What is the Sampling Theorem? State the theorem.
History– As stated in FSO Chapter 2 Overview Video, the sampling theorem provides the basics for all digital audio. Although the authorsdidn’t have audio in mind it allows us to capture and produce virtuously any sound with life like clarity. The Sampling Theorem, in layman’s terms is simply the scientific basis for digital audio.
Theorem• The Sampling Theorem allows us to capture and reproduce audio using the same means as capturing and reproducing motion pictures. A sample; sequence of frames reconstructs the visual motion just as a sequence of samples reconstructs the audio signal. A signal can be reconstructed from a series of evenly-spaced measurements, or samples, as long as the signal contains no frequencies higher than half the sampling rate. Of course, Shannon one of the people credited for this theorem put it in mathematical terms, with the fundamentals being the same.• The Sampling Theorem tells us that, within certain limitations, we can analyze a sound, store it or transmit it digitally, and reproduce it accurately.
• The Nyquist-Shannon Sampling Theorem has manyIt’s implications to uses and implications today. The fundamentals laid outthe field of audio by it are used in all digital recordings today.production. Recording and Transmission • The Nyquist Theorem has greatly changed the way audio is recorded and shared. Audio is recorded and reconstructed in a string of 1s and 0s known as binary. This binary is then converted back to an analog signal whenever we want to play the sound back. A more in-depth look at how this works is explained on the Digital/Analog and Analog/Digital Conversion page of the wiki. Encoding • Encoding is used for storing audio files into a codec for smaller file sizes. They can either be lossless, meaning that they reproduce the exact same sound with no loss in quality; or they may be lossy, which means that some minor quality loss will occur. Commonly used codec’s include MP3 (lossy), Apple’s M4A or AAC (lossy), and FLAC (lossless) Quality Loss • Lossy encoders cause varying degrees of quality loss in audio files. Although there is a loss in quality, if a file is encoded correctly and at a high enough bitrate, the loss should be negligible. The actual loss of data is dependent on the target bitrate selected at the time of encoding. For example, while the average listener probably wont be able to discern a Wave from a 320 kbps MP3 file, the loss will be instantly recognizable with a 64 kbps MP3.
Although the Nyquist-Shannon Sampling has become standard in the recording industry, it is not without faults. There are still a few key limitations that affect its usage.
Pre-Recording-PreparationsAddressingAliasing • As aliasing is caused by audio exceeding the Nyquist Frequency, logically recording at a higher sample rate would reduce the chance of aliasing. Since human hearing cant exceed 20 kHz, any file recorded at or above 44.1 kHz should be unaffected by aliasing.One of the main limitations, Anti-Aliasing-Filtersaliasing, is addressed bypreparing ahead of time or • The other method of addressing aliasing is by running the recorded audio through an anti-aliasing filter. The anti-by using an anti-aliasing aliasing filter is a type of low-pass filter that cuts out any signal not within its range. This is used to block outfilter. signals that would exceed the Nyquist Frequency and cause aliasing. Dither • Quantization error is the other main limitation that is faced in digital recording. The method by which we diminish the effects of quantization error is known as "dither". Dithering is accomplished by adding a low-level noise to the recording. Although it brings a low-level hiss, it reduces the distortion caused by quantization error. • The noise is of a level less than the least-significant bit before rounding to 16-bits. The noise has the effect of spreading errors across the entire audio spectrum. Because of the nature of dithering, it should always be added last when mastering. This is because any change in audio could have an effect on the dithers ability to reduce distortion.
How these remedies are implemented in A/D and D/A conversion!• We can measure the energy, or amplitude of the signal by using successive measurements, with each measurement giving us more accurate results.• Oversampling is the technique of sampling an analogue stream at a much, much higher rate, or frequency. This technique does not use successive measurement to determine the absolute value of the energy/amplitude of the sample. Instead, it measures the relative value of the sample to either a modulating triangle wave, as in PWM, or against the previously measured sample of the same stream as in PDM.• Both of these techniques, as compared to the low sampling rate found in 16 bit or 24 bit PCM allow for sampling rates at such a high frequency that the nyquist frequency is subsequently many times higher than any range of audible frequencies that we now have lessened the effect of using dither and a real world low pass or anti-aliasing filter.