Periodic Signals

**Strings (S _{i}P_{j}A_{jk}) = S_{7}P_{2}A_{21} Base Sequence = 12735 String Sequence = 12735 - 2 - 21 **

Expressions Of Pj Problems

Periodic Signals

Math

Pj Problems - Overview

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Signaling is a ubiquitious phenomenon in the societies of cognitive beings.

(a) What is a signal?

(b) What is a periodic signal?

(c) Identify the following periodic waveforms:

(i)

(ii)

(iii)

**The string**:

S_{7}P_{1}A_{17} (Containership -Location)

S_{7}P_{2}A_{21} (Identity - Physical Properties).
**The math**:

Pj Problem of interest is of type *identity*.

(a) *Signals* are essentially indicators of *presence* (the *being* in a location). Thereafter, one is interested in the *identity*, analyses and possible uses of the *presence*. In electric circuits, the voltage and current signals indicate the *presence* of voltage and current sources. The identities of voltage and current signals are expressed as *waveforms* (graphical representations of functions).

(b) *Periodic Signals* are *time-dependent cyclic signals*. In other words, there is an interval called the *period* of the signal during which the signal repeats the same waveform pattern.

Mathematically, an arbitrary signal x(t) is periodic if it can be expressed as follows:

x(t) = x(t + nT)-------(1)

Where n = 1, 2, 3 ...; and T is the *period* (the time it takes to complete 1 cycle) of x(t).

The *frequency* (f) of *x(t)* in cycles/sec (Hertz) is:

*f = 1/T*-------(2)

(ci) The waveforms in c(i) are cosine and sine periodic signals (sinusoids).

The red waveform is a cosine function. Its general form is:

v(t) = V_{max}cosω----(3)

Where the radian frequency (angular velocity), ω = 2πf.

The signal expressed in (3) is often the reference sinusoid.

A sinusiodal signal has the general form:

v(t) = V_{max}cos(ωt + θ)-----(4)

The sinusiodal signal expressed in equation (4) is said to *lead* the sinusiodal signal expressed in equation (3) by *phase angle, θ*.

The *phase angle θ* = *ωτ*

Where τ is the *time shift*

The *Blue* waveform is a sine function. Its general form is:
*v(t) = V _{max}sinωt*----(5)

The relationship between the cosine signal and the sine signal is:

V

Since the cosine signal

The relationship can also be expressed as follows:

V

Since the sine signal

(cii) The waveforms in c(ii) is a periodic pulse signal.

(ciii) The waveform in c(iii) is a periodic sawtooth signal.

The *point* **.** is a mathematical abstraction. It has negligible size and a great sense of position. Consequently, it is front and center in abstract existential reasoning.

Single Variable Functions

Conics

Ordinary Differential Equations (ODEs)

Vector Spaces

Real Numbers

Separation Of Variables As Solution Method For Homogeneous Heat Flow Equation

Newton And Fourier Cooling Laws Applied To Heat Flow Boundary Conditions

Fourier Series

Derivation Of Heat Equation For A One-Dimensional Heat Flow

The Universe is composed of *matter* and *radiant energy*. *Matter* is any kind of *mass-energy* that moves with velocities less than the velocity of light. *Radiant energy* is any kind of *mass-energy* that moves with the velocity of light.

Periodic Table

Composition And Structure Of Matter

How Matter Gets Composed

How Matter Gets Composed (2)

Molecular Structure Of Matter

Molecular Shapes: Bond Length, Bond Angle

Molecular Shapes: Valence Shell Electron Pair Repulsion

Molecular Shapes: Orbital Hybridization

Molecular Shapes: Sigma Bonds Pi Bonds

Molecular Shapes: Non ABn Molecules

Molecular Orbital Theory

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