is all the power applied to the system, unlike what is done in the -tone calculation), and on the DC output power (where we ignore the other, higher frequency product, which is at f). In the -tone test, as shown before, inputs of ACos( ω t ) and ACos( ω t ) result in an IM product of A Cos( ( ω ) t). The peak voltage of one input is A, k ω while the RMS = A. Likewise, the peak voltage of the IM output is k A while the RMS is k A. The IP calculation from the -tone test is then IP-tone = (Pin) IM (0log( A )) 0 log( (5) 4 IP-tone = k A ) In the -tone test, the single input is ACos( ω t ), resulting in a A DC IM output voltage of k. The peak input voltage is A, while the RMS = A. If we were to (recklessly) apply the classical IP calculation with these quantities, we would obtain IP-tone = (Pin) IM (0log( A )) 0 log( (6) 4 4 IP-tone = k A ) The second term in (6) is 3 db lower than the second term in (5). This is the same 3 db difference already identified in the first part of this paper. Therefore, if we wish to use the results from a -tone test to calculate IP using IP = (Pin) IM, we must first add 3 db to the measurement of the DC IM product. Then the result will match that of the -tone test. Therefore, the correct IP equation for a -tone test, where Pin is the power of the single input tone and IM is the power of the DC output, follows: IP = (Pin) (IMDC + 3 db) (7) Using the IM data presented in the earlier graph, we calculate the IP according to IP = (Pin) IMACRMS for the -tone test, and IP = (Pin) (IMDC + 3 db) for the -tone test. The results are plotted in Figure 5. We find the calculated IP values to generally agree. For consistency, we must use the same equation to determine the IP requirement itself. The GSM AM suppression specification (from GSM 05.05) sets the IP requirement for a DCR. For the GSM900 band, the single blocker applied is 3 dbm, while the desired signal is at 99 dbm. In order to keep the DC product below 9 dbc, the IIP must be: IIPREQ = (Pin) (IMDC + 3 db) where the IM level is referred to the antenna IIP REQ = (-3 dbm) ((-99 dbm-9 dbm) + 3 db) IIP 900MHz REQ = +43 dbm at the antenna Then, in a receiver with 3 db of front end loss due to switches and filters, the IIP requirement at the LNA input becomes IIP 900MHz REQ = +40 dbm at the Low Noise Amplifier (LNA) input Likewise for Digital Cellular Systems (DCS) and Personal Communications System (PCS) receivers with 3 db front end loss, the IIP requirements at the LNA inputs are + 4 dbm and + 44 dbm, respectively. IIP vs. Blocker Input Level IIP ( dbm ) Block e r Le ve l (dbm ) Single tone Tw o tone 073A 5_0690 Figure 5. Calculated IP Values for -Tone and -Tone Tests 4 Conexant Proprietary 073A Preliminary Data Subject to Change July, 00

5 CX7407 Two-Tone vs. Single-Tone Measurement CX7407 IP-Compensation Circuit Calibration Using Tones or Tone The Conexant CX7407 DCR implements an IP-compensation circuit that, once calibrated at a single blocker amplitude, compensates for nd-order non-linearity at all amplitudes, until the system approaches compression. It relies on the nonlinearity, maintaining a nd-order characteristic, an assumption, which obviously breaks down at compression. The IP compensation circuit (patents pending) does not involve or resemble AC coupling of the signal. Therefore it does not cause a DC notch in the signal. It also does not exhibit a frequency rolloff in its ability to reduce nd-order products. It suppresses both the DC IM due to a single blocker frequency, and the AC IM due to two blocker frequencies. This also makes it suitable for suppressing IM due to amplitude-modulated blockers. Here we show the CX7407 IP performance for both -tone and -tone blocker inputs. The IP compensation circuit of the CX MHz RX path was calibrated at the center of the AMPS band, that is, 88.5 MHz, using a single blocker frequency of 30 dbm at the LNA input, at +3 MHz offset. After calibration, the IP at this point measured +7.8 dbm. Then the IM products were measured over a range of blocker levels from 46 dbm to 4 dbm. This was done for both -tone and -tone cases, for example, tone at 46 dbm, vs. tones at 46 dbm each. The -tone test was repeated with tone separations of 5 khz and 30 khz, set 3 MHz from the receiver channel. Figure 6 and Figure 7 show the measured IM outputs and the calculated IP. In the IM output plot, limit lines are included that show the receiver s desired output in dbvrms (due to a 99 dbm signal at antenna, or 0 dbm at LNA) and a maximum DC IM at 9 dbc. The specification line stops at 34 dbm since we assume 3 db of front-end loss, while the GSM AMsuppression test uses a 3 dbm blocker at the antenna. Neither the DC nor AC IM products violate the 9 dbc limit. In the IIP plot, the fixed 9 dbc IM limit is translated to an IIP limit that scales with the blocker amplitude, reaching the previously derived value of +40 dbm at the GSM AM-suppression-test point. The IIP calculated at the LNA input, whether from the -tone or -tone test, stays well above the limit. The particular shapes of the curves should be noted. In the IM plot, at lower blocker levels, the -tone IM product is still roughly 3 db higher than the -tone DC product, and these products are both very small and nearly constant due to the action of the IP compensation circuit. But, as the blockers increase, the products begin to rise as the higher orders of nonlinearity start to become significant. There is a local minimum in the DC IM curve at a blocker level of 30 dbm, precisely because this is the point where the system was calibrated. The -tone IM plot shows a far less pronounced minimum at a 6 db lower blocker level. At these higher blocker amplitudes, where higher order products become significant, the AC and DC results stray away from the 3 db rule, as the system is optimized at one particular amplitude. This amplitude occurs at only one point along the -tone-test waveform. Output DC (dbv) and AC (dbvrms) Output DC & AC(rms) vs. Blocker Input Level MHz Desired Output from -99dBm Signal (dbv rms) DC IM Limit (-9d Bc) Blocker Level (dbm ) at LNA Si ngl e to ne Two Tone - 5K Two Tone - 30K 073A 6_0690 Figure 6. Measured IM Outputs 073A Conexant Proprietary 5 July, 00 Preliminary Data Subject to Change

6 Two-Tone vs. Single-Tone Measurement CX IIP vs. Blocker Input Level MHz 70.0 IIP ( dbm ) at LNA Practical IIP Reqmt for -9dBc IM AM Suppr. Sp ec Point Blocker Level (dbm ) at LNA Si ngl e t one Two Tone - 5 K Two Tone - 30 K 073A 7_0690 Figure 7. Calculated IP In the IIP plot, the IP as calculated from the -tone test reaches a very high peak at a blocker level of 30 dbm, corresponding to the IM minimum at the same blocker level. Again, this is because this is the point where the calibration was done. The IP calculated from the -tone test also reaches a peak, but at a 6 db lower blocker level. This can be explained because the -tone test generates an IM peak voltage that is equal to the -tone test s DC IM voltage, when the -tone test is done with 6 db lower blocker inputs. The best case occurs when the AC IM waveform peak reaches the point at which the system was calibrated. Extending the 9 dbc line in the IM plot, Figure 6, to where it intersects the IM curves, we find that the DC IM reaches 9 dbc at a blocker level of 7 dbm at the LNA, and the AC IM does so for blocker levels of 3.5 dbm at the LNA. With the calibration point unchanged, the tests were repeated with the receiver tuned at 869 MHz and 894 MHz, each time keeping the blocker(s) offset at 3 MHz. The results are shown in Table. They show that the IP calibration can be performed at midband with the results holding up well to the band edges. Table. Results of IP Calibration at Midband Frequency MHz 88.5 MHz MHz -Tone Blocker Input at LNA for 9 dbc DC IM -8.5 dbm -7.0 dbm -4.5 dbm -Tone Blocker Inputs at LNA for 9 dbc AC IM -3.0 dbm -3.5 dbm -9.5 dbm Conclusion Care must be taken when relating the results of -tone and - tone tests for IP. With all tones applied being equal in amplitude, the -tone test produces an AC RMS voltage that is 3 db higher than the DC voltage produced by the -tone test. This must be taken into account consistently when using a -tone test to determine both IP requirements and measured performance. The IP compensation mechanism of the Conexant CX7407 suppresses the AC and DC IM products equally well for blocker amplitudes up to about 34 dbm at the LNA input (when calibrated using a 30 dbm single tone). At higher blocker amplitudes, the AC and DC results stray away from the 3 db rule, as the IP compensation begins dealing with non-linearity beyond nd -order. Nonetheless, the CX7407 passes the GSM AM-suppression test with significant margin, and still passes even when a second blocker is added at the same amplitude. 6 Conexant Proprietary 073A Preliminary Data Subject to Change July, 00

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