Calculating the total power when combining two or more CW signals is often associated with the analysis of a balanced amplifier as shown here:
In this example the two amplifiers have equal gains of 10 dB. Assuming that the input splitter and output combiner 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, the lower amplifier has a gain of -100 dB:
The resulting power at the output shows a 6 dB drop in power when compared to the situation where the amplifier gains are equal. Intuition states that there should only be a 3 dB drop in power. This intuition comes from the 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 signals.
This dilemma in the total power of two combined signals is a result of overlooking whether the signals are combining in a coherent fashion or whether the signals are combining in a non-coherent fashion. This article explains the difference between coherent and non-coherent signal combination and how to calculate the resulting power after combining these two classes of signals.
Average Power Calculation
First, a review of average power calculation single tone and multi-tone signals. For a single tone signal, consider a time domain voltage signal, v(t), placed across a resistor, R. The instantaneous power dissipated 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 frequencies, 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:
The average power, Pave, shown in the green legend is a computed value using numerical integration on the displayed waveform.
Signals at Different Frequencies
When combining signals at two different frequencies, these two signals add in a non-coherent fashion where the powers of the individual signals add. Signals that combine in a non-coherent fashion are considered uncorrelated. In the following example, two signals each with 10 mW average power are added resulting in a combined power of 20 mW (13.01dBm):
Independent noise sources being random also combine in a non-coherent fashion 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 of 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 where the frequencies of the two signals are not the same, first convert power 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, the combined signal power calculation is:
Signal 1: Power = 0 dBm. Convert to mW, Power = 10(0/10) = 1 mW
Signal 2: Power = -10 dBm. Convert to mW, Power = 10(-10/10) = 0.1 mW
Combined signal power = 1 mW + 0.1 mW = 1.1 mW. Convert to dBm, Power = 10*log10(1.1) = 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 signals is termed coherent signal combination.
First, represent each signal in phasor form:
Add the voltages of individual signals and then compute the average power. Peak voltage is computed from
Let signal 1 have power of 0 dBm with phase of 0 deg
Let signal 2 have power of -10 dBm with phase of 30 deg
Phasor Notation: 0.316*cos(0) +j*0.316*sin(0) = 0.3162 + j0
Phasor Notation: 0.1*cos(30) +j*0.1*sin(30) = 0.0866 + j0.05
Using the computer simulation of this example results in the same combined 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 phase, but their amplitudes are different. The x-axis is the amplitude difference between the two signals and the y-axis is the combined signal power difference 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 amplitude, but are at different phases. The x-axis is the phase difference between 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 above the power of each individual signal. The computer simulations confirm this result:
Power Addition Using a Power Combiner
When using power combiners, the gain of the power combiner needs to be accounted for. For lossless power combiners, the gain:
In this example, two 0 dBm signals at the same frequency and phase are combined. 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 power of 9.54 dBm. With the -4.77 dB gain of the combiner, the combiner’s 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, the final output power is -0.62 dBm:
Balanced 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 out, 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 is such that the signals at the hybrid combiner add in-phase.
The following plot shows the signal power (blue trace) and noise power density (red trace) at each stage of the balanced amplifier
Following the signal power through the chain, the signal power at the outputs 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 (coherent power addition) and with a -3 dB gain of the output combiner the final output power is:
+7 dBm signal power at each amplifier output + 6 dB due to coherent power addition -3 dB due to combiner gain = +10 dBm
Noise generated in each amplifier, however, is not coherent. In other words, 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 combiner gain of -3 dB, the noise power at the output is -161 dBm/Hz
(1) Simon Haykin, (1983). ”Communication Systems”