This course studies signals. A voice recording is a signal. So is a stock
price written down once a day, a radar echo, or a row of pixel brightness
values. The course fourier-analysis builds the machinery that splits a
signal into simple waves. This course picks up where that leaves off. It
asks what happens once a signal exists only at a finite set of instants.
That is what happens the moment a computer stores it.
Here is the arc of the whole course, stated once. We start with discrete-time signals and systems. Then comes the z-transform, and then the discrete Fourier transform. Next we design digital filters that do something useful to a signal. We close with a hard fact of real hardware: a signal has to be stored in a fixed number of bits.
Two courses come before this one. fourier-analysis supplies the
transform machinery: Fourier series, the Fourier transform, convolution,
sampling, and the DFT. probability is needed from chapter 12 on, where
this course's own work on random signals starts. This chapter re-teaches,
at working depth, the handful of facts from each that later chapters lean
on.
Picture a video of a weight swinging on a string. Next to it, picture a chart that someone fills in by hand once a minute. The video is continuous. Pause it at any instant at all, and the weight is at some exact spot. The hand-drawn chart is discrete. It has a value at minute 1 and a value at minute 2. In between it has nothing at all, because nobody wrote anything down there. Neither picture is "more real" than the other. They are just two ways to track how something changes.
A continuous-time signal is the video of the swinging weight. Books write it x(t), said out loud as "x of t". That means the value x at the exact time t. A discrete-time signal is the once-a-minute chart. Books write it x[n], said as "x of n". That means the n-th number in the list. The square brackets are the tip-off: only whole-numbered steps exist here, so there is no such thing as step 1.5.
Why does this split earn its own vocabulary? Why not leave it as an offhand remark? Because a computer can only ever store the second kind. A computer holds a finite list of numbers, never the first kind. Everything this course builds sits in that gap. You start with a continuous-time signal, such as a real voice or a real radar echo. You turn it into a discrete-time signal that a computer can hold. Then you ask what you gained, and what you lost for good.
A signal is a function of one or more independent variables that carries information. A voltage that varies over time is one. So is a sound-pressure wave, or a sequence of daily stock prices. In this course the independent variable is time, or a sample index standing in for time. One split matters more here than almost anywhere else in mathematics. Does that variable range over the real numbers, or only over the integers?
A continuous-time signal x(t) is defined for every real number t in its domain. Between any two instants lie infinitely many more, each with its own value. A discrete-time signal x[n] is defined only at integer values of the index n. It is a sequence of numbers, with nothing at all defined "between" one index and the next.
Both kinds of signal split again. A deterministic signal has an explicit rule or formula that fixes every value, such as x(t) = \cos(2\pi t). For a random signal, no formula predicts the next value exactly. Only its statistics can be pinned down. Chapters 12–13 return to that case. Every signal this course studies before chapter 12 is deterministic. For the first eleven chapters, the discrete-or-continuous split is the one that governs everything else.
One signal, two pictures
Drag the amplitude and frequency sliders. The dashed curve is the continuous-time signal x(t); the solid dots are the discrete-time signal x[n] -- the SAME cosine, sampled at each integer n. Watch how the dots alone still trace out the dashed curve's shape, at least for now (chapter 6 shows when that stops being true).
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Worked example
Classifying a signal by its independent variable
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1
A thermostat logs the room temperature once every hour, forever. Is the signal that comes out continuous-time or discrete-time? One question decides it. Does the signal have a value at every real-numbered instant, or only at set instants with gaps between them?
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2
The log has a value at hour 0, hour 1, hour 2, and so on. It logs nothing at hour 1.5. Nothing there would even have a meaning, since the device only checks on the hour.
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3
The answer is \boxed{\text{discrete-time}} We write it as T[n], the temperature at the n-th hour, rather than T(t). The square brackets show this directly, and they are the subject of the next section.
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Worked example
A signal that looks discrete but isn't
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1
A sound engineer says "the microphone's output." Before any gear touches it, is that signal continuous-time or discrete-time? Sound pressure at the mic is a physical quantity. It exists at every instant, whether or not anything is measuring it yet.
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2
The raw sound-pressure wave has a well-defined value at every real-numbered instant of time. It skips none of them.
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3
The answer is \boxed{\text{continuous-time}} It turns discrete-time only once an analog-to-digital converter samples it, which is the topic of chapter 6. It is easy to mix up "the signal I saved" with "the sound that made it." That mix-up is common and easy to avoid. What you saved is discrete-time. The sound itself was continuous-time.
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Where this shows up
Cell-phone and consumer-electronics DSP
Every step inside a modern phone call plays out this continuous-to-discrete story in hardware and software together. A continuous-time sound-pressure wave hits the microphone. An analog-to-digital converter turns that wave into a discrete-time signal. A digital signal processor squeezes the numbers down and sends them out. At the far end, a digital-to-analog converter turns the numbers back into a continuous-time sound wave. A speaker can then play that wave out loud. Almost none of the "processing" happens to the continuous-time signal itself. It happens to the discrete-time sequence of numbers, which is what this course spends its length on.
Check your understanding (counts toward your progress)
A signal is defined for every real number t in its domain. What kind of signal is it?
Right on try — credit. Right — the answer had already been shown, so no credit this time. Next try is worth Answer shown after three tries — this one no longer earns credit. You chose to see the answer — this one earns no credit.
A discrete-time signal is defined only at integer indices, not at every real number.
Random vs. deterministic is a separate classification from continuous vs. discrete.
Periodicity is a separate property from whether the signal's argument is continuous or discrete.
Why
A continuous-time signal x(t) is defined for every real number t — the defining property of the continuous case.
Before any microphone samples it, is ordinary acoustic sound pressure continuous-time or discrete-time?
Right on try — credit. Right — the answer had already been shown, so no credit this time. Next try is worth Answer shown after three tries — this one no longer earns credit. You chose to see the answer — this one earns no credit.
Perception doesn't create the physical quantity — the pressure wave exists at every instant regardless of who's listening.
Sound pressure varying over time is a textbook example of a signal.
Why
The physical pressure wave is defined at every real-numbered instant; it only becomes discrete-time once something samples it.
Ask whether the physical quantity itself has gaps, or only the act of measuring it does.
A sensor logs a value once per second, forever. Its recorded signal is best described as
Right on try — credit. Right — the answer had already been shown, so no credit this time. Next try is worth Answer shown after three tries — this one no longer earns credit. You chose to see the answer — this one earns no credit.
A once-per-second log has no value between readings — that's the discrete-time case, not continuous.
The signal is well-defined at every second; it just isn't defined between seconds, which is exactly what discrete-time means.
A signal is one or the other, never both — the independent variable is either every real number or only integers.
Why
The log has values only at separated instants (once per second), the defining property of a discrete-time signal.
Practice: Continuous-time or discrete-time? (2 problems · doesn’t affect your progress)
A weather station records rainfall totals once per day. Continuous-time or discrete-time?
Right on try — credit. Right — the answer had already been shown, so no credit this time. Next try is worth Answer shown after three tries — this one no longer earns credit. You chose to see the answer — this one earns no credit.
Once-per-day readings have gaps between them — nothing recorded at, say, 2:30pm — which is the discrete-time signature.
Why
A once-per-day reading is defined only at separated instants (each day), the defining property of a discrete-time signal.
Discrete-time
The voltage across a capacitor as it charges, before any measurement touches it. Continuous-time or discrete-time?
Right on try — credit. Right — the answer had already been shown, so no credit this time. Next try is worth Answer shown after three tries — this one no longer earns credit. You chose to see the answer — this one earns no credit.
The physical voltage exists at every instant of the charging process, whether or not it's being measured.
Why
The physical voltage is a real quantity defined at every real-numbered instant during charging — continuous-time, like the raw acoustic-pressure example in the lesson.
Continuous-time