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Study notes

Qwestrum Engineering360 · Electrical & Electronics · Signals & Systems

Fourier Series and Transform

The Fourier series represents a periodic signal as a sum of harmonics (a discrete line spectrum), while the Fourier transform extends this to aperiodic signals (a continuous spectrum), revealing frequency content.

Exam tip: keep SI units consistent end-to-end, write the governing relation symbolically before substituting, and sanity-check magnitude and sign.

Key formulas & points

Skim these first — then read the full notes below.

Line spectrum for periodic; continuous spectrum for aperiodic
Symmetry: even → cosine terms; odd → sine terms
Convolution in time ↔ multiplication in frequency

Topic details

Introduction

A periodic signal is a weighted sum of harmonics at integer multiples of ω₀; the coefficients c_n form a discrete line spectrum. Symmetry simplifies the work: even signals have only cosine terms, odd signals only sine terms, and half-wave symmetry removes even harmonics.

Scope in B.Tech and GATE syllabus

For aperiodic signals the period tends to infinity, the lines merge into a continuous spectrum, and the Fourier transform X(jω) describes the amplitude and phase at each frequency.

Key relations & formulas

Formulas (Indian textbook notation)

$FS: x(t) = Σ c_{n} e^(jn\omega_{0}t); c_{n} = (\frac{1}{T})\int x(t) e^(-jn\omega_{0}t) dt$

Formulas (Indian textbook notation)

$FT: X(j\omega) = \int x(t) e^(-j\omega t) dt$

Formulas (Indian textbook notation)

$Parseval: \int |x|^{2} dt = (\frac{1}{2\pi})\int |X(j\omega)|^{2} d\omega$

Notation and sign conventions

Relation 1 —

FS: x

Formulas (Indian textbook notation)

$FS: x(t) = Σ c_{n} e^(jn\omega_{0}t); c_{n} = (\frac{1}{T})\int x(t) e^(-jn\omega_{0}t) dt$

Write this relation with symbols exactly as in Signals & Systems — Oppenheim & Willsky before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.

Relation 2 —

FT: X

Formulas (Indian textbook notation)

$FT: X(j\omega) = \int x(t) e^(-j\omega t) dt$

Write this relation with symbols exactly as in Signals & Systems — Oppenheim & Willsky before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.

Relation 3 —

Parseval: \int |x|^{2} dt =

Formulas (Indian textbook notation)

$Parseval: \int |x|^{2} dt = (\frac{1}{2\pi})\int |X(j\omega)|^{2} d\omega$

Write this relation with symbols exactly as in Signals & Systems — Oppenheim & Willsky before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.

Fundamentals and definitions

Key transform properties turn hard time-domain operations into easy frequency-domain ones: convolution in time becomes multiplication in frequency, and a time shift becomes a linear phase term. These make filtering and system analysis tractable.

Governing relations in practice

Parseval’s theorem equates total energy in time and frequency, allowing energy to be computed in whichever domain is simpler.

Design and analysis considerations

For a square wave the Fourier series contains only odd harmonics with amplitudes falling as 1/n, illustrating why sharp edges need many high-frequency components.

Assumptions and validity limits

State assumptions explicitly before using any relation for fourier series and transform — steady state, uniform properties, linear elastic material, ideal gas, incompressible flow, etc., as applicable.

Wrong assumptions invalidate the entire solution even when the formula is correct. In Signals & Systems viva and GATE descriptive questions, listing valid assumptions often earns separate marks.

Step-by-step problem approach

1. Read the question and list given data with SI units (common in Signals & Systems papers).
2. Draw a neat labelled diagram where applicable — examiners in Indian universities award diagram marks even when arithmetic slips.
3. Identify which relation from this topic applies to fourier series and transform.
4. Use equation 1:

FS: x

.
5. Use equation 2:

FT: X

.
6. Substitute values, compute, and verify units and sign (direction).
7. State conclusion in one line — e.g. safe/unsafe, stable/unstable, feasible/infeasible.

Applications & exam relevance

Fourier Series and Transform appears in communications and control. In Indian electrical curricula this topic is tested because it connects theory to continuous and discrete signals.

GATE and semester exams often combine fourier series and transform with earlier units — revise prerequisites before attempting mixed problems.

Industry interview panels sometimes ask: "Where did you use fourier series and transform?" — answer with a lab, mini-project, or plant visit example if possible.

Common mistakes in exams

• Including even harmonics for a signal with half-wave symmetry
• Forgetting the 1/T factor in the Fourier-series coefficient
• Confusing the discrete (series) and continuous (transform) spectra
• Missing the linear phase term for a time-shifted signal

Quick revision checklist

Before attempting fourier series and transform problems, confirm you can:

1. Line spectrum for periodic; continuous spectrum for aperiodic
2. Symmetry: even → cosine terms; odd → sine terms
3. Convolution in time ↔ multiplication in frequency

Revise the solved examples in Signals & Systems — Oppenheim & Willsky and one previous-year GATE or university paper for this unit.

Worked examples

Try the problem first — open the solution when you are ready to check.

Fundamental of a square wave

Problem

A symmetric square wave of amplitude ±A has a Fourier series with only odd harmonics of amplitude (4A/π)(1/n). Find the amplitude of the fundamental and the third harmonic for A = 5.

Solution

Fundamental (n = 1): amplitude = 4A/π = 4×5/π = 20/3.1416 = 6.37.
Third harmonic (n = 3): amplitude = (4A/π)(1/3) = 6.37/3 = 2.12.
Even harmonics are absent; amplitudes decay as 1/n.

Conceptual check — Fourier Series and Transform

Problem

In a Signals & Systems semester or GATE paper you are asked: "State the main assumption, the governing relation, and one practical consequence of fourier series and transform." What should a complete answer include?

Exams & GATE

Oppenheim — find Fourier coefficients of square wave.

📖 Standard books (India)

Signals & Systems — Oppenheim & Willsky
Read: Syllabus unit
Laplace, Fourier, and LTI systems

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