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What is the Sampling Theorem in signal processing?
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The Sampling Theorem states that a continuous signal can be completely represented by its samples if the sampling frequency is at least twice the highest frequency of the signal, also known as the Nyquist rate.
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Who formulated the Sampling Theorem, and when?
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The Sampling Theorem was formulated by Claude Shannon in 1949, although it was prefigured by Harry Nyquist and others earlier.
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What is the Nyquist rate?
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The Nyquist rate is the minimum sampling rate required to avoid aliasing, equal to twice the highest frequency present in the signal.
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Explain the concept of aliasing in signal processing.
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Aliasing is the phenomenon that occurs when a signal is sampled below the Nyquist rate, causing different signal frequencies to become indistinguishable or 'aliased' in the sampled data.
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How can aliasing be prevented when sampling a signal?
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Aliasing can be prevented by using a low-pass filter to remove frequencies higher than half the sampling rate before sampling the signal.
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What is meant by 'Nyquist frequency'?
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The Nyquist frequency is half the sampling rate of a discrete signal processing system and represents the highest frequency that can be accurately represented.
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Why is the Sampling Theorem important in digital signal processing?
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The Sampling Theorem is crucial because it provides the foundation for converting continuous signals to discrete form without loss of information, which is essential for digital processing and storage.
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Define 'over-sampling' in the context of the Sampling Theorem.
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Over-sampling occurs when a signal is sampled at a rate significantly higher than the Nyquist rate, which can be used to improve signal resolution and reduce noise.
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What happens if a continuous signal contains frequency components beyond the Nyquist limit?
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Frequency components beyond the Nyquist limit cause aliasing in the sampled signal, which distorts the original signal’s representation.
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How is the Nyquist criterion applied in real-world signal processing applications?
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In practice, the Nyquist criterion is applied by ensuring sampling rates are set above the Nyquist rate and by using anti-aliasing filters to remove high-frequency noise.
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Can the Sampling Theorem be applied to non-periodic signals?
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Yes, the Sampling Theorem can be applied to non-periodic signals by considering the signal’s bandwidth and ensuring the sampling rate is twice the signal's maximum frequency component.
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What is a practical example of a system or device that utilizes the Sampling Theorem?
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A practical example is the digital audio player, which uses the Sampling Theorem to convert analog audio signals into digital form accurately.
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Describe 'quantization' in the context of signal sampling.
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Quantization is the process of mapping a continuous range of values (such as a sample of an analog signal) to a finite range of discrete levels.
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What role does the Sampling Theorem play in telecommunications?
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The Sampling Theorem allows telecommunications systems to efficiently encode continuous signals into digital signals, which are more robust for transmission over long distances.
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How does the concept of 'band-limited signals' relate to the Sampling Theorem?
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A band-limited signal is one that contains no frequency components higher than a certain cutoff frequency, which ensures that it can be sampled without aliasing if the sampling rate meets the Nyquist criterion.
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