# How does one describe the “frequency response” of a mixer?

Suppose one uses a vector network analyzer to measure the $$S_{21}$$ parameter of an RF mixer over a frequency range. Would it be appropriate to call this the “frequency response” of the mixer? I find this confusing because mixers are not LTI systems, and it is therefore meaningless to speak of the frequency response of a mixer.

But, from what I understand, experimentally measuring the $$S_{21}$$ parameter through a LTI system such as a lowpass filter, amplifier, or cable precisely yields the frequency response of the system. In that case, what does the $$S_{21}$$ parameter through a mixer (not LTI) correspond to?

• – MBaz Mar 26 '19 at 14:24

May be you should clarify what that $$S_{21}$$ parameter (a two-port parameter ?) is, if that's something important about the answer. But otherwise, as you have already answered yourself, a mixer in its most basic form:

$$y(t) = g(t) x(t) = \cos(\omega_0 t) x(t)$$

is not an LTI system hence it does not have an impulse response $$h(t)$$ or frequency response $$H(\omega) = \int_{-\infty}^{\infty} h(t) e^{-j \omega t} dt$$, (Fourier transform of $$h(t)$$).

Note that, you can experiment with the mixer by sending it a test signal at a particular frequency $$\omega_1$$. Then the output will be :

$$y(t) = \cos(\omega_0 t) \cos(\omega_1 t) = 0.5 \cos( (\omega_0 + \omega_1) t) + 0.5 \cos( (\omega_1-\omega_0) t)$$

which is a pair of cosine waves at different frequencies than $$\omega_0$$; a manifestation of non-LTI systems. Hence when such a setup is constructed, what's measured will not be the frequency response.

• My apologies. Yes, it is a two-port transmission parameter through the mixer; taking one terminal as the input, another as the output, and fixing all others at DC, and calculating the ratio of the output signal to the input signal. – user33568 Mar 26 '19 at 8:37
• ok assuming that the input terminal and output terminal obeys the I/O relationship as shown above, than the answer is same... – Fat32 Mar 26 '19 at 8:58