tags: mus-407 digital-audio audio audio-programming

Digital Delay Line

A digital delay line (DDL) is a type of [audio signal] processing that digitally implements a [delay line]. It stores a sequence of [audio samples] in memory and outputs them after a period of time.

• can be any period of time

Mixing delayed output with input produces a variety of effects:

• echo
• [multi-tap delay]
• [comb filters] & resonators
• [phasing] & [flanging]
• [chorus]
• [pitch-shifting and harmonization]
• [reverberation], simulation of room acoustics

These effects are split between [fixed delays] and [variable delays].

Analog vs Digital Delay Lines

Analog delay:

• implemented via reel-to-reel tape deck
• delay time determined by tape speed & physical distance between record/play heads
• multiple delays possible with multiple play heads and/or multiple tape decks

Digital Delay:

• any number of samples in length
• quantity and length of delays limited by available memory
• flexible in terms of physical space

Signal Flow

Basic DDL (inherently stable):

Basic DDL with [feedback] (unstable if delayed signal [amplitude] > 1):

Delays in Parallel/Series

Parallel: multiple copies are individually sent through multiple delays; outputs are summed

Series: signal is sent through a succession of delays; input/output are usually mixed at each stage

Implementation

Uses a [circular queue] or circular buffer

• a block of memory are treated as if the end and beginning are connected (visually a circle)
• list of sequential memory locations for audio samples
• has a write pointer at and least one read pointer
• read pointer also called a tap
• pointers advance through a buffer, R lags behind W by # of samples corresponding to delay time
• pointers "wrap" around back to beginning when they reach the end of the queue

Digital Filters vs. Digital Delays

In terms of design, digital [filters] & delays are essentially distinguishable

• delay an input signal by some amount
• mix delayed output with input

Filters require delay/mixing in order to cancel/reinforce certain [frequencies].

Consider a simple [lowpass filter]:

$$y[n] = 0.5 \times x[n] + 0.5 \times x[n-1]$$

• x = input signal
• y = output signal
• n = sample index
• x[n] = input signal at nth sample
• y[n] = output signal at nth sample

Performs averaging function on consecutive sample pairs

Creates a slight smoothing effect on [waveform] shape, thus attenuating higher frequencies.

For a waveform at [Nyquist frequency], waveform will be completely nullified (consecutive samples are equal and opposite, yielding a zero average)

Longer-term averaging function (ten incremental sample delays in parallel) creates a stronger smoothing effect, lowering the [cutoff frequency]:

$$y[n] = 0.1 \times x[n] + 0.1 \times x[n-1] + ... + 0.1 \times x[n-9]$$

Digital filters and digital delays are closely related in design and result:

• some types of delay lines can noticeably affect a sound's [spectrum]
• some filters can introduce [phase]/timing shifts
  • capable of smearing [transients]

General naming distinction:

• delay time measured in [samples] → filter
• delay time measured in seconds/milliseconds → delay line
  • related to [linear superposition] and related [wave-interference] properties

Sources

• https://ccrma.stanford.edu/~jos/pasp/Delay_Lines.html