# Digital Audio: Part 3 - The Mathematics Of Sampling

Digital audio relies completely on sampling and no treatment of the subject can be complete without looking at how it works.

Fig.1. Sampling can be thought of as the audio waveform amplitude modulating a series of narrow pulses or needles, to produce a pulse-amplitude-modulated (PAM) signal.

Fig.2. Here, two different waveforms are represented by the same set of samples. Which is correct depends on the associated filters.

Fig.3. All sampling processes must be preceded and followed by appropriate filters. It is not meaningful to discuss sampling without considering the action of the filters.

Fig.4. When there is contra-rotation, as shown here, the horizontal processes are in opposition and cancel out, whereas the vertical process results in a sine wave having equal parts of positive and negative frequency. Modulation causes one sine wave to carry another so that negative frequencies subtract to produce lower side bands.

Fig.5. a) An un-modulated set of sampling needles has a spectrum like this. b) In conventional sampling, the base band is repeated as upper and lower sidebands around multiples of the sampling rate. c) The limit is reached at Fs/2, the Nyquist Frequency, when the lower sideband touches the baseband. d) An alternative sampling system in which the base band is band-limited. The lower side band is now below the baseband in the spectrum. The baseband can still be recovered with a band-pass filter.

Fig.6. In a reconstruction filter the zero crossings in the filter impulses align with adjacent samples so the output waveform effectively joins the samples into a continuous waveform identical within the available bandwidth to the original.

Fig.7. Transform duality holds that a square in either the frequency or time domains results in a sin/x in the other domain.

# Audio Post For ‘Billions’ With Eric Hirsch & Gregg Swiatlowski At Goldcrest NY

The multi award winning team at Goldcrest share their creative insight and technique through an exploration of the subtle soundscape for Billions.

# Podcast Studio Design: Getting The Room Right

Podcasting is an increasingly popular pastime in the U.S., with an estimated 120 million podcast listeners in 2021. Back in 2006, only 22 percent of the adult population was aware of podcasting. By 2021, this figure had risen to 78 percent. Podcasting is becoming an…

# Broadcast Audio Workflow: Part 2 - Entertainment With An Audience

We continue our discussion of broadcast audio workflow with multi-award winner Robert Edwards. We look at the many challenges that come when a live audience is added to the broadcast mix.

# Electricity: Part 7 - Lithium Batteries

Lithium batteries are all the rage on account of their low weight and high capacity. But how good are they really?

# Cloud-Native Audio Mixers - Current Developments In Virtualized Broadcast Audio Mixing

As the wider broadcast industry picks up the pace with virtualized, cloud-native production systems we take a look at what audio vendors currently have available and what may be on the horizon.