Signals and Systems

Continuous and discrete time, energy and power, periodicity, even and odd parts, and the three transformations of time

The unit step, the impulse and its sifting property, Euler's formula, phasors, and their discrete-time counterparts

Memory, causality, stability, invertibility, time invariance and linearity, and how systems connect

The impulse response, the convolution sum, flip and slide, FIR and IIR systems, and causality and stability read from h[n]

The convolution integral, the RC circuit, step responses, how widths and areas combine, and why deconvolution is hard

Linear constant-coefficient equations, zero-input and zero-state responses, characteristic roots, first- and second-order systems and block diagrams

Complex exponentials as eigenfunctions, the synthesis and analysis equations, orthogonality, and the coefficients of square, sawtooth and triangle waves

Dirichlet conditions, mean-square convergence, the 9 % overshoot at a jump, Parseval's relation and smoothing the partial sums

The frequency response at the harmonics, a square wave through an RC filter, total harmonic distortion, and why linear systems never create new frequencies

From series to transform, the rectangular pulse and its sinc, exponentials and Gaussians, magnitude and phase, duality and line spectra

Shifts, scaling and modulation, the convolution theorem, Parseval's relation and the time–bandwidth limit

Decibels and decades, first- and second-order factors, straight-line asymptotes, resonance and Q

Why ideal filters cannot be built, filter specifications, Butterworth and Chebyshev responses, order, and distortionless transmission

Impulse-train sampling, spectral copies, the Nyquist rate, who found the theorem, bandlimited signals in practice and bandpass sampling

Folding and the alias formula, the wagon wheel, anti-aliasing filters, oversampling and aliasing in discrete time

Sinc interpolation, the zero-order hold and its droop, linear interpolation, decimation, interpolation and polyphase resampling

The spectrum of a sequence, its period of 2π, normalised frequency, FIR frequency responses, moving averages and the DTFT of samples

The DFT as samples of the DTFT, frequency bins, the DFT matrix, circular convolution, the radix-2 FFT and N² against N log N

Why off-bin tones leak, the rectangular window's transform, Hann, Hamming, Blackman and Kaiser windows, zero padding and Welch's method

The short-time Fourier transform, spectrograms, the window-length trade-off, chirps and instantaneous frequency, and a first look at wavelets

From Fourier to Laplace, regions of convergence, the common pairs, the derivative and convolution properties, and the initial- and final-value theorems

Partial fractions for distinct, repeated and complex poles, residues, differential equations with initial conditions, and zero-input and zero-state responses

Transfer functions, poles and zeros, how pole positions set the response, stability, second-order specifications and dominant poles

Closed-loop transfer functions, sensitivity and disturbance rejection, gain traded for bandwidth, stabilisation, the root locus, and gain and phase margins

Definition and region of convergence, the common pairs, the unit circle and the DTFT, the shift property, and the map z = e^{sT} from the s-plane

H(z) from difference equations, poles inside the unit circle, inverse transforms with residuez, FIR and IIR filters, and impulse responses from poles

Reading H(e^{jΩ}) from a pole–zero plot, zeros on the unit circle, resonators and notches, all-pass filters and minimum phase

The ideal impulse response, truncation and the window method, Kaiser's formula, linear phase, and equiripple design by Parks–McClellan

Analog prototypes — Butterworth, Chebyshev and elliptic — the bilinear transform, frequency prewarping, order against sharpness, and phase

Direct forms, second-order sections, coefficient quantisation, round-off and limit cycles, overflow, and real-time block processing

DSB-SC from the modulation property, AM with a carrier, the modulation index, envelope and coherent detection, and frequency-division multiplexing

Frequency and phase modulation, the modulation index, Bessel sidebands, Carson's rule, the noise advantage, and demodulation

Correlation and the matched filter, pseudo-random codes, processing gain, direct-sequence spread spectrum, and GPS acquisition in code phase and Doppler