Finding Minimum Sampling Rate: Worked Examples

Finding the Minimum Sampling Rate: Worked Examples

Example 1: Single Sinusoid

\(x(t) = 5\cos(2\pi \cdot 1000 t)\)

• Maximum frequency: $F_{\text{max}} = 1000$ Hz
• Nyquist rate: $F_s \ge 2000$ Hz

Example 2: Sum of Sinusoids

\(x(t) = 3\cos(2\pi \cdot 500 t) + 2\sin(2\pi \cdot 1200 t)\)

• Frequencies: 500 Hz and 1200 Hz
• $F_{\text{max}} = 1200$ Hz
• Nyquist rate: $F_s \ge 2400$ Hz

Example 3: Product of Sinusoids (Modulation)

\(x(t) = \cos(2\pi \cdot 1000 t) \cdot \cos(2\pi \cdot 10000 t)\) Using $\cos A \cos B = \frac{1}{2}[\cos(A+B) + \cos(A-B)]$:

• Frequencies: 9000 Hz and 11000 Hz
• $F_{\text{max}} = 11000$ Hz
• Nyquist rate: $F_s \ge 22000$ Hz

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