We have an X axis and a Y axis.
Let us also create an axis that represents the angle theta.
The green line rotates by the angle theta.
For each value of theta, the X coordinate represents the cosine of theta,
and the Y coordinate represents the sine of theta.
The angle theta can range anywhere between negative infinity and positive infinity.
This two dimensional pattern is what we refer to as a sine wave.
We can change the sine wave’s amplitude,
phase,
and frequency.
Suppose we have two sine waves that are identical
except for the fact that they have different amplitudes.
If we add these two sine waves together,
the result can be represented graphically as shown.
Now, suppose that the two sine waves also have different phases.
The sum of these two wave forms can be represented graphically like this.
So long as the two sine waves have the same frequency, their sum will always be
another sine wave of the exact same frequency, but with a different amplitude and phase.
Now, suppose that the two sine waves also have a different frequency.
If we add together sine waves of different frequencies,
then the resultant waveform is no longer a sine wave.
We can add together three sine waves.
We can add together four sine waves.
We can add together five sine waves.
In fact, we can add together an infinite number of sine waves.
Here, by adding together an infinite number of sine waves,
we have produced a pattern that looks like this.
Now, let’s add together a different set of sine waves.
By adding an infinite number of sine waves,
we have produced this other pattern that looks like this.
As it turns out, every possible waveform and function in existence
can be generated by just adding together different sets of sine waves.
In the two examples shown here,
the sum of all the sine waves happened to be repeating waveforms.
In these cases, only certain frequencies of sine waves were needed,
and each of these sine waves had a measurable amplitude.
Non-repeating waveforms can also be generated by adding sine waves together,
but in these cases, sine waves of every possible frequency may be needed,
and each of these sine waves have an amplitude that is infinitely small.
When an infinite number of sine waves with infinitely small amplitudes
are added together, the result can be a waveform that we can see and measure.
This is the same way in which if we add together an infinite number of infinitely
thin sheets of paper, although each sheet of paper by itself has zero volume,
their combination can be an object with an actual volume that we can see and measure.
Some sheets of paper can be larger than others.
Although the volume of each sheet of paper by itself is zero,
when we have an infinite number of sheets of paper added together,
we will be able to measure the object’s density,
and the density in some parts of the object
will be higher than the density in other parts of the object.
Similarly, when we add together an infinite number of sine waves
of infinitely small amplitudes, we will be able to measure the density of frequencies,
and this density of frequencies will be higher around some frequencies than others.
This is what we refer to as the frequency spectrum of a waveform.
All signals and waveforms have a frequency spectrum.
When signals and waveforms interact with physical objects,
their frequency spectrum is altered.
By understanding how their frequency spectrums are altered,
we can understand how the signals and waveforms are altered.
The signals and waveforms that we see in real life have a beginning and an end.
However, each of these signals and waveforms can be thought of as the combination
of an infinite number of sine waves, and each sine wave has no beginning or ending.
It is just that all these sine waves exactly cancel each other out at all times
except during the time that the signal is present.
Therefore, all signals that we see in real life can be thought of as the combination
of an infinite number of sine waves that have always been present
since the beginning of time, and which will continue to exist through all eternity.
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