Signal Freq Domain L1

Why is time-domain analysis sometimes insufficient?

Because complex signals can hide frequency components that are not obvious in time domain :contentReference[oaicite:0]{index=0}

What is the purpose of Fourier Analysis?

To decompose signals into sinusoidal components to understand their frequency content :contentReference[oaicite:1]{index=1}

Explain the key difference between Fourier Series and Fourier Transform.

Fourier Series applies to periodic signals while Fourier Transform applies to aperiodic signals :contentReference[oaicite:2]{index=2}

State the Fourier Transform equation.

G(ω) = ∫ f(t)e^{-jωt} dt :contentReference[oaicite:3]{index=3}

State the inverse Fourier Transform.

f(t) = (1/2π) ∫ G(ω)e^{jωt} dω :contentReference[oaicite:4]{index=4}

Explain what Fourier Transform physically represents.

It shows how much of each frequency exists in a signal

Explain why aperiodic signals require Fourier Transform.

Because they cannot be represented by repeating sinusoids :contentReference[oaicite:5]{index=5}

Explain how Fourier Transform is derived from Fourier Series.

By letting the period approach infinity :contentReference[oaicite:6]{index=6}

What does G(ω) represent?

Frequency spectrum of the signal

Explain why Fourier Transform produces continuous spectrum.

Because aperiodic signals require continuous frequency range

What is the key difference between discrete and continuous spectrum?

Periodic signals → discrete lines

Explain meaning of magnitude spectrum.

Shows amplitude of each frequency component

Explain meaning of phase spectrum.

Shows phase shift of each frequency component

State Parseval’s theorem.

Total signal power equals sum of squares of Fourier coefficients :contentReference[oaicite:7]{index=7}

Why is Parseval’s theorem important?

Links time-domain energy to frequency-domain energy

Explain Fourier Transform of square pulse.

Produces sinc-shaped frequency spectrum :contentReference[oaicite:8]{index=8}

State result for square pulse FT.

G(ω) = 2τ sin(ωτ)/ω :contentReference[oaicite:9]{index=9}

Explain effect of pulse width on spectrum.

Narrow pulse → wide spectrum

Define unit impulse function δ(t).

Infinitely narrow signal with unit area :contentReference[oaicite:11]{index=11}

Explain physical meaning of impulse function.

Represents idealised instantaneous event

State key property of impulse function.

∫ f(t)δ(t−a)dt = f(a) :contentReference[oaicite:12]{index=12}

Explain why impulse function is useful in sampling.

It allows discrete signals to be represented mathematically

Define convolution.

y(t) = ∫ h(τ)u(t−τ)dτ :contentReference[oaicite:13]{index=13}

Explain physical meaning of convolution.

Measures how one signal modifies another

Explain convolution process.

Flip one function

State convolution theorem.

Time-domain convolution equals frequency-domain multiplication :contentReference[oaicite:14]{index=14}

Why is convolution theorem important?

It simplifies system analysis in frequency domain

Explain effect of system on signal using convolution.

Output = input convolved with system response

Explain convolution with impulse.

Shifts the signal by impulse location :contentReference[oaicite:15]{index=15}

Define discrete signal.

Signal sampled at discrete time intervals :contentReference[oaicite:16]{index=16}

Why are real-world signals discrete?

Because computers sample signals at finite intervals

Explain sampled signal representation.

f(t) = Σ xn δ(t−nT) :contentReference[oaicite:17]{index=17}

Explain Fourier Transform of sampled signal.

Becomes sum of exponentials weighted by samples :contentReference[oaicite:18]{index=18}

Define Discrete Fourier Transform (DFT).

Transforms finite discrete signal into frequency domain

State DFT equation.

X(k) = Σ xn e^{-j2πkn/N}

Explain meaning of DFT output.

Frequency components at discrete frequencies

Define sampling frequency.

Rate at which signal is sampled

Explain Nyquist frequency.

Maximum frequency that can be correctly captured = fs/2

Define aliasing.

High frequencies appearing as lower frequencies due to undersampling

Explain cause of aliasing.

Sampling rate too low relative to signal frequency

Explain how to prevent aliasing.

Use sampling rate > 2× highest frequency

Define frequency resolution.

Smallest distinguishable frequency difference

State resolution formula.

Δf = fs / N

Explain how to improve frequency resolution.

Increase number of samples

Define spectral leakage.

Energy spreading across frequency bins

Explain cause of leakage.

Signal not periodic within sampling window

Explain windowing.

Applying function to reduce leakage

Why does windowing help?

Reduces discontinuities at boundaries

Define Fast Fourier Transform (FFT).

Efficient algorithm to compute DFT

Why is FFT important?

Reduces computation from O(N²) to O(N log N)

Explain practical importance of FFT.

Allows real-time signal processing

Explain trade-off between time and frequency resolution.

Better frequency resolution requires longer time window

Explain relation between time compression and frequency spread.

Short signals have wide frequency content :contentReference[oaicite:19]{index=19}

Explain why sharp signals need many frequencies.

High frequency components required to represent rapid changes :contentReference[oaicite:20]{index=20}

Explain physical meaning of frequency domain.

Represents signal in terms of oscillatory components

Explain difference between amplitude and power spectrum.

Amplitude shows magnitude

Explain effect of increasing N in FFT.

Improves frequency resolution but increases computation

Explain importance of Fourier methods in engineering.

Used in signal processing control vibration and sensors :contentReference[oaicite:21]{index=21}