Digital Audio

The mystery behind sampling and reconstruction
Spring 2019 - Audio Tech Talk Series
January 22, 2019

What is audio?

audio_wave

Analog Audio

analog

Digital Audio

stairsteps
stem

Sampling Theorem

analog
stem

Sampling Theorem

A continuous time signal (analog) can be completely represented by its samples and can be recovered back when the sampling frequency \( f_s \) is greater than or equal to the twice the highest frequency component \( f_m \) of the signal. i.e.

\( f_s\geq2*f_m \)

Examples

  • Human speech (100 Hz - 4,000 Hz)
  • Music (20 Hz - 20,000 Hz)
  • Temperatue sensor (~0 Hz - 0.25 Hz)

 

Reconstruction

There is only ONE bandlimited signal that fits the sampled points.

reconstruction
reconstruction2

Caveats

  • Sampled signal MUST be bandlimited -> Aliasing
  • Finite bit depth (e.g. 16 bits) -> Quantization noise

Pulse Code Modulation (PCM)

Sampling Rate and Bit Depth

stem

Bit depth = number of discrete levels -> Dynamic range

Sampling rate = number samples per second -> Bandwidth

Sampling Rate

\(f_s = 44.1 \) kHz -> \( f_m = 22.05 \) kHz

\( f_s = 22.05 \) kHz -> \( f_m = 11.025 \) kHz

\( f_s = 11.025 \) kHz -> \( f_m = 5.5125 \) kHz

Aliasing

aliasing_sines

Anti-Aliasing Filters

aa_filters
mix

1 kHz + 8 kHz @ 16bit 44.1 kHz

mix

1 kHz + 8 kHz @ 16bit 11.025 kHz

mix

1 kHz + 8 kHz @ 16bit 11.025 kHz (No Anti-Aliasing filter)

\(f_s = 22.05 \) kHz -> \( f_m = 11.025 \) kHz

\( f_s = 22.05 \) kHz -> \( f_m = 11.025 \) kHz (no anti-aliasing filter)

Bit Depth

16 bit -> 96 dB

8 bit -> 48 dB

Difference between 16 bit and 8 bit (with dither)

Difference between 16 bit and 8 bit (no dither)

Quantization Noise

quantization_noise

Dither

1 kHz sine wave @ 16 bit 44.1 kHz

1 kHz sine wave @ 8 bit 44.1 kHz (with dither)

1 kHz sine wave @ 8 bit 44.1 kHz (no dither)

Analog comparison

Format Dynamic range Effective Bit Depth
Cassette 40 dB 6 bits
Vinyl 60 dB 10 bits
Reel-to-Reel 80 dB 13 bits
CD 96 dB 16 bits
HD Audio 144 dB 24 bits

* These are estimates -> analog hardware performance varies

Further Reading

  • Sigma-Delta converter (PDM)
  • Error-correction codes (EFM, Reed-Solomon, etc.)
  • Perceptual Audio Coding (MP3, AAC, etc.)
  • Relevant ECE Courses
    • ECE 3300: Signals & Systems
    • ECE 4270: Communcation Systems
    • ECE 3170: Random Signal Analysis
    • ECE 4670: Digital Signal Processing

Next Talk - February 5

Spectral Analysis

Decomposing audio with algorithms