Intel® Integrated Performance Primitives (Intel® IPP) Developer Guide and Reference

ID 790148
Date 10/31/2024
Public
Document Table of Contents

Triangle-Generating Functions

This section describes the functions that generate a periodic signal with a triangular wave form (referred to as “triangle”) of a given frequency, phase, magnitude, and asymmetry.

A real periodic signal with triangular wave form x[n] (referred to as a real triangle) of a given frequency rFreq, phase value phase, magnitude magn, and asymmetry h is defined as follows:

x[n] = magn * cth(2π* rFreq*n + phase), n = 0, 1, 2,...

A complexl periodic signal with triangular wave form x[n] (referred to as a complex triangle) of a given frequency rFreq, phase value phase, magnitude magn, and asymmetry h is defined as follows:

x[n] = magn * [cth(2π* rFreq*n + phase) + j * sth(2π* rFreq*n + phase)], n = 0, 1, 2,...

The cth () function is determined as follows:

H = π + h



cth (α + k* 2π) = cth (α), k = 0, ±1, ±2, ...

When H = π , asymmetry h = 0, and function cth() is symmetric and a triangular analog of the cos() function. Note the following equations:

cth (H/2 + k*π) = 0, k = 0, ±1, ±2, ...

cth (k* 2π) = 1, k = 0, ±1, ±2, ...

cth (H + k* 2π) = -1, k = 0, ±1, ±2, ...

The sth () function is determined as follows:



sth (α + k* 2π) = sth (α), k = 0, ±1, ±2, ...

When H = π , asymmetry h = 0, and function sth() is symmetric and a triangular analog of the sine function. Note the following equations:

sth (α) = cth (α + (3π + h)/2) , k = 0, ±1, ±2, ...

sth (k* π) = 0, k = 0, ±1, ±2, ...

sth ((π -h)/2 + k* 2π) = 1, k = 0, ±1, ±2, ...

sth ((3π +h)/2 + k* 2π) = -1, k = 0, ±1, ±2, ...