Signals can be scaled, shifted in time, reflected, and combined through addition and multiplication. These operations are essential for modeling system inputs and understanding how signal transformations propagate through systems.
From signal classification, you know that a signal is simply a function — a mapping from an independent variable (usually time) to a value. The operations in this topic are the basic algebraic manipulations you can perform on that function. They seem abstract at first, but each has a concrete physical meaning that will serve you throughout systems analysis.
Time shifting replaces t with t − t₀, producing x(t − t₀). If t₀ > 0, the signal is delayed — the event that occurred at time 0 now occurs at time t₀. If t₀ < 0, the signal is advanced (occurs earlier). Physically, a delayed signal might represent a sound wave arriving t₀ seconds late after traveling a distance, or a sensor with processing delay. The key rule is: to delay a signal by t₀, replace t with (t − t₀) everywhere in the signal's expression. Students often get confused about direction — subtracting inside the function shifts right (later), adding shifts left (earlier).
Time scaling replaces t with at, producing x(at). If a > 1, the signal is compressed — it runs through its entire pattern in less time (higher playback speed). If 0 < a < 1, the signal is stretched (slower playback). If a = −1, you get time reversal: x(−t), the signal reflected about the time axis. In discrete time, only integer downsampling and upsampling are valid, making time scaling more constrained. Amplitude scaling multiplies the entire signal by a constant: A·x(t). This scales the signal's magnitude without touching its timing — amplification (A > 1) or attenuation (0 < A < 1).
The operations combine. x(at − t₀) is both scaled and shifted — but order matters. Evaluate carefully: x(at − b) = x(a(t − b/a)), so the effective time shift is b/a, not b. Getting this wrong is the most common mistake when working with transformed signals. The right mental model is: first write the transformation as x(a(t − t₀)), then you can read off the scale factor a and the time shift t₀ independently.
Addition and multiplication of signals combine two signals sample-by-sample (or instant-by-instant for continuous signals). Addition models superposition — two sound waves arriving simultaneously sum their pressures. Multiplication models modulation — multiplying a low-frequency message signal by a high-frequency carrier shifts the message to a new frequency band for transmission. You will use all of these operations constantly when you study convolution and the Fourier transform: convolution is built from shifts, and the Fourier transform decomposes signals into complex exponentials that are themselves the building blocks that scaling and shifting act on most cleanly.