### Problem

Why are there differences between the output of MIXER_B/MIXER_F when mixing a real signal and mixing the equivalent complex envelope signal.

### Solution

There are two main reasons for differences between the output of MIXER_B/MIXER_F when mixing a real signal and mixing the equivalent complex envelope signal. The spur synthesis model used by MIXER_B and MIXER_F has limitations when applied to real signals. This is because the models synthesize the spurs based on a single tone model, which in turn relies upon complex arithmetic. When working with a complex envelope signal, the magnitude of individual spurs can be controlled because raising a complex envelope representation of a single tone signal to a given power (which, along with a scaling and multiplication by another single tone raised to a power, is essentially what the spur generation involves) results in another single tone with a frequency that is the power times the original frequency. For example, the third harmonic of a 1 GHz tone represented by:

s(t) = A exp^{(j2Π·1e9·t)}

is simply:

s(t)^{3} = A^{3}exp^{(j2Π·3·1e9·t)}

When working with real signals, however, raising a real single tone signal to a given power generates not only the harmonic at that power, but terms at every second lower harmonic. For example, raising a tone to the fifth power generates tones at the first and third harmonics as well as the fifth. While the spur synthesis model compensates for these additional terms with a single tone at the reference LO and input power levels, it is unable to compensate at different power levels or more complex signals due to the lack of phase information in the real signal.

The second cause of differences between the mixers operating on real signals versus complex envelope signals is again related to the spur synthesis model and the reliance on complex arithmetic. However in this case the larger deviation from expected results occurs with the complex envelope signals. When a real two tone input signal is raised to a power, it generates intermodulation products at various combinations of the frequencies of the two input signals. For example, raising a two-tone real signal of frequencies f1 and f2 to the third power generates terms at the frequencies 3f1, 3f2, 2f1-f2, 2f1+f2, 2f2-f1, and 2f2+f1.

On the other hand, raising a complex two-tone signal of frequencies f1 and f2 only generates terms at the frequencies 3f1, 3f2, 2f1+f2, and 2f2+f1:

(A1·exp^{(j2Π·f1·t)} + A2·exp^{(j2Π·f2·t))3} = (A1·A2)^{3}(exp(j2Π·f1·t) + (exp(j2Π·f2·t))^{3}

Ignoring the A1 x A2 term:

exp(j2Π·f1·t) + (exp(j2Π·f2·t))^{3} = (exp(j2Π·f1·t) + (exp(j2Π·f2·t))·(exp(j2Π·2f1·t) + 2·exp(j2Π·(f1+f2)·t) + exp(j2Π·2f2·t))

= exp(j2Π·3f1·t) + 3·exp(j2Π·(2f1+f2)·t) + 3·exp(j2Π·(f1+2f2)·t) + exp(j2Π·3f2·t)

The net result is that the mixer operating on a complex envelope signal will not generate the m1-m2 and m2-m1 terms of the intermodulation products of the input signal, while the mixer operating on a real signal will generate those terms. Note that these are the intermodulation products of the input signal, not of the mixing between the input signal and the LO.