The Nyquist-Shannon Sampling Theorem is a cornerstone of digital signal processing. It tells us how fast we need to sample a signal to capture all its information, setting the minimum sampling rate at twice the highest frequency in the signal.

This theorem is crucial for avoiding aliasing, where high frequencies get mistaken for low ones. It's used in everything from audio recording to data acquisition, ensuring we can accurately represent and reconstruct analog signals in the digital world.

Nyquist-Shannon Sampling Theorem

Theorem Statement and Implications

• The Nyquist-Shannon sampling theorem states that a continuous-time signal can be perfectly reconstructed from its samples if the sampling frequency is at least twice the highest frequency component in the signal
• Establishes the minimum sampling rate required to avoid aliasing and ensures that the original signal can be recovered from its sampled version without loss of information
• The sampling frequency, denoted as $fs$, must be greater than or equal to $2B$, where $B$ is the bandwidth of the signal (the highest frequency component present in the signal)
• If the sampling frequency is less than $2B$, aliasing occurs, causing high-frequency components to be misinterpreted as low-frequency components, leading to signal distortion and loss of information
• Assumes that the signal is bandlimited, meaning that it has a finite bandwidth and no frequency components beyond a certain limit

Aliasing and Bandlimited Signals

• Aliasing is a phenomenon that occurs when the sampling rate is insufficient to capture the highest frequency components in the signal
• When aliasing occurs, high-frequency components of the signal are misinterpreted as lower-frequency components, resulting in distortion and loss of information
• To avoid aliasing, the signal must be bandlimited, meaning that it has a finite bandwidth and no frequency components beyond a certain limit
• Bandlimiting a signal involves filtering out high-frequency components that exceed the Nyquist frequency (half the sampling rate) before sampling
• Anti-aliasing filters, such as low-pass filters, are used to bandlimit the signal and ensure that the Nyquist-Shannon sampling theorem is satisfied

Minimum Sampling Rate

Calculating the Minimum Sampling Rate

• To avoid aliasing, the sampling rate must be at least twice the highest frequency component present in the signal, as stated by the Nyquist-Shannon sampling theorem
• The minimum sampling rate, denoted as $fs_{min}$, is equal to $2B$, where $B$ is the bandwidth of the signal
• If the signal's highest frequency component is known, the minimum sampling rate can be calculated by multiplying that frequency by 2
• For example, if a signal has a bandwidth of 10 kHz, the minimum sampling rate required to avoid aliasing would be 20 kHz ($fs_{min} = 2 \times 10$ kHz $= 20$ kHz)

Oversampling and Practical Considerations

• When designing a sampling system, it is common practice to use a sampling rate slightly higher than the minimum required to provide a margin of safety and account for any practical limitations
• Oversampling involves using a sampling rate that is significantly higher than the minimum required by the Nyquist-Shannon sampling theorem
• Oversampling helps to reduce aliasing, improve signal-to-noise ratio (SNR), and simplify the design of anti-aliasing filters
• Practical considerations, such as the characteristics of the analog-to-digital converters (ADCs) and the available storage or transmission bandwidth, may influence the choice of sampling rate

Sampling Rate vs Bandwidth

Relationship between Sampling Rate and Bandwidth

• The sampling rate and signal bandwidth are directly related, as the minimum sampling rate required to avoid aliasing depends on the signal's bandwidth
• As the signal bandwidth increases, the minimum required sampling rate also increases to ensure that the Nyquist-Shannon sampling theorem is satisfied
• If the sampling rate is fixed, the maximum allowable signal bandwidth is limited to half the sampling rate ($B_{max} = fs / 2$)
• When the sampling rate is increased, the maximum allowable signal bandwidth also increases, allowing for the accurate representation of higher-frequency components in the sampled signal

Undersampling and Bandwidth Limitation

• Conversely, if the sampling rate is decreased, the maximum allowable signal bandwidth decreases, potentially leading to aliasing if the signal contains frequency components above the Nyquist frequency (half the sampling rate)
• Undersampling occurs when the sampling rate is lower than the minimum required by the Nyquist-Shannon sampling theorem, resulting in aliasing and loss of information
• To avoid undersampling, the signal bandwidth must be limited to half the sampling rate or less
• Bandwidth limitation can be achieved through the use of anti-aliasing filters, which remove frequency components above the Nyquist frequency before sampling

Sampling Theorem Applications

Audio and Video Recording

• In audio recording, the theorem is used to determine the minimum sampling rate required to capture the full range of human hearing (approximately 20 Hz to 20 kHz), resulting in common sampling rates such as 44.1 kHz for CD-quality audio
• Higher sampling rates, such as 96 kHz or 192 kHz, are used in professional audio recording to capture a wider frequency range and provide better resolution
• In video recording, the theorem is applied to determine the minimum sampling rate needed to capture the desired frame rate and resolution without aliasing artifacts
• Common video sampling rates include 24 frames per second (fps) for cinema, 30 fps for NTSC television, and 25 fps for PAL television

Data Acquisition and Digital Communication

• In data acquisition systems, such as sensors and measurement devices, the theorem is used to select an appropriate sampling rate based on the bandwidth of the signals being measured, ensuring accurate representation of the data
• Oversampling is often employed in data acquisition to improve signal-to-noise ratio (SNR) and reduce the effects of quantization noise
• In digital communication systems, the theorem is applied to determine the minimum sampling rate required for the accurate representation and transmission of analog signals over digital channels, such as in modems and wireless communication systems
• Pulse-code modulation (PCM) is a common sampling technique used in digital communication, where the analog signal is sampled at a fixed rate and quantized into discrete levels for transmission
• When applying the sampling theorem, it is important to consider the practical limitations of the hardware, such as the maximum sampling rate supported by the analog-to-digital converters (ADCs) and the available storage or transmission bandwidth