The
receiver's intermodulation spurious response attenuation is a measure
of the receiver's ability to receive a modulated input RF signal
frequency on its assigned channel frequency in the presence of two
interfering CW tones. These tones are separated from the assigned input
signal frequency such that the nth-order mixing of the two undesired
signals can occur in the nonlinear elements of the receiver, producing
a third signal in the band of the desired signal. The receiver's
protection against spurious response interference is a measure of the
receiver's ability to discriminate between the input signal at the
assigned frequency and an undesired signal at any other frequency to
which it is responsive.
The 2nd- and
3rd-order intercept points of a receiver represent the two most
important linearity specifications of a certain radio circuit or
system. They allow us to predict the receiver's intermodulation (IM)
performance, which describes the radio's susceptibility to interference
for adjacent or nearby users. In this article, a modification of the
traditional receiver intercept point (IP) cascading equation is
presented for both the 2nd- and 3rd-order IM cases. The mathematical
derivation for each of the 2nd-order (IP2) and 3rd-order (IP3)
intercept point cascading equations incorporates the effect of adding
selectivity (S) between receiver stages in order to improve the overall
2nd- and 3rd-order input intercept points IIP2 and IIP3, respectively.
Note: In this article, all variables in uppercase letters refer to dB
or dBm units and all variables in lowercase letters refer to linear
units.
In wireless specifications related to the recommended minimum
performance standards for the mobile station, the receiver's
intermodulation (IM) performance is specified technically under two
main topics: the receiver's intermodulation spurious response
attenuation and the receiver's protection against spurious response
interference. The receiver's intermodulation spurious response
attenuation is a measure of the receiver's ability to receive a
modulated input RF signal frequency on its assigned channel frequency
in the presence of two interfering CW tones. These tones are separated
from the assigned input signal frequency such that the nth-order mixing
of the two undesired signals can occur in the nonlinear elements of the
receiver, producing a third signal in the band of the desired signal.
The receiver's protection against spurious response interference is a
measure of the receiver's ability to discriminate between the input
signal at the assigned frequency and an undesired signal at any other
frequency to which it is responsive.
Interference Due to 3rd-Order IM Products
As a result of 3rd-order mixing in a receiver front end, two off-channel CW tones at frequencies f1 and f2 introduce a 3rd-order intermodulation product at a frequency equal to (2f1 - f2), which can fall in the on-channel signal passband (Figure 1a).
This in-band 3rd-order intermodulation (IM3) product reduces the
carrier-to-interference ratio (C/I) at the receiver's demodulator.
Based on a linear 3:1 slope (Figure 1b), the level of this input
IM3 product (IIM3, in dBm) can be determined using an equation that
includes the receiver overall input IP3 (IIP3, in dBm) and the input
power level of each of the two off-channel CW tones (PI, in dBm) [1]:
Figure
1. IM3 product as an in-band interferer due to two off-channel CW tones
(a) and 3rd-order intercept point (IP) definition (b).
Figure 2 represents conventional dual-conversion superheterodyne
receiver architecture. In this type of receiver architecture, the IM3
products due to the off-channel CW interferers are generated in the
low-noise amplifier (LNA), the first mixer, the IF amplifier, the
second mixer, and the IF limiting amplifier. All the IM3 products add
up at the demodulator input, resulting in an equivalent in-band IM3
product at the receiver input (IIM3). This IM3 product, acting as an
in-band interferer, can be reduced by minimizing the 3rd-order IM
contribution from the IF amplifier, the second mixer, and the IF
limiting amplifier. This is done by introducing a certain amount of IF
selectivity (S) to these off-channel interferers in the IF filter
following the first mixer (IF filter #1). Note that filter selectivity
(S) refers to the attenuation in the IF filter #1 rejection band at the
off-channel interferers' frequencies relative to the on-channel filter
passband insertion loss (IL). Thus, the total rejection (R) (in dB) in
the IF filter stopband at the off-channel CW tones' frequencies can be
defined as R = -(IL + S). The IF filter selectivity (S) reduces the
3rd-order distortion and dynamic range requirements of the subsequent
receiver blocks. As a result, the overall receiver's IIP3 can be
optimized in order to minimize the equivalent in-band IIM3 product and
thus meet the required receiver baseband (C/I) ratio.
Figure 2. Conventional superheterodyne dual-conversion receiver.
Modified 3rd-Order Input Intercept Point (IIP3) Cascading Equation
In Figure 3,
the dual-conversion receiver architecture, shown in Figure 2, is
divided into three blocks: the RF block, the IF filter #1, and the IF
block. The RF block, or block #1, consists of the receiver RF stages
preceding the first IF filter. The IF block, or block #2, consists of
the receiver's IF stages following the first IF filter. Block #1 has a
gain G1 at RF and an equivalent 3rd-order input intercept point IIP31. Block #2 has a gain G2 at IF and an equivalent 3rd-order input intercept point IIP32. The power level of each of the two off-channel CW interferers present at the receiver input is assumed to be equal to PI. P1 is the level of the two off-channel CW tones' input to block #1. P2
is the level of the two off-channel CW tones translated to IF and input
to block #2. IIM3 is the total IM3 distortion power due to the two
off-channel CW tones, referred to the receiver input. IIM31 is the total IM3 distortion power generated in block #1, referred to its input. IIM32 is the total IM3 distortion power generated in block #2, referred to its input.
Figure
3. Block diagram for deriving the modified IP3 cascading equation,
which incorporates the effect of adding selectivity (S) to receiver
stages at the two off-channel CW tones' frequencies. Power symbols in
dBm, and gain in dB.
In the following derivation, the input IIM3 distortion voltage of block
#1 and the input IM3 distortion voltage of block #2, divided by the
voltage gain of preceding stages, are added exactly in-phase as a
worst-case condition to get the total IM3 distortion voltage, referred
to the receiver input. Assuming a system characteristic impedance of 1, we can write the following:
where the square root converts the IM3 powers to voltages. The terms iim3, iim31, and iim32 are in linear power units (watts or milliwatts). Also, G1 (dB) = 10.log10(g1) and IL (dB) = 10.log10(il).
Equation (1) can be rearranged into the following equation:
Equation (3) defines the input IP3 for the whole receiver.
Equation (3) can be rewritten in linear power units (milliwatts, or mW),
instead of dBm:
Similar to what we did in equation (3), we can define IIP31
and IIP32 for block #1 and block #2, respectively, as:
Knowing that P1(dBm) = PI and P2(dBm) = PI +(G1-IL-S), we
obtain from equations (5) and (6):
As we have done for equation (3), equations (7) and (8)
can be rewritten in linear power units, instead of dBm, to result in equations
(9) and (10), respectively:
where S(dB) = 10.log10(s) and IL(dB) = 10.log10(il). Note
that S(dB) and IL(dB) are positive numbers.
Going back to equation (2) and dividing it by (pI)1/2,
If we substitute each term in equation (11) by equivalent
terms using equations (4), (9), and (10), and after simplifying by removing
the term pI, we obtain the following modified IIP3 cascading equation:
As we can see from equation (12), by choosing a highly selective IF
filter (s>>1), we can minimize the effect of the IF block input
IP3 (IIP32) on the overall receiver input IP3 (IIP3), which becomes almost defined by the RF block IIP3 (IIP31). It is worth noting that, in a cascaded system analysis, the IF block input IP3 (IIP32)
should be replaced with an equivalent input IP3, which incorporates the
effect of introducing selectivity ahead of the IF block. This
equivalent IIP32 can be written as:
A more generalized equation predicting the overall input
IP3 of a receiver chain with M cascaded stages can be formulized based on equation (12). Each stage has a linear gain (gn), an input IP3 (iip3n, in watts), and a selectivity (sn) at the two off-channel CW tones' frequencies, which introduce an in-band IM3 product:
where Sn(dB) = 10.log10(sn).
Note: This equation simplifies to the classic equation for intercept
point calculation of a cascade of M stages when setting sn to 1, which translates into setting selectivity Sn to 0dB [1].
Interference Due to 2rd-Order IM Products
Receiver spurious responses are frequencies that are different from the
on-channel RF signal frequency, yet that still produce an output
interferer in the receiver passband if encountered at a sufficiently
high level. One of these spurious response frequencies is at the
half-IF frequency. This half-IF spurious response results in a
2nd-order intermodulation product (IM2), which occurs in the receiver
RF front end. Its level can be predicted from the 2nd-order intercept
point (IP2) of the receiver RF front end, which is defined up to and
including the first mixer in the receiver chain (Figure 2). For
high-side injection in the first mixer (Figure 4a), a CW tone at the input of the receiver, which is offset from the LO frequency by -fIF/2, is downconverted to IF by the (-2.fCW + 2.fLO)
IM product, occurring in the first mixer [1,2]. As for low-side
injection, a CW tone, which is offset from the LO frequency by +fIF/2, is downconverted to IF by the (2.fCW - 2.fLO) IM product. Based on a linear 2:1 slope (Figure 4b),
the level of this input IM2 product (IIM2, in dBm) can be determined
using an equation that includes the receiver RF front-end input IP2
(IIP2, in dBm) and the power level of the input half-IF CW tone (PI, in dBm) [1]:
Figure
4. IM2 product as an in-band interferer due to half-IF spurious
response (a) and 2nd-order intercept point (IP) definition (b).
This in-band IM2 product due to the half-IF spurious response
can be reduced by minimizing the 2rd-order IM contribution from the
first mixer. This is done by introducing a certain amount of RF
selectivity (S) to this off-channel interferer in the RF filters
preceding the first mixer (RF filters #1 and #2). Note that filter
selectivity (S) refers to the attenuation in the RF filter rejection
band at the spurious response frequency relative to the on-channel
filter passband insertion loss (IL). The RF filter selectivity (S)
reduces the 2nd-order distortion and dynamic range requirements of the
first mixer. As a result, the overall receiver's RF front-end IIP2 can
be optimized in order to minimize the equivalent in-band IIM2 product
due to the half-IF tone, thus meeting the required receiver baseband
(C/I) ratio.
Modified 2rd-Order Input Intercept Point (IIP2) Cascading Equation
In Figure 5, the dual-conversion receiver RF front end is
divided into three blocks: RF filter #2, block #1 (which consists of
all stages preceding RF filter #2), and block #2 (which follows RF
filter #2 and includes the first mixer). Block #1 has a gain G1 at RF and an equivalent 2nd-order input intercept point IIP21. Block #2 has a gain G2 at RF and an equivalent 2nd-order input intercept point IIP22. The power level of each half-IF CW tone present at the receiver input is assumed to be equal to PI. P1 is the level of the half-IF CW tone input to block #1. P2
is the level of the half-IF CW tone input to block #2. IIM2 is the
total IM2 distortion power due to the half-IF CW tone, referred to the
receiver input. IIM21 is the total IM2 distortion power generated in block #1, referred to its input. IIM22 is the total IM2 distortion power generated in block #2, referred to its input.
Figure
5. Block diagram for deriving the modified IP2 cascading equation,
which incorporates the effect of adding RF selectivity (S) to receiver
RF front-end stages at the half-IF spurious frequency. Power symbols in
dBm, and gain in dB.
In the following derivation, the input IM2 distortion voltage of block
#1 and the input IM2 distortion voltage of block #2, divided by the
voltage gain of preceding stages, are added exactly in-phase as a
worst-case condition to get the total IM2 distortion voltage, referred
to the receiver input. Assuming a system characteristic impedance of 1, we can write the following:
where the square root converts the IIM2 powers to voltages. The terms iim2, iim21, and iim22 are in linear power units (watts or milliwatts). Also, G1 (dB) = 10.log10(g1) and IL (dB) = 10.log10(il).
Equation (15) can be rearranged into the following equation:
Equation (17) defines the input IP2 for the whole receiver.
Equation (17) can be rewritten in linear power units (milliwatts, or mW), instead of dBm:
Similar to what we did with equation (17), we can define
IIP21 and IIP22 for block #1 and block #2, respectively, as:
Knowing that P1(dBm) = PI and P2(dBm)
= PI + (G1 - IL - S), we obtain from equations (19)
and (20):
As we have done for equation (17), equations (21) and (22)
can be rewritten in linear power units, instead of dBm, to result in equations
(23) and (24), respectively:
where S(dB) = 10.log10(s) and IL(dB) = 10.log10(il). Note
that S(dB) and IL(dB) are positive numbers.
Returning to equation (16) and dividing it by (pI)1/2,
If we substitute each term in equation (25) by equivalent
terms using equations (18), (23), and (24), and after simplifying by removing
the term pI, we obtain the following modified IIP2 cascading equation:
As we can see from equation (12), by choosing a highly selective RF
filter (s>>1), we can minimize the effect of the first-mixer
block input IP2 (IIP22) on the overall receiver RF front-end
input IP2 (IIP2). It is worth noting that, in a cascaded system
analysis, the first-mixer block input IP2 (IIP22) should be
replaced with an equivalent input IP2, which incorporates the effect of
introducing selectivity in the RF filter. This equivalent IIP22 can be written as:
A more generalized equation to predict the overall input IP2 of a
receiver's RF front-end chain with M cascaded stages can be formulized
based on equation (26). Each stage has a linear gain (gn), an input IP2 (iip2n, in watts), and a selectivity (sn) at the half-IF CW tone frequency, which introduces an in-band IM2 product:
where Sn(dB) = 10.log10(sn).
References
- S. Maas, Microwave Mixers, Norwood, MA; Artech House, 1993.
- P. Vizmuller, RF Design Guide, Norwood, MA; Artech House, 1995.
A similar version of this article appeared in the December 1997 issue of RF Design.
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APP 749: May 17, 2001
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