7.02 Symmetry Conditions & Waveform Analysis | 7.04 Parseval’s Theorem & Convergence Conditions
7.03 Exponential Fourier Series & Complex Spectra
Core Idea
While the Trigonometric Fourier Series decomposes a periodic signal into real-valued sines and cosines, the Complex Exponential Fourier Series uses Euler’s relation to represent periodic waveforms using complex exponential functions. This formulation is much more compact and mathematically elegant because it consolidates both amplitude and phase information into a single set of complex coefficients {each representing a phasor at a specific harmonic frequency}. These coefficients extend across both positive and negative frequencies, yielding a symmetrical Double-Sided Complex Spectrum that serves as the perfect mathematical bridge to the continuous Fourier Transform.
1. The Physics of Complex Exponential Harmonics
A continuous-time periodic signal with a fundamental period repeats itself for all time: . Its fundamental angular frequency is defined as: \Omega_0 = rac{2\pi}{T_0} \quad ext{[rad/s]}
Under the complex exponential framework, we construct a set of harmonically related complex exponential signals of the form:
1.1 The Orthogonality of Complex Exponentials
Just as sines and cosines form an orthogonal basis, the complex exponential set is orthogonal over any full period :
eq m \end{cases}$$ This means that multiplying any harmonic by the complex conjugate of another harmonic and integrating over a period yields zero if their frequencies differ, indicating that harmonically related complex exponentials share **zero mutual information**. --- ## 2. The Core Mathematical Formulation The **Complex Exponential Fourier Series** represents a periodic signal $x(t)$ as an infinite linear combination of these orthogonal complex exponential phasors: $$x(t) = \sum_{n=-\infty}^{\infty} C_n e^{j n \Omega_0 t} \quad ext{--- [The Synthesis Equation]}$$ To calculate the complex Fourier coefficients $C_n$, we use the corresponding analysis integral over any convenient single period interval of duration $T_0$: $$C_n = rac{1}{T_0} \int_{T_0} x(t) e^{-j n \Omega_0 t} \, dt \quad ext{--- [The Analysis Equation]}$$ ### 2.1 Tabular Breakdown of Formulation Variables | Variable | Mathematical Role | Physical Meaning | Dimension / Unit | | :--- | :--- | :--- | :--- | | **$x(t)$** | Left-hand side function | Continuous-time periodic wave | V, A, etc. | | **$C_n$** | Complex scaling multiplier | Harmonic phasor coefficient | Same as $x(t)$ | | **$n$** | Summation integer index | Harmonic index number (positive/negative) | Unitless | | **$\Omega_0$** | Fundamental angular frequency | Spacing between adjacent spectral lines | rad/s | | **$e^{j n \Omega_0 t}$** | Harmonically related basis function | Symmetrical rotating vector in complex plane | Unitless | --- ## 3. Rigorous Trigonometric-to-Exponential Derivation This is a **high-yield 6-mark theoretical derivation** that is frequently tested in Section B of ECE 2107 exams. ### The Proof: 1. **State the Trigonometric Fourier Series Baseline:** $$x(t) = rac{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) \quad ext{--- (Equation 1)}$$ 2. **Substitute Euler's Identities:** Replace the real-valued trigonometric terms with their complex exponential equivalents using Euler's formulas: $$\cos(n \Omega_0 t) = rac{e^{j n \Omega_0 t} + e^{-j n \Omega_0 t}}{2}$$ $$\sin(n \Omega_0 t) = rac{e^{j n \Omega_0 t} - e^{-j n \Omega_0 t}}{2j}$$ 3. **Incorporate into the Series expansion:** $$x(t) = rac{a_0}{2} + \sum_{n=1}^{\infty} a_n \left( rac{e^{j n \Omega_0 t} + e^{-j n \Omega_0 t}}{2} ight) + \sum_{n=1}^{\infty} b_n \left( rac{e^{j n \Omega_0 t} - e^{-j n \Omega_0 t}}{2j} ight)$$ 4. **Group the terms by exponent sign:** Recall that $ rac{1}{j} = -j$. Re-algebraize the coefficients of the positive exponentials ($e^{j n \Omega_0 t}$) and the negative exponentials ($e^{-j n \Omega_0 t}$): $$x(t) = rac{a_0}{2} + \sum_{n=1}^{\infty} \left( rac{a_n - j b_n}{2} ight) e^{j n \Omega_0 t} + \sum_{n=1}^{\infty} \left( rac{a_n + j b_n}{2} ight) e^{-j n \Omega_0 t}$$ 5. **Define the Symmetrical Complex Coefficients $C_n$:** To simplify this expression into a single consolidated sum from $-\infty$ to $+\infty$, we define: * **For $n = 0$ (The DC offset):** $$C_0 = rac{a_0}{2}$$ * **For $n > 0$ (Positive harmonics):** $$C_n = rac{a_n - j b_n}{2}$$ * **For $n < 0$ (Negative harmonics):** Let $m = -n > 0$: $$C_n = C_{-m} = rac{a_m + j b_m}{2} = C_m^*$$ 6. **Assemble the Symmetrical Compact Sum:** By mapping these definitions, the negative exponent sum merges with the positive exponent sum to yield the single unified expression: $$x(t) = \sum_{n=-\infty}^{\infty} C_n e^{j n \Omega_0 t} \quad ext{(Q.E.D.)}$$ --- ## 4. Complex Spectral Plots (Magnitude & Phase) Because the complex coefficients $C_n$ are generally complex numbers, they must be represented in polar form to be plotted physically: $$C_n = |C_n| e^{j ngle C_n}$$ * **Magnitude Spectrum ($|C_n|$):** A plot of the magnitude of each harmonic phasor versus frequency $n \Omega_0$. For any real-valued signal, the magnitude spectrum is **strictly symmetric (even)**: $$|C_{-n}| = |C_n|$$ * **Phase Spectrum ($ngle C_n$):** A plot of the phase angle of each harmonic phasor versus frequency $n \Omega_0$. For any real-valued signal, the phase spectrum is **strictly anti-symmetric (odd)**: $$ngle C_{-n} = -ngle C_n$$ ``` Magnitude Spectrum |Cn| Phase Spectrum ngle Cn | | +pi/2 (90 deg) 0.75 (DC) | o | | | o | o 0.216 +----+----o---> n | | | -pi/2| | --+---+---+-----> n (Harmonics) (o) | -1 0 1 | ``` --- ## 5. High-Yield Worked Examples (The Exam Killers) ### 5.1 The 12-Mark Half-Wave Rectified Cosine [PYQ 2021] **Question:** Determine the complex exponential Fourier series for a half-wave rectified cosine signal with amplitude $A$ and fundamental period $T_0$. ``` Amplitude = A _ _ _ _ / \ / -----------+ +-----------+ +----------- -T0/4 T0/4 3T0/4 5T0/4 <-------- Period T0 ----------> ``` #### Step-by-Step Solution: 1. **Formulate the Piecewise Mathematical Model:** For a single period centered around the origin $t = 0$: $$x(t) = egin{cases} A \cos(\Omega_0 t), & - rac{T_0}{4} \le t \le rac{T_0}{4} \ 0, & - rac{T_0}{2} \le t < - rac{T_0}{4} \quad ext{and} \quad rac{T_0}{4} < t \le rac{T_0}{2} \end{cases}$$ 2. **Evaluate the Complex Fourier Integral:** Using the analysis equation over the interval $[- rac{T_0}{2}, rac{T_0}{2}]$: $$C_n = rac{1}{T_0} \int_{-T_0/4}^{T_0/4} A \cos(\Omega_0 t) e^{-j n \Omega_0 t} \, dt$$ Substitute $\cos(\Omega_0 t) = rac{e^{j \Omega_0 t} + e^{-j \Omega_0 t}}{2}$: $$C_n = rac{A}{2 T_0} \int_{-T_0/4}^{T_0/4} \left( e^{j \Omega_0 t} + e^{-j \Omega_0 t} ight) e^{-j n \Omega_0 t} \, dt$$ $$C_n = rac{A}{2 T_0} \int_{-T_0/4}^{T_0/4} \left[ e^{-j(n-1)\Omega_0 t} + e^{-j(n+1)\Omega_0 t} ight] \, dt$$ 3. **Evaluate for the Specific Case $n = \pm 1$:** Substitute $n = 1$: $$C_1 = rac{A}{2 T_0} \int_{-T_0/4}^{T_0/4} \left( 1 + e^{-j 2 \Omega_0 t} ight) \, dt$$ $$C_1 = rac{A}{2 T_0} \left[ t + rac{e^{-j 2 \Omega_0 t}}{-j 2 \Omega_0} ight]_{-T_0/4}^{T_0/4}$$ Recall that $\Omega_0 = rac{2\pi}{T_0} \implies 2 \Omega_0 \left( rac{T_0}{4} ight) = \pi$: $$C_1 = rac{A}{2 T_0} \left[ rac{T_0}{2} + rac{e^{-j \pi} - e^{j \pi}}{-j 2 \Omega_0} ight]$$ Since $e^{-j \pi} = e^{j \pi} = -1 \implies e^{-j\pi} - e^{j\pi} = 0$: $$C_1 = rac{A}{2 T_0} \left( rac{T_0}{2} + 0 ight) = rac{A}{4}$$ Since the signal is real and even, we are guaranteed that $C_{-1} = C_1^* = rac{A}{4}$. 4. **Evaluate for general $n eq \pm 1$:** $$C_n = rac{A}{2 T_0} \left[ rac{e^{-j(n-1)\Omega_0 t}}{-j(n-1)\Omega_0} + rac{e^{-j(n+1)\Omega_0 t}}{-j(n+1)\Omega_0} ight]_{-T_0/4}^{T_0/4}$$ Substitute the upper and lower limits $\Omega_0 \left( rac{T_0}{4} ight) = rac{\pi}{2}$: $$C_n = rac{A}{2 T_0 \Omega_0} \left[ rac{e^{-j(n-1)\pi/2} - e^{j(n-1)\pi/2}}{-j(n-1)} + rac{e^{-j(n+1)\pi/2} - e^{j(n+1)\pi/2}}{-j(n+1)} ight]$$ Substitute $T_0 \Omega_0 = 2\pi$ and simplify the complex fractions using $ rac{e^{j heta} - e^{-j heta}}{2j} = \sin( heta)$: $$C_n = rac{A}{2\pi} \left[ rac{\sin((n-1)\pi/2)}{n-1} + rac{\sin((n+1)\pi/2)}{n+1} ight]$$ Applying trigonometric angle-addition identities: * $\sin\left( rac{n\pi}{2} - rac{\pi}{2} ight) = -\cos\left( rac{n\pi}{2} ight)$ * $\sin\left( rac{n\pi}{2} + rac{\pi}{2} ight) = \cos\left( rac{n\pi}{2} ight)$ $$C_n = rac{A \cos(n\pi/2)}{2\pi} \left[ - rac{1}{n-1} + rac{1}{n+1} ight]$$ $$C_n = rac{A \cos(n\pi/2)}{2\pi} \left[ rac{-(n+1) + (n-1)}{n^2 - 1} ight] = rac{A \cos(n\pi/2)}{2\pi} \left[ rac{-2}{n^2 - 1} ight]$$ $$\mathbf{C_n = rac{A \cos(n\pi/2)}{\pi(1 - n^2)}} \quad (n eq \pm 1)$$ 5. **Write out the Final Complex Exponential Fourier Series:** $$x(t) = rac{A}{4} e^{j \Omega_0 t} + rac{A}{4} e^{-j \Omega_0 t} + \sum_{n=-\infty, n eq \pm 1}^{\infty} \left[ rac{A \cos(n\pi/2)}{\pi(1 - n^2)} ight] e^{j n \Omega_0 t}$$ --- ### 5.2 The 3-Mark Spectral Identification Question [PYQ 2018] **Question:** The complex exponential Fourier representation of a signal $x(t)$ over the period $(0, T)$ is: $$x(t) = \sum_{n=-\infty}^{\infty} rac{3}{4+(n\pi)^2} e^{j 3 n \pi t}$$ Determine: (i) the fundamental period $T$, and (ii) plot the double-sided amplitude and phase spectrum. #### Step-by-Step Solution: 1. **Identify the Fundamental Frequency:** Compare the exponent term of the given equation directly to the standard Synthesis Equation: $$e^{j n \Omega_0 t} = e^{j 3 n \pi t} \implies n \Omega_0 t = 3 n \pi t \implies \Omega_0 = 3\pi ext{ rad/s}$$ 2. **Calculate the Period $T$:** $$\Omega_0 = rac{2\pi}{T} \implies 3\pi = rac{2\pi}{T} \implies \mathbf{T = rac{2}{3} ext{ s}}$$ 3. **Calculate Spectral Magnitude Values $|C_n|$:** The complex coefficient is purely real and positive for all $n$: $$C_n = rac{3}{4+(n\pi)^2} \implies |C_n| = rac{3}{4+(n\pi)^2}$$ * For $n=0$: $|C_0| = rac{3}{4} = \mathbf{0.75}$ * For $n = \pm 1$: $|C_{\pm 1}| = rac{3}{4+\pi^2} pprox rac{3}{4+9.87} pprox \mathbf{0.216}$ * For $n = \pm 2$: $|C_{\pm 2}| = rac{3}{4+4\pi^2} pprox rac{3}{4+39.48} pprox \mathbf{0.069}$ 4. **Determine the Phase Spectrum $ngle C_n$:** Since $C_n$ is purely real and positive for all $n$, the phase angle is exactly zero: $$ngle C_n = 0 ext{ rad} \quad ( ext{for all } n)$$ --- ## 6. ECE 2108 Laboratory MATLAB Representation In **ECE 2108 Lab Experiment 3**, you reconstruct complex exponential series by running finite harmonic loop summations. Below is the script used to reconstruct an exponential decay signal $f(t) = e^{-0.5 t}$ over $[0, \pi]$ using its complex coefficients: ```matlab % ECE 2108 Lab 3: Exponential Fourier Series Reconstruction clc; clear all; close all; N = 1000; % Time vector resolution t = linspace(0, pi, N); % One full cycle sum_signal = 0.504; % DC Component (C0) M = 64; % Number of harmonics to sum for j = 1:M % Summing complex conjugate phasor pairs to yield a real-valued output sum_signal = sum_signal + ((0.504*2)/(1+16*j^2)) * (cos(2*j*t) + 4*j*sin(2*j*t)); end % Plotting results figure; plot(t, sum_signal, 'r', 'LineWidth', 2); hold on; plot(t, exp(-0.5*t), 'g--', 'LineWidth', 2); grid on; xlabel('Time (sec)'); ylabel('Amplitude'); legend('Fourier Reconstruction', 'Ideal Exponential'); title(['Reconstruction using ' num2str(M) ' Harmonics']); ``` --- ## 7. Common Mistakes That Cost Marks > [!danger] **The Positive exponent Analysis Trap** > > A very common student slip is using a positive exponent $e^{j n \Omega_0 t}$ inside the analysis integral for $C_n$. Remember: **The Synthesis equation (summation) uses a positive exponent**, while **the Analysis equation (integral) must use a negative exponent $e^{-j n \Omega_0 t}$**. Writing a positive exponent in the integral swaps the values of $C_n$ and $C_{-n}$, causing a cascade of errors. > [!warning] **The Real-Valued Phase Jump Failure** > > When a coefficient $C_n$ evaluates to a purely real, **negative** number (for example, $C_2 = -0.15$), students often plot its phase as $0^\circ$. A negative real number has a magnitude of $+0.15$, but its phase is $\pm 180^\circ$ (or $\pm \pi$ rad). Ignoring this phase inversion results in a **zero-mark** grading for the phase spectrum. --- ## 8. PYQ Bank — Verbatim Questions & Answer Plans ### 8.1 PYQ 2021 [3 Marks] **Question:** Determine the complex exponential Fourier series for the signal illustrated below: Half-wave rectified cosine. * **Answer Plan:** 1. Define the piecewise integration limits for a single period centered at the origin as shown in **Section 5.1**. 2. Write the complex Fourier coefficient integral using a negative exponent. 3. Evaluate the separate cases for $n = \pm 1$ and general $n eq \pm 1$, proving $C_n = rac{A \cos(n\pi/2)}{\pi(1 - n^2)}$ [5.1]. 4. Assemble the final synthesis summation [5.1]. ### 8.2 PYQ 2018 [3 Marks] **Question:** The complex exponential Fourier representation of a signal $x(t)$ over period $(0, T)$ is $x(t) = \sum_{n=-\infty}^{\infty} rac{3}{4+(n\pi)^2} e^{j3n\pi t}$. Determine (i) the value of period $T$; (ii) plot the double-sided amplitude and phase spectrum. * **Answer Plan:** 1. Extract the fundamental frequency $\Omega_0 = 3\pi$ rad/s by comparing exponents [5.2]. 2. Solve for $T = 2/3$ s using the period formula [5.2]. 3. Calculate the magnitude coefficients for $n = 0, \pm 1, \pm 2$ to plot the symmetric double-sided amplitude spectrum [5.2]. 4. Identify the phase spectrum as a flat $0^\circ$ line because all coefficients are real and positive [5.2]. --- ## 9. Self-Check Before Moving On - [ ] Can you mathematically derive the complex coefficient conversion formulas ($C_n = rac{a_n - jb_n}{2}$) starting from trigonometric integrals? - [ ] Do you know how to determine the phase of a negative real Fourier coefficient? - [ ] Can you calculate the fundamental period $T$ of a complex Fourier series directly from its exponent term? [5.2] - [ ] Do you know how to plot double-sided magnitude and phase spectra, highlighting their respective symmetries? --- *Source: (k.Deergha Rao) signals and systems.pdf (Chapter 3), 01 Fourier Series.pdf, ECE 2108 Signal & Systems-1 (1).pdf (Experiment 3).*