香农公式和奈式准则英文

### Shannon Formula and Nyquist Criterion in English

#### Shannon's Theorem (Shannon Capacity)

The Shannon-Hartley theorem, often referred to as the Shannon capacity, establishes a relationship between channel bandwidth \( W \), signal power \( S \), noise power spectral density \( N_0 \), and the maximum data rate \( C \) that can be achieved over a communication channel with an arbitrarily low error probability. Mathematically, this is expressed as:

\[ C = W \log_2\left(1 + \frac{S}{N}\right) \]

where:
- \( C \) represents the channel capacity or maximum achievable data rate,
- \( W \) denotes the bandwidth of the channel in Hertz,
- \( S/N \) stands for the signal-to-noise ratio.

This equation indicates how much information can be transmitted through a noisy channel without errors given its bandwidth and the level of background noise present within it[^1].

#### Nyquist Criterion

Nyquist’s first theorem addresses pulse code modulation systems where pulses are used to represent binary digits during transmission across channels free from thermal agitation effects—essentially ideal conditions devoid of random fluctuations caused by heat energy at absolute zero temperature. According to his findings published back in 1928,

For distortion-free transmission, the sampling frequency must be greater than twice the highest frequency component contained within the original analog waveform; otherwise known as the Nyquist Rate. Specifically stated mathematically for baseband signaling:

\[ f_s > 2f_m \]

Where \( f_s \) refers to the sample rate while \( f_m \) signifies the message signal's upper limit frequency. This ensures accurate reconstruction post-transmission assuming perfect filtering techniques applied both pre-and-post conversion processes involved when converting continuous signals into discrete sequences suitable for digital processing environments[^3].

Additionally, regarding idealized models such as those characterized purely theoretically under laboratory settings rather than practical implementations subject to real-world constraints like interference sources not accounted hereunder discussion parameters set forth earlier concerning theoretical foundations laid out originally by Harry Nyquist himself pertaining specifically towards telegraphy applications initially but later generalized further beyond just these contexts alone[^2].

--related questions--

1. What factors influence the accuracy of reconstructing an analog signal after digitization?
2. How does increasing the signal-to-noise ratio affect the channel capacity according to Shannon's formula?
3. In what scenarios might one prefer using oversampling rates higher than the minimum required by the Nyquist criterion?
4. Can you explain why maintaining proper phase relationships among sampled points matters in preventing aliasing issues during reconstructions efforts following conversions undertaken via ADC/DAC operations typically found inside modern electronics devices today?