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The highest frequency component of a speech signal needed for telephonic communications is about 3.1 kHz. What is the suitable value for the sampling rate?
- a)1 kHz
- b)2 kHz
- c)4 kHz
- d)8 kHz
Correct answer is option 'D'. Can you explain this answer?
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The highest frequency component of a speech signal needed for telephon...
The Sampling Theorem
The sampling theorem states that in order to accurately reconstruct a continuous-time signal from its samples, the sampling rate must be at least twice the highest frequency component of the signal. This is known as the Nyquist rate.
In the case of telephonic communications, the highest frequency component of a speech signal is about 3.1 kHz. Therefore, to accurately capture and transmit the speech signal, the sampling rate must be at least twice this frequency, i.e., 2 * 3.1 kHz = 6.2 kHz.
Available Options
Let's evaluate the available options for the suitable value of the sampling rate:
a) 1 kHz: This sampling rate is below the Nyquist rate, as it is less than 6.2 kHz. Therefore, it is not suitable for accurately capturing the speech signal.
b) 2 kHz: Similar to option a), this sampling rate is also below the Nyquist rate and cannot accurately capture the speech signal.
c) 4 kHz: Again, this sampling rate is below the Nyquist rate and insufficient for accurately capturing the speech signal.
d) 8 kHz: This sampling rate is twice the highest frequency component of the speech signal (3.1 kHz * 2 = 6.2 kHz). Therefore, it satisfies the Nyquist rate and is suitable for accurately capturing the speech signal.
Conclusion
To accurately capture the highest frequency component of a speech signal for telephonic communications, a suitable value for the sampling rate is 8 kHz (option d). This sampling rate ensures that the signal is adequately sampled, allowing for faithful reconstruction during transmission and reception.
The sampling theorem states that in order to accurately reconstruct a continuous-time signal from its samples, the sampling rate must be at least twice the highest frequency component of the signal. This is known as the Nyquist rate.
In the case of telephonic communications, the highest frequency component of a speech signal is about 3.1 kHz. Therefore, to accurately capture and transmit the speech signal, the sampling rate must be at least twice this frequency, i.e., 2 * 3.1 kHz = 6.2 kHz.
Available Options
Let's evaluate the available options for the suitable value of the sampling rate:
a) 1 kHz: This sampling rate is below the Nyquist rate, as it is less than 6.2 kHz. Therefore, it is not suitable for accurately capturing the speech signal.
b) 2 kHz: Similar to option a), this sampling rate is also below the Nyquist rate and cannot accurately capture the speech signal.
c) 4 kHz: Again, this sampling rate is below the Nyquist rate and insufficient for accurately capturing the speech signal.
d) 8 kHz: This sampling rate is twice the highest frequency component of the speech signal (3.1 kHz * 2 = 6.2 kHz). Therefore, it satisfies the Nyquist rate and is suitable for accurately capturing the speech signal.
Conclusion
To accurately capture the highest frequency component of a speech signal for telephonic communications, a suitable value for the sampling rate is 8 kHz (option d). This sampling rate ensures that the signal is adequately sampled, allowing for faithful reconstruction during transmission and reception.
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The highest frequency component of a speech signal needed for telephon...
Nyquist Sampling Theorem:
A continuous-time signal can be represented in its samples and can be recovered back when sampling frequency fs is greater than or equal to twice the highest frequency component of the message signal, i.e.
fs ≥ 2fm
Therefore when we want to convert continuous signals to discrete signals, the sampling must be done at the Nyquist rate.
Calculation:
Given that,
fm = 3.1 kHz
⇒ fs ≥ 2fm
⇒ fs ≥ 2 × 3.1 = 6.4 kHz
A continuous-time signal can be represented in its samples and can be recovered back when sampling frequency fs is greater than or equal to twice the highest frequency component of the message signal, i.e.
fs ≥ 2fm
Therefore when we want to convert continuous signals to discrete signals, the sampling must be done at the Nyquist rate.
Calculation:
Given that,
fm = 3.1 kHz
⇒ fs ≥ 2fm
⇒ fs ≥ 2 × 3.1 = 6.4 kHz
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The highest frequency component of a speech signal needed for telephonic communications is about 3.1 kHz. What is the suitable value for the sampling rate?a)1 kHzb)2 kHzc)4 kHzd)8 kHzCorrect answer is option 'D'. Can you explain this answer?
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