Question E7E08:
What are the principal frequencies that appear at the output of a mixer?  

Choices:  
A: Two and four times the input frequency  
B: The square root of the product of input frequencies  
C: The two input frequencies along with their sum and difference frequencies  
D: 1.414 and 0.707 times the input frequency    

Correct Answer is  C: The two input frequencies along with their sum and difference frequencies  

Correct answer: C — The two input frequencies along with their sum and difference frequencies

Why C is correct:
- An ideal mixer multiplies two signals. Using the identity
  $$\cos(2\pi f_1 t)\cos(2\pi f_2 t) = \tfrac{1}{2}\left[\cos 2\pi(f_1+f_2)t + \cos 2\pi|f_1-f_2|t\right],$$
  the product contains new components at $f_1+f_2$ and $|f_1-f_2|$.
- In real mixers, some of the original inputs leak through (limited isolation), so the principal outputs are typically at $f_1$, $f_2$, $f_1+f_2$, and $|f_1-f_2|$.

Why the others are incorrect:
- A: “Two and four times the input frequency” describes harmonics from nonlinear distortion or frequency multipliers, not the sum/difference products of mixing.
- B: “Square root of the product of input frequencies” has no meaning in standard frequency conversion; mixers do not produce $f=\sqrt{f_1 f_2}$.
- D: “1.414 and 0.707 times the input frequency” are amplitude scaling factors ($\sqrt{2}$ and $1/\sqrt{2}$), not new frequencies from mixing.

Useful formulas and rules of thumb:
- Sum and difference:
  $$f_{\text{sum}} = f_1 + f_2,\quad f_{\text{diff}} = |f_1 - f_2|.$$
- Real mixers also leak inputs: $f_1$ and $f_2$ often appear at the output.
- Spur rule of thumb for strong nonlinearity: additional products at $m f_1 \pm n f_2$ (integers $m,n$), but the principal products are the sum and difference.