The first week motivates forthcoming material by illustrating how signals can be modeled abstractly as functions on sets. The emphasis is on characterizing the domain and the range, not on characterizing the function itself. The startup sequence of a voiceband data modem is used as an illustration, with a supporting applet that plays the very familiar sound of the handshake of V32.bis modem, and examines the waveform in both the time and frequency domain. The domain and range of the following signal types is given: sound, images, position in space, angles of a robot arm, binary sequences, word sequences, and event sequences. Since often the first day of the first week is a holiday, and the first class needs to be largely devoted to logistics, there is really only one lecture in which to do this.

The second week introduces systems as functions that map functions (signals) into functions (signals). Again, it should focus not on how the function is defined, but rather on what is the domain and range. Block diagrams are defined as a visual syntax for composing functions. Applications considered are DTMF signaling, modems, digital voice, and audio storage and retrieval. These all share the property that systems are required to convert domains of functions. For example, to transmit a digital signal through the telephone system, the digital signal has to be converted into a signal in the domain of the telephone system (i.e., a bandlimited audio signal).

Week 3 is when the students get seriously into Matlab. The first lecture in this week is therefore devoted to the problem of relating declarative and imperative descriptions of signals and systems. This sets the framework for making the intellectual connection between the labs and the mathematics.
The rest of the week is devoted to introducing the notion of state and state machines. State machines are described by a function \set{update} that, given the current state and input, returns the new state and output. In anticipation of composing state machines, the concept of \textit{stuttering} is introduced. This is a slightly difficult concept to introduce at this time because it has no utility until you compose state machines. But introducing it now means that we don't have to change the rules later when we compose machines.

The fourth week deals with nondeterminism and equivalence in state machines. Equivalence is based on the notion of simulation, so simulation relations and bisimulation are defined for both deterministic and nondeterministic machines. These are used to explain that two state machines may be equivalent even if they have a different number of states, and that one state machine may be an abstraction of another, in that it has all input/output behaviors of the other (and then some).

This week is devoted to composition of state machines. The deep concepts are synchrony, which gives a rigorous semantics to block diagrams, and feedback. The most useful concept to help subsequent material is that feedback loops with delays are always well formed.

We consider linear systems as state machines where the state is a vector of reals. Difference equations and differential equations are shown to describe such state machines. The notions of linearity and superposition are introduced, and convolution is touched upon (to be dealt with in more detail later). This week is a good time for the first midterm, with coverage including chapters 1-4, and appendix A of the reader, except sections 4.7 and 4.8 (i.e., leaving out feedback, since until this week, they haven't worked any problems involving feedback).

Matrices and vectors are used to compactly describe systems with linear and time-invariant state updates. Impulses and impulse response are introduced. The deep concept here is linearity, and the benefits it brings, specifically being able to write the state and output response as a convolution sum. We also begin to develop frequency domain concepts, using musical notes as a way to introduce the idea that signals can be given as sums of sinusoids.

This week introduces frequency domain concepts and the Fourier series. Periodic signals are defined, and Fourier series coefficients are calculated by inspection for certain signals. The frequency domain decomposition is motivated by the linearity of systems considered last week (using the superposition principle), and by psychoacoustics and music.

In this week, we consider linear, time-invariant (LTI) systems, and introduce the notion of frequency response. We show that a complex exponential is an eigenfunction of an LTI system. The Fourier series is redone using complex exponentials, and frequency response is defined in terms of this Fourier series, for periodic inputs.

The use of complex exponentials is further explored, and phasors and negative frequencies are discussed. The concept of filtering is introduced, with the terms lowpass, bandpass, and highpass, with applications to audio and images. Composition of LTI systems is introduced, with a light treatment of feedback.

We describe signals as sums of weighted impulses and then use linearity and time invariance to derive convolution. FIR systems are introduced, with a moving average being the prime example. Implementation of FIR systems in software and hardware is discussed, and signal flow graphs are introduced. Causality is defined.

We relate frequency response and convolution, building the bridge between time and frequency domain views of systems. We introduce the DTFT and the continuous-time Fourier transform and derive various properties. These transforms are described as generalizations of the Fourier series where the signal need not be be periodic.

This week wraps up the discussion of Fourier transform and then introduces sampling and aliasing as a major application of these techniques. Emphasis is on intuitive understanding of aliasing and its relationship to the periodicity of the DTFT. The Nyquist-Shannon sampling theorem is stated and related to this intuition, but its proof is not emphasized.

This week wraps up the discussion of sampling and aliasing and begins a review that focuses on how to apply the techniques of the course in practice. Filter design is considered with the objective of illustrating how frequency response applies to real problems, and of enabling educated use of filter design software. The modem startup sequence example is considered again in some detail, zeroing in detection of the answer tone to illustrate design tradeoffs.

This week develops applications that combine techniques of the course. The precise topics depend on the interests and expertise of the instructors, but we have specifically covered the following:
Speech analysis and synthesis, using a historical Bell Labs recording of the Voder and Vocoder from 1939 and 1940 respectively, and explaining how the methods illustrated there (parametric modeling) are used in today's digital cellular telephones.
Digital audio, with emphasis on encoding techniques such as MP3. Psychoacoustic concepts such as perceptual masking are related to the frequency domain ideas in the course.
Vehicle automation, with emphasis on feedback control systems for automated highways. The use of discrete magnets in the road and sensors on the vehicles provides a superb illustration of the risks of aliasing.