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Multi-Tone Nyquist Rate

The Nyquist rate for the signal x(t) = [1 + cos(2000πt) + sin(4000πt)] is

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Answer & Explanation

Correct AnswerA. 4000

Quick Explanation

cos(2000πt) has frequency 1000 Hz and sin(4000πt) has frequency 2000 Hz because ω=2πf. The highest frequency is therefore 2000 Hz, so the Nyquist rate is 2fmax=4000 samples/s. Hence option A.

Formula / Key Relation

fmax=4000π/(2π)=2000 Hz; fs,Nyquist=2fmax=4000 samples/s

Multi-Tone Nyquist Rate explanatory diagram

Detailed Explanation

Correct Answer

A — 4000

Concept and Reasoning

cos(2000πt) has frequency 1000 Hz and sin(4000πt) has frequency 2000 Hz because ω=2πf. The highest frequency is therefore 2000 Hz, so the Nyquist rate is 2fmax=4000 samples/s. Hence option A.

Calculation / Key Relation

fmax=4000π/(2π)=2000 Hz

fs,Nyquist=2fmax=4000 samples/s

Exam Takeaway

Convert angular frequency to hertz before doubling.

Common Mistake

Do not treat the coefficient 4000π as frequency in hertz.

Quick Trick

Convert angular frequency to hertz before doubling.

Common Mistake

Do not treat the coefficient 4000π as frequency in hertz.

Exam Tip

Convert angular frequency to hertz before doubling.

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Multi-Tone Nyquist Rate

Quick Explanation

Formula / Key Relation

Detailed Explanation

Correct Answer

Concept and Reasoning

Calculation / Key Relation

Exam Takeaway

Common Mistake

Quick Trick

Common Mistake

Exam Tip

Previous Exam Appearances

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