Download Two-tone vs. Single-tone Measurement of 2nd

Application Note
Two-Tone vs. Single-Tone Measurement of 2nd-Order Non-linearity and
IP2 Performance of Direct Conversion Receivers
This application note describes how to find the Second Order
Intercept Point (IP2) from 1-tone and 2-tone tests of Direct
Conversion Receivers (DCRs). It also presents measurement
results for the GSM900 receive (RX) path, as used in the AMPS
band, for Skyworks first-generation of DCRs.
Relation Between 2-Tone and 1-Tone Tests for 2ndOrder Non-linearity
Second order non-linearity is an important phenomenon in
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
DC offset, both with single tone or two tones. A two-tone
condition is illustrated in Figure 1 and Figure 2.
Two input tones (f1, f2)
f1
f2
The relationship between f1, f2 and f3, f4 is:
f 3 = f 2 − f1
Likewise for f4:
f 4 = f 2 + f1
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) = k1 vi(t) + k2 vi2(t) + k3 vi3(t) + k4 vi4(t) + k5 vi5(t) +
(1)
For a two-tone case then:
Vi(t) = A Cos(ω1t) + B Cos(ω2t)
(2)
Inserting (2) into (1) and using the well-known trigonometric
equalities:
Figure 1. Two-Tone Condition: Input
f1
f3
f2
f4
f3 = f2 - f1
f4 = f2 + f1
Figure 2. Two-Tone Condition: Output
Application Note
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Two-Tone vs. Single-Tone Measurement
1
1
3
3
5
15
15
Vo(t) = 2 k2 A2 + 2 k2 B2 + { k1 A + 4 k3 A3 + 2 k3 AB2 + 8 k5 A5 + 4 k5 A3B2 + 8 k5 AB4 } Cos(ω1t) +
3
3
5
15
15
{ k1 B + 4 k3 B3 + 2 k3 A2B + 8 k5 B5 + 4 k5 A2B3 + 8 k5 A4B} Cos(ω2t) +
1
3
1
1
3
1
{ 2 k2 A2 + 2 k4 A4 + 2 k4 A2 B2} Cos(2ω1t) + { 2 k2 B2 + 2 k4 B4 + 2 k4 A2 B2} Cos(2ω2t) +
3
3
3
3
{k2 AB + 2 k4 A3B + 2 k4 AB3} Cos((ω1 + ω2) t ) + {k2 AB + 2 k4 A3B + 2 k4 AB3} Cos((ω2 - ω1) t ) +
5
5
1
5
5
1
{4 k3A3 + 16 k5A5 + 4 k5A3B2} Cos(3ω1t) + {4 k3B3 + 16 k5B5 + 4 k5A2B3} Cos(3ω2t) +
5
15
3
{ 4 k3 A2B + 4 k5A4B + 8 k5 A2B3} Cos((2ω1 ± ω2) t ) +
5
15
3
{ 4 k3 AB2 + 4 k5AB4 + 8 k5 A3B2} Cos((ω1 ± 2ω2) t ) +
1
3
1
3
3
2 2
2 k4 A B Cos((3ω1 ± ω2) t ) + 2 k4 AB Cos((ω1 ± 3ω2) t ) + 4 k4 A B Cos((2ω1 + 2ω2) t ) +
1 4
1 4
k
4A Cos(4ω1t) + k4B Cos(4ω2t)
8
8
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 are:
1
1
k 2 A2 + k 2 B 2
2
2
The difference term is:
3
3

3
3
k 2 AB + k 4 A B + k 4 AB Cos ((ω 2 − ω 1 )t )
2
2


Under a single tone condition, B is set to zero and the DC
1
created is k 2 A 2
2
For a 2-tone test and only 2nd-order non-linearity, the higher
order terms are ignored and coefficient of the output is then
k 2 AB . This coefficient is the peak amplitude of the difference
frequency output. In calculating IP2, we are concerned with
1
k 2 AB .
power, so we want to know the RMS output; it is
2
2
(3)
The ratio of the output of the 1-tone test to the output of the 2tone test is then:
1
k2 A2
2
R=
1
k 2 AB
2
(4)
The numerator is the DC output voltage from the 1-tone test,
and the denominator is the ACRMS voltage from the 2-tone test.
Remember that in the 1-tone test, the input amplitude is A, while
in the 2-tone test the amplitudes are A and B. Assuming equal
tones for the 2-tone test (B = A), then
R=
1
2
, or –3 dB.
This simply says that the root mean square (RMS) AC signal
created from a 2-tone test is 3 dB higher than the DC offset
created from a 1-tone test, when the 1-tone test uses
the same amplitude as one of the two tones. In other words:
DC voltage (1-tone) = AC RMS voltage (2-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
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Two-Tone vs. Single-Tone Measurement
Output DC ( dBv) and AC (dBvrm s )
Output DC and AC(r ms) vs. Blocker Input Level
0.0
- 10.0
- 20.0
- 30.0
- 40.0
- 50.0
- 46
- 44
- 42
-40
Blocke r Le ve l (dBm )
- 38
- 36
-34
-32
Single tone
-30
Tw o to ne
Figure 3. Output DC and AC(rms) vs. Blocker Input Level
This classical IP2 equation has a resemblance to the much
more often used IP3 equation for a 2-tone test, which is:
2(IP3) = 3(Pin) – IM3
Pout
IP2
IP2 Calculation From 2-Tone and 1-Tone
Measurements
By convention, in any 2-tone test (including Third-Order
Intercept Point (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 IM2 power at the difference frequency
f2 - f1, rather than the sum of powers at both f2 - f1 and f2 + f1.
From the slopes in Figure 4, we can see that IP2 can be
calculated from one set of measurements as:
IP2 = 2(Pin) – IM2
where all quantities are in dBm.
101731E
July 31, 2006
o rd
er
p
Pin
2nd
In Figure 4, we plot the fundamental and 2nd-order output
powers vs. input power for a generic system that has some 2ndorder non-linearity. In Figure 4, the gain is normalized to unity so
Input IP2 = Output IP2 (IIP2 = OIP2). In practice, we usually
refer all quantities to the input.
Fu
nd
am
rod
en
uct
ta
l
2:1
slo 1:1
p e sl
op
e
2nd 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 Second Order Intermodulation Product
(IM2) output in dBVDC to a 60 kHz IM2 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 1.4 dB at 60 kHz. Therefore, the 60 kHz output
should be 1.6 dB higher than the DC. This is indeed the case
over most of the tested range.
IM2
IP2
Pin
Figure 4. Fundamental and 2nd Order
Output Powers vs. Input Power
The next question is how to correctly calculate IP2 when a
1-tone test is done and the IM2 product is DC. The answer must
be the same as that found in the 2-tone test.
First, we must choose the conventions for the 1-tone test. We
calculate IP2 based on the power of the single applied tone
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Two-Tone vs. Single-Tone Measurement
first add 3 dB to the measurement of the DC IM2 product. Then
the result will match that of the 2-tone test.
(even though this is all the power applied to the system, unlike
what is done in the 2-tone calculation), and on the DC output
power (where we ignore the other, higher frequency product,
which is at 2f1).
Therefore, the correct IP2 equation for a 1-tone test, where Pin is
the power of the single input tone and IM2 is the power of the
DC output, follows:
In the 2-tone test, as shown before, inputs of ACos (ω 1t ) and
ACos (ω 2 t ) result in an IM2 product of
(
IP2 = 2(Pin) – (IM2DC + 3 dB)
)
k 2 A 2 Cos (ω 2 − ω 1 ) t . The peak voltage of one input is A,
while the RMS =
1
2
Using the IM2 data presented in the earlier graph, we calculate
the IP2 according to IP2 = 2(Pin) – IM2ACRMS for the 2-tone test,
and IP2 = 2(Pin) – (IM2DC + 3 dB) for the 1-tone test. The results
are plotted in Figure 5. We find the calculated IP2 values to
generally agree.
A . Likewise, the peak voltage of the IM2
output is k 2 A 2 while the RMS is
1
k 2 A 2 . The IP2
2
calculation from the 2-tone test is then IP22-tone = 2(Pin) – IM2
IP22-tone = 2(10log(
1 2
1
A )) − 10 log( k 22 A 4 )
2
2
(5)
In the 1-tone test, the single input is ACos(ω 1t ) , resulting in a
DC IM2 output voltage of
while the RMS =
1
k2 A2 . The peak input voltage is A,
2
For consistency, we must use the same equation to determine
the IP2 requirement itself. The GSM AM suppression
specification (from GSM 05.05) sets the IP2 requirement for a
DCR. For the GSM900 band, the single blocker applied is
–31 dBm, while the desired signal is at –99 dBm. In order to
keep the DC product below –9 dBc, the IIP2 must be:
IIP2REQ = 2(Pin) – (IM2DC + 3 dB)
1
where the IM2 level is referred to the antenna
A . If we were to (recklessly) apply the
2
classical IP2 calculation with these quantities, we would obtain
IIP2 REQ = 2(-31 dBm) – ((-99 dBm-9 dBm) + 3 dB)
IIP2 900MHz REQ = +43 dBm at the antenna
IP21-tone = 2(Pin) – IM2
IP21-tone = 2(10log(
(7)
1 2
1
A )) − 10 log( k 22 A4 )
2
4
(6)
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
1-tone test to calculate IP2 using IP2 = 2(Pin) – IM2, we must
Then, in a receiver with 3 dB of front end loss due to switches
and filters, the IIP2 requirement at the LNA input becomes:
IIP2 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 IIP2 requirements at the LNA inputs are + 42 dBm and
+ 44 dBm, respectively.
IIP2 vs . Bloc ke r Input Leve l
40.0
IIP2 ( dBm )
35.0
30.0
25.0
20.0
- 46
-44
- 42
Blo ck e r L e ve l (d Bm )
-40
- 38
-3 6
-34
Sin gle ton e
-3 2
-30
Tw o tone
Figure 5. Calculated IP2 Values for 1-Tone and 2-Tone Tests
4
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Two-Tone vs. Single-Tone Measurement
Skyworks IP2-Compensation Circuit Calibration Using
2 Tones or 1 Tone
Skyworks DCRs implement an IP2-compensation
circuit that, once calibrated at a single blocker amplitude,
compensates for 2nd-order non-linearity at all amplitudes, until
the system approaches compression. It relies on the nonlinearity, maintaining a 2nd-order characteristic, an assumption,
which obviously breaks down at compression.
The IP2 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 2nd-order products. It
suppresses both the DC IM2 due to a single blocker frequency,
and the AC IM2 due to two blocker frequencies. This also makes
it suitable for suppressing IM2 due to amplitude-modulated
blockers.
Here we show the IP2 performance for both 1-tone
and 2-tone blocker inputs. The IP2 compensation circuit of the
900MHz RX path was calibrated at the center of the
AMPS band, that is, 881.5 MHz, using a single blocker
frequency of –30 dBm at the LNA input, at +3 MHz offset. After
calibration, the IP2 at this point measured +71.8 dBm.
Then the IM2 products were measured over a range of blocker
levels from –46 dBm to –24 dBm. This was done for both 1-tone
and 2-tone cases; for example, 1 tone at –46 dBm, vs. 2 tones
at –46 dBm each. The 2-tone test was repeated with tone
separations of 25 kHz and 30 kHz, set 3 MHz from the receiver
channel. Figure 6 and Figure 7 show the measured IM2 outputs
and the calculated IP2.
In the IM2 output plot, limit lines are included that show the
receiver’s desired output in dBVrms (due to a –99 dBm signal at
antenna, or –102 dBm at LNA) and a maximum DC IM2 at
–9 dBc. The specification line stops at –34 dBm since we
assume 3 dB of front-end loss, while the GSM 05.05 AMsuppression test uses a –31 dBm blocker at the antenna.
Neither the DC nor AC IM2 products violate the –9 dBc limit.
In the IIP2 plot, the fixed –9 dBc IM2 limit is translated to an IIP2
limit that scales with the blocker amplitude, reaching the
previously derived value of +40 dBm at the GSM 05.05
AM-suppression-test point. The IIP2 calculated at the LNA input,
whether from the 1-tone or 2-tone test, stays well above the
limit.
The particular shapes of the curves should be noted. In the IM2
plot, at lower blocker levels, the 2-tone IM2 product is still
roughly 3 dB higher than the 1-tone DC product, and these
products are both very small and nearly constant due to the
action of the IP2 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 IM2 curve at a blocker level of –30 dBm, precisely
because this is the point where the system was calibrated. The
2-tone IM2 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 2-tone-test waveform.
Output DC & AC(rm s) vs. Blocker Input Level - 881.5MHz
Out put DC (dBv) and AC ( dBv rm s )
0.0
-10.0
-20.0
Desired Output f rom - 99dBm Signal ( dBV rms)
-30.0
DC IM 2 Limit (-9d Bc
-40.0
-50.0
-60.0
-70.0
-46
-42
-38
Block e r Le ve l (dBm ) at LNA
-34
-3 0
- 26
Single to ne
Two Tone - 25K
Two Tone - 30K
Figure 6. Measured IM2 Outputs
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5
Two-Tone vs. Single-Tone Measurement
IIP2 vs. Blocker Input Level - 881.5MHz
80.0
IIP2 (dBm ) at LNA
70.0
60.0
50.0
mt
Re q
IIP2 M 2
l
a
I
tic
Pr ac r - 9dBc
fo
40.0
AM Supp r.
Sp ec Point
30.0
20.0
10.0
-46
-42
- 38
-34
-30
- 26
Single t one
Two Tone - 25 K
Two Tone - 30 K
Block e r Le ve l (dBm ) a t LNA
Figure 7. Calculated IP2
In the IIP2 plot, the IP2 as calculated from the 1-tone test
reaches a very high peak at a blocker level of –30 dBm,
corresponding to the IM2 minimum at the same blocker level.
Again, this is because this is the point where the calibration was
done. The IP2 calculated from the 2-tone test also reaches a
peak, but at a 6 dB lower blocker level. This can be explained
because the 2-tone test generates an IM2 peak voltage that is
equal to the 1-tone test’s DC IM2 voltage, when the 2-tone test
is done with 6 dB lower blocker inputs. The best case occurs
when the AC IM2 waveform peak reaches the point at which the
system was calibrated.
Extending the –9 dBc line in the IM2 plot, Figure 6, to where it
intersects the IM2 curves, we find that the DC IM2 reaches –9
dBc at a blocker level of –27 dBm at the LNA, and the AC IM2
does so for blocker levels of –31.5 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 1.
They show that the IP2 calibration can be performed at midband
with the results holding up well to the band edges.
Table 1. Results of IP2 Calibration at Midband
Frequency
869.0 MHz
881.5 MHz
894.0 MHz
1-Tone Blocker Input at LNA for –9 dBc DC IM2
-28.5 dBm
-27.0 dBm
-24.5 dBm
2-Tone Blocker Inputs at LNA for –9 dBc AC IM2
-32.0 dBm
-31.5 dBm
-29.5 dBm
Conclusion
Care must be taken when relating the results of 2-tone and
1-tone tests for IP2. With all tones applied being equal in
amplitude, the 2-tone test produces an AC RMS voltage that is
3 dB higher than the DC voltage produced by the 1-tone test.
This must be taken into account consistently when using a
1-tone test to determine both IP2 requirements and measured
performance.
6
The IP2 compensation mechanism in Skyworks 1st-generation
DCRs suppresses the AC and DC IM2 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 IP2 compensation begins dealing with non-linearity
beyond 2nd-order. Nonetheless, the DCRs pass the
GSM 05.05 AM-suppression test with significant margin, and
still pass even when a second blocker is added at the same
amplitude.
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101731E
July 31, 2006
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