7.03 Exponential Fourier Series & Complex Spectra | Chapter 8: Continuous-Time Fourier Transform (CTFT)
7.04 Parseval’s Theorem & Convergence Conditions
Core Idea
For any periodic signal, the continuous-time Fourier series represents a transition from the time domain to the discrete frequency domain. This note establishes the twin pillars of this transition: Dirichlet’s Conditions, which dictate when a periodic signal can be mathematically represented as a valid Fourier series, and Parseval’s Power Theorem, which mathematically guarantees the conservation of average power across this domain boundary. Together, they provide the analytical confidence required to process physical power signals in the frequency domain.
1. Dirichlet’s Convergence Conditions
While any passive RLC circuit or physical communication channel responds only to finite-energy waveforms, mathematicians spent decades debating which abstract functions could be decomposed into infinite sinusoids. Jean Baptiste Joseph Fourier originally asserted that any periodic signal has a Fourier series—a claim that is technically false without restrictions.
To restore mathematical rigor, Peter Gustav Lejeune Dirichlet established three sufficient conditions. If a periodic signal with fundamental period satisfies these Dirichlet Conditions, its Fourier series is guaranteed to converge to at all points of continuity.
1.1 Condition 1: Absolute Integrability
The signal must be absolutely integrable over any single fundamental period :
- Physical Significance: This condition guarantees that the area under the absolute envelope of the signal is finite.
- Mathematical Necessity: Looking at the Fourier analysis equation C_n = rac{1}{T_0} \int_{T_0} x(t) e^{-jn\Omega_0 t} \, dt, we can bound the magnitude of the coefficients:
ight| \le rac{1}{T_0} \int_{T_0} |x(t)| |e^{-jn\Omega_0 t}| , dt = rac{1}{T_0} \int_{T_0} |x(t)| , dt$$ Since the integral of is bounded, each Fourier coefficient (and ) is mathematically guaranteed to be a finite, bounded number.
- Counterexample: The function x(t) = rac{1}{t} for with period violates this condition because \int_{0}^{1} rac{1}{t} \, dt = \ln(t) ig|_0^1 = \infty. It cannot be represented by a Fourier series.
1.2 Condition 2: Finite Extrema (Maxima and Minima)
The signal must possess a finite number of local maxima and minima during any single fundamental period .
- Physical Significance: The signal cannot oscillate with infinite frequency within a finite time interval.
- Counterexample: Consider the signal x(t) = \sin\left( rac{2\pi}{t} ight) for with period . As , the frequency approaches infinity, creating an infinite number of zero-crossings and extrema in the interval . This signal cannot be modeled with a Fourier series because it is physically impossible to construct or approximate using a countable sum of harmonically related sinusoids.
1.3 Condition 3: Finite Discontinuities
The signal must have a finite number of discontinuities in any finite interval of time, and each of these discontinuities must be finite in magnitude.
- Physical Significance: The signal cannot have infinite step-jumps.
- The Convergence Rule: At any point of jump discontinuity , the Fourier series summation does not converge to either the top or bottom of the step. Instead, it converges precisely to the statistical average of the left and right-hand limits: \lim_{N o \infty} S_N(t_d) = rac{x(t_d^+) + x(t_d^-)}{2}
2. Parseval’s Power Conservation Theorem
2.1 The Concept of Power Conservation
From an engineering perspective, the continuous-time Fourier series splits a complex signal into separate frequency channels. Parseval’s Theorem mathematically proves the Conservation of Power. It guarantees that a signal’s total average power is identical whether calculated by integrating its amplitude-squared over time (oscilloscope view) or by summing the squared magnitudes of its discrete frequency components (spectrum analyzer view).
Parseval's Identity (General Form)
For a periodic power signal with period and complex exponential Fourier coefficients :
P = rac{1}{T_0} \int_{T_0} |x(t)|^2 \, dt = \sum_{n=-\infty}^{\infty} |C_n|^2
2.2 Rigorous Mathematical Proof (Complex Exponential Form)
Let be a complex-valued periodic signal with period . The average power is defined in the time domain as:
P = rac{1}{T_0} \int_{T_0} |x(t)|^2 \, dt = rac{1}{T_0} \int_{T_0} x(t) \cdot x^*(t) \, dt \quad ext{--- (Equation 1)}
where denotes the complex conjugate.
- Express the conjugate signal using its complex Fourier Series expansion: Since , taking the complex conjugate of both sides yields:
ight)^* = \sum_{n=-\infty}^{\infty} C_n^* e^{-jn\Omega_0 t} \quad ext{--- (Equation 2)}$$
- Substitute Equation 2 into the power integral (Equation 1):
ight] dt$$
- Interchange the order of summation and integration: (This step is mathematically guaranteed to be valid because Dirichlet’s conditions ensure uniform convergence of the series).
ight] \quad ext{--- (Equation 3)}$$
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Identify the Fourier Analysis term: Recall the fundamental analysis equation defining the complex coefficients: C_n = rac{1}{T_0} \int_{T_0} x(t) e^{-jn\Omega_0 t} \, dt The term inside the brackets of Equation 3 is exactly equal to !
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Simplify to get the final Parseval identity: P = \sum_{n=-\infty}^{\infty} C_n^* \cdot C_n = \sum_{n=-\infty}^{\infty} |C_n|^2 \quad lacksquare
2.3 The Trigonometric Equivalent Form
To express Parseval’s identity using real-valued trigonometric coefficients (), we use the algebraic relationships between complex and trigonometric representations:
- For : C_0 = rac{a_0}{2} \implies |C_0|^2 = \left( rac{a_0}{2} ight)^2.
- For : C_n = rac{a_n - jb_n}{2} \implies |C_n|^2 = rac{a_n^2 + b_n^2}{4}.
- Symmetry check: Since is real, |C_{-n}|^2 = |C_n|^2 = rac{a_n^2 + b_n^2}{4}.
Expanding the complex sum yields:
ight)$$ $$\sum_{n=-\infty}^{\infty} |C_n|^2 = \left( rac{a_0}{2} ight)^2 + \sum_{n=1}^{\infty} \left( rac{a_n^2 + b_n^2}{4} + rac{a_n^2 + b_n^2}{4} ight)$$ $$\sum_{n=-\infty}^{\infty} |C_n|^2 = \left( rac{a_0}{2} ight)^2 + rac{1}{2} \sum_{n=1}^{\infty} \left( a_n^2 + b_n^2 ight)$$ Therefore, the **Trigonometric Parseval Identity** is: $$P = rac{1}{T_0} \int_{T_0} [x(t)]^2 \, dt = \left( rac{a_0}{2} ight)^2 + rac{1}{2} \sum_{n=1}^{\infty} \left(a_n^2 + b_n^2 ight)$$ --- ## 3. Physical Interpretation of Power Components When Parseval's identity is applied to electrical systems, it decomposes total signal power into clear, measurable components: | Mathematical Component | Signal Processing Name | Physical Meaning (E.g., Voltage Signal across $1\,\Omega$) | | :--- | :--- | :--- | | $$\left( rac{a_0}{2} ight)^2 = |C_0|^2$$ | **DC Power** | Power dissipated by the constant baseline offset of the signal. | | $$ rac{1}{2}a_n^2$$ | **Cosine Harmonic Power** | Power contribution of the $n$-th cosine component. | | $$ rac{1}{2}b_n^2$$ | **Sine Harmonic Power** | Power contribution of the $n$-th sine component. | | $$|C_n|^2 + |C_{-n}|^2 = rac{a_n^2 + b_n^2}{2}$$ | **Harmonic Channel Power** | Total average power contained in the $n$-th active physical frequency slot. | --- ## 4. High-Yield Worked Examples (The Exam Killers) ### 4.1 Example 1: The 2018 Complex Spectral Power Allocation [PYQ 2018 - 3 Marks] **Question:** The complex exponential Fourier representation of a periodic signal $x(t)$ over a period $(0, T)$ is given by: $$x(t) = \sum_{n=-\infty}^{\infty} rac{3}{4+(n\pi)^2} e^{j3n\pi t}$$ Determine: 1. The value of fundamental period $T$. 2. What percentage of total power is contained in the first five terms of the series, given that the maximum (total) power of the signal is $P_{ ext{total}} = 0.7 ext{ W}$? #### Step-by-Step Analytical Solution: ##### Part 1: Finding the Fundamental Period $T$ The standard synthesis equation for the complex exponential Fourier series is: $$x(t) = \sum_{n=-\infty}^{\infty} C_n e^{jn\Omega_0 t}$$ Comparing this directly with the given equation: $$jn\Omega_0 t = j3n\pi t \implies \Omega_0 = 3\pi ext{ rad/s}$$ The fundamental angular frequency is related to the period $T$ by $\Omega_0 = rac{2\pi}{T}$. Therefore: $$T = rac{2\pi}{\Omega_0} = rac{2\pi}{3\pi} = \mathbf{ rac{2}{3} ext{ seconds}} pprox 0.667 ext{ s}$$ --- ##### Part 2: Calculating Percentage Power in the First Five Terms The "first five terms" of the symmetric complex Fourier series correspond to the harmonic indices: $$n = 0, \quad n = 1, \quad n = -1, \quad n = 2, \quad n = -2$$ Let us evaluate the complex coefficient $C_n = rac{3}{4+n^2\pi^2}$ for each of these indices: 1. **For $n = 0$ (DC component):** $$C_0 = rac{3}{4 + 0} = 0.75 \implies |C_0|^2 = (0.75)^2 = \mathbf{0.56250 ext{ W}}$$ 2. **For $n = \pm 1$ (1st Harmonic):** $$C_1 = C_{-1} = rac{3}{4 + \pi^2} pprox rac{3}{4 + 9.86960} = rac{3}{13.86960} pprox 0.21630$$ $$|C_1|^2 = |C_{-1}|^2 pprox (0.21630)^2 = \mathbf{0.04679 ext{ W}}$$ 3. **For $n = \pm 2$ (2nd Harmonic):** $$C_2 = C_{-2} = rac{3}{4 + 4\pi^2} pprox rac{3}{4 + 39.47842} = rac{3}{43.47842} pprox 0.06900$$ $$|C_2|^2 = |C_{-2}|^2 pprox (0.06900)^2 = \mathbf{0.00476 ext{ W}}$$ Now, apply Parseval's identity over these 5 discrete terms to find the partial power $P_5$: $$P_5 = \sum_{n=-2}^{2} |C_n|^2 = |C_0|^2 + 2|C_1|^2 + 2|C_2|^2$$ $$P_5 pprox 0.56250 + 2(0.04679) + 2(0.00476)$$ $$P_5 pprox 0.56250 + 0.09358 + 0.00952 = \mathbf{0.66560 ext{ W}}$$ We are given that the total maximum power of the signal is $P_{ ext{total}} = 0.7 ext{ W}$. The percentage of power contained within the first five terms is: $$\% ext{ Power} = rac{P_5}{P_{ ext{total}}} imes 100\% = rac{0.66560}{0.70000} imes 100\% = \mathbf{95.09\%}$$ * **Exam Note:** A single DC offset and the first two harmonics contain over $95\%$ of the total signal power! This demonstrates why physical systems can be safely band-limited without losing vital signal information. --- ### 4.2 Example 2: Equivalence Proof of Trigonometric & Exponential Power Calculations **Question:** A periodic voltage signal has the following non-zero trigonometric Fourier coefficients: $a_0 = 4 ext{ V}$, $a_2 = 2 ext{ V}$, and $b_2 = -2 ext{ V}$. 1. Calculate the total average power using the trigonometric Parseval equation. 2. Convert these coefficients to complex exponential form and calculate the power. Prove that both methods yield identical results. #### Step-by-Step Analytical Solution: ##### Method 1: Trigonometric Power Calculation The trigonometric Parseval power formula is: $$P = \left( rac{a_0}{2} ight)^2 + rac{1}{2} \sum_{n=1}^{\infty} (a_n^2 + b_n^2)$$ Substitute the given parameters ($a_0 = 4$, $a_2 = 2$, $b_2 = -2$): $$P = \left( rac{4}{2} ight)^2 + rac{1}{2}\left(a_2^2 + b_2^2 ight)$$ $$P = (2)^2 + rac{1}{2}\left(2^2 + (-2)^2 ight)$$ $$P = 4 + rac{1}{2}(4 + 4) = 4 + rac{1}{2}(8) = 4 + 4 = \mathbf{8 ext{ Watts}}$$ --- ##### Method 2: Complex Exponential Power Calculation First, convert the trigonometric coefficients to complex exponential coefficients using standard conversion formulas: 1. **For $n = 0$:** $$C_0 = rac{a_0}{2} = rac{4}{2} = 2 \implies |C_0|^2 = 2^2 = \mathbf{4 ext{ W}}$$ 2. **For $n = 2$:** $$C_2 = rac{a_2 - jb_2}{2} = rac{2 - j(-2)}{2} = 1 + j \implies |C_2|^2 = (1)^2 + (1)^2 = \mathbf{2 ext{ W}}$$ 3. **For $n = -2$:** $$C_{-2} = C_2^* = 1 - j \implies |C_{-2}|^2 = (1)^2 + (-1)^2 = \mathbf{2 ext{ W}}$$ All other $C_n$ coefficients are zero. Summing these exponential components yields: $$P = \sum_{n=-\infty}^{\infty} |C_n|^2 = |C_0|^2 + |C_2|^2 + |C_{-2}|^2$$ $$P = 4 + 2 + 2 = \mathbf{8 ext{ Watts}}$$ Both methods yield **exactly $8 ext{ W}$**, verifying the absolute mathematical equivalence between trigonometric and exponential power formulas. --- ## 5. Common Mistakes That Cost Marks > [!danger] **The Trigonometric $1/2$ Factor Omission Trap** > > In examinations, a very common mistake is forgetting the $1/2$ multiplier when calculating power from trigonometric coefficients: > $$P = \left( rac{a_0}{2} ight)^2 + \sum_{n=1}^{\infty} \left(a_n^2 + b_n^2 ight) \quad \mathbf{[WRONG!]}$$ > Sinusoidal waves of peak amplitude $A$ have an RMS value of $A/\sqrt{2}$, meaning their average power is $A^2/2$. Forgetting this factor will double your AC power terms and result in a **zero-mark** penalty on derivations. > [!warning] **Complex Exponential Half-Sided Summation Error** > > When calculating power in the complex exponential domain, students often sum only the positive indices: > $$P = |C_0|^2 + \sum_{n=1}^{\infty} |C_n|^2 \quad \mathbf{[WRONG!]}$$ > The complex exponential Fourier series uses a **double-sided spectrum**. You must include both $C_n$ and its negative conjugate partner $C_{-n}$ to account for $100\%$ of the signal energy. --- ## 6. PYQ Bank — Verbatim Questions & Answer Plans ### 6.1 PYQ 2022/2018 [8 Marks/7 Marks] **Question:** Define Fourier series. State necessary and sufficient conditions (Dirichlet conditions) for the existence of the Fourier series representation for a signal. * **Answer Plan:** 1. Define **Fourier Series** as the representation of a continuous-time periodic signal as an infinite sum of harmonically related sinusoidal harmonics. 2. Write the trigonometric Fourier series formula, clearly labeling $a_0, a_n, b_n$, and $\omega_0$. 3. State the three **Dirichlet Conditions** verbatim as written in **Section 1**: * Absolute integrability over one period. * Finite number of maxima and minima over one period. * Finite number of discontinuities over one period. 4. Provide the mathematical equations for absolute integrability and the convergence behavior at step discontinuities. ### 6.2 PYQ 2020/2019 [4 Marks] **Question:** State and prove Parseval’s identity for Fourier series. * **Answer Plan:** 1. State **Parseval's Theorem** verbally: the total average power of a periodic signal is equal to the sum of the average powers of its individual harmonic components. 2. Write down both the trigonometric and complex exponential Parseval mathematical equations. 3. Reproduce the step-by-step mathematical proof using the conjugate substitution method as shown in **Section 2.2**. --- ## 7. Self-Check Before Moving On - [ ] Can you state the absolute integrability condition equation of Dirichlet from memory? [1.1] - [ ] Do you know what value the Fourier series converges to at a jump discontinuity of magnitude $10$ (it converges to the exact midpoint of the step)? [1.3] - [ ] Can you mathematically prove Parseval's identity using the complex exponential representation in under 3 minutes? [2.2] - [ ] Why is there a $1/2$ scaling factor in the trigonometric Parseval equation but not in the complex exponential form? [2.3] --- *Source: (k.Deergha Rao) signals and systems.pdf (Chapter 3), 01 Fourier Series.pdf, Rabiul sir class note.pdf.*