Ontological Layer: Temporal Elasticity

Multirate DSP

Changing the Speed of Digital Information Without Destroying the Signal

The Rate Shifter

Sometimes the sampling rate dictated by hardware does not match the rate required for efficient algorithmic processing or transmission. Realm 13 provides the tools to alter the discrete fabric of time itself. We learn to safely throw samples away (Decimation) by first applying Anti-Alias filters, and to create new samples from nothing (Interpolation) by stuffing zeros and applying Anti-Imaging filters. Combining these allows us to seamlessly convert signals between any rational rates.

INPUT RATE fs → UPSAMPLE (INSERT ZEROS) → LOW-PASS FILTER → DOWNSAMPLE (DROP) → OUTPUT RATE fs(L/M)

The 14 Multirate Nodes

Ontological Mapping: Interpolation, Decimation, and Polyphase

CHAMBER 01

Why Multirate DSP?

Matching disparate systems (Audio/Comm)
Reducing computational load
Oversampled ADC/DAC architectures
More efficient narrowband filtering

CHAMBER 02

Decimation (↓M)

Retaining every M-th sample
y[n] = x[Mn]
Lowers the sampling frequency
Reduces data bandwidth and processing

CHAMBER 03

Decimation Aliasing

Dropping samples lowers Nyquist limit
High frequencies fold back into baseband
Must enforce: f_max < f_s / (2M)
The necessity of Anti-Alias filtering

CHAMBER 04

Upsampling (↑L)

Inserting L-1 zeros between samples
Increases the sampling frequency
Does not add new information initially
Prepares sequence for interpolation

CHAMBER 05

Anti-Imaging Filter

Zero-insertion compresses the spectrum
Creates high-frequency spectral "images"
Low-pass filter removes these replicas
Smoothes the zero-stuffed time sequence

CHAMBER 06

Full Interpolation

Upsampling (↑L) + Anti-Imaging Filter
Turns zeros into correct intermediate values
Sinc interpolation in time domain
Gain scaling required (multiply by L)

CHAMBER 07

Combined Operations

Cascading Interpolation then Decimation
Order matters (Interpolate first retains info)
Enables non-integer rate changes
Overall factor: f_out = f_in × (L/M)

CHAMBER 08

Rational Conversion

Converting rates by factor L/M
Example: 48 kHz to 44.1 kHz (L=147, M=160)
Single monolithic filter architecture
Highly efficient implementation basis

CHAMBER 09

Multirate Filtering

One filter handles anti-aliasing AND anti-imaging
Cutoff frequency ωc = min(π/L, π/M)
Filter operates at the highest intermediate rate
Stringent transition band requirements

CHAMBER 10

Polyphase Structures

Avoids multiplying by zero (Upsampling)
Avoids calculating discarded samples (Decimation)
Decomposes filter H(z) into parallel phases
Massive computational savings

CHAMBER 11

Multistage Conversion

Factoring large rate changes (e.g., M=12 → 3,2,2)
Reduces steep filter transition requirements
Lowers overall computational cost
Relaxed specifications for early stages

CHAMBER 12

Multirate Applications

Audio Sample Rate Converters (ASRC)
Software Defined Radio (SDR) decimation
Oversampled Delta-Sigma ADCs / DACs
Image resizing and resolution scaling

CHAMBER 13

Multirate Classification

Integer Decimation vs Integer Interpolation
Rational Conversion (L/M)
Direct Implementation vs Polyphase
Single-stage vs Multistage execution

FINAL PROTOCOL

14 — The Rate Shifter Laboratory

1. IN: 48 kHz. TARGET: 44.1 kHz

2. UPSAMPLE L=147 (New fs = 7.056 MHz)

3. FILTER ωc = min(π/147, π/160)

4. DECIMATE M=160 → OUT: 44.1 kHz

[1] SET RATIO L/M [2] ZERO-STUFF [3] APPLY LOW-PASS [4] DROP SAMPLES [5] MEASURE ALIASING

You have mastered the elasticity of digital time. By understanding how to carefully expand and contract sequences using Interpolation and Decimation, you can bridge disparate digital worlds and optimize computational performance. This prepares us for the final theoretical frontier: Realm 14, Advanced DSP and Application Domains.

MULTIRATE DIAGNOSTIC RUNNING