Most of digital signal processing is one trick: break a complicated signal into simple pieces, push each piece through the system, and add the answers back up. This course makes that trick physical. You drag samples on an honest stem plot and watch a system answer; you discover why only linear systems can be split apart; you meet the impulse response — the single fingerprint that decides everything a system does — and then a flip-and-slide machine that turns it into convolution. From there: echoes, discrete calculus, low- and high-pass filters, the algebra of cascaded systems, and correlation finding a known pulse buried in noise. After Smith, The Scientist and Engineer’s Guide to DSP, chapters 5–7. No prerequisites beyond curiosity; honest enough for an engineering course.
A signal records how one quantity varies with another; a system is any process that turns an input signal into an output. Drag one, watch the other.
Linearity = homogeneity + additivity, and DSP also needs shift-invariance. Live-test any system against all three probes and read the verdict.
Break a signal into simple pieces, push each through a linear system, and add the answers back — synthesis, decomposition, and the divide-and-conquer foundation of all DSP.
The four ways to split a signal into simpler pieces that re-sum to it — impulse, step, even/odd and interlaced — and why each one matters.
Send in a single unit spike, the delta — out comes the system’s fingerprint, the impulse response h[n]. Scale and slide the spike; the response just scales and slides.
Every input sample is a scaled, shifted impulse — so it scatters a scaled, shifted copy of the impulse response into the output. Pile all the copies up and you have convolution, seen from the input side.
Flip the impulse response, slide it across the input, multiply-accumulate — and the output appears one sample at a time. Convolution, made mechanical.
Now that the impulse response IS the system, design it by hand: identity, gain, delay and echo are simple patterns of deltas — and [1,−1] is the discrete derivative, the running sum its integral.
A low-pass kernel is a hump of positive taps that averages neighbours — noise melts away. δ minus that hump is its high-pass twin: the slow drift vanishes, the detail stays.
How systems combine: cascades convolve into one response, order is free, parallel branches add — and convolving a pulse with itself enough times yields a Gaussian.
Slide a known pulse across a noisy return — without the flip. The cross-correlation peaks exactly where the target hides: the matched filter, the optimal way to find a waveform buried in noise. Radar, made visible.