Signals & Systems
ECE-Y425 · Spring semester
Electrical & Computer Engineering Dept., University of Patras
About the course
This course introduces the mathematical language engineers use to describe, analyse and manipulate signals and the systems that act on them. Starting from the basic taxonomy of signals and the properties of linear time-invariant (LTI) systems, it builds up convolution and correlation, the Fourier series and transform, sampling and aliasing, and the discrete Fourier transform together with the FFT.
The second half turns these tools into practice: spectral leakage and windowing, frequency-domain filtering, the Z-transform, FIR and IIR filter design, real-time filtering, and an introduction to random processes, power spectral densities and practical denoising. Every lecture comes with a practical component in which the ideas are implemented and checked on real data.
Syllabus
- Lec. 1Introduction to Signals: Energy, Power & the dB Scale
- Lec. 2LTI Systems Essentials
- Lec. 3Convolution & Correlation
- Lec. 4From Fourier Series to the Fourier Transform
- Lec. 5Sampling, Aliasing & a DTFT Preview
- Lec. 6From DTFT to DFT: Why Finite Data Matters
- Lec. 7DFT and FFT
- Lec. 8Spectral Leakage & Frequency Resolution
- Lec. 9Windowing: Theory & Practice
- Lec. 10Ideal Filters & Frequency-Domain Filtering
- Lec. 11Z-transform
- Lec. 12FIR Filter Design
- Lec. 13IIR Filters
- Lec. 14From Filter Design to Real-Time Filtering
- Lec. 15Random Processes: SSS/WSS & Correlation
- Lec. 16Spectral Analysis of Stochastic Signals
- Lec. 17Noise Models, SNR & Practical Denoising
Lecture notes
The notes are written per topic, so one document often covers several lectures.
Lectures & materials, week by week
Slides and notes for each lecture, grouped by teaching week.
Week 1
Lecture 1
Introduction to Signals: Energy, Power & the dB Scale
- Taxonomy: CT/DT, periodic, deterministic/random, real/complex
- Basic signals (impulse/step/sinusoid/exponential/chirp)
- Energy, power, RMS, intro to Parseval idea
- dB/log plots, dynamic range intuition
- Practical: build/plot signals; compute E/P/RMS; dB scaling
Slides
Notes
Lecture 2
LTI Systems Essentials
- Introduction to systems; taxonomy
- System properties
- Stability
- LTI definition
- Zero-input response
- Zero-state response
- Impulse response
- Convolution from LTI perspective
- Convolution CT/DT, properties
Slides
Shared notes · Lectures 2 & 3: Introduction to Systems
Notes
Week 2
Lecture 3
Convolution & Correlation
- Convolution CT/DT, properties
- Correlation (auto/cross), relation to convolution
- Matched-filter intuition (preview for later noise topics)
- Practical: implement conv/corr; delay estimation in noise
Slides
Shared notes · Lectures 2 & 3: Introduction to Systems
Notes
Lecture 4
From Fourier Series to the Fourier Transform
- FS as projection on sinusoids, Gibbs effect quickly
- Transition to FT: limit of FS, core properties/pairs
- Duality + basic magnitude/phase meaning
- Practical: spectrum of simple waveforms + phase demo
Slides
Shared notes · Lectures 4 & 6: Fourier Transform
Notes
Week 3
Lecture 5
Sampling, Aliasing & a DTFT Preview
- Sampling theorem, Nyquist, reconstruction idea
- Aliasing in time/frequency with pictures
- DTFT concept as “true” discrete-time spectrum
- Practical: downsample audio, see aliasing
Slides
Notes
Lecture 6
From DTFT to DFT: Why Finite Data Matters
- Finite-length observation → periodic extension
- Frequency bins and grid; intuition for discrete spectrum
- Setup for DFT next lecture
- Practical: take short signals and predict spectral effects
Slides
Shared notes · Lectures 4 & 6: Fourier Transform
Notes
Week 4
Lecture 7
DFT and FFT
- DFT definition, matrix/projection view
- FFT algorithm idea + complexity
- Interpreting bins, normalization conventions
- Practical: first real FFT lab; identify tones
Slides
Notes
Lecture 8
Spectral Leakage & Frequency Resolution
- Leakage from time truncation
- Resolution vs record length; Rayleigh limit
- Dynamic range limits
- Practical: two-close-sinusoids experiment
Slides
Shared notes · Lectures 8 & 9: Spectral Leakage, Frequency Resolution, Dynamic Range and Windowing
Notes
Week 5
Lecture 9
Windowing: Theory & Practice
- Window tradeoffs: main-lobe/side-lobe
- Hann/Hamming/Blackman/Kaiser
- Coherent gain + amplitude correction
- Practical: compare windows on same data
Slides
Shared notes · Lectures 8 & 9: Spectral Leakage, Frequency Resolution, Dynamic Range and Windowing
Notes
Lecture 10
Ideal Filters & Frequency-Domain Filtering
- Ideal LP/HP/BP/BS via DFT perspective
- Why brick-wall is noncausal/infinite
- Frequency masking and overlap-add idea
- Practical: filter by multiplying spectra
Slides
Shared notes · Lectures 10, 12, 13 & 14: Filtering
Notes
Week 6
Lecture 11
Z-transform
- From Difference Equations to System Function
- Poles, Zeros, and Stability (Geometric View)
- Frequency Response as Unit-Circle Evaluation
- Brief Laplace Connection (Continuous-Time Analog)
Slides
Notes
Lecture 12
FIR Filter Design
- FIR basics, linear phase
- Windowed-sinc design
- Specs: ripple, transition band, order
- Practical: design FIR for given specs, validate in FFT
Slides
Shared notes · Lectures 10, 12, 13 & 14: Filtering
Notes
Week 7
Lecture 13
IIR Filters
- IIR structure, recursion, stability
- Poles/zeros intuition (expand on Z-transform)
- Butterworth/Chebyshev idea
- Practical: compare FIR vs IIR for same specs
Slides
Shared notes · Lectures 10, 12, 13 & 14: Filtering
Notes
Lecture 14
From Filter Design to Real-Time Filtering
- Why plain convolution is an offline operation
- How can we do online convolution?
- FIR case
- Zero-phase filters (offline)
- IIR case
- SOS preview
Slides
Shared notes · Lectures 10, 12, 13 & 14: Filtering
Notes
Week 8
Lecture 15
Random Processes: SSS/WSS & Correlation
- Random processes, ensemble vs time averages
- SSS, WSS, autocorrelation meaning
- AR intuition as “signal generator”
- Practical: simulate AR processes
Slides
Shared notes · Lectures 15 & 16: Random Processes
Notes
Lecture 16
Spectral Analysis of Stochastic Signals
- PSD definition from autocorrelation (Wiener–Khinchin)
- Periodogram limits, Welch method
- Bias/variance tradeoff
- What if the process is not stationary? Short-time Fourier transform
- Spectrogram
- Practical: PSD estimation & spectrogram on noisy data
Slides
Shared notes · Lectures 15 & 16: Random Processes
Notes
Shared notes · Lectures 16 & 17: Spectral Analysis of Stochastic Signals & Denoising Pipeline
Notes
Week 9
Lecture 17
Noise Models, SNR & Practical Denoising
- White/pink/brown noise
- SNR in linear & dB
- Spectral subtraction / bandlimiting
- Practical: denoise real recordings
Slides
Shared notes · Lectures 16 & 17: Spectral Analysis of Stochastic Signals & Denoising Pipeline
Notes