User:Darjohn

What is the Sampling Theorem??
With the rapid advancements in data acquisition technology and introduction of micro-computers, there is a greater need to understand how and why sound and frequency works. The Sampling Theorem is a result in the field of information theory, particularly with regards to telecommunications and signal processing. Sampling is the process of converting a signal into a numeric sequence. Claude Shannon’s version of the theorem states that “If a function x contains no frequency higher than a B hertz (Hz), it is completely determined by giving its ordinates at a series of points space 1/2B seconds apart.” The theorem is commonly called the Nyquist sampling theorem, it is also known as Nyquist–Shannon theorem. It is often just referred to simply as The Sampling Theorem. The most commonly used limitation is the Nyquist sampling theorem.

In essence, the theorem shows that a bandwidth analog signal that has been sampled can be perfectly reconstructed from an infinite sequence of samples if the sampling rate exceeds 2B samples per second, where B is the highest frequency in the original signal. If a signal contains a component at exactly B hertz, then samples spaced at exactly 1/(2B) seconds do not completely determine the signal, Shannon's statement notwithstanding. This sufficient condition can be weakened, as discussed at Sampling of non-baseband signals below.

More recent statements of the theorem are sometimes careful to exclude the equality condition; that is, the condition is if x(t) contains no frequencies higher than or equal to B; this condition is equivalent to Shannon's except when the function includes a steady sine wave component at exactly frequency B.

The theorem assumes a role of any real-world situation, as it only applies to signals that are sampled for infinite time; any time-limited x(t) cannot be perfectly band limited. Perfect reconstruction is mathematically possible for the idealized model but only an approximation for real-world signals and sampling techniques, albeit in practice often a very good one. The theorem also leads to a formula for reconstruction of the original signal. The proof of the theorem leads to an understanding of the aliasing that can occur when a sampling system does not satisfy the conditions of the theorem. The sampling theorem doesn’t tell us when certain conditions and procedures are not exactly met, but it does suggest non-ideality, analytical framework that can be studied The theorem describes two processes in signal processing 1). A sampling process, in which a continuous time signal is converted to a discrete time signal, 2). A reconstruction process, in which the original continuous signal is recovered from the discrete time signal.

All of these processes can lead to various other effects such as downsampling, non-uniform sampling, aliasing, and anti-aliasing. In aliasing, for a sine wave of half of the sampling frequency the sine wave will alias to another of the same frequency, but with a different phase and amplitude. Imagine looking at the wheels of a fast moving car slowly rotating backwards. That would be the best visual example.

With this theorem we can now clarify and get a better idea of how to control sound and how to get a cleaner, smoother sound for the future. And with the analog-to-digital conversion taking over, it will reduce the time and work taken for sound manipulation.

A/D and D/A Conversion
Everybody wants to try to create an analog-versus-digital debate, when in fact, they can work hand and hand. Some people believe that no DAW (Digital Audio Workstation) will ever sound as good as tape. No virtual instrument will be as good as a real analog synth. And some believe the exact opposite. However, the truth is that analog and digital audio is inexorably linked. The real instrument sound in the physical world is analog, while all audio distribution is digital. The most important step is to get the signal back and forth between the two.

The classic conversion of analog to digital is known as successive approximation. It compares the input voltage with a series of smaller voltages. The D/A conversion uses series of resistors of different values in a ladder fashion. Imagine a row of light bulbs of different intensities being turned on in different combinations to get different levels of illumination. By turning the resistors on in various combinations, the correct output voltage will be achieved. One of the greatest challenges, though, is the necessity of brickwall filters to prevent aliasing on input and imaging on output. Modern D/A converters often use more than one-bit for their output stage. This allows for a cheaper Nyquist filter.

Many A/D converters can start by taking one-bit samples at a rate that is multiple times higher than the regular rate. The most common one-bit design is referred to as a sigma-delta converter. If the input is larger than the reference a one is input on the binary code, if otherwise, a zero is input on the binary code. The simplicity of the sigma-delta converter allows it to run at a higher sampling rate. However, getting more samples than are required, it creates what is called oversampling. For example, if sampling takes place at 64 times the regular sampling rate of 44.1 kHz, frequencies as high as 1.4 can be captured without aliasing. Once the one-bit data has been stored a decimation filter converts it to the standard digital representation. We have now gone over the A/D and D/A conversion. The basic theme here is that with the design of A/D and D/A converters, you can create a quality sound that is cost effective. Hopefully, they are now little less intimidating. To think, something as simple as one-bit, is all you need for conversion.