Fourier Transform, Fourier Series, and frequency spectrum

Summary

This script explores the concept of sine waves, illustrating how they can be manipulated in amplitude, phase, and frequency. It explains that adding sine waves of the same or different frequencies results in new waveforms, with identical frequencies summing to another sine wave, while different frequencies create complex patterns. The script delves into the idea that any waveform can be constructed from an infinite combination of sine waves, even non-repeating ones, by using an infinite number of sine waves with infinitesimal amplitudes. It concludes by likening this to the accumulation of paper sheets to form a tangible volume, drawing a parallel to the frequency spectrum and its importance in understanding signal interactions with physical objects.

Takeaways

• 📏 The script introduces the concept of a sine wave, which is a pattern formed by the rotation of a line representing the angle theta, with its X and Y coordinates being the cosine and sine of theta, respectively.
• 🔄 The angle theta in a sine wave can vary from negative to positive infinity, creating a continuous two-dimensional wave pattern.
• 🔧 The properties of a sine wave, such as amplitude, phase, and frequency, can be modified, affecting the shape and characteristics of the wave.
• 🔄 When two sine waves with different amplitudes are added, the result is a new sine wave with a combined amplitude, maintaining the same frequency.
• 🔄 Adding sine waves with different phases results in a wave with a new phase shift, but the frequency remains unchanged.
• 🔄 The sum of sine waves with different frequencies does not result in a sine wave, but rather a more complex waveform.
• ∞ By combining an infinite number of sine waves, complex patterns and waveforms can be created, illustrating the principle of Fourier series.
• 📊 The frequency spectrum of a waveform is represented by the density of frequencies, which can be higher around some frequencies than others, and is crucial for understanding signal interactions.
• 📈 Non-repeating waveforms can be generated by adding sine waves of every possible frequency, each with an infinitely small amplitude.
• 📚 The script suggests that every waveform or function can be represented as a sum of sine waves, highlighting the fundamental role of sine waves in signal analysis.
• 🌐 Real-life signals and waveforms can be thought of as combinations of sine waves that have always existed and will continue to exist eternally, with their frequency spectrums being altered upon interaction with physical objects.

Q & A

What is the relationship between the angle theta and the coordinates on the sine wave?

For each value of theta, the X coordinate on the sine wave represents the cosine of theta, and the Y coordinate represents the sine of theta.

What are the three characteristics of a sine wave that can be altered?

The three characteristics of a sine wave that can be altered are amplitude, phase, and frequency.

What happens when two sine waves with different amplitudes but the same frequency are added together?

When two sine waves with different amplitudes but the same frequency are added, the result is a sine wave with a different amplitude but the same frequency.

How does the phase difference between two sine waves affect their sum?

If two sine waves have different phases, their sum can be represented graphically with a different phase shift, but the frequency remains the same.

What is the result of adding sine waves of different frequencies together?

When sine waves of different frequencies are added together, the resultant waveform is no longer a sine wave.

Why can an infinite number of sine waves produce a wide variety of patterns?

An infinite number of sine waves can produce a wide variety of patterns because each sine wave can have different frequencies, amplitudes, and phases, allowing for complex combinations.

What is the significance of repeating waveforms in the context of adding sine waves?

Repeating waveforms result when the sum of sine waves contains only certain frequencies with measurable amplitudes, which can be easily identified and analyzed.

How are non-repeating waveforms generated by adding sine waves?

Non-repeating waveforms are generated by adding sine waves of every possible frequency, where each sine wave has an infinitely small amplitude.

What is the concept of the frequency spectrum of a waveform?

The frequency spectrum of a waveform refers to the distribution of frequencies and their amplitudes within the waveform, which can be measured and analyzed.

How does the frequency spectrum change when signals and waveforms interact with physical objects?

When signals and waveforms interact with physical objects, their frequency spectrum is altered, which can be studied to understand how the signals and waveforms are affected.

What is the philosophical implication of considering all signals as combinations of infinite sine waves?

The philosophical implication is that all signals we observe in real life can be thought of as combinations of sine waves that have always been present and will continue to exist eternally, highlighting the continuous and eternal nature of waveforms.

Outlines

00:00

📚 Understanding Sine Waves and Their Combinations

This paragraph introduces the concept of sine waves, which are two-dimensional patterns formed by the coordinates representing the cosine and sine of an angle theta. It explains how the amplitude, phase, and frequency of a sine wave can be altered. The paragraph further demonstrates that by adding two identical sine waves with different amplitudes and phases, the result is a new sine wave with a unique amplitude and phase, but the same frequency. It also discusses the addition of sine waves with different frequencies, which results in a waveform that is no longer a simple sine wave. The paragraph concludes by illustrating the addition of multiple sine waves, emphasizing that any waveform can be created by combining sine waves.

05:00

🌌 The Infinite Possibilities of Sine Waves

This paragraph delves into the idea that an infinite number of sine waves can be combined to create various patterns, including non-repeating waveforms. It explains that while repeating waveforms require specific frequencies with measurable amplitudes, non-repeating waveforms might involve sine waves of every possible frequency, each with an infinitely small amplitude. The analogy of stacking infinitely thin sheets of paper to create a volume is used to illustrate how an infinite sum of sine waves can result in a measurable waveform. The concept of the frequency spectrum is introduced, highlighting its importance in understanding how signals and waveforms interact with physical objects and how they are altered.

10:01

🌀 The Eternal Nature of Sine Waves

This paragraph explores the philosophical and theoretical implications of sine waves, suggesting that all real-life signals and waveforms can be viewed as combinations of an infinite number of sine waves that have always existed and will continue to exist eternally. It emphasizes that while these signals have a beginning and an end in our perception, they can be conceptualized as the sum of sine waves without a start or finish, which only cancel each other out except during the signal's presence. The paragraph concludes by inviting viewers to explore more about mathematics through other videos on the channel and encourages subscription for updates.

15:04

📢 Invitation to Learn More Mathematics

This final paragraph serves as a call to action, inviting viewers to seek more information about mathematics through additional videos on the channel. It also prompts viewers to subscribe for notifications about new video releases, ensuring they stay updated with the latest content.

Keywords

💡Sine Wave

A sine wave is a mathematical curve that describes a smooth periodic oscillation. In the video, it is used to illustrate how signals can be represented as a combination of sine waves. The sine wave's properties, such as amplitude, phase, and frequency, can be manipulated to create various waveforms.

💡Amplitude

Amplitude refers to the height of the wave from its central axis to its peak. It represents the strength or intensity of the wave. In the video, changing the amplitude of sine waves alters the resulting waveform when multiple sine waves are added together.

💡Phase

Phase is the position of a point in time on a waveform cycle. It determines the horizontal shift of the wave. The video discusses how different phases of sine waves affect the resulting waveform when they are summed together.

💡Frequency

Frequency is the number of cycles a wave completes in a given time period. It determines the wave's pitch or tone. The video highlights how combining sine waves with different frequencies results in a waveform that is no longer a sine wave.

💡Theta

Theta (θ) is a variable representing an angle in trigonometry. In the video, it is used to describe the rotation of a green line on a graph, where the X coordinate represents the cosine of θ and the Y coordinate represents the sine of θ.

💡Frequency Spectrum

The frequency spectrum is a representation of the different frequencies present in a waveform. It shows how the density of frequencies varies. The video explains that all signals and waveforms have a frequency spectrum, which is altered when they interact with physical objects.

💡Waveform

A waveform is a graphical representation of a signal's variation over time. The video discusses how different waveforms can be generated by adding together sine waves with varying amplitudes, phases, and frequencies.

💡Infinite Sine Waves

This concept refers to the idea that any waveform can be created by summing an infinite number of sine waves with infinitely small amplitudes. The video uses this to illustrate how complex waveforms can be constructed from simple sine waves.

💡Non-repeating Waveforms

Non-repeating waveforms do not repeat at regular intervals. The video explains that these waveforms can also be generated by adding sine waves, but in this case, sine waves of every possible frequency may be needed.

💡Mathematical Representation

This term refers to the use of mathematical concepts and equations to describe and analyze waveforms and signals. The video uses mathematical representations to explain how combining sine waves can generate various waveforms and how this understanding can be applied in real life.

Highlights

Introduction of the concept of a sine wave, defined by the cosine and sine of angle theta for each X and Y coordinate respectively.

Explanation of the three key properties of sine waves: amplitude, phase, and frequency.

Demonstration of how adding two sine waves with different amplitudes results in a new sine wave with a combined amplitude.

Illustration of the effect of adding sine waves with different phases, resulting in a shifted waveform.

Statement that sine waves with the same frequency, when added, always result in another sine wave of the same frequency but with altered amplitude and phase.

Observation that adding sine waves of different frequencies does not result in a sine wave, indicating the importance of frequency matching.

Concept of adding multiple sine waves to create complex waveforms.

The idea that an infinite number of sine waves can be added together to produce a wide variety of patterns.

Every possible waveform and function can be generated by adding different sets of sine waves, highlighting the fundamental role of sine waves in signal analysis.

Description of how repeating waveforms are created by adding sine waves with specific frequencies and measurable amplitudes.

Introduction of non-repeating waveforms which require sine waves of every possible frequency with infinitely small amplitudes.

Analogy of adding infinitely thin sheets of paper to explain the concept of creating a measurable waveform from an infinite number of sine waves with small amplitudes.

Explanation of the frequency spectrum of a waveform and how it represents the density of frequencies.

Discussion on how signals and waveforms interact with physical objects and how their frequency spectrums are altered.

Philosophical perspective that real-life signals can be thought of as combinations of infinite sine waves that have always been present.

Invitation to explore more about mathematics through other videos on the channel and a call to subscribe for updates.

Transcripts

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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