Trigonometric Fourier Series (Example 1)

Main ideas / concepts conveyed

• The video demonstrates how to compute a trigonometric Fourier series for a specific waveform: a rectangular (piecewise constant) signal (X_T).
• It emphasizes the importance of analyzing waveform symmetry to simplify calculations:
  • If the waveform has zero net area over one period, the DC (average) term is zero.
  • If the waveform is even (mirror symmetric about the (y)-axis), all sine terms vanish ((b_n=0)), leaving only cosine harmonics.
• Then it computes the remaining cosine Fourier coefficients (a_n) using integration over the fundamental period.
• Finally, it substitutes the coefficient pattern back into the Fourier series to obtain the full expansion.

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Step-by-step methodology

1) Write the general trigonometric Fourier series form

Start from: [ X_T = a_0 + \sum_{n=1}^{\infty} a_n \cos(n\Omega_0 T) + \sum_{n=1}^{\infty} b_n \sin(n\Omega_0 T) ]

where (a_n) and (b_n) are Fourier coefficients and (\Omega_0) is the fundamental angular frequency.

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2) Analyze the given waveform to simplify terms

DC term (a_0)

• The waveform is described as having zero net area over one period (positive area cancels negative area).
• Therefore: [ a_0 = 0 ]

Sine coefficients (b_n)

• The waveform is described as mirror image about the (y)-axis, i.e., it is an even signal.

• Even signals satisfy: [ X(-T)=X(T) ]

• For even signals: [ b_n = 0 ]

So the Fourier series simplifies to: [ X_T = \sum_{n=1}^{\infty} a_n \cos(n\Omega_0 T) ]

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3) Determine the fundamental period (T_0)

The waveform repeats over an interval of 4 seconds. Examples given:

• From (-1) to (3): (T_0 = 3 - (-1) = 4)
• From (-3) to (1): (T_0 = 1 - (-3) = 4)

Instruction emphasized: Choose an integration interval that truly covers one full repeat (not a misleading interval).

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4) Compute the fundamental angular frequency (\Omega_0)

Use: [ \Omega_0 = \frac{2\pi}{T_0} ] With (T_0=4): [ \Omega_0 = \frac{2\pi}{4} = \frac{\pi}{2} ]

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5) Set up the formula for (a_n)

Use: [ a_n = \frac{2}{T_0}\int_{\text{one period}} X(T)\cos(n\Omega_0 T)\, dT ]

With (T_0=4), (\Omega_0=\pi/2), and choosing ([-1,3]): [ a_n = \frac{2}{4}\int_{-1}^{3} X(T)\cos\left(\frac{n\pi}{2}T\right)dT ] which simplifies to: [ a_n = \frac{1}{2}\int_{-1}^{3} X(T)\cos\left(\frac{n\pi}{2}T\right)dT ]

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6) Split the integral according to piecewise values of the waveform

The waveform is given as:

• (X(T)=1) for (-1 \le T \le 1)
• (X(T)=-1) for (1 \le T \le 3)

So: [ a_n=\frac12\left(\int_{-1}^{1} 1\cdot \cos\left(\frac{n\pi}{2}T\right)dT +\int_{1}^{3} (-1)\cos\left(\frac{n\pi}{2}T\right)dT\right) ]

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7) Perform substitution to integrate

Let: [ \theta = \frac{n\pi}{2}T ] Then: [ dT = \frac{2}{n\pi}d\theta ]

Update limits based on (T):

• For (T=-1): (\theta=-\frac{n\pi}{2})
• For (T=1): (\theta=\frac{n\pi}{2})
• For (T=3): (\theta=\frac{3n\pi}{2})

And use: [ \int \cos\theta\, d\theta = \sin\theta ]

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8) Use trigonometric parity cases (even vs odd (n))

The value of (a_n) depends on whether (n) is even or odd.

Even (n)

• It concludes (a_n=0).
• Reason used: terms like (\sin(n\pi)) become (0) because they are integer multiples of (\pi).
• A notation clarification is mentioned:
  • Use (\sin(m\pi)=0) where (m) is an integer.

Odd (n)

Split into:

• (n=1,5,9,13,\dots) (i.e., (n \equiv 1 \pmod 4))
• (n=3,7,11,15,\dots) (i.e., (n \equiv 3 \pmod 4))

Then:

• If (n=1,5,9,\dots), [ a_n = \frac{4}{n\pi} ]

• If (n=3,7,11,\dots), [ a_n = -\frac{4}{n\pi} ]

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9) Substitute coefficients into the Fourier cosine series

Since only cosine terms remain and even (n) contribute nothing, the expansion is: [ X_T = \sum_{n=1}^{\infty} a_n \cos\left(\frac{n\pi}{2}T\right) ]

with:

• (a_n=0) for even (n)
• (a_n=\pm \frac{4}{n\pi}) for odd (n), alternating sign for (n=1,3,5,7,\dots)

The video lists the first few terms, then summarizes the alternating pattern. Examples:

• (a_1 = \frac{4}{\pi})
• (a_3 = -\frac{4}{3\pi})
• (a_5 = \frac{4}{5\pi})
• (a_7 = -\frac{4}{7\pi})

Leading to: [ X_T=\frac{4}{\pi}\left(\cos\left(\frac{\pi}{2}T\right) -\frac{1}{3}\cos\left(\frac{3\pi}{2}T\right) +\frac{1}{5}\cos\left(\frac{5\pi}{2}T\right) -\frac{1}{7}\cos\left(\frac{7\pi}{2}T\right) +\cdots\right) ]

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Lessons / takeaways emphasized

• Analyze symmetry before doing heavy calculations:
  • Even (\Rightarrow b_n=0)
  • Zero average over a period (\Rightarrow a_0=0)
• Correct choice of integration limits (one fundamental period) is crucial.
• The core work is computing the coefficient integrals; without them, the closed-form coefficients cannot be obtained.

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Speakers / sources featured

• Single speaker: The instructor/lecturer narrating the solution (no name provided in subtitles).
• No external sources (e.g., books, papers, other speakers) are explicitly cited in the provided subtitles.