Receiver Intermodulation and Dynamic Range
On the opposite end from small signals (i.e., those near the noise
floor), strongly received signals can also limit the performance
of receivers. Isn’t this statement paradoxical? On the contrary: as we’ll see in this lecture, two strong signals that are near to each other in frequency can create spurious receiver outputs.
These new spurs are produced when unintentional mixing occurs
in a receiver amplifier or mixer. This unintentional mixing
happens in the receiver “front end” (e.g., RF pre-amplifier
and/or RF mixer) when:
1. Two or more signals are received that are close together in
frequency, and
2. One or more of these signals is so strong that circuit
components in the receiver front end no longer behave as
intended.
Examples of this latter situation include large input signals that
drive the semiconductor devices in a mixer into nonlinear
behavior leading to unintended mixing, or a small-signal
amplifier driven into nonlinear behavior so that it is no longer a
linear amplifier.
As discussed next, it is possible in such a circumstance that
unintended audio output signals will be produced, in addition to
the intended signal. This is, by definition, a spurious output, or
spur. These particular spurs are called intermodulation products.
It is important to realize that these spurs are different from those
considered earlier in this course (i.e., image frequencies and
mixer products). Mathematics of Intermodulation Products
To understand the origin of intermodulation products (IPs),
consider the Taylor series expansion of a “weakly nonlinear”
output voltage

The IPs occur when V_{i} is the sum of two (or more) strong
signals that are close together in frequency:
V_{i} =V cos(?_{1}t) +V_{2}cos(?_{2}t)
Let’s choose V_{1} =V_{2} =V and 1 2 ?_{1} 2 , as in the text.
We will now substitute (14.42) into (14.41) and expand each of
the HO terms. We will assume that V_{1} and V_{2} are large enough
that the HO terms in (14.41) are appreciable in size to G_{v}V_{i}.
• Second-order products:
V_{i} =[V cos(?_{1}t) +V(?_{2}t)]^{2}
Using Mathematica, we can symbolically expand and then
simplify this expression: From this result, we see that

Now, let’s imagine that we have two strong signals at
f _{1}= 7.030 MHz and f_{2} = 7.040 MHz
as in Prob. 35. IP spurs should then be located at
2f_{1} = 14.060 MHz
2f_{2} = 14.080 MHz,
|f_{1} – f_{2}| = 0.010 MHz
f_{1} + f_{2} = 14.070 MHz
All of these IP spurs are located far from the intended RF
signals and would be well attenuated by the RF Filter.
• Third-order products:
V_{i}^{3} =[V cos(?_{1} t) +Vcos(?_{2}t)] ^{3}
Again using Mathematica:
Using f_{1} and f_{2} defined above, then the IP spurs are located at:
f_{1} = 7.030 MHz (actually one fundamental)
f_{2} = 7.040 MHz (the other fundamental)
3f_{1} = 21. 090 MHz
3f_{2} = 21.120 MHz
2f_{2} - f_{1} = 7.050 MHz (very near input f)
2f_{1} - f_{2} = 7.020 MHz (again, very near input f)
2f_{2} + f_{1} = 21.110 MHz
2f_{1} + f_{2} = 21.100 MHz
The two spurs near the intended operational frequencies of the
receiver are defined in the text as
f _{3}= 2 f _{1}? f _{2}
and
f_{3} = 2 f_{2} ? f _{1}
These two particular third-order IPs are often troublesome
because they can be close in value to the frequencies that
created them (when f_{1} ? f_{2} is small). If f_{1} and f_{2} fall within the
passband of the RF Filter, then it is conceivable offending IPs
at f_{3} and/or f_{3} could pass through the IF Filter. In such a situation, these IPs cannot be filtered out by the receiver and spurious outputs will occur.
• Fourth-order products:
V _{i}^{4}= [Vcos(?_{1}t) +Vcos(?_{}t)] ^{4}
Again using Mathematica:
Using f_{1} and f_{2} defined above, these IP spurs are located at:
2f_{1} = 14.060 MHz
2f_{2} = 14.080 MHz
4f_{1} = 28.120 MHz
4f_{2} = 28.160 MHz
3f_{2} - f_{1} = 14.090 MHz
3f_{1} - f_{2} = 14.050 MHz
3f_{2} + f_{1} = 28.150 MHz
3f_{1} + f_{2} = 28.130 MHz
|2f_{1} - 2f_{2}| = 0.020 MHz
2f_{1} + f_{2} = 21.110 MHz
|f_{1} – f_{2}| = 0.010 MHz
f_{1} + f_{2} = 14.070 MHz
None of these IPs are close to the receiver’s input bandwidth,
so they are easily filtered out by the RF Filter.
• Fifth-order products:
V_{i}^{5} = [Vcos(?_{1} t) +Vcos(?_{2}t)] ^{5}
You will determine these IP frequencies in Prob. 35. As stated
in the text, the two fifth-order IP spurs that can cause trouble
in the NorCal 40A are
f _{5}= 3 f_{1} ? 2 _{2}f
and
f _{5}= 3 f_{2} ? 2 f_{1}
In the example here using the f_{1}and f_{2} specified above, then
f _{5}= 7.010 MHz and f_{5} = 7.060 MHz
These two spurs are within the passband of the receivertoo
close to the input signal frequency to be filtered out by the RF
Filter. Together with f_{3} and f _{3}, this is more bad news!
• Higher-order products:
No other IP spurs are close to the input frequency, or
generally do not have appreciable signal level.
Lastly, IP spurs are always present in a receiver. However, only
when the input signals are sufficiently strong do the IPs rise
above the noise floor and, perhaps, become large enough to
cause audio output. Are IP Spurs Really a Problem?
Intermodulation product spurs can be a real problem in a
receiver. This is probably truer today than it was 30 years ago.
There are two primary reasons:
1. Solid-state amplifiers are easier to drive into nonlinear
behavior than tube amplifiers. From (14.41), we see that
IP spurs are due to nonlinear behavior.
2. There are more RF signals today such as wireless PCS,
radio stations, microwave datalinks, etc. If you are too
close to a transmitter, your receiver “front end” may be
driven to nonlinearity.
If you were concerned with jamming an adversary’s radio or
radar, perhaps you could take advantage of IP spurs. How? Effects of Intermodulation Products
The effects of intermodulation products are illustrated below in
Fig. 14.9. The slope of the “signal” and “intermodulation” plots
are approximately equal to the order of the dominant IP.

To understand this last statement, first note that the average
signal power is expressed as
Next from

and using (1) we can deduce that for a particular IP of order n:
V_{nrms} ?V_{irms}^{n}
This important fact can be verified from (14.43) in the text
where V ?V _{2} , from (2) above where V ?V_{3} , etc. Consequently, substituting (6) into (5) we find

For a log-log plot of this output power versus P_{i}, we need to rearrange (7) so that P_{i} is explicitly present:

Simplifying gives

This is, of course, an equation for a straight line. The slope of
this P versus P_{i} curve equals n, which is the order of the IP. Hence, we have proven the conjecture. Dynamic Range
The minimum detectible intermodulation input (MDI) is the
input signal that produces a total output signal = 2P_{n} (signal + noise) for the dominant IP.
Then, by definition
Dynamic range = MDI – MDS [W]
and is illustrated above in
Dynamic range is:
?? The range of useful input signal power levels for a receiver.
?? Limited by noise for small signals and by receiver frontend
nonlinearities for large signals.
Good receivers have a dynamic range ?100 dB, or so. Effects of Antenna Noise on Dynamic Range
Antenna noise can have a marked effect on dynamic range. In
the NorCal 40A, the antenna noise is approximately 30 dB
above the receiver noise:

From this plot, with:
?? the slope of the signal curve equal to 1:1 (linear power
amplification), and
?? the slope of the IP curve equal to 3:1 (dominant third-order
IP),
then increasing the noise floor by 30 dB due to the antenna noise
causes the:
1. MDS to increase by 30 dB 1/1= 30dB
2. MDI to increase by 30 dB 1 /3 =10dB
Therefore, with the antenna attached to the receiver, the
dynamic range decreases by 20 dB!
Interestingly, we can retrieve some of the dynamic range by
introducing attenuation at the front end, though you will
sacrifice receiver sensitivity and you may decrease the loudness
of the signal.For example, if we add 15 dB of attenuation at the front end of the receiver, then
1. MDS decreases by 15 dB (= 15 dB 1/1)
2. MDI decreases by 5 dB (= 15 dB 1/3)
Consequently, the dynamic range increases by 15-5 = 10 dB. In
practice, you can use your RF Gain pot to improve dynamic
range if you need to.
Of course, the best way to increase dynamic range is with a
better mixer design (lower noise, less susceptibility to IP). This
would involve a more complicated design and likely more
expense.