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The above discussion has already shown the sampling theorem in an informal and intuitive way that could easily be refined into a formal proof. However, the original proof of the sampling theorem, which will be given here, provides the interesting observation that the samples of a signal with period ${T}_{s}$ provide Fourier series coefficients for the original signal spectrum on $(-\pi /{T}_{s},\pi /{T}_{s})$ .
Let $x$ be a $(-\pi /{T}_{s},\pi /{T}_{s})$ bandlimited signal and ${x}_{s}$ be its samples with sampling period ${T}_{s}$ . We can represent $x$ in terms of its spectrum $X$ using the inverse continuous time Fourier transfrom and the fact that $x$ is bandlimited. The result is
This representation of $x$ may then be sampled with sampling period ${T}_{s}$ to produce
Noticing that this indicates that ${x}_{s}\left(n\right)$ is the $n\mathrm{th}$ continuous time Fourier series coefficient for $X\left(\omega \right)$ on the interval $(-\pi /{T}_{s},\pi /{T}_{s})$ , it is shown that the samples determine the original spectrum $X\left(\omega \right)$ and, by extension, the original signal itself.
Another way to show the sampling theorem is to derive the reconstruction formula that gives the original signal $\tilde{x}=x$ from its samples ${x}_{s}$ with sampling period ${T}_{s}$ , provided $x$ is bandlimited to $(-\pi /{T}_{s},\pi /{T}_{s})$ . This is done in the module on perfect reconstruction. However, the result, known as the Whittaker-Shannon reconstruction formula, will be stated here. If the requisite conditions hold, then the perfect reconstruction is given by
where the sinc function is defined as
From this, it is clear that the set
forms an orthogonal basis for the set of $(-\pi /{T}_{s},\pi /{T}_{s})$ bandlimited signals, where the coefficients of a $(-\pi /{T}_{s},\pi /{T}_{s})$ signal in this basis are its samples with sampling period ${T}_{s}$ .
The Nyquist-Shannon Sampling Theorem and the Whittaker-Shannon Reconstruction formula enable discrete time processing of continuous time signals. Because any linear time invariant filter performs a multiplication in the frequency domain, the result of applying a linear time invariant filter to a bandlimited signal is an output signal with the same bandlimit. Since sampling a bandlimited continuous time signal above the Nyquist rate produces a discrete time signal with a spectrum of the same form as the original spectrum, a discrete time filter could modify the samples spectrum and perfectly reconstruct the output to produce the same result as a continuous time filter. This allows the use of digital computing power and flexibility to be leveraged in continuous time signal processing as well. This is more thouroughly described in the final module of this chapter.
The properties of human physiology and psychology often inform design choices in technologies meant for interactin with people. For instance, digital devices dealing with sound use sampling rates related to the frequency range of human vocalizations and the frequency range of human auditory sensativity. Because most of the sounds in human speech concentrate most of their signal energy between 5 Hz and 4 kHz, most telephone systems discard frequencies above 4 kHz and sample at a rate of 8 kHz. Discarding the frequencies greater than or equal to 4 kHz through use of an anti-aliasing filter is important to avoid aliasing, which would negatively impact the quality of the output sound as is described in a later module. Similarly, human hearing is sensitive to frequencies between 20 Hz and 20 kHz. Therefore, sampling rates for general audio waveforms placed on CDs were chosen to be greater than 40 kHz, and all frequency content greater than or equal to some level is discarded. The particular value that was chosen, 44.1 kHz, was selected for other reasons, but the sampling theorem and the range of human hearing provided a lower bound for the range of choices.
The Nyquist-Shannon Sampling Theorem states that a signal bandlimited to $(-\pi /{T}_{s},\pi /{T}_{s})$ can be reconstructed exactly from its samples with sampling period ${T}_{s}$ . The Whittaker-Shannon interpolation formula, which will be further described in the section on perfect reconstruction, provides the reconstruction of the unique $(-\pi /{T}_{s},\pi /{T}_{s})$ bandlimited continuous time signal that samples to a given discrete time signal with sampling period ${T}_{s}$ . This enables discrete time processing of continuous time signals, which has many powerful applications.
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