Signal processing is the study of signals and systems
Signal (n): A detectable physical quantity . . . by which messages or information can be transmitted (Merriam-Webster)
Signals carry information Examples: • • • •
Speech signals transmit language via acoustic waves Radar signals transmit the position and velocity of targets via electromagnetic waves Electrophysiology signals transmit information about processes inside the body Financial signals transmit information about events in the economy
Systems manipulate the information carried by signals
Week 1: Discrete-Time Signals Signals are (potentially infinitely-long) vectors that live in a vector space Norm measures the strength of a signal x: kxk2 , kxk1 , kxk∞ The inner product hx, yi measures the similarity between two signals x and y The Cauchy Schwarz Inequality calibrates inner product similarity measurements and enables powerful ways and means to detect and classify signals 0 ≤ hx, yi ≤ kxk2 kyk2 Discrete-time sinusoids ejωn are lovely but have two non-intuitive properties: • They alias • Most are not periodic
Week 2: Discrete-Time Systems Linear systems, time-invariant systems, linear time-invariant (LTI) systems Linear systems are matrices that map an input vector to an output vector LTI systems are characterized by their impulse response h Infinite-length signals • LTI systems are (infinitely large) Toeplitz matrices • Convolution: y = h ∗ x
Finite-length/periodic signals • LTI systems are circulent matrices • Circular convolution: y = h ~ x
A system is BIBO stable if and only if its impulse response khk1 < ∞
Week 3: Discrete Fourier Transform (DFT) Fourier representation (basis) for finite-length/periodic signals Eigenvectors of circulent LTI systems are the harmonic sinusoids ej
Fast Fourier Transform (FFT) DFT diagonalizes LTI systems (circulent convolution) y =h~x
DFT
←→
Y [k] = H[k] X[k]
Week 4: Discrete-Time Fourier Transform (DTFT) Fourier representation for infinite-length signals Defined as the limit of the DFT as the signal length → ∞ Eigenvectors of Toeplitz LTI systems are the sinusoids ejωn Discrete-Time Fourier Transform (DTFT) (Analysis)