Fourier Series introduction

1. Diagram
   Observation

   The vertical axis is labeled y and the plotted yellow waveform is labeled $y=f(t)$.

$y=f(t)$

The periodic square-wave function being represented as a sum of trigonometric terms.

t is time in seconds; the displayed graph shows one period from 0 to $2\pi$ and repeats.

1. Audio
   Observation

   The speaker says the horizontal axis is in terms of time, in terms of seconds.

2. Diagram
   Observation

   The horizontal axis is labeled $t(s)$, with tick marks at $2\pi$ and $4\pi$.

t

Time variable measured in seconds.

Real-valued time variable shown on the horizontal axis.

1. Audio
   Observation

   The speaker says its period is equal to $2\pi$ and mentions units of seconds per cycle.

2. Formula
   Observation

   The board writes $T = 2\pi \frac{sec}{cycle}$.

T

Period of the square-wave function $f(t)$.

Positive time interval per cycle; here $T=2\pi$ seconds per cycle.

1. Audio
   Observation

   The speaker says frequency is going to be one over that and can also be described as hertz.

2. Formula
   Observation

   The board writes freq = $\frac{1}{2\pi} \frac{cycles}{sec}$.

freq

Frequency of the periodic function, reciprocal of the period.

Cycles per second for this example; numerically $\frac{1}{2\pi}$ when $T=2\pi$.

1. Audio
   Observation

   The speaker calls it a baseline constant that will shift the function up or down and says it is based on the average value over one period.

2. Formula
   Observation

   The expression begins $f(t)=a_0$+.

$a_0$

Baseline constant term in the proposed Fourier-series representation.

Constant coefficient associated with the average value of $f(t)$ over one period.

1. Audio
   Observation

   The speaker says let's take a sub 1 times cosine of t.

2. Formula
   Observation

   The board writes + $a_1\cos(t)$.

$a_1$

Coefficient multiplying the first cosine term in the proposed expansion.

Constant coefficient for the $\cos(t)$ component.

1. Audio
   Observation

   The speaker says plus b sub 1 of sine of t.

2. Formula
   Observation

   The board writes + $b_1\sin(t)$.

$b_1$

Coefficient multiplying the first sine term in the proposed expansion.

Constant coefficient for the $\sin(t)$ component.

1. Formula
   Observation

   The expression begins as $f(t)=a_0+a_1\cos(t)+b_1\sin(t)$ and is later extended with higher harmonics.

2. Audio
   Observation

   The speaker refers to representing a function by weighted sines and cosines.

$f(t)$

The periodic target function being represented as a sum of trigonometric terms.

Periodic function of time t; in the example shown, period $T=2\pi$.

1. Diagram
   Observation

   Yellow label $y=f(t)$ points to the square-wave graph.

$y=f(t)$

Graph label identifying the plotted waveform as the function f of t.

Coordinate plane with horizontal axis $t(s)$ and vertical axis y.

1. Formula
   Observation

   Written as the first term in $f(t)=a_0+a_1\cos(t)+b_1\sin(t)$.

2. Audio
   Observation

   The speaker discusses weights that tell how much each component is involved.

1. The video does not explicitly name $a_0$ as the constant or DC term in this clip.

$a_0$

Constant coefficient in the Fourier-series representation.

Real-valued coefficient.

1. Formula
   Observation

   Appears in $a_1\cos(t)$.

2. Audio
   Observation

   The speaker says if $a_1$ is much larger than $b_1$, the function has more cosine of t than sine of t.

$a_1$

Weighting coefficient for the fundamental cosine term $\cos(t)$.

Real-valued coefficient.

1. Formula
   Observation

   Appears in $b_1\sin(t)$.

2. Audio
   Observation

   Compared directly with $a_1$ when explaining relative contribution of cosine and sine.

$b_1$

Weighting coefficient for the fundamental sine term $\sin(t)$.

Real-valued coefficient.

1. Audio
   Observation

   The speaker describes the graph as a square wave and says it is a periodic function that completes one cycle every $2\pi$ seconds.

2. Diagram
   Observation

   A yellow rectangular pulse train is drawn on axes labeled y and $t(s)$, with repeated pulses aligned to tick marks at $2\pi$ and $4\pi$.

The video introduces a square-wave function $y=f(t)$ whose graph repeats after a fixed time interval. The spoken description identifies the period as $2\pi$ seconds, matching the visible repetition of the rectangular pulses.

1. The function is periodic.

2. One full cycle takes $2\pi$ seconds in this example.

1. Audio
   Observation

   The speaker says frequency is going to be one over that and can also be described as hertz.

2. Formula
   Observation

   The board writes freq = $\frac{1}{2\pi} \frac{cycles}{sec}$.

For this square wave, the frequency is defined as the reciprocal of the period. Since the period is $2\pi$ seconds per cycle, the frequency is written as $\frac{1}{2\pi}$ cycles per second, equivalently hertz.

1. The function has a well-defined period T.

2. Units are cycles per second when T is measured in seconds.

1. Square wave as a periodic function

1. Audio
   Observation

   The speaker asks whether a periodic function like this can be represented as an infinite sum of sines and cosines of different periods or frequencies.

The central question posed in the clip is whether the given periodic square wave can be rewritten as a sum of sinusoidal components. This motivates the later written expression beginning with a constant term plus cosine and sine terms.

1. The target function is periodic.

2. The proposed representation uses sines and cosines with different periods or frequencies.

1. Square wave as a periodic function

1. Audio
   Observation

   The speaker says there will be some baseline constant that shifts it up or down and that it is based on the average value of the function over one period.

2. Formula
   Observation

   The expression starts as $f(t)=a_0$+.

In the proposed expansion, $a_0$ is introduced as a constant offset. The speaker explicitly links this term to the average value of the function over one period, distinguishing it from the oscillatory sine and cosine terms.

1. The representation is being built term by term.

2. $a_0$ accounts for vertical shift or mean level over one period.

1. Goal of representing a periodic function by sines and cosines

1. Audio
   Observation

   The speaker adds $a_1\cos(t)$ and then $b_1\sin(t)$, asking why these are chosen.

2. Formula
   Observation

   The board shows $f(t)=a_0+a_1\cos(t)+b_1\sin(t)$.

After the constant term, the video writes the first oscillatory pieces as $a_1\cos(t)$ and $b_1\sin(t)$. These are presented as the initial periodic components in the attempted representation of $f(t)$.

1. The original function has period $2\pi$ in this setup.

2. The chosen basic trigonometric arguments are t.

1. Baseline constant $a_0$ in the series
2. Square wave as a periodic function

1. Formula
   Observation

   The board builds $f(t)=a_0+a_1\cos(t)+a_2\cos(2t)+a_3\cos(3t)+\cdots+b_1\sin(t)+b_2\sin(2t)+b_3\sin(3t)+\cdots$.

2. Audio
   Observation

   The speaker describes representing something by essentially weighted sines and cosines.

3. Caption evidence
   Observation

   At 01:24-01:28 the title 'Fourier Series' is written on screen.

This clip introduces a Fourier series as an infinite trigonometric expansion of a periodic function. The visible construction starts from a constant term plus one cosine and one sine at the base frequency, then adds higher harmonics $\cos(2t)$, $\cos(3t)$, $\sin(2t)$, $\sin(3t)$, and so on indefinitely. The coefficients $a_n$ and $b_n$ act as weights controlling how much each harmonic contributes.

1. Applied here to a periodic function with period $2\pi$.

2. The series is presented formally with ellipses rather than a fully justified convergence statement.

1. Period and frequency of the displayed waveform
2. Effect of replacing t by 2t on frequency

1. Diagram
   Observation

   The graph is annotated $T=2\pi$ sec/cycle and freq=$1/(2\pi)$ cycles/sec.

2. Audio
   Observation

   The speaker refers to periods of $2\pi$ and later compares frequencies of $\cos(t)$ and $\cos(2t)$.

The square wave shown on the board has period $2\pi$ seconds per cycle. Its frequency is therefore $1/(2\pi)$ cycles per second. This establishes the base oscillation scale for the trigonometric terms that follow.

1. Refers specifically to the plotted waveform in this clip.

1. Formula
   Observation

   The term $\cos(2t)$ is added after $\cos(t)$.

2. Audio
   Observation

   The speaker says, 'This has a frequency of 1 over 2 pi. This has twice the frequency. This has a frequency of 1 over pi.'

The clip uses $\cos(2t)$ to show that doubling the argument of the cosine doubles the frequency relative to $\cos(t)$. Since the base term $\cos(t)$ corresponds to frequency $1/(2\pi)$, the term $\cos(2t)$ corresponds to frequency $1/\pi$. The same harmonic logic is then extended to $\cos(3t)$, $\sin(2t)$, and $\sin(3t)$.

1. Discussed for the period-2\pi setting shown on the board.

1. Period and frequency of the displayed waveform

1. Audio
   Observation

   The speaker says Fourier series were interesting in the study of differential equations because many differential equations are easy to solve with sines and cosines but less obvious for more general functions like a square wave.

The video motivates Fourier series by explaining that differential equations often become easier to solve when the forcing or input can be expressed using sines and cosines. If a more complicated periodic function such as a square wave can be rewritten as a sum of sinusoids, then one may obtain more general solutions to differential equations.

1. Presented as motivation rather than a worked theorem or proof.

1. Fourier series as a weighted sum of sines and cosines

1. Audio
   Observation

   The speaker says this is really the foundation of signal processing and is very useful.

1. The sentence is cut off before the speaker elaborates.

Near the end of the clip, the speaker states that representing functions as sums of sinusoids is foundational to signal processing. The clip does not develop this application further before ending.

1. Only a brief introductory claim is given in this excerpt.

1. Fourier series as a weighted sum of sines and cosines

1. Formula
   Observation

   The board displays $f(t)=a_0+a_1 \cos (t)+a_2 \cos (2t)+a_3 \cos (3t)$+… + $b_1 \sin (t)+b_2 \sin (2t)+b_3 \sin (3t)$+…

2. Audio
   Observation

   The speaker says these coefficients are weights on cosines and sines.

The clip presents a periodic function $f(t)$ as a sum of a constant term plus cosine and sine terms with integer multiples of t inside the trigonometric arguments. The coefficients $a_n$ and $b_n$ multiply those cosine and sine terms.

1. The displayed example is for a periodic function.

2. The video does not state convergence conditions or the general period normalization beyond the shown example.

1. $f(t)$
2. $a_0$
3. $a_1$
4. $a_2$
5. $a_3$
6. $b_1$
7. $b_2$
8. $b_3$

1. Formula
   Observation

   The board writes $T = 2π$ sec/cycle and freq = $1/(2π)$ cycles/sec.

2. Diagram
   Observation

   The square wave repeats along the t-axis with marks at 2π and 4π.

For the plotted periodic signal, the board identifies one cycle as lasting 2π seconds and states the corresponding frequency as $1/(2π)$ cycles per second. This matches the reciprocal relation between period and frequency.

1. Applies to the specific periodic signal drawn on the board.

2. The unit labels are explicitly written on the board.

1. T
2. freq
3. t

1. Audio
   Observation

   The speaker states that the function completes one cycle every $2\pi$ seconds and writes the period as $2\pi$ sec/cycle.

2. Formula
   Observation

   The board writes $T = 2\pi \frac{sec}{cycle}$.

The displayed periodic square wave has period $T=2\pi$ seconds per cycle.

1. The graph shown is the function under discussion.

2. The horizontal axis is time in seconds.

For the specific function $y=f(t)$ drawn in this clip.

1. Audio
   Observation

   The speaker says frequency is one over the period and can be described as hertz.

2. Formula
   Observation

   The board writes freq = $\frac{1}{2\pi} \frac{cycles}{sec}$.

If the period is $T=2\pi$ seconds per cycle, then the frequency is $\frac{1}{2\pi}$ cycles per second.

1. The function is periodic with period $T=2\pi$.

2. Frequency is defined as the reciprocal of period.

For this example function and the stated definition of frequency.

1. Audio
   Observation

   The speaker asks why start with cosine of t and sine of t, then says that if the original function has a period of $2\pi$, it would make sense to involve functions that have...

2. Formula
   Observation

   The written terms use $\cos(t)$ and $\sin(t)$.

1. The sentence explaining the reason is cut off before completion, so the full justification is not audible in this clip.

Because the original function has period $2\pi$, the speaker indicates that it makes sense to begin with trigonometric functions whose natural period matches that choice.

1. The original function has period $2\pi$.

2. The candidate basis functions are $\cos(t)$ and $\sin(t)$.

For the setup in this clip where the base period is $2\pi$.

1. Audio
   Observation

   The speaker says that $a_1\cos(t)+b_1\sin(t)$ by itself is not going to describe the function because it would look like a very clean sinusoid, not like a square wave.

2. Diagram
   Observation

   The graph on screen is a square wave labeled $y=f(t)$.

The combination $a_1\cos(t)+b_1\sin(t)$ alone produces a sinusoidal shape, so it cannot by itself equal the square wave shown on the board.

1. The target function is the displayed square wave with period $2\pi$.

2. Only the fundamental-frequency cosine and sine terms are used.

For the specific square-wave example shown in this clip.

1. Audio
   Observation

   The speaker says they will add sinusoids of frequencies that are multiples of these frequencies.

2. Formula
   Observation

   The board is extended with $\cos(2t)$, $\cos(3t)$, $\sin(2t)$, $\sin(3t)$, and ellipses.

1. The clip asserts the need for higher harmonics but does not prove convergence or exact equality for the square wave.

To approximate the square wave beyond a simple sinusoid, the series must include sinusoids whose frequencies are integer multiples of the fundamental frequency.

1. The function being modeled is periodic with period $2\pi$.

2. The representation uses sine and cosine terms.

Stated for the construction being built in this clip; no universal theorem is proved here.

1. Audio
   Observation

   The speaker says these infinite series were explored originally by Fourier and are called Fourier series.

2. Caption evidence
   Observation

   The title 'Fourier Series' is written on screen.

Infinite series that represent functions by weighted sines and cosines are called Fourier series, after Fourier.

Historical naming statement given in the lecture.

1. Audio
   Observation

   The speaker says a lot of differential equations are easy to solve when you involve sines and cosines, but not as obvious when you have more general functions like a square wave.

1. No specific differential equation is shown or solved in this clip.

Many differential equations are easier to solve when the involved functions are sines and cosines than when they are more general periodic functions.

1. The discussion concerns solving differential equations.

Broad qualitative claim made by the lecturer without a formal proof in this excerpt.

1. Audio
   Observation

   The speaker says, “if a1 is much bigger than a2, then that tells you that that function contains a lot more of the 1 over 2 pi hertz frequency than the 1 over pi frequency.”

2. Audio
   Observation

   The speaker adds that maybe a3 is bigger than a1 or a2.

1. The spoken frequency labels are tied to the displayed series, but the clip does not write those frequency values separately on the board.

In the displayed Fourier-series setup, if $a_1$ is much larger than $a_2$, then the function contains more of the $1/(2π)$ hertz frequency than of the $1/π$ frequency; similarly, $a_3$ may dominate $a_1$ or $a_2$.

1. The function is represented in the displayed form with cosine coefficients $a_1$, $a_2$, $a_3$.

2. The comparison is about the coefficients multiplying $\cos (t)$, $\cos (2t)$, and $\cos (3t)$.

For the example coefficients discussed in the clip.

1. Audio
   Observation

   The speaker says Fourier series and eventually Fourier transforms will get us into the frequency domain where we can start doing some signal processing.

The clip states that Fourier series, and later Fourier transforms, move analysis from the time domain into the frequency domain, enabling signal processing.

1. The discussion concerns periodic functions represented by Fourier series.

General claim made verbally about the role of Fourier series and Fourier transforms.

1. Audio
   Observation

   The speaker first identifies the period as $2\pi$ seconds per cycle, then says frequency is one over that.

2. Formula
   Observation

   The board shows $T = 2\pi \frac{sec}{cycle}$ followed by freq = $\frac{1}{2\pi} \frac{cycles}{sec}$.

1. Expression

   $$T = 2\pi \frac{sec}{cycle}$$
   Explanation

   The period of the square wave is read from the graph and stated verbally as one cycle every $2\pi$ seconds.

   Justification

   Direct observation of the repeating waveform and the speaker's explicit statement.

   Shown in the video
2. Expression

   $$freq = \frac{1}{T}$$
   Explanation

   Frequency is taken as the reciprocal of period.

   Justification

   Stated definition in the audio: frequency is going to be one over that.

   Shown in the video
3. Expression

   $$freq = \frac{1}{2\pi} \frac{cycles}{sec}$$
   Explanation

   Substituting $T=2\pi$ gives the numerical frequency for this example.

   Justification

   Algebraic substitution into the reciprocal relation.

   Derived from the video

For the displayed square wave, the period is $2\pi$ seconds per cycle and the corresponding frequency is $\frac{1}{2\pi}$ cycles per second.

1. Audio
   Observation

   The speaker asks whether $f(t)$ can be written as the sum of sines and cosines, then adds a baseline constant and the first cosine and sine terms.

2. Formula
   Observation

   The board progressively writes $f(t)=a_0+a_1\cos(t)+b_1\sin(t)$.

1. Expression

   $$f(t)=?$$
   Explanation

   The starting point is the given periodic function whose representation is being sought.

   Justification

   The speaker explicitly asks whether this $f(t)$ can be rewritten as a sum of sines and cosines.

   Shown in the video
2. Expression

   $$f(t)=a_0+\cdots$$
   Explanation

   A constant baseline term is added first to account for vertical shift or average level.

   Justification

   The speaker says there will be some baseline constant based on the average value over one period.

   Shown in the video
3. Expression

   $$f(t)=a_0+a_1\cos(t)+\cdots$$
   Explanation

   The first oscillatory term chosen is a cosine with argument t.

   Justification

   The speaker says let's take a sub 1 times cosine of t.

   Shown in the video
4. Expression

   $$f(t)=a_0+a_1\cos(t)+b_1\sin(t)$$
   Explanation

   A sine term with argument t is then added alongside the cosine term.

   Justification

   The speaker says plus b sub 1 of sine of t.

   Shown in the video

The clip builds the opening form of a Fourier-type expansion as $f(t)=a_0+a_1\cos(t)+b_1\sin(t)$, motivated by representing a periodic function using sinusoidal components.

1. Formula
   Observation

   The expression grows step by step from $f(t)=a_0+a_1\cos(t)+b_1\sin(t)$ to include +$a_2\cos(2t)$, +$a_3\cos(3t)$, +$b_2\sin(2t)$, +$b_3\sin(3t)$, followed by ellipses.

2. Audio
   Observation

   The speaker explains the weights, notes that one harmonic is insufficient, and then adds higher multiples of the base frequency.

1. This is a constructive explanation, not a rigorous derivation of coefficients or convergence.

1. Expression

   $$f(t)=a_0+a_1\cos(t)+b_1\sin(t)$$
   Explanation

   Start with a constant term plus the fundamental cosine and sine components.

   Justification

   These are the first terms visibly written on the board.

   Shown in the video
2. Expression

   $$a_1\gg b_1 \Rightarrow \text{more }\cos(t)\text{ than }\sin(t)$$
   Explanation

   Compare the sizes of $a_1$ and $b_1$ to interpret the relative contribution of the two fundamental sinusoids.

   Justification

   The speaker explicitly says that if $a_1$ is much larger than $b_1$, the function has more cosine of t than sine of t.

   Shown in the video
3. Expression

   $$a_1\cos(t)+b_1\sin(t)\ \text{is still only sinusoidal}$$
   Explanation

   Recognize that the fundamental pair alone cannot reproduce the square-wave shape shown on the graph.

   Justification

   The speaker contrasts a very clean sinusoid with the displayed square wave.

   Shown in the video
4. Expression

   $$+a_2\cos(2t)$$
   Explanation

   Add the second harmonic cosine term.

   Justification

   Written on the board and described verbally as another weighting coefficient times cosine of 2t.

   Shown in the video
5. Expression

   $$+a_3\cos(3t)+\cdots$$
   Explanation

   Continue the cosine series through higher integer multiples of the base frequency.

   Justification

   The speaker says he will keep going on forever, and the board shows the pattern with ellipses.

   Shown in the video
6. Expression

   $$+b_2\sin(2t)+b_3\sin(3t)+\cdots$$
   Explanation

   Mirror the same harmonic extension for the sine terms.

   Justification

   The speaker says he will do the same thing with the sines, and those terms are written on the board.

   Shown in the video

The clip builds the general Fourier-series template as an infinite weighted sum of constant, cosine, and sine harmonics.

1. Audio
   Observation

   The speaker says, 'This has a frequency of 1 over 2 pi. This has twice the frequency. This has a frequency of 1 over pi.'

2. Formula
   Observation

   The newly added term is $a_2\cos(2t)$.

1. Expression

   $$T=2\pi$$
   Explanation

   Use the period stated for the displayed waveform.

   Justification

   The board explicitly labels $T=2\pi$ sec/cycle.

   Shown in the video
2. Expression

   $$freq(\cos(t))=\frac{1}{2\pi}$$
   Explanation

   Identify the base frequency associated with the fundamental term.

   Justification

   The speaker directly states this frequency while discussing the graph and series.

   Shown in the video
3. Expression

   $$freq(\cos(2t))=2\cdot\frac{1}{2\pi}=\frac{1}{\pi}$$
   Explanation

   Doubling the argument doubles the frequency.

   Justification

   The speaker explicitly says the new term has twice the frequency and equals 1 over pi.

   Shown in the video

In the period-2\pi setting of the clip, $\cos(2t)$ has frequency $1/\pi$ cycles per second.

1. Formula
   Observation

   The board writes $T = 2π$ sec/cycle and freq = $1/(2π)$ cycles/sec.

2. Audio
   Observation

   The speaker refers to the $1/(2π)$ hertz frequency and the $1/π$ frequency when discussing coefficients.

1. The clip does not explicitly derive the $1/π$ label in words; it is inferred from the displayed series argument 2t together with the board’s period/frequency notation.

1. Expression

   $$T=2\pi\ \frac{\text{sec}}{\text{cycle}}$$
   Explanation

   The board identifies the period of the drawn periodic signal.

   Justification

   Directly written on the board.

   Shown in the video
2. Expression

   $$\text{freq}=\frac{1}{T}=\frac{1}{2\pi}\ \frac{\text{cycles}}{\text{sec}}$$
   Explanation

   The fundamental frequency is the reciprocal of the period.

   Justification

   Standard period-frequency relation, consistent with the formula written on the board.

   Derived from the video
3. Expression

   $$\cos(t),\ \cos(2t),\ \cos(3t),\ldots$$
   Explanation

   The displayed series uses trigonometric arguments t, 2t, 3t, … .

   Justification

   Directly visible in the Fourier-series formula on the board.

   Shown in the video
4. Expression

   $$\frac{1}{2\pi},\ \frac{1}{\pi},\ \frac{3}{2\pi},\ldots$$
   Explanation

   These correspond respectively to the fundamental and higher harmonics associated with the displayed cosine terms.

   Justification

   Derived by combining the board’s period/frequency notation with the integer multiples appearing inside cos(nt).

   Derived from the video

For the displayed example, the coefficient $a_1$ is associated with frequency $1/(2π)$, $a_2$ with $1/π$, and $a_3$ with $3/(2π)$.

1. Audio
   Observation

   The speaker says they will review trigonometry, especially integrating trig functions, then solve for the coefficients, and see how well they approximate f.

1. Expression

   $$$$
   Explanation

   Review trigonometry, especially integration of trigonometric functions.

   Justification

   Stated verbally as the next step in future videos.

   Shown in the video
2. Expression

   $$$$
   Explanation

   Solve for the coefficients $a_n$ and $b_n$.

   Justification

   Stated verbally as the next step after the trigonometry review.

   Shown in the video
3. Expression

   $$$$
   Explanation

   Check how well the resulting series approximates the original function f.

   Justification

   Stated verbally as the final goal of the upcoming procedure.

   Shown in the video

The clip outlines a future workflow: use trigonometric integration to determine the Fourier coefficients and then assess the approximation quality for f.

1. Diagram
   Observation

   A yellow square-wave graph labeled $y=f(t)$ is drawn on axes with $t(s)$, $2\pi$, and $4\pi$ marked.

2. Audio
   Observation

   The speaker describes the graph as a square wave with period $2\pi$ seconds and asks whether it can be represented as a sum of sines and cosines.

3. Formula
   Observation

   The board writes $T = 2\pi \frac{sec}{cycle}$, freq = $\frac{1}{2\pi} \frac{cycles}{sec}$, and $f(t)=a_0+a_1\cos(t)+b_1\sin(t)$.

Given the displayed periodic square wave $y=f(t)$, identify its period and frequency and begin writing it as a sum of trigonometric terms.

1. The graph is a square wave labeled $y=f(t)$.

2. The horizontal axis is time t in seconds.

3. The waveform repeats every $2\pi$ seconds.

State the period T.；Compute the frequency.；Write the first terms of a proposed sinusoidal expansion.

1. Expression

   $$T = 2\pi \frac{sec}{cycle}$$
   Explanation

   Read one full cycle from the repeating rectangular pattern.

   Justification

   The speaker explicitly says the function completes one cycle every $2\pi$ seconds.

   Shown in the video
2. Expression

   $$freq = \frac{1}{2\pi} \frac{cycles}{sec}$$
   Explanation

   Take the reciprocal of the period to obtain frequency.

   Justification

   The speaker defines frequency as one over the period.

   Derived from the video
3. Expression

   $$f(t)=a_0+a_1\cos(t)+b_1\sin(t)$$
   Explanation

   Begin the representation with a constant term plus the first cosine and sine terms.

   Justification

   The speaker introduces a baseline constant based on average value, then adds $a_1\cos(t)$ and $b_1\sin(t)$.

   Shown in the video

$T=2\pi \frac{sec}{cycle}$,$\quad$ freq=$\frac{1}{2\pi}\frac{cycles}{sec}$,$\quad f(t)=a_0+a_1\cos(t)+b_1\sin(t)\text{ (initial form shown)}$

The answer matches the formulas visibly written on the board during the clip.

1. Diagram
   Observation

   A yellow square wave is drawn on axes labeled y and $t(s)$, with marks at $2\pi$ and $4\pi$ and annotations $T=2\pi$ sec/cycle and freq=$1/(2\pi)$ cycles/sec.

2. Formula
   Observation

   The lower-left expression is built as a trigonometric series intended to represent $y=f(t)$.

3. Audio
   Observation

   The speaker contrasts a clean sinusoid with a square wave and motivates adding harmonics.

1. The numerical amplitude of the square wave is not labeled on the y-axis.

2. The clip does not compute the actual Fourier coefficients for this square wave.

Represent the displayed periodic square wave $y=f(t)$ using weighted sines and cosines.

1. The graph is a square wave with period $T=2\pi$.

2. The frequency is labeled as $1/(2\pi)$ cycles/sec.

3. The series begins with $f(t)=a_0+a_1\cos(t)+b_1\sin(t)$.