2. What is the Sampling
Theorem?
State the theorem.

3. History– As stated in FSO Chapter 2 Overview Video, the sampling
theorem provides the basics for all digital audio. Although the authors
didn’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.

4. 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.

5. • The Nyquist-Shannon Sampling Theorem has many
It’s implications to uses and implications today. The fundamentals laid out
the 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 won't be able to
discern a Wave from a 320 kbps MP3 file, the loss will be
instantly recognizable with a 64 kbps MP3.

6. 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.

7. Pre-Recording-Preparations
Addressing
Aliasing • 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 can't exceed 20 kHz, any file recorded at or
above 44.1 kHz should be unaffected by aliasing.
One of the main limitations,
Anti-Aliasing-Filters
aliasing, is addressed by
preparing 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 out
filter. 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 dither's
ability to reduce distortion.

8. 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.