DSP is what happens after you sample. Every audio plug-in, every voice assistant, every cellphone modem, every modern radar, every spectrum analyzer, every MRI machine, every Wi-Fi chip, every earbud's noise canceller, every disk-drive read channel, and every side-channel power-analysis attack on a chip is, at its core, a Discrete Fourier Transform, a digital filter, or a convolution. The math comes from Chapter 3 (Signals & Systems); the implementations now run on dedicated DSP chips, FPGAs, GPUs, or general-purpose CPUs with SIMD. This chapter teaches you to design and reason about those implementations.
In Chapter 3 we built up the math of signals and systems: continuous- and discrete-time signals, Fourier, sampling, LTI systems, convolution, Laplace, Z-transform. That chapter ended with the Z-transform and a promise that we would put it to work. This is that work. We will turn the Fourier transform into something a computer can actually compute (the DFT, then the FFT). We will design real digital filters (FIR and IIR), choosing windows and analog prototypes the way a working engineer does. We will run signals between sample rates (multirate DSP) the way every modern audio interface and every sigma-delta ADC does internally. And we will look at the silicon: the architectural tricks that let DSP chips multiply-and-add at full clock rate without missing a beat.
By the end you should be able to look at a DSP problem (filter this audio, demodulate this OFDM signal, align these power traces, reduce this 192 kHz audio to 48 kHz), reach for the right tool, predict its trade-offs, and write the code or pick the chip.
This chapter assumes you have Chapter 3 fresh. If sampling, the unit circle, and convolution feel hazy, skim that chapter first. The ideas here build directly on top.