gen_qn: Generate a Quantisation Noise (QN) or Rounding Error Sequence

In SMAC-Group/simts: Time Series Analysis Tools

Generate a Quantisation Noise (QN) or Rounding Error Sequence

Description

Simulates a QN sequence given Q^2.

Usage

gen_qn(N, q2 = 0.1)

Arguments

N	An integer for signal length.
q2	A double that contains autocorrection.

Value

A vec containing the QN process.

Process Definition

Quantization Noise (QN) with parameter Q^2 \in R^{+}. With i.i.d Y_t \sim U(0,1) (i.e. a standard uniform variable), this process is defined as:

X_t = \sqrt{12Q^2}(Y_{t}-Y_{t-1})

Generation Algorithm

To generate the quantisation noise, we follow this recipe: First, we generate using a random uniform distribution:

U_k^*\sim U\left[ {0,1} \right]

Then, we multiple the sequence by \sqrt{12} so:

{U_k} = \sqrt{12} U_k^*

Next, we find the derivative of {U_k}

{{\dot U}_k} = \frac{{{U_{k + \Delta t}} - {U_k}}}{{\Delta t}}

In this case, we modify the derivative such that: {{\dot U}_k}\Delta t = {U_{k + \Delta t}} - {U_k}

Thus, we end up with:

{x_k} = \sqrt Q {{\dot U}_k}\Delta t

{x_k} = \sqrt Q \left( {{U_{k + 1}} - {U_k}} \right)

SMAC-Group/simts documentation built on Sept. 4, 2023, 5:25 a.m.