Date: Thu, 28 Mar 2024 19:15:30 +0000 (GMT)
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Introduction
Calculating the total power when comb=
ining two or more CW signals is often associated with the analysis of a bal=
anced amplifier as shown here:
=
In this example the two amplifiers ha=
ve equal gains of 10 dB. Assuming that the input splitter and output combin=
er are lossless and with an input signal with 0 dBm power, the output power=
is +10 dBm (See the blue annotations in the system diagram).
A dilemma occurs when the amplifiers have different gains. Shown here, &=
nbsp;the lower amplifier has a gain of -100 dB:
The resulting power at the output shows a 6 dB drop in power when compar=
ed to the situation where the amplifier gains are equal. Intuition states t=
hat there should only be a 3 dB drop in power. This intuition comes from th=
e thinking that two equal power signals when combined have a total combined=
power that is twice as high (or 3.01 dB) as the power of one of the signal=
s.
This dilemma in the total power of two combined signals is a result of o=
verlooking whether the signals are combining in a coherent fashion or wheth=
er the signals are combining in a non-coherent fashion. This article explai=
ns the difference between coherent and non-coherent signal combination and =
how to calculate the resulting power after combining these two classes of s=
ignals.
Average Power C=
alculation
First, a review of average power calculation single tone and multi-tone =
signals. For a single tone signal, consider a time domain voltage signal, <=
em>v(t), placed across a resistor, R. The instantaneous power dissipat=
ed in the resistor is defined as(1):
For a sinusoidal voltage signal, the average power is defined as:
where T is the duration of one period of the sinusoidal signal. However,=
for more complicated signals, such as adding two sinusoids at different fr=
equencies, the more general calculation for average power must be used=
(1):
Here is a worked example of a single sinusoidal signal:
Using the computer to perform this calculation gives the same results:=
p>
The average power, Pave, shown in the green legend is a computed value u=
sing numerical integration on the displayed waveform.
Signal Addition=
Signals at Different Frequencies
When combining signals at two different frequencies, these two signals a=
dd in a non-coherent fashion where the powers of the individual signals add. Signals that combine in a non-coher=
ent fashion are considered uncorrelated. In the following example, two sign=
als each with 10 mW average power are added resulting in a combined power o=
f 20 mW (13.01dBm):
Independent noise sources being random also combine in a non-coherent fa=
shion where the powers of the independent =
noise signals add.
A chart of uncorrelated signal addition of signals is shown here:
The x-axis is the power difference in dB of the two individual signals. =
The y-axis is the power difference of the combined signal above the power o=
f the larger of the two individual signals.
Two equal power signals adding in a non-coherent fashion results in 3.01=
dB power increase in the combined signal. For unequal amplitude signals wh=
ere the frequencies of the two signals are not the same, first convert powe=
r expressed on a log scale to power expressed on a linear scale. Next, add =
the linear powers and then convert to power expressed on a log scale.
Here is a worked example of combining two unequal amplitude signals when=
the frequencies of the signals are not the same. Consider one signal with =
a power level of 0 dBm and the other signal that has a power of -10 dBm, th=
e combined signal power calculation is:
Signal 1: Power =3D 0 dBm. Convert =
to mW, Power =3D 10(0/10) =3D 1 mW
Signal 2: Power =3D -10 dBm. Conver=
t to mW, Power =3D 10(-10/10) =3D 0.1 mW
Combined signal power =3D 1 mW + 0.1 mW =
=3D 1.1 mW. Convert to dBm, Power =3D 10*log10(1.1) =3D 0.41 dBm=
Signals at the Same Frequency
When combining signals where the frequencies of the signals are the same=
, then vector addition of the individual signal voltages is required before=
computing the power of the combined signal. Signals at the same frequency =
are considered to be correlated and the vector addition on the individual s=
ignals is termed coherent signal combination.
First, represent each signal in phasor form:
Add the voltages of individual signals and then compute the average powe=
r. Peak voltage is computed from
Worked example:
Let signal 1 have power of 0 dBm with pha=
se of 0 deg
Let signal 2 have power of -10 dBm with p=
hase of 30 deg
Signal 1:
Phasor Notation: 0.316*cos(0) +j*0.316*si=
n(0) =3D 0.3162 + j0
Signal 2:
Phasor Notation: 0.1*cos(30) +j*0.1*sin(3=
0) =3D 0.0866 + j0.05
Combined signal:
Using the computer simulation of this example results in the same combin=
ed signal average power:
Here are two charts showing the addition of two signals that are at the =
same frequency:
The left chart shows the results when the two signals have the same phas=
e, but their amplitudes are different. The x-axis is the amplitude differen=
ce between the two signals and the y-axis is the combined signal power diff=
erence above the higher power input signal. For instance if one signal is 0=
dBm and the other is -10 dBm, the combined power is 2.39 dBm.
The right chart shows the results when the two signals have the same amp=
litude, but are at different phases. The x-axis is the phase difference bet=
ween the two signals and the y-axis is the combined signal power difference=
above the higher power signal.
From the charts, if the two input signals are at the same frequency with=
equal amplitudes and phases, then the combined output power is 6.02 dB abo=
ve the power of each individual signal. The computer simulations confirm th=
is result:
Power =
Addition Using a Power Combiner
When using power combiners, the gain of the power combiner needs to be a=
ccounted for. For lossless power combiners, the gain:
In this example, two 0 dBm signals at the same frequency and phase are c=
ombined. The two powers add to be 6.02 dBm followed by the -3.01 dB gain of=
the combiner, the resulting combiner output power is 3.01 dBm:
For a 3-way combiner, the combiner gain is -4.77 dB. Adding three 0 dBm =
signals that are at the same frequency and phase results in a combined powe=
r of 9.54 dBm. With the -4.77 dB gain of the combiner, the combiner=E2=80=
=99s output power is 4.77 dBm:
For two signals at the same frequency, one with 0 dBm and the other with=
-10 dBm, the combined power is 2.39 dBm. With combiner gain of -3.01 dB, t=
he final output power is -0.62 dBm:
Balance=
d Amplifier Signal and Noise
As mentioned above, balanced amplifiers are an example of an application=
that is dependent upon coherent combination of signals. And as it turns ou=
t, this application is also dependent on non-coherent combination of noise.=
Shown is a typical balanced amplifier showing signal power annotations:
This example assumes that the splitter and combiner are lossless and so =
their gains are -3.01 dB. The phasing of the hybrid splitter and combiner i=
s such that the signals at the hybrid combiner add in-phase.
The following plot shows the signal power (blue trace) and noise power d=
ensity (red trace) at each stage of the balanced amplifier
Following the signal power through the chain, the signal power at the ou=
tputs of the two amplifiers is +7 dBm. Since these signal combine in-phase =
and they are at the same frequency and power, the signals add 6 dB (coheren=
t power addition) and with a -3 dB gain of the output combiner the final ou=
tput power is:
+7 dBm signal power at each amplifier output + 6 dB due to coheren=
t power addition -3 dB due to combiner gain =3D +10 dBm
Noise generated in each amplifier, however, is not coherent. In other wo=
rds, noise power of the two amplifiers is uncorrelated with each other. As =
such, the noise powers in the output combiner add 3 dB. The noise power at =
the output of each amplifier is -161 dBm/Hz. The two noise powers add in a =
non-coherent fashion for a combined power of -158 dBm/Hz. With the output c=
ombiner gain of -3 dB, the noise power at the output is -161 dBm/Hz
(1) Simon Haykin, (1983). =E2=80=9DCommunication Systems=E2=
=80=9D
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