﻿ Periodic Signals

Periodic Signals

Strings (SiPjAjk) = S7P2A21     Base Sequence = 12735     String Sequence = 12735 - 2 - 21

Expressions Of Pj Problems
Periodic Signals
Math

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:
S7P1A17 (Containership -Location)
S7P2A21 (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) = Vmaxcosω----(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) = Vmaxcos(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) = Vmaxsinωt----(5)

The relationship between the cosine signal and the sine signal is:
Vmaxcosωt = Vmaxsin(ωt + π/2)----(6).
Since the cosine signal leads the sine signal by π/2.
The relationship can also be expressed as follows:
Vmaxsinωt = Vmaxcos(ωt - π/2)----(7).
Since the sine signal lags the cosine signal by π/2.

(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.
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Blessed are they that have not seen, and yet have believed. John 20:29