Signals and Systems
Convolution, transforms and sampling are all one idea seen from different angles: a system is completely described by what it does to one impulse. These topics run real signals through real systems rather than restating the identities.
Start from the beginning →8 topics you can watch now, 12 still to come.
Foundations
What counts as a signal, what counts as a system, and the two properties that make either one tractable.
- Signals, Systems, and Why LTI MattersStart here. Push four different inputs through the same box and find out which boxes let you predict the fourth from the first three.
- Continuous and Discrete TimeTwo independent splits, constantly confused: one is about the variable, the other about the value — and all four combinations exist.
- The Standard SignalsImpulse, step and ramp are one family related by calculus — and the exponential earns its place for a reason nothing else can claim.
- Energy and power signals
- Even, odd and periodic decomposition
Time-domain analysis
The impulse response, and the one operation that uses it.
- The Impulse ResponseWhy one measurement describes a system completely — built up from impulses rather than asserted.
- Convolution, GraphicallySlide one shape past another and watch the overlap area rise, peak and fall — that area is the output.
- Properties of convolution
- Causality and BIBO stability
Fourier analysis
The same signal, rewritten as a list of frequencies.
- Fourier transform and its properties
- Magnitude and phase spectra
- Parseval's theorem
Sampling
Turning a continuous signal into numbers without losing it.
- The Sampling TheoremSampling replicates the spectrum. Keep the copies apart and reconstruction is exact — not approximate.
- AliasingRaise a tone past Nyquist and watch it fold back down. Not distortion — two signals becoming indistinguishable.
- Reconstruction and the anti-aliasing filter in detail
Laplace and Z transforms
Two generalisations that turn calculus into algebra.
- Laplace transform and the region of convergence
- Transfer functions and poles and zeros
- Z transform for discrete systems
- Stability from the pole locations
About Signals and Systems
Signals and systems is the subject that teaches you to stop looking at time. Its central claim is that a complicated signal can be rewritten as a sum of simple sinusoids, and that many systems which are awkward to describe in time become almost trivial once you do.
That change of view is what a Fourier series is, and it is genuinely a change of view rather than a calculation trick. A square wave and its harmonic sum are the same object seen two ways. Learning to move between the two is the skill the rest of communication, control and signal processing assumes you already have.
It is unusually rewarding to animate, because a partial sum converging on a waveform as harmonics are added shows in seconds what a page of algebra states slowly.
What to know first
- Calculus, including integration by parts
- Complex numbers in polar form, and comfort with exponentials of imaginary arguments
Where it gets used
- Understanding what a filter does to a signal before building one
- Reading a spectrum plot and identifying which component is which
- Working out why sampling too slowly produces a convincingly wrong signal