7.00 Continuous-Time Fourier Series Compact Review

Chapter 7: Continuous-Time Fourier Series (CTFS) - Compact Review

7.01 Trigonometric Fourier Series Representation

*(Target: Theory Descriptive / Numerical Solving)*

• Concept: Represents a real-valued, periodic, non-sinusoidal signal as an infinite linear combination of harmonically related sines and cosines alongside a constant DC offset.
• Governing Formulas: $f(t)=\frac{a_0}{2}+{\sum }_{n=1}^{\infty }{a}_n\cos (n{\omega }_{0}t)+{\sum }_{n=1}^{\infty }{b}_n\sin (n{\omega }_{0}t)$ where the fundamental angular frequency is: ${\omega }_0=\frac{2\pi }{T_0}$
• Euler-Fourier Coefficients: $a_0=\frac{2}{T_0}{\int }_0^{T_0}f(t)dt$ $a_n=\frac{2}{T_0}{\int }_0^{T_0}f(t)\cos (n{\omega }_{0}t)dt$ $b_n=\frac{2}{T_0}{\int }_0^{T_0}f(t)\sin (n{\omega }_{0}t)dt$
• DC Term Note: The constant term in the series is $\frac{a_0}{2}$, which represents the exact statistical mean or average value of the signal over one fundamental period.

---

7.02 Dirichlet Conditions for Convergence

*(Target: Theory Descriptive / 5-to-8-Mark Question)*

• Concept: The sufficient (but not necessary) conditions that a periodic signal $f(t)$ must satisfy to guarantee a valid, convergent Fourier series representation.
• The Three Conditions:
  1. Absolute Integrability: $f(t)$ must be absolutely integrable over any single period: ${\int }_{T_0}|f(t)|dt<\infty$ {This guarantees that all coefficients $a_0,a_n,b_n$ remain finite}.
  2. Bounded Maxima and Minima: $f(t)$ must possess a finite number of local maxima and minima within any single period.
  3. Finite Discontinuities: $f(t)$ must contain only a finite number of discontinuities within any finite time interval, and each discontinuity must be of finite height.

---

7.03 Waveform Symmetry Shortcuts (Symmetry Conditions)

*(Target: Theory Descriptive / Shortcut Rules)*

• Concept: Physical symmetries in the time-domain waveform force specific Fourier series coefficients to exactly zero, drastically reducing calculation times.

Symmetry Type	Mathematical Condition	Coefficients Set to Zero	Active Terms in Series	Integration Shortcut
Even Symmetry	$f(t)=f(-t)$	$b_n=0$ for all $n$	DC ($a_0$) and Cosine terms ($a_n$) only	$a_n=\frac{4}{T_0}{\int }_0^{T_0/2}f(t)\cos (n{\omega }_{0}t)dt$
Odd Symmetry	$f(t)=-f(-t)$	$a_0=0$, $a_n=0$ for all $n$	Sine terms ($b_n$) only	$b_n=\frac{4}{T_0}{\int }_0^{T_0/2}f(t)\sin (n{\omega }_{0}t)dt$
Half-Wave Symmetry	$f\left(t-\frac{T_0}{2}\right)=-f(t)$	$a_0=0$, $a_n=b_n=0$ for even $n$	Odd harmonics only ($n=1,3,5,\dots$)	Integrates over half period, scaled by 2
Even Quarter-Wave	Even + Half-Wave	$b_n=0$ for all $n$; $a_n=0$ for even $n$	Cosine terms with odd harmonics only	$a_n=\frac{8}{T_0}{\int }_0^{T_0/4}f(t)\cos (n{\omega }_{0}t)dt$
Odd Quarter-Wave	Odd + Half-Wave	$a_0=0$, $a_n=0$ for all $n$; $b_n=0$ for even $n$	Sine terms with odd harmonics only	$b_n=\frac{8}{T_0}{\int }_0^{T_0/4}f(t)\sin (n{\omega }_{0}t)dt$

---

7.04 Mathematical Proof: Odd Functions Contain Sine Terms Only

*(Target: Mathematical Proof / 6-Mark Derivation)*

• Starting Premise: Let $f(t)$ be a periodic function with odd symmetry, satisfying $f(-t)=-f(t)$. The cosine coefficient is defined as: $a_n=\frac{2}{T_0}{\int }_{-T_0/2}^{T_0/2}f(t)\cos (n{\omega }_{0}t)dt$
• Proof (derivation): We split the integration boundary into two equal symmetric half-intervals: $a_n=\frac{2}{T_0}\left[{\int }_{-T_0/2}^{0}f(t)\cos (n{\omega }_{0}t)dt+{\int }_0^{T_0/2}f(t)\cos (n{\omega }_{0}t)dt\right]$ Apply the variable substitution $u=-t$ (hence $dt=-du$) exclusively to the first integral: ${\int }_{-T_0/2}^{0}f(t)\cos (n{\omega }_{0}t)dt={\int }_{T_0/2}^{0}f(-u)\cos (-n{\omega }_{0}u)(-du)={\int }_0^{T_0/2}-f(u)\cos (n{\omega }_{0}u)du$ {Since $f(-u)=-f(u)$ due to odd parity, and $\cos (-x)=\cos (x)$ due to even parity}. Substituting this back into the overall coefficient equation yields: $a_n=\frac{2}{T_0}\left[{\int }_0^{T_0/2}-f(t)\cos (n{\omega }_{0}t)dt+{\int }_0^{T_0/2}f(t)\cos (n{\omega }_{0}t)dt\right]=0$
• Conclusion: Thus, $a_0=0$ and $a_n=0$ for all $n\ge 1$, proving that odd functions are composed strictly of sine term coefficients ($b_n$).

---

7.05 Complex Exponential Fourier Series (Complex Spectra)

*(Target: Theory Descriptive / Formula Alignment)*

• Concept: Uses Euler’s complex exponential identity ($e^{j\theta }=\cos \theta +j\sin \theta$) to represent periodic signals in a mathematically elegant, compact format. This format acts as the direct bridge to continuous frequency-domain transforms.
• Synthesis Equation: $f(t)={\sum }_{n=-\infty }^{\infty }{C}_n{e}^{jn{\omega }_{0}t}$
• Analysis Equation: $C_n=\frac{1}{T_0}{\int }_{-T_0/2}^{T_0/2}f(t)e^{-jn{\omega }_{0}t}dt$
• Trigonometric to Exponential Coefficient Conversion:
  • DC Term ($n=0$): $C_0=\frac{a_0}{2}$
  • Positive Harmonics ($n>0$): $C_n=\frac{a_n-j{b}_n}{2}$
  • Negative Harmonics ($n<0$): $C_{-n}=C_n^{\ast }=\frac{a_n+j{b}_n}{2}$ {Note: For real-valued signals, $C_{-n}$ is always the complex conjugate of $C_n$}.

---

7.06 Symmetrical Properties of Complex Coefficients ($C_n$)

*(Target: Theory Descriptive / Spectral Interpretation)*

• Because $C_n$ is a complex number, it is represented as a phasor containing magnitude and phase: $C_n=|C_n|e^{j{\theta }_n}$.
• Real and Even Signals: If $f(t)$ is real and even, then $C_n$ is purely real and even ($C_n=C_n^{\ast }=C_{-n}$). Phase is 0 or $\pi$.
• Real and Odd Signals: If $f(t)$ is real and odd, then $C_n$ is purely imaginary and odd ($C_n=-C_n^{\ast }=-C_{-n}$).
• Symmetry of Real-Valued Signals: For any real-valued physical signal $f(t)$:
  • Magnitude Spectrum: $|C_n|$ is perfectly symmetric (even) about the origin: $|C_{-n}|=|C_n|$
  • Phase Spectrum: $\angle C_n={\theta }_n$ is perfectly anti-symmetric (odd) about the origin: ${\theta }_{-n}=-{\theta }_n$

---

7.07 Parseval’s Theorem for Fourier Series (Conservation of Power)

*(Target: Mathematical Proof / 8-Mark Question)*

• Statement: The total average power $P_{avg}$ of a periodic power signal is conserved and remains identical whether computed in the time domain or by summing the power spectral densities of its individual frequency harmonics.
• Governing Identity: $P_{avg}=\frac{1}{T_0}{\int }_{-T_0/2}^{T_0/2}|f(t){|}^{2}dt={\sum }_{n=-\infty }^{\infty }|C_n{|}^2={\left(\frac{a_0}{2}\right)}^2+\frac{1}{2}{\sum }_{n=1}^{\infty }\left(a_n^2+b_n^2\right)$
• Proof of Parseval’s Power Identity (derivation): Representing average power in the time domain: $P_{avg}=\frac{1}{T_0}{\int }_{-T_0/2}^{T_0/2}f(t)f^{\ast }(t)dt$ Substitute the exponential Fourier series synthesis equation for the conjugate signal $f^{\ast }(t)={\sum }_{n=-\infty }^{\infty }{C}_n^{\ast }{e}^{-jn{\omega }_{0}t}$: $P_{avg}=\frac{1}{T_0}{\int }_{-T_0/2}^{T_0/2}f(t)\left[{\sum }_{n=-\infty }^{\infty }{C}_n^{\ast }{e}^{-jn{\omega }_{0}t}\right]dt$ Interchanging the order of summation and integration: $P_{avg}={\sum }_{n=-\infty }^{\infty }{C}_n^{\ast }\left[\frac{1}{T_0}{\int }_{-T_0/2}^{T_0/2}f(t)e^{-jn{\omega }_{0}t}dt\right]$ Recognizing that the bracketed integral is the exact definition of $C_n$: $P_{avg}={\sum }_{n=-\infty }^{\infty }{C}_n^{\ast }{C}_n={\sum }_{n=-\infty }^{\infty }|C_n{|}^2$ Substituting trigonometric identities ($C_0=\frac{a_0}{2}$, and $|C_n{|}^2+|C_{-n}{|}^2=\frac{a_n^2+b_n^2}{2}$) yields: $P_{avg}={\left(\frac{a_0}{2}\right)}^2+\frac{1}{2}{\sum }_{n=1}^{\infty }\left(a_n^2+b_n^2\right)$

---

7.08 Approximation Errors: Bessel’s Inequality & Mean Square Error

*(Target: Theory Descriptive / Formula Interpretation)*

• N-th Partial Sum: Reconstructing a signal with a finite number of harmonics ($N$): $S_N(t)=\frac{a_0}{2}+{\sum }_{n=1}^N{a}_n\cos (n{\omega }_{0}t)+{\sum }_{n=1}^N{b}_n\sin (n{\omega }_{0}t)$
• Mean Square Error (MSE): The remaining error energy in a truncated approximation: $E_N=\frac{1}{T_0}{\int }_{-T_0/2}^{T_0/2}{\left(f(t)-S_N(t)\right)}^{2}dt$
• The MSE Identity: $E_N=\frac{1}{T_0}{\int }_{-T_0/2}^{T_0/2}|f(t){|}^{2}dt-\left[{\left(\frac{a_0}{2}\right)}^2+\frac{1}{2}{\sum }_{n=1}^N(a_n^2+b_n^2)\right]$
• Bessel’s Inequality: Since the mean square error must always be non-negative ($E_N\ge 0$): ${\left(\frac{a_0}{2}\right)}^2+\frac{1}{2}{\sum }_{n=1}^N(a_n^2+b_n^2)\le \frac{1}{T_0}{\int }_{-T_0/2}^{T_0/2}|f(t){|}^{2}dt$ {Physical Meaning: The power of a finite harmonic approximation is always strictly bounded by the total true power of the actual signal}.

---

7.09 The Gibbs Phenomenon

*(Target: Theory Descriptive / 5-Mark Question)*

• Definition: At any point of jump discontinuity in a piecewise continuous periodic function, the reconstructed Fourier series approximation displays a distinct high-frequency ringing overshoot near the discontinuity.
• Core Concepts:
  • The maximum overshoot amplitude does not decay to zero even as the number of terms $N$ approaches infinity.
  • The peak overshoot converges to approximately 8.95% (nearly 9%) of the height of the jump discontinuity.
  • Increasing the number of terms $N$ compresses the physical width of the overshoot oscillations toward zero, but the peak amplitude remains constant.

---

7.10 Common Mistakes That Cost Marks

Critical Exam Pitfalls

• DC Term $a_0$ vs. $a_0/2$ Scaling: Mixing up the scaling when writing the final series. Remember: if you define $a_0=\frac{2}{T_0}\int f(t)dt$, you must write the series with $\frac{a_0}{2}$ as the first term. Alternatively, if you write the series with $a_0$ first, your integration formula must be $a_0=\frac{1}{T_0}\int f(t)dt$.
• The $1/2$ Factor in Parseval’s Power: Forgetting the multiplier of $1/2$ on the harmonic power sum when converting from trigonometric coefficients: $P_{avg}=C_0^2+\frac{1}{2}\sum (a_n^2+b_n^2)$.
• Complex Exponent Conjugation Signs: Forgetting that the forward complex coefficient $C_n$ utilizes a negative exponent ($e^{-jn{\omega }_{0}t}$), whereas the reconstruction synthesis sum utilizes a positive exponent ($e^{jn{\omega }_{0}t}$).
• Harmonic Bounds Violation: Applying trigonometric symmetries without verifying if the signal is purely even or odd. If a signal is asymmetrical, both $a_n$ and $b_n$ terms must be computed.

---

7.11 PYQ Bank — Verbatim Questions & Answer Plans

Q1: Define Fourier Series and state Dirichlet conditions [KUET 2022, 2018, 2017]

• Answer Plan:
  1. State the verbal definition of Fourier Series.
  2. Provide the standard trigonometric representation equations along with Euler-Fourier coefficient integrals.
  3. Formally list the three Dirichlet conditions (Absolute integrability, finite extrema, finite discontinuities).

Q2: Justify that odd functions have only sine term coefficients [KUET 2018, 2017]

• Answer Plan:
  1. Define odd symmetry mathematically: $f(-t)=-f(t)$.
  2. Set up the integral for $a_n$ and split boundaries into $[-T_0/2,0]$ and $[0,T_0/2]$.
  3. Perform variable substitution $u=-t$ and show that the two integrals cancel each other out to zero, proving $a_n=0$.

Q3: State and prove Parseval’s Identity for Fourier Series [KUET 2020, 2019]

• Answer Plan:
  1. Formulate the physical meaning of Parseval’s identity (power conservation between time and frequency domains).
  2. Write out the exponential and trigonometric power equations.
  3. Exhaustively write out the step-by-step proof using complex conjugates and interchanging integration and summation.

---

7.12 Self-Check Before Moving On

• Can you write down the Trigonometric Fourier Series equations and Euler-Fourier coefficient integrals from memory?
• Do you know how to convert trigonometric coefficients ($a_n,b_n$) directly into complex exponential coefficients ($C_n$)?
• Can you outline the exact steps to prove that an odd function contains no cosine terms?
• Do you understand the physical significance of Parseval’s theorem in terms of real-world physical power?
• Can you describe what happens to the overshoot of a Gibbs phenomenon as the harmonic order $N\to \infty$?

Source: Signals and Systems (K. Deergha Rao) Chapter 3, Continuous-Time Signals and Systems (Senior Notes Ch 7).