What physical process varies?
Sound, light, motion, voltage?
Signal meaning depends on the phenomenon being measured.
Side 66
Signal
Processing
A study of how measurements become usable information. Signal processing represents signals across time and frequency, separates structure from noise, and designs transformations that preserve what matters.
sample→represent→transform→filter→infer
06signal lenses
05filter ideas
05noise questions
66Side
A signal is variation carrying information.
Voltage, sound pressure, light intensity, acceleration and neural activity all become signals when measured as functions of time or space.
Time, space or another coordinate?
Choose representation.
The same data can reveal different structure in different domains.
How large is the signal?
Scale + units.
Amplitude may represent energy, intensity, displacement or another quantity.
How quickly does it vary?
Cycles per unit time.
Periodic and oscillatory structure is often easier to interpret spectrally.
How are oscillations aligned?
Relative timing.
Phase can determine cancellation, reinforcement and system delay.
Sampling turns continuous variation into discrete data.
The sampling rate determines which frequencies can be represented without ambiguity.
Measurements per second.
Higher rates preserve faster variation but increase data volume.
Need enough samples for the bandwidth.
Frequencies above half the sample rate can fold into false lower frequencies.
Different signals become indistinguishable.
Undersampling creates artifacts that cannot be removed after capture.
Amplitude becomes discrete levels.
Finite resolution introduces quantization error.
Observe finite segments.
Window choice changes spectral leakage and resolution.
Remove frequencies before sampling.
Analog filtering protects the digital representation from out-of-band content.
Fourier analysis trades waveform shape for frequency composition.
Complex signals can be represented as combinations of sinusoids.
Spectrum
Amplitude by frequency.
Spectral views reveal periodic components hidden in time traces.
Transform
Map time ↔ frequency.
Fourier transforms preserve information while changing representation.
FFT
Compute the transform efficiently.
The fast Fourier transform reduces the computational cost of discrete spectral analysis.
Resolution
Longer observation separates nearby frequencies.
Time and frequency resolution trade against one another.
Leakage
Finite windows spread spectral energy.
Windowing controls but does not eliminate this effect.
Filters preserve some components and suppress others.
Filter design expresses what counts as signal and what counts as unwanted variation.
| Filter | Passes | Suppresses | Typical use |
|---|---|---|---|
| Low-pass | Slow variation | High frequencies | Smoothing |
| High-pass | Fast variation | Low-frequency drift | Edge/change emphasis |
| Band-pass | Selected frequency range | Outside band | Channel or feature isolation |
| Notch | Most frequencies | Narrow interference band | Remove line hum / known interference |
| Adaptive | Changes with observed conditions | Time-varying noise | Echo/noise cancellation |
Convolution describes how linear time-invariant systems respond.
If the response to an impulse is known, the response to any input can be constructed from shifted and weighted copies.
Probe the system.
The impulse response summarizes an LTI system’s behavior.
Local transformation rule.
Convolution kernels smooth, sharpen, differentiate or detect patterns.
Slide and sum.
Each output sample combines neighboring input samples according to the kernel.
Convolution becomes multiplication.
Fourier representation can make filtering computationally and conceptually simpler.
Real-time systems cannot use future input.
Causal filters depend only on present and past samples.
Filtering shifts information in time.
Phase response matters when timing relationships must be preserved.
Inference begins when measurements are uncertain.
Noise cannot always be removed; often the goal is to estimate the underlying signal probabilistically.
SNR
Compare desired signal power with noise power.
Averaging
Reduce uncorrelated noise by combining repeated observations.
Correlation
Detect known structure inside noisy measurements.
Estimation
Infer hidden parameters or states from imperfect observations.
Prediction
Use models and prior samples to estimate future or missing values.
Signals and SystemsOppenheim & Willsky · signal representation
Discrete-Time Signal ProcessingOppenheim & Schafer · digital methods
Understanding Digital Signal ProcessingRichard Lyons · practical foundation
Statistical Digital Signal Processingestimation and noise