1.2 Time-Domain Analysis of Signals

Signal Operations

Time-domain analysis is about understanding how signals behave as functions of time. Every operation you perform on a signal in this domain, whether shifting, scaling, or convolving, has a direct physical interpretation. Mastering these operations is essential because they form the foundation for analyzing how systems respond to inputs.

Basic Signal Transformations

Time-shifting moves a signal along the time axis without changing its shape. Shifting $x(t)$ by a constant $t_0$ produces $x(t - t_0)$.

• Positive $t_0$ shifts the signal to the right (delay)
• Negative $t_0$ shifts the signal to the left (advance)
• Example: If $x(t) = \sin(t)$, then $x(t-2) = \sin(t-2)$ is the same sine wave delayed by 2 seconds

Amplitude scaling multiplies the signal by a constant $a$, producing $ax(t)$.

• If $|a| > 1$, the signal is amplified; if $0 < |a| < 1$, it's attenuated
• If $a < 0$, the signal is also inverted (flipped vertically)
• Example: $2\cos(t)$ has twice the amplitude of $\cos(t)$

Time reversal flips the signal about the vertical axis, producing $x(-t)$.

• Example: If $x(t) = e^{-t}u(t)$ (a decaying exponential for $t \geq 0$), then $x(-t) = e^{t}u(-t)$ is a growing exponential for $t \leq 0$

Combining operations requires care about order. Amplitude scaling commutes freely with time-shifting and reversal, so $2[x(t-3)]$ and $[2x(t)]_{t \to t-3}$ give the same result. However, time-shifting and time-scaling (compression/expansion) do not generally commute. When you have a combined argument like $x(at + b)$, the safest approach is to always factor it as $x(a(t + b/a))$: first shift by $b/a$, then scale by $a$. Alternatively, apply the operations in the order dictated by the substitution variable directly.

Signal Symmetry

Even and Odd Components

Any signal $x(t)$ can be uniquely decomposed into an even part and an odd part. This decomposition is useful because many signal properties and system analyses simplify when you work with symmetric components separately.

Even component:

$x_e(t) = \frac{x(t) + x(-t)}{2}$

This part is symmetric about $t = 0$, meaning $x_e(t) = x_e(-t)$.

Odd component:

$x_o(t) = \frac{x(t) - x(-t)}{2}$

This part is antisymmetric, meaning $x_o(t) = -x_o(-t)$. Note that every odd signal must equal zero at $t = 0$.

The original signal is always the sum of its components: $x(t) = x_e(t) + x_o(t)$.

Example: For $x(t) = e^{t}$:

• $x_e(t) = \frac{e^{t} + e^{-t}}{2} = \cosh(t)$
• $x_o(t) = \frac{e^{t} - e^{-t}}{2} = \sinh(t)$

Some signals are purely even or purely odd:

• $\cos(t)$ is even: $\cos(-t) = \cos(t)$
• $\sin(t)$ is odd: $\sin(-t) = -\sin(t)$
• A purely even signal has $x_o(t) = 0$; a purely odd signal has $x_e(t) = 0$

Signal Energy and Power

Energy and Power Calculations

Energy and power give you two different ways to measure the "size" of a signal. The distinction between them is what lets you classify signals into meaningful categories.

Energy of a continuous-time signal captures the total squared content across all time:

$E = \int_{-\infty}^{\infty} |x(t)|^2 \, dt$

Example: For $x(t) = e^{-|t|}$, you get $E = \int_{-\infty}^{\infty} e^{-2|t|} \, dt = 2\int_{0}^{\infty} e^{-2t} \, dt = 1$. (The total energy is 1, not 2.)

Power captures the time-averaged squared content, which is meaningful for signals that persist forever:

$P = \lim_{T \to \infty} \frac{1}{2T} \int_{-T}^{T} |x(t)|^2 \, dt$

Example: For $x(t) = \cos(t)$, the power is $P = \frac{1}{2}$, which you can verify using the identity $\cos^2(t) = \frac{1 + \cos(2t)}{2}$ and noting that the $\cos(2t)$ term averages to zero.

Discrete-time versions use summations:

• Energy: $E = \sum_{n=-\infty}^{\infty} |x[n]|^2$
• Power: $P = \lim_{N \to \infty} \frac{1}{2N+1} \sum_{n=-N}^{N} |x[n]|^2$

Signal Classification

Signals fall into three categories based on energy and power:

A signal cannot be both an energy signal and a power signal. If energy is finite, the time-average of that finite quantity over an infinite interval goes to zero, so $P = 0$. If power is finite and nonzero, the signal must persist indefinitely, making total energy infinite.

This classification matters practically: energy signals are natural for transient analysis and detection problems, while power signals are the right framework for steady-state and communication system analysis.

Convolution of Signals

Convolution Definition and Properties

Convolution describes how a linear time-invariant (LTI) system transforms an input signal into an output signal. If you know the system's impulse response $h(t)$, you can find the output for any input using convolution.

Continuous-time convolution:

$y(t) = x(t) * h(t) = \int_{-\infty}^{\infty} x(\tau) \, h(t - \tau) \, d\tau$

Discrete-time convolution:

$y[n] = x[n] * h[n] = \sum_{k=-\infty}^{\infty} x[k] \, h[n-k]$

The intuition: you're decomposing the input into weighted, shifted impulses, then summing up the system's response to each one.

Key properties:

• Commutativity: $x(t) * h(t) = h(t) * x(t)$. You can choose which signal to flip and slide.
• Associativity: $(x * h_1) * h_2 = x * (h_1 * h_2)$. Cascaded systems can be combined into one equivalent impulse response.
• Distributivity: $x * (h_1 + h_2) = x * h_1 + x * h_2$. Parallel systems add.
• Identity: $x(t) * \delta(t) = x(t)$. Convolving with the impulse gives back the original signal.

LTI System Response

The impulse response $h(t)$ completely characterizes an LTI system. It's the output you get when the input is a unit impulse $\delta(t)$. Once you have $h(t)$, convolution gives you the output for any input.

Example: Consider an LTI system with impulse response $h(t) = e^{-t}u(t)$ (a first-order system, like an RC circuit). If the input is a unit step $x(t) = u(t)$:

$y(t) = \int_{0}^{t} (1) \cdot e^{-(t-\tau)} \, d\tau = (1 - e^{-t})u(t)$

The output starts at zero and gradually rises toward 1, which matches the charging behavior of a capacitor in an RC circuit driven by a step voltage.

Convolution Techniques

Graphical convolution (step-by-step):

1. Choose one signal to flip: replace $h(\tau)$ with $h(-\tau)$

2. Shift the flipped signal by $t$: you now have $h(t - \tau)$

3. Multiply $x(\tau) \cdot h(t - \tau)$ for each value of $\tau$

4. Integrate (or sum) the product over all $\tau$ to get $y(t)$ at that specific $t$

5. Repeat for different values of $t$ to build the full output

This method gives you strong visual intuition for how the overlap between the two signals changes with time. It works well for piecewise signals like rectangular pulses.

Analytical convolution evaluates the integral directly using known closed-form expressions. For example, convolving $e^{-at}u(t)$ with $e^{-bt}u(t)$ (where $a \neq b$) yields $\frac{e^{-bt} - e^{-at}}{a - b} u(t)$. This approach gives exact answers but requires you to carefully track integration limits, especially when signals have different regions of support.

Numerical convolution discretizes the signals and computes the convolution sum directly. This is the practical choice when signals come from measurements or when closed-form integration isn't feasible. The tradeoff is potential discretization error and higher computational cost.