Example 1 Find the smallest positive period of the function $f(x)=\cos \sqrt{2} x+\sin \frac{3}{8} \sqrt{2} x$.
The minimum positive period of the function $\mathrm{y}=\cos \sqrt{2} \mathrm{x}$ is $\mathrm{T}_{1}=\frac{2 \pi}{\sqrt{2}}=3 \times \frac{\sqrt{2}}{3} \pi$, and the minimum positive period of the function $\mathrm{y}=\sin \frac{3}{8} \sqrt{2} \mathrm{x}$ is $\mathrm{T}_{2}=\frac{2 \pi}{\frac{3}{8} \sqrt{2}}=8 \times \frac{\sqrt{2}}{3} \pi$. According to Theorem $1^{\prime}$, the minimum positive period of the function $\mathrm{f}(\mathrm{x})$ is
$$
\mathrm{T}=3 \times 8 \times \frac{\sqrt{2}}{3} \pi=8 \sqrt{2} \pi
$$

Similarly, the minimum positive period of the function $\mathrm{f}(\mathrm{x})=\sin (2 \mathrm{k} \pi \mathrm{x})-\cos (2 \mathrm{~m} \pi \mathrm{x})$ (where $\mathrm{m}, \mathrm{k} \in \mathrm{N}, \mathrm{m}, \mathrm{k}>1$ and $(\mathrm{m}, \mathrm{n})=1$) is $\mathrm{T}=1$.