Freq Domain L1

Why is frequency domain analysis used instead of time domain?

It reveals hidden frequency components in signals that are not obvious in time domain :contentReference[oaicite:0]{index=0}

Explain the key idea behind Fourier analysis.

Any signal can be represented as a sum of sinusoids or complex exponentials :contentReference[oaicite:1]{index=1}

Why is Fourier analysis powerful in engineering?

It allows complex signals to be broken down into simpler frequency components for analysis :contentReference[oaicite:2]{index=2}

Give engineering applications of Fourier analysis.

Signal processing vibration analysis heat transfer and communications :contentReference[oaicite:3]{index=3}

Explain what it means to "look inside" a signal.

To identify the frequency components that make up the signal :contentReference[oaicite:4]{index=4}

List main Fourier methods.

Fourier series Fourier transform DFT and FFT :contentReference[oaicite:5]{index=5}

Why is FFT widely used?

It efficiently computes frequency components for digital signals :contentReference[oaicite:6]{index=6}

Explain complex exponential representation of signals.

Signals can be expressed as e^(jθ) which simplifies analysis and computation :contentReference[oaicite:7]{index=7}

State Euler’s formula.

e^(jθ) = cosθ + j sinθ :contentReference[oaicite:8]{index=8}

Explain why complex form is useful.

It simplifies multiplication division and frequency analysis

Convert cosine into exponential form.

cos(x) = (e^(jx) + e^(-jx)) / 2 :contentReference[oaicite:9]{index=9}

Convert sine into exponential form.

sin(x) = (e^(jx) - e^(-jx)) / (2j) :contentReference[oaicite:10]{index=10}

Explain physical meaning of complex signals.

They represent magnitude and phase of sinusoidal components

Explain concept of signal approximation.

A signal can be approximated by scaling and combining basis functions :contentReference[oaicite:11]{index=11}

What is projection in signal approximation?

The contribution of one function in representing another :contentReference[oaicite:12]{index=12}

Explain why error vector must be perpendicular in approximation.

This minimises error magnitude giving best approximation :contentReference[oaicite:13]{index=13}

Define orthogonality in vectors.

Two vectors are orthogonal if their dot product is zero :contentReference[oaicite:14]{index=14}

Why does orthogonality matter in signal decomposition?

It ensures components are independent and do not overlap

Explain dot product physically.

Measures alignment between two vectors

State dot product formula in 2D.

V1·V2 = x1x2 + y1y2 :contentReference[oaicite:15]{index=15}

When is dot product zero?

When vectors are perpendicular (orthogonal)

Extend orthogonality concept to functions.

Two functions are orthogonal if their integral product is zero :contentReference[oaicite:16]{index=16}

State orthogonality condition for functions.

∫ f1(t)f2(t) dt = 0 :contentReference[oaicite:17]{index=17}

Explain meaning of non-zero integral in functions.

Functions share components and are not independent

Explain why orthogonal functions are useful.

They allow clean decomposition of signals into independent parts

Are sine and cosine orthogonal?

Yes over a full period :contentReference[oaicite:18]{index=18}

Is square wave orthogonal to sine?

No it contains sinusoidal components :contentReference[oaicite:19]{index=19}

Explain why square wave contains sinusoids.

It can be approximated as a sum of sinusoidal harmonics

Explain projection coefficient λ.

It determines how much of one function contributes to another :contentReference[oaicite:20]{index=20}

State formula for projection coefficient.

λ = (∫f1(t)f2(t)dt) / (∫f2^2(t)dt) :contentReference[oaicite:21]{index=21}

Explain physical meaning of λ.

Amplitude scaling for best approximation

Why is error minimised using projection?

Because orthogonality ensures minimum squared error

Explain why integration is done over one period.

Periodic signals repeat so one period fully represents behaviour :contentReference[oaicite:22]{index=22}

State period-frequency relationship.

ω = 2π / T :contentReference[oaicite:23]{index=23}

Explain result λ = 4/π for square wave approximation.

It is the best-fit amplitude for first sinusoidal component :contentReference[oaicite:24]{index=24}

Explain why adding more sinusoids improves approximation.

It reduces error by capturing more signal features :contentReference[oaicite:25]{index=25}

Explain general signal decomposition.

f(t) = λ1f1(t) + λ2f2(t) + ... + error :contentReference[oaicite:26]{index=26}

What happens as more components are added?

Error approaches zero

Explain basis functions in Fourier analysis.

Orthogonal sinusoids used to build signals

Why must basis functions be orthogonal?

To ensure independent contributions and accurate decomposition

Explain final takeaway of Fourier analysis.

Any signal can be represented as sum of orthogonal sinusoidal components :contentReference[oaicite:27]{index=27}