Signal Processing

1. Definition

• Signal Processing = techniques for analyzing, modifying, and extracting information from signals.
• A signal is any function that conveys information about a phenomenon, often expressed as a function of time or space.
  • Continuous-time signal: $x(t)$
  • Discrete-time signal: $x[n]$

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2. Examples of Signals

• Audio signals: speech, music waveforms
• Image signals: pixel intensity patterns (2D signals)
• Biomedical signals: ECG (heart), EEG (brain), EMG (muscle)
• Communication signals: radio waves, Wi-Fi, 5G data
• Sensor signals: accelerometer, gyroscope, temperature

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3. Main Goals of Signal Processing

• Filtering: remove noise, extract useful parts
• Compression: reduce data size (e.g., MP3, JPEG)
• Feature Extraction: detect patterns (e.g., speech recognition)
• Transformation: convert signal into another domain for easier analysis (e.g., frequency domain)
• Restoration: enhance degraded signals (e.g., denoising, deblurring images)

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4. Core Techniques

1. Time-domain methods
   • Direct manipulation of signals in time sequence
   • Example: moving average filter
2. Frequency-domain methods
   • Use Fourier Transform (FT) to analyze frequencies
   • Example: filtering out high-frequency noise
3. Time-Frequency methods
   • For signals whose frequency content changes over time
   • Example: Short-Time Fourier Transform (STFT), Wavelet Transform

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5. Important Transforms

• Fourier Transform (FT): time ↔ frequency
• Discrete Fourier Transform (DFT) & FFT (Fast Fourier Transform)
• Laplace Transform: useful for continuous-time system analysis
• Z-Transform: for discrete-time system analysis
• Wavelet Transform: local time-frequency analysis

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6. Digital Signal Processing (DSP)

• In modern systems, most signal processing is done digitally.
• Steps:
  1. Sampling: convert analog → discrete signal (Nyquist-Shannon theorem applies)
  2. Quantization: map continuous values → discrete values
  3. Processing: digital filtering, compression, analysis
  4. Reconstruction: convert back if needed

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7. Applications

• Audio processing: speech recognition, noise cancellation, music compression
• Image processing: edge detection, face recognition, medical imaging
• Communication systems: modulation/demodulation, error correction
• Biomedical: ECG denoising, EEG pattern recognition
• Radar & Sonar: object detection and tracking
• IoT / Sensors: smoothing noisy measurements

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8. Connection to Data Science / ML

• Many ML tasks involve preprocessing signals:
  • Speech → MFCC (Mel-frequency cepstral coefficients)
  • Images → convolution filters extract features
  • EEG/ECG → time-frequency features for classification
• Deep learning (CNNs, RNNs, Transformers) can be seen as advanced signal processing pipelines.

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Summary

• Signal Processing = study of how to represent, filter, transform, and analyze signals.
• Covers time-domain, frequency-domain, time-frequency methods.
• Key tools: Fourier, Wavelet, Z-transform, DSP techniques.
• Applications: audio, image, biomedical, communications, radar, IoT.
• Plays a fundamental role in data science and machine learning when working with sequential or high-dimensional data.