Trigonometric Fourier Series

The Trigonometric Fourier Series is one of the most fundamental tools in signals and systems analysis. It allows any periodic signal to be expressed as an infinite sum of sine and cosine terms, each at a harmonic frequency of the fundamental. Understanding this representation is essential for GATE aspirants and forms the backbone of spectral analysis in communication systems.

Core Concept Explanation

Any signal that repeats itself after a fixed time interval T is called a periodic signal. The Fourier Series states that such a signal can be perfectly reconstructed by summing an infinite set of sinusoids, each being an integer multiple of the fundamental frequency. This is not just a mathematical convenience but reflects a deep physical truth: any linear, time-invariant system responds independently to each frequency component, so knowing the spectrum of a signal means knowing how any such system will behave.

The fundamental frequency is defined as ω₀ = 2π/T radians per second, where T is the period of the signal. The nth harmonic has frequency nω₀. The first harmonic (n=1) is the fundamental, the second harmonic has twice the frequency, and so on. Each harmonic contributes a cosine term with coefficient aₙ and a sine term with coefficient bₙ.

The DC component a₀/2 represents the average value of the signal over one complete period. A signal with equal positive and negative areas over a cycle has zero DC value. In circuit terms, this is the steady offset around which the signal oscillates. All three components — DC, cosine terms, and sine terms — together rebuild the original periodic waveform exactly, provided enough harmonics are included.

It is important to note that the coefficients aₙ and bₙ are obtained by exploiting the orthogonality property of sinusoids. When two sinusoids of different integer frequencies are multiplied and integrated over a full period, the result is zero. This means multiplying x(t) by cos(nω₀t) and integrating isolates only the cosine component at that specific harmonic, giving aₙ directly.

Mathematical Expression

The complete trigonometric Fourier series representation is written as:

x(t) = a₀/2 + Σ [aₙ cos(nω₀t) + bₙ sin(nω₀t)] where the summation runs from n = 1 to infinity.

The three coefficient formulas are derived from orthogonality. The integral limits span any complete period, conventionally from 0 to T or from −T/2 to T/2. The factor 2/T normalizes the result so that the coefficient represents true amplitude.

a₀ = (2/T) ∫₀ᵀ x(t) dt

aₙ = (2/T) ∫₀ᵀ x(t) cos(nω₀t) dt

bₙ = (2/T) ∫₀ᵀ x(t) sin(nω₀t) dt

The amplitude of the nth harmonic is given by Cₙ = √(aₙ² + bₙ²), and the phase angle is φₙ = arctan(−bₙ/aₙ). These two quantities together constitute the amplitude and phase spectra of the signal.

Practical Understanding

In communication engineering, the Fourier series tells us how much power a signal carries at each harmonic frequency. When a periodic signal passes through a filter, the filter acts on each harmonic independently. Only those harmonics within the filter's passband survive at the output. This is why understanding the coefficient magnitudes and their decay rate with n is critical when designing filters and amplifiers.

For smooth signals, the coefficients aₙ and bₙ decrease rapidly with increasing n. For signals with sharp discontinuities such as square waves or sawtooth waves, the coefficients decay slowly (as 1/n), meaning many harmonics are needed to reconstruct the sharp edges. This slow decay is directly related to the Gibbs phenomenon, where truncating the series at any finite n causes overshoot near discontinuities.

Given:
A full-wave rectified cosine signal x(t) = |cos(πt/T)| with period T = 1 s
ω₀ = 2π/T = 2π rad/s

Why this formula applies:
The signal is periodic with known period, so Fourier coefficients are computed
by integrating over one period using the standard formulas.

Formula:
a₀ = (2/T) ∫₀ᵀ x(t) dt
aₙ = (2/T) ∫₀ᵀ x(t) cos(nω₀t) dt

Substitution:
For x(t) = |cos(πt)| with T = 1:
a₀ = 2 ∫₀¹ |cos(πt)| dt = 2 × [sin(πt)/π]₀^0.5 × 2 = 4/π ≈ 1.273

aₙ = 2 ∫₀¹ |cos(πt)| cos(2nπt) dt

Using product-to-sum identity and symmetry:
aₙ = (4/π) × [1/(1 - 4n²)] × (-1)^(n+1)  for n ≥ 1

Calculation for n = 1:
a₁ = (4/π) × [1/(1 - 4)] = (4/π) × (-1/3) = -4/(3π) ≈ -0.424

Calculation for n = 2:
a₂ = (4/π) × [1/(1 - 16)] = (4/π) × (-1/15) ≈ -0.085

Final Answer:
a₀/2 ≈ 0.637 (DC level)
a₁ ≈ -0.424 (fundamental cosine amplitude)
a₂ ≈ -0.085 (2nd harmonic amplitude)
bₙ = 0 for all n (even function, confirmed by symmetry)

Exam Tip: In GATE, if x(t) is an even function, all bₙ = 0 automatically. If x(t) is an odd function, all aₙ = 0 (including a₀). Never compute both sets of coefficients for a signal with clear symmetry — identify the symmetry first and eliminate half the work.

Quick Revision

• The Trigonometric Fourier Series expresses a periodic signal as x(t) = a₀/2 + Σ[aₙcos(nω₀t) + bₙsin(nω₀t)] where ω₀ = 2π/T.
• Coefficient a₀ = (2/T)∫x(t)dt gives the DC (average) value. The factor 2 keeps convention consistent with aₙ formula.
• aₙ and bₙ are found using orthogonality of cosines and sines. aₙ = (2/T)∫x(t)cos(nω₀t)dt, bₙ = (2/T)∫x(t)sin(nω₀t)dt.
• Harmonic amplitude: Cₙ = √(aₙ² + bₙ²). Phase: φₙ = arctan(−bₙ/aₙ). These form the amplitude and phase spectra.
• Even functions have bₙ = 0. Odd functions have aₙ = 0 and a₀ = 0. Half-wave symmetric functions have only odd harmonics.
• Slow 1/n coefficient decay indicates sharp discontinuities in the signal and is associated with the Gibbs phenomenon near edges.
• Common GATE trap: Confusing a₀ with a₀/2. The series uses a₀/2 as the DC term, but the formula computes a₀ directly — always write a₀/2 in the series expansion.