- Exponential Fourier Series Definition: The exponential Fourier series is defined as a method to represent a periodic signal using complex exponentials.
- Representation of Periodic Signal: eriodic signals can be represented in both continuous and discrete time domains using Fourier series.
- Amplitude and Phase Spectra: The amplitude spectrum shows the magnitude of the Fourier coefficients, while the phase spectrum shows their angles.
- Symmetry in Spectra: For real signals, the amplitude spectrum is even, and the phase spectrum is odd.
- Parseval’s Theorem: This theorem explains that the average power of a periodic signal equals the sum of the squares of its Fourier coefficients.
Fourier Series at a Glance
A continuous-time signal x(t) is periodic if a positive value T exists for which the following relation holds for every t:
The smallest such T, when it exists, is the fundamental period. A periodic signal satisfying suitable convergence conditions, including the sufficient Dirichlet’s Conditions, can be represented by a FOURIER SERIES of harmonically related basis functions.
Two common Fourier Series forms are equivalent when their coefficients are converted consistently.
- Exponential Fourier Series
- Trigonometric Fourier Series
The trigonometric form uses real sine and cosine coefficients. The exponential form uses complex coefficients and often makes algebra, system analysis and symmetry more compact.
Exponential Fourier Series
This page covers three parts of Exponential Fourier Series analysis:
- Representation of Periodic Signal.
- Amplitude and Phase Spectra of a Periodic Signal.
- Power Content of a Periodic Signal.
Representation of Periodic Signal
Fourier series apply to two distinct signal classes:
- Continuous Time Domain.
- Discrete Time Domain.
Continuous Time Domain
The complex Exponential Fourier Series representation of a continuous-time periodic signal x(t) with fundamental period To is:
Each Ck is a Complex Fourier Coefficient calculated by:
The integral ∫0T0 can cover any complete period. Common limits include 0 to T0 and -T0/2 to T0/2. Shifting the integration interval by a full period does not change the coefficient.
To derive the analysis equation, multiply the synthesis series by e(-jlω0t) and integrate both sides over one period.
After interchanging summation and integration where convergence permits:
Orthogonality makes the integral zero when k ≠ l. When k = l, the exponential product is one and the integral equals the period.
Equation (4) therefore reduces to:
The zero-index coefficient C0 is the average value of x(t) over one period.
For a real-valued x(t), the coefficients have conjugate symmetry:
Here, * denotes complex conjugation.
Discrete Time Domain
The discrete-time Fourier series uses a finite set of unique complex exponentials over one sequence period.
For a periodic sequence x[n] with fundamental period No, the synthesis equation is:
The Ck coefficients follow the analysis equation:
There are N0 unique coefficients. Extending the index beyond one block repeats the coefficient sequence with period N0. Discrete-time exponential orthogonality gives the derivation.
Amplitude and Phase Spectra of a Periodic Signal
Write each complex Fourier coefficient Ck in polar form:
A plot of |Ck| at angular frequency kω0 is the magnitude spectrum. A plot of phase Фk at the same harmonic frequencies is the phase spectrum. Because k is an integer, a continuous-time periodic signal has discrete line spectra rather than a continuous spectral curve.
For a real periodic signal, C-k = Ck*, so:
The magnitude spectrum is even. The phase is antisymmetric modulo 2π where the coefficient is nonzero; phase is undefined at zero magnitude. The original reference to an odd function of 0 should read angular frequency ω.
Power Content of a Periodic Signal
The average Power Content of a Periodic Signal is the one-period mean of |x(t)|²:
For the stated complex exponential Fourier-series normalization, Parseval’s relation is:
This identity equates time-domain average power with the sum of |Ck|². For real signals, |x(t)|² equals x(t)²; complex signals require the magnitude squared.
