Academic Domain: Signals, Systems & DSP
The Signal
The Mathematics of Information in Motion
The Architecture of Transformation
A signal is information changing through time or space. A system is what transforms it. Here, we cross the mathematical bridge from continuous physical reality into the discrete, calculable domain of Digital Signal Processing—allowing us to extract meaning from the noise of the universe.
REALITY →
SIGNAL →
SYSTEM →
TIME DOMAIN →
FREQUENCY DOMAIN →
FOURIER →
LAPLACE / Z-TRANSFORM →
SAMPLING →
DIGITAL REPRESENTATION →
DSP →
FILTERING →
FFT →
ANALYSIS
The 15 Realms of Processing
Signals, Systems & DSP Architecture
REALM 01
Nature of a Signal
- Continuous vs Discrete-time
- Analog vs Digital
- Deterministic vs Random
- Elementary Signals & Transformations
The System (LTI)
- Linearity & Time Invariance
- Causality & Stability
- Impulse & Step Response
- Continuous & Discrete Convolution
The Frequency World
- Sinusoids & Complex Exponentials
- Magnitude & Phase Response
- Frequency-selective Systems
- Ideal vs Practical Filters
Fourier's World
- Fourier Series (Periodic signals)
- Continuous-Time Fourier Transform
- Discrete-Time Fourier Transform
- Spectral Content & Bandwidth
The Laplace Domain
- Bilateral & Unilateral Transform
- Region of Convergence (ROC)
- Poles, Zeros & Transfer Function
- Stability Analysis in s-Domain
The Z-Domain
- Z-Transform Definitions
- ROC for Discrete Systems
- Difference Equation Solutions
- Pole-Zero Stability Interpretation
Sampling the Real World
- Nyquist-Shannon Theorem
- Aliasing & Spectral Overlap
- Anti-aliasing & Oversampling
- Ideal vs Practical Reconstruction
DSP & Representation
ADC
DSP
DAC
- Complete DSP Architecture
- Quantization Process & Error
- Signal-to-Quantization Noise
- Resolution & Bit Depth
Digital Filtering
- FIR vs IIR Structures
- Windowing Methods (FIR)
- Analog Prototype Transform (IIR)
- Direct, Cascade, Parallel Forms
The DFT & FFT
- Discrete Fourier Transform
- Frequency Bins & Computation
- Fast Fourier Transform (Radix-2)
- Decimation in Time/Frequency
Digital Convolution
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h
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x
x
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- Linear vs Circular Convolution
- Sequence System Output
- Relationship to DFT
- Filtering & Pattern Matching
Operations & Analysis
- Auto/Cross Correlation
- Energy & Power Spectral Density
- White vs Colored Noise Models
- SNR & Noise Filtering
Advanced DSP
- Multirate DSP (Decimation/Interp)
- Short-Time Fourier Transform
- Wavelets & Multiresolution
- Adaptive Filtering (LMS/RLS)
Applications of DSP
- Audio Processing & Compression
- Image Filtering & Enhancement
- Biomedical Signals (ECG/EEG)
- Seismic & Environmental Analysis
15 — The Signal Laboratory
[1] VISUALIZATION [2] LTI ANALYSIS [3] TRANSFORMS [4] SAMPLING [5] FILTER DESIGN [6] FFT COMPUTATION
The convergence of mathematical theory and computational practice. Utilizing software environments to visualize spectra, analyze pole-zero stability, design discrete filters, and process real-world signals.