Signals to learning / Keep this handy

Notation without mystery.

Read symbols as a compact description of operations. Each lesson introduces the symbols it uses; this page is a quick return point.

How to read the equations

Σ means sum the indexed terms; Π means multiply them.
floor(a) rounds down to the greatest integer no larger than a.
x(t) uses continuous time; x[n] uses integer sample index.
ŷ is a predicted value; ᵀ denotes transpose, not exponentiation.
θ, φ, ω are angles or angular frequencies with units stated locally.
η is a learning rate; λ is a penalty strength; ρ is a decay factor in the memory lab.

Shape conventions

A tensor is an array with several axes. Always state the layout: [batch, time, channels] differs from [batch, channels, time]. In the STFT calculator, output uses [batch, frames, bins, real/imaginary]. No universal convention can be assumed across libraries.

Units and boundary policies

Hz means cycles per second; radians/sample is angular frequency on the sample grid. FP32 real values occupy 4 bytes; a pair representing a complex value occupies 8 bytes in the traffic model. MB here means 10⁶ bytes, not MiB. Time origins, zero extension, centring and filter delay are part of a reproducible specification.

Glossary

Signal: A quantity indexed by time, position or another independent variable. State the axes and units.
Sample / x[n]: One value in an indexed sequence. Its physical timestamp depends on acquisition information.
Sample rate fₛ: Measurements per second (Hz or samples/s). Time spacing is 1/fₛ.
Frequency f / period T: Cycles per second and seconds per cycle. T=1/f for f>0.
Phase φ: A cycle’s angular offset. Radians and degrees are different units: 2π rad=360°.
Vector / norm: Ordered coordinates and their length. Euclidean norm is √Σxᵢ².
Matrix / transpose: A rectangular array representing, for example, a linear map. Transpose swaps row and column indices.
Dot product: Σqᵢkᵢ: a weighted sum sensitive to both lengths and alignment.
Cosine similarity: Dot product divided by the product of norms; undefined if either norm is zero.
Complex number: a+ib, where i²=−1. Real and imaginary parts provide two coordinates; magnitude is √(a²+b²).
Derivative / gradient: Local sensitivity of a function; a gradient collects partial derivatives for several inputs.
Probability / expectation: Nonnegative probabilities sum to one for exhaustive discrete outcomes. Expectation is the probability-weighted mean.
Logarithm: The inverse of exponentiation. log or ln in ML loss here means natural log; log₁₀ is explicitly written for decibels.
Power / RMS: Mean-square power is mean(x²) over a stated record; RMS is its square root. Physical watts require an appropriate physical model.
SNR / decibels: 10 log₁₀(signal power/noise power). Equal-shaped amplitude ratios use 20 log₁₀ instead.
Nyquist frequency: Half the sample rate. Reconstruction requires an appropriate band-limit and filtering assumptions.
Aliasing: Different continuous/input frequencies become indistinguishable after sampling; phase/sign conventions still matter.
Quantisation / clipping: Rounding to representable amplitude levels versus saturating outside the allowed range.
DFT / FFT: The finite discrete Fourier transform versus algorithms that calculate it efficiently.
DC / bin spacing: DC is zero frequency. For an N-point transform, frequency grid spacing is fₛ/N.
Window / leakage: A finite-record weighting and the spread of spectral coefficients due to observation/window boundaries.
STFT / hop: Transforms of successive windowed frames; hop is the number of samples between frame starts.
LTI / causality: Linear time-invariant systems preserve scaled sums and shifts. Causality separately requires no future-input dependence.
Convolution / correlation: Convolution uses x[n−k] under the course convention. CNN layers commonly apply cross-correlation without kernel reversal.
FIR / IIR: Finite impulse response versus potentially infinite impulse response, often implemented using feedback recurrences.
Frequency response / group delay: An LTI system’s complex gain by frequency; group delay is negative phase slope with respect to angular frequency.
State / BIBO stability: State carries history. Bounded-input bounded-output stability requires every bounded input to produce bounded output under the stated conditions.
Feature / target / parameter: Measured or derived input; desired output; adjustable model coefficient. Do not confuse them.
Loss / residual / MSE: A fitting objective; prediction minus target; average squared residual.
Training / validation / test: Parameter fitting; selection and tuning; final assessment of a fixed procedure.
Leakage: Information unavailable at prediction time or improperly drawn from evaluation data influences fitting or selection.
Logit / sigmoid / softmax: Unbounded score; binary probability map; normalisation of finite scores into a categorical probability distribution.
Cross-entropy: Negative log-probability of the target under the model, with an explicit logarithm base.
Precision / recall: TP/(TP+FP) and TP/(TP+FN); undefined denominators require a declared reporting policy.
Activation / ReLU: A nonlinear element; ReLU(x)=max(0,x). Its derivative at zero needs an implementation convention.
Backpropagation / optimiser: Compute loss gradients through a graph; then use those gradients to update parameters.
Regularisation / generalisation: A restriction, penalty or training procedure intended to limit fitting; performance beyond the fitting examples.
Embedding / attention: A feature representation; query/key scoring followed by normalised weighted retrieval of values.
RoPE / state-space model: Position-dependent paired rotations; input-driven recurrent state with an output readout. They are not interchangeable mechanisms.
Conditioning / numerical stability: Sensitivity of the problem to inputs; sensitivity of an implementation to arithmetic errors.
Throughput / latency / traffic: Work completed per time; elapsed time for a result; bytes moved. Different quantities need different measurements.

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