Signals Fundamentals: quick revision, 18 bites

Master Signal Transformations

Shift, flip, and scale systematically.

To sketch transformed signals like $x(at+b)$, always **factor** the argument first: $x(a(t + b/a))$. Then apply operations in this order: **scale** (multiply $t$ by $a$), **shift** (add/subtract $b/a$), and **flip** (if $a$ is negative). This prevents common sketching errors.

Takeaway. Factor the argument first; then scale, shift, and flip in that order.

Unit Step Function

The on/off switch for signals.

The continuous-time unit step $u(t)$ is 0 for $t<0$ and 1 for $t\geq 0$. The discrete-time step $u[n]$ is 0 for $n<0$ and 1 for $n\geq 0$. It is used to **gate** or **truncate** other signals.

Takeaway. The step function turns signals on at a specified time.

Unit Ramp Function

The step's integral — a linear rise.

The unit ramp $r(t)$ equals $t$ for $t\geq 0$ and 0 for $t<0$. It is the **integral** of the step function. Ramp signals increase linearly with time after they turn on.

Takeaway. A ramp is a linearly increasing signal that starts at zero.

Sketching Shifted Steps

Find where the argument crosses zero.

To sketch $u(-t+4)$, find the **zero crossing** of the argument: $-t+4=0 \Rightarrow t=4$. Since the coefficient of $t$ is negative, the step is flipped. So $u(-t+4)$ is 1 for $t<4$ and 0 for $t\geq 4$.

Takeaway. Set the step argument to zero to find the transition point.

Rectangular Pulse Function

A window of width Π.

The rectangular pulse $\Pi(t)$ is 1 for $|t| < 1/2$ and 0 otherwise. The expression $\Pi((t-2)/4)$ represents a pulse **centered** at $t=2$ with **width** 4. Adding $\Pi(t-2)$ gives a pulse centered at 2 with width 1.

Takeaway. Pulse width and center are read directly from the argument form.

Ramp with Step Multiplication

Truncating a ramp with a step.

The signal $r(t) \cdot u(-t+3)$ is the ramp $r(t)$ **multiplied** by a step that is 1 for $t<3$. This means the ramp is active only for $t<3$. For $t<0$, $r(t)=0$, so the signal is 0. For $0 \leq t < 3$, it equals $t$. For $t\geq 3$, it is 0.

Takeaway. Multiplying by a flipped step truncates the ramp at the step's edge.

Combining Ramps for Piecewise Signals

Sum shifted ramps to build shapes.

The signal $r(t) - 2r(t-2) + r(t-4)$ creates a **triangular** shape. For $0 \leq t < 2$: only $r(t)$ is active → slope 1. For $2 \leq t < 4$: $r(t) - 2r(t-2)$ → slope -1. For $t \geq 4$: all terms cancel → value 0.

Takeaway. Weighted ramp sums create piecewise-linear signals with controlled slopes.

Discrete Step Differences

Difference of steps creates a window.

For $x[n] = u[-n+2] - u[-n-2]$: The first step is 1 for $n \leq 2$. The second is 1 for $n \leq -2$. Subtracting gives 1 only where the first is 1 and the second is 0, i.e., for $-1 \leq n \leq 2$.

Takeaway. Subtracting two flipped steps yields a finite discrete pulse.

Periodic Signal Definition

Repeats exactly after a fixed time.

A continuous-time signal $x(t)$ is **periodic** if there exists a positive $T_0$ such that $x(t + T_0) = x(t)$ for all $t$. The smallest such $T_0$ is the **fundamental period**.

Takeaway. Periodicity means exact repetition after the fundamental period $T_0$.

Period of Cosine and Sine

Frequency determines the period.

For $x(t) = \cos(\omega_0 t)$ or $\sin(\omega_0 t)$, the fundamental period is $T_0 = 2\pi/\omega_0$. For $\cos(6\pi t)$, $\omega_0 = 6\pi$, so $T_0 = 2\pi/(6\pi) = 1/3$ seconds.

Takeaway. Period is $2\pi$ divided by the angular frequency.

Period of Complex Exponentials

Euler's formula links to cos/sin.

For $x(t) = e^{j\omega_0 t}$, the fundamental period is also $T_0 = 2\pi/|\omega_0|$. For $e^{j8\pi t}$, $\omega_0 = 8\pi$, so $T_0 = 2\pi/(8\pi) = 1/4$ seconds.

Takeaway. Complex exponentials are periodic with period $2\pi/|\omega_0|$.

Period of Sum of Sinusoids

Find the LCM of individual periods.

For $x(t) = 2\cos(50\pi t) + 3\sin(25\pi t)$: First term has $T_1 = 2\pi/(50\pi) = 1/25$. Second has $T_2 = 2\pi/(25\pi) = 2/25$. The fundamental period is the **least common multiple** (LCM) of $1/25$ and $2/25$, which is $2/25$ seconds.

Takeaway. The sum is periodic if the ratio of periods is rational; period is their LCM.

Non-Periodic Signals

Constants and steps break periodicity.

The signal $x(t) = 2u(t) + 3\cos(2\pi t)$ is **not periodic** because the step $u(t)$ is not periodic. Adding a non-periodic component to a periodic one makes the entire signal non-periodic.

Takeaway. Any signal containing a non-periodic component like a step is non-periodic.

Period of Sum of Complex Exponentials

Compare frequencies to find the period.

For $x(t) = 3e^{j(4\pi t + \pi/3)} + 4e^{-j(2t + \pi/2)}$: First term has $\omega_1 = 4\pi$, so $T_1 = 1/2$. Second has $\omega_2 = 2$, so $T_2 = \pi$. Since $T_1/T_2 = (1/2)/\pi = 1/(2\pi)$ is **irrational**, the sum is **not periodic**.

Takeaway. If the ratio of individual periods is irrational, the sum is non-periodic.

Signal with Constant Offset

Constants are periodic with any period.

For $x(t) = 2 + \sin(4\pi t)$: The constant 2 is periodic with any period. The sine term has $T = 2\pi/(4\pi) = 1/2$. Therefore, the sum is periodic with fundamental period $T_0 = 1/2$ seconds.

Takeaway. Adding a constant to a periodic signal does not change its fundamental period.

Check Periodicity Quickly

Rational ratio means periodic.

Takeaway. Test rationality of period ratios to determine periodicity.

Remember: Scale-Shift-Flip

SSF order prevents sketching errors.

When transforming $x(t)$ to $x(at+b)$, always remember **SSF**: first **S**cale, then **S**hift, then **F**lip (if needed). This order guarantees correct sketches every time.

Takeaway. Scale, Shift, Flip — in that order — for accurate signal sketches.

Why Periodicity Matters

Periodic signals enable Fourier analysis.

Periodic signals can be represented as sums of sinusoids (Fourier series). This is fundamental to analyzing LTI systems, filtering, and communications. Knowing the fundamental period is the first step.

Takeaway. Periodicity is the gateway to frequency-domain analysis.