At its heart, a signal is just information in motion. Your voice traveling through the air is a signal. The electrical pulses carrying this text to your screen are a signal. They all convey information from one point to another.

A system is anything that acts on a signal to change it. A microphone is a system that turns the sound waves of your voice (one signal) into electrical waves (another signal). A radio is a system that pulls a specific music signal out of the air and plays it through a speaker.

Signals come in a few key varieties. The first distinction is about time: is the signal defined at every single moment, or only at specific points in time?

Continuous-Time vs. Discrete-Time A continuous-time signal is a smooth, unbroken flow of information. Think of the temperature in a room over a day. It changes constantly, and you could measure it at any instant. We represent its variable with parentheses, like $x(t)$.

A discrete-time signal is a series of snapshots. It's only defined at specific, separate moments. For example, if you check the temperature only once every hour, you get a discrete-time signal. We use square brackets for this, like $x[n]$, where $n$ is an integer representing the sample number (the first sample, second sample, and so on).

The second distinction is about the signal's value: can it be any value within a range, or only specific, predefined values?

Analog vs. Digital An analog signal can have any value within a range. A light dimmer is a good example; you can set it to any level of brightness between fully off and fully on.

A digital signal can only take on a finite number of specific values. A standard light switch is digital: it's either on or off, with no in-between. In computing, these values are typically represented as 1s and 0s.

Most real-world signals are continuous and analog (like sound). To process them with computers, we convert them into discrete-time, digital signals through a process called sampling and quantization.

Just like signals, systems have classifications that help us understand and predict their behavior. The two most important properties are linearity and time-invariance.

1. Scaling (or Homogeneity): If you double the input signal's strength, the output signal's strength also doubles. A simple stereo amplifier is mostly linear; turn the volume knob to make the input twice as strong, and the output sound becomes twice as strong.
2. Superposition (or Additivity): The system's response to two signals added together is the same as adding up the individual responses to each signal. If you play a guitar note and a piano note through a speaker (system), the sound you hear is the sum of what each would sound like played alone.

A guitar distortion pedal is a classic example of a nonlinear system. It doesn't just make the signal louder; it fundamentally changes its shape, adding new frequencies. Doubling the input doesn't simply double the output.

The other key property is whether a system's behavior changes over time.

If you clap your hands in a concert hall, you'll hear an echo. If you go back tomorrow and clap in the exact same way, you'll hear the exact same echo. The concert hall is a time-invariant system.

A time-variant system, however, changes its behavior. An example is a mobile radio in a moving car. As the car moves, the radio signal strength and quality change due to buildings and hills, even if the radio station's broadcast is constant. The system (the path from the broadcast tower to the car) is changing over time.

We can manipulate signals using basic math. These operations are the building blocks for more complex signal processing.

Addition and Multiplication Adding two signals, $x_1(t) + x_2(t)$, combines them. This is what happens in a mixer when two audio tracks are played together. Multiplying a signal by a constant, like $2x(t)$, is called scaling or amplification. It changes the signal's amplitude, making it stronger or weaker.

Time-Shifting Time-shifting slides a signal along the time axis. We write this as $x(t - t_0)$.

• If $t_0$ is positive, the signal is delayed. For example, $x(t - 2)$ is the original signal $x(t)$ shifted two seconds to the right (happening later).
• If $t_0$ is negative, the signal is advanced. The signal $x(t + 2)$ is the original signal shifted two seconds to the left (happening earlier).

These foundational concepts of signals, systems, and their basic operations are the language we use to describe and analyze everything from audio effects to medical imaging.