Fourier Series is a mathematical tool that represents a periodic signal as a weighted sum of harmonically related sinusoids (or complex exponentials).

Definition: Any periodic signal with period that satisfies the Dirichlet conditions can be expressed as:

where is the fundamental angular frequency and are the Fourier coefficients.

Significance:

• It decomposes a complex periodic signal into simple sinusoidal components.
• Enables frequency domain analysis of signals.
• Reveals the harmonic content (frequency spectrum) of a signal.
• Simplifies analysis of LTI systems, since sinusoids are eigenfunctions of such systems.
• Forms the foundation for the Fourier Transform and other spectral analysis tools.

Exponential Fourier Series Representation:

A continuous-time periodic signal with fundamental period can be written as:

Derivation of :

Multiply both sides by and integrate over one period:

Using the orthogonality property:

Only the term survives, giving:

Therefore:

This is called the analysis equation, while the summation is the synthesis equation.

The Dirichlet conditions are a set of sufficient conditions that guarantee the convergence of the Fourier series of a periodic signal .

Condition 1 — Absolute Integrability:
Over any period, must be absolutely integrable:

This guarantees that all coefficients are finite.

Condition 2 — Finite Number of Maxima and Minima:
In any finite interval, must have a finite number of maxima and minima (bounded variation).

Condition 3 — Finite Number of Discontinuities:
In any finite interval, must have a finite number of finite discontinuities.

Key Points:

• These are sufficient but not necessary conditions.
• Most practical signals satisfy them.
• At points of discontinuity, the Fourier series converges to the average of the left and right limits:
  

Gibbs Phenomenon refers to the peculiar behavior of the Fourier series near points of discontinuity of a signal.

Explanation:

• When a signal with a jump discontinuity (e.g., a square wave) is approximated by a finite number of Fourier series terms, an overshoot (ripple) appears near the discontinuity.
• As more terms are added, the ripples get compressed toward the discontinuity but the peak overshoot does not diminish — it stays at approximately 9% of the jump height.

Key Characteristics:

• The overshoot magnitude remains constant (~9%) regardless of the number of harmonics .
• As , the width of the overshoot region approaches zero.
• The energy in the error goes to zero, so convergence still holds in the mean-square sense, but not uniformly.

Practical Implication:

• Causes ringing artifacts in reconstructed signals (e.g., in filtering and signal reconstruction).
• Can be reduced using window functions (e.g., Hamming, Hanning windows).

Linearity Property:

If and are two periodic signals with the same period , having Fourier coefficients and respectively, then:

Proof:

The Fourier coefficients of are:

Substitute :

Split the integral:

Hence the Fourier series is a linear operation.

Time-Shifting Property:

If , then a time-shifted signal has:

Proof:

Let . Its Fourier coefficients are:

Substitute , :

Interpretation: A time shift changes only the phase of each coefficient, not its magnitude. Hence — the magnitude spectrum is unchanged.

Parseval's Theorem:

For a periodic signal with Fourier coefficients , the average power is:

Proof:

Start with the average power:

Substitute for :

Interchange sum and integral:

Physical Significance:

• The total average power in the time domain equals the sum of powers of all harmonic components in the frequency domain.
• represents the power contributed by the harmonic.
• It confirms energy/power conservation between the two domains.

Given: A symmetric periodic square wave with period .

Fourier Coefficient Calculation:

For (DC component):

For :

Since :

Observation:

• The coefficients follow a sinc function envelope.
• are real (due to even symmetry of ).
• The spectrum is discrete with harmonics spaced at .

Conjugate Symmetry Property:

If is a real-valued periodic signal with Fourier coefficients , then:

Explanation / Proof:

Since is real, . The coefficients are:

Taking the complex conjugate:

Consequences:

• Magnitude is even:
• Phase is odd:
• Real part is even, imaginary part is odd.

Special Cases:

• If is real and even, then is real and even.
• If is real and odd, then is purely imaginary and odd.

Comparison of Fourier Series Forms:

Relation Between Coefficients:

Conclusion: The exponential form is preferred in advanced analysis due to its compactness and mathematical elegance, while the trigonometric form gives a more intuitive picture of magnitude and phase.

Time-Scaling Property:

If has period and Fourier coefficients , then (with ) is periodic with period and fundamental frequency .

Key Result: The Fourier coefficients remain unchanged, but the harmonic frequencies are scaled:

Proof:

Start with:

Replace with :

Interpretation:

• The coefficients are identical.
• Only the fundamental frequency changes from to .
• Time compression () spreads out the spectrum in frequency.

Frequency Spectrum:

The frequency spectrum of a periodic signal is a graphical representation of its Fourier coefficients plotted against frequency (or harmonic number ).

Since a periodic signal contains only discrete harmonic frequencies, the spectrum is a discrete (line) spectrum.

Two Components:

1. Magnitude Spectrum:

• A plot of versus frequency .
• Shows the strength/amplitude of each harmonic component.
• For real signals, it is an even function: .

2. Phase Spectrum:

• A plot of versus frequency .
• Shows the phase angle of each harmonic.
• For real signals, it is an odd function: .

Key Points:

• The spectrum exists only at discrete frequencies (multiples of ).
• Together, magnitude and phase spectra completely characterize the signal in the frequency domain.
• Useful for understanding harmonic content, bandwidth, and filtering.

Given:

The original sine wave has period , but the rectified signal has period . Let the new fundamental frequency be .

Fourier Coefficients:

Using and , integrating over :

Evaluating the integral yields:

Observations:

• The DC component (): .
• All coefficients are real (since is even).
• Only even harmonics of the original frequency appear.
• The amplitudes decrease as , indicating fast convergence.

Differentiation Property:

If , then:

Explanation: Differentiating the synthesis equation term by term:

Each coefficient is multiplied by . This emphasizes high-frequency components (larger ).

Integration Property:

If (with for finite result), then:

Explanation: Integration divides each coefficient by , which attenuates high-frequency components and produces a smoother signal.

Note: For integration to yield a periodic signal, the DC term must be zero; otherwise the integral grows unbounded.

Multiplication Property:

If and (both with the same period ), then their product has coefficients given by the discrete convolution of the individual coefficients:

Proof:

Express both signals in Fourier series form:

Let :

Comparing with :

Interpretation: Multiplication in the time domain corresponds to convolution in the frequency (coefficient) domain.

Software Simulation of Frequency Spectrum:

Software tools like MATLAB, Octave, or Python (NumPy/Matplotlib) are used to compute and visualize the Fourier coefficients of periodic signals.

General Procedure:

1. Define the signal: Create a time vector and generate the periodic signal samples over one or more periods.
2. Compute coefficients: Numerically evaluate the analysis equation using integration or the FFT algorithm.
3. Extract magnitude and phase: Compute and .
4. Plot spectra: Use stem plots to display the discrete magnitude and phase spectra.

Example (Python):
python
import numpy as np
import matplotlib.pyplot as plt

T = 1.0 # period
fs = 1000 # sampling rate
t = np.arange(0, T, 1/fs)
x = np.sign(np.sin(2np.pit/T)) # square wave

FFT for coefficients

X = np.fft.fft(x)/len(x)
k = np.fft.fftfreq(len(x), 1/fs)

plt.stem(k[:20], np.abs(X[:20]))
plt.xlabel('Frequency (Hz)')
plt.ylabel('|a_k|')
plt.title('Magnitude Spectrum')
plt.show()

Advantages:

• Fast computation via FFT.
• Easy visualization of harmonic content.
• Allows study of Gibbs phenomenon by varying the number of harmonics.

Reconstruction Using Partial Sums:

A square wave can be reconstructed by summing its harmonics. The Fourier series of an odd symmetric square wave (amplitude ) contains only odd harmonics:

Partial Sum: Using a finite number of harmonics:

Effect of Increasing N:

• : A single sinusoid — a rough approximation.
• : The waveform begins to flatten and resemble a square shape.
• Large : The approximation improves; edges become sharper.

Key Observations:

• Adding more harmonics improves accuracy in the flat regions.
• Near discontinuities, the Gibbs overshoot (~9%) persists regardless of .
• The mean-square error decreases as .

Simulation Insight: Plotting for increasing clearly shows both improved reconstruction and the persistent Gibbs ripples near edges.

Mean-Square Convergence vs Pointwise Convergence:

Explanation:

• Mean-square convergence is a weaker, energy-based criterion. It is satisfied by nearly all practical signals since it only requires .
• Pointwise convergence demands the series equal the signal value at each individual point. At jump discontinuities, the series converges to , so exact pointwise equality fails there.

Conclusion: For signals with discontinuities, mean-square convergence holds even though uniform convergence does not — the Gibbs overshoot has zero energy in the limit.

Given: An impulse train (Dirac comb) with period :

Computing Fourier Coefficients:

Within one period (say to ), there is a single impulse at :

Using the sifting property of the impulse :

Fourier Series Representation:

Observations:

• All coefficients are equal and real ().
• The spectrum is a uniform impulse train in frequency.
• This important result shows that an impulse train in time corresponds to an impulse train in frequency — fundamental to sampling theory.

Complex Exponentials as Eigenfunctions:

An eigenfunction of a system is an input signal for which the output is a scaled version of the same input. The scaling factor is the eigenvalue.

Demonstration:

Consider an LTI system with impulse response . Apply the input . The output is the convolution:

The output is the same exponential scaled by , the system's frequency response. Thus is an eigenfunction with eigenvalue .

Usefulness for Fourier Analysis:

• Since a periodic signal can be decomposed as , the output is simply:
  
• Each harmonic is independently scaled by the frequency response.
• This transforms the difficult convolution operation into simple multiplication in the frequency domain, greatly simplifying LTI system analysis.