Ferkans — Interactive Telecom Tutor

Signals Are the Language of Communications

Every wireless system transmits, receives, and processes signals. A signal is simply a function of time (or space, or frequency) that carries information. This chapter develops the mathematical tools for analysing signals and the systems that process them — the foundation on which every subsequent chapter builds.

Definition:
Signal Classification

A continuous-time (CT) signal is a function $x : \mathbb{R} \to \mathbb{C}$ .

Energy signal: $E_x = \int_{-\infty}^{\infty} |x(t)|^2\,dt < \infty$ . Finite total energy, zero average power. Example: a single pulse.
Power signal: $P_x = \lim_{T \to \infty} \frac{1}{2T} \int_{-T}^{T} |x(t)|^2\,dt \in (0, \infty)$ . Infinite energy but finite average power. Example: a sinusoid.
Deterministic: completely specified by a mathematical formula.
Random (stochastic): described by probability distributions (treated in Section 4.6 and Chapter 2).

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Definition:
Dirac Delta Function

The Dirac delta $\delta(t)$ is the distribution satisfying

$\int_{-\infty}^{\infty} \delta(t)\,\varphi(t)\,dt = \varphi(0)$

for every continuous test function $\varphi$ . Key properties:

Sifting: $\int_{-\infty}^{\infty} x(t)\,\delta(t - t_0)\,dt = x(t_0)$
Scaling: $\delta(at) = \frac{1}{|a|}\delta(t)$
Fourier pair: $\delta(t) \leftrightarrow 1$

Strictly, $\delta(t)$ is not a function but a distribution (generalised function). We use it freely as engineers do, with the understanding that all integrals involving $\delta$ are shorthand for distributional pairings.

Definition:
Linear Time-Invariant (LTI) System

A system $\mathcal{H}$ mapping input $x(t)$ to output $y(t)$ is:

Linear: $\mathcal{H}\{a x_1 + b x_2\} = a\,\mathcal{H}\{x_1\} + b\,\mathcal{H}\{x_2\}$
Time-invariant: if $y(t) = \mathcal{H}\{x(t)\}$ , then $\mathcal{H}\{x(t - t_0)\} = y(t - t_0)$

An LTI system is completely characterised by its impulse response $h(t) = \mathcal{H}\{\delta(t)\}$ .

Definition:
Convolution

The output of an LTI system with impulse response $h(t)$ and input $x(t)$ is the convolution:

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

Properties:

Commutative: $x * h = h * x$
Associative: $x * (h_1 * h_2) = (x * h_1) * h_2$
Identity: $x * \delta = x$

Convolution in time corresponds to multiplication in frequency: $Y(f) = X(f) H(f)$ . This is the single most important property for frequency-domain analysis.

Convolution Animation

Watch the "flip and slide" interpretation of convolution. The impulse response $h(\tau)$ is flipped and slid across the input $x(\tau)$ , and the output $y(t)$ is the area under the product.

Parameters

Signal pair

Definition:
Stability and Causality

An LTI system is:

BIBO stable (bounded-input bounded-output) iff $\int_{-\infty}^{\infty} |h(t)|\,dt < \infty$ (i.e., $h \in L^1$ ).
Causal iff $h(t) = 0$ for $t < 0$ (the output depends only on present and past inputs).

All physical communication systems are causal. Stability ensures that finite-power inputs produce finite-power outputs.

Example: RC Low-Pass Filter

An RC circuit has impulse response $h(t) = \frac{1}{RC} e^{-t/RC}\,u(t)$ where $u(t)$ is the unit step. Verify it is causal and BIBO stable. Find the output when the input is $x(t) = u(t)$ (unit step).

Solution

Causality

$h(t) = 0$ for $t < 0$ (due to $u(t)$ ). $\checkmark$

BIBO stability

$\int_0^\infty \frac{1}{RC} e^{-t/RC}\,dt = \frac{1}{RC} \cdot RC = 1 < \infty$ . $\checkmark$

Step response

$y(t) = \int_0^t \frac{1}{RC} e^{-\tau/RC}\,d\tau = \left[-e^{-\tau/RC}\right]_0^t = 1 - e^{-t/RC}$ for $t \geq 0$ .

This is the classic exponential rise to the steady-state value of 1. $\blacksquare$

Theorem: Transfer Function

For an LTI system with impulse response $h(t)$ , the transfer function (frequency response) is

$H(f) = \int_{-\infty}^{\infty} h(t)\,e^{-j2\pi f t}\,dt = \mathcal{F}\{h(t)\}$

and the input-output relation in the frequency domain is

$Y(f) = H(f)\,X(f).$

This is the convolution theorem: convolution in time becomes multiplication in frequency.

Complex exponentials $e^{j2\pi f t}$ are eigenfunctions of LTI systems: the system scales each frequency component by $H(f)$ without mixing frequencies.

Proof

Eigenfunction property

$\mathcal{H}\{e^{j2\pi f_0 t}\} = \int_{-\infty}^{\infty} h(\tau) e^{j2\pi f_0 (t-\tau)}\,d\tau = e^{j2\pi f_0 t} \underbrace{\int_{-\infty}^{\infty} h(\tau) e^{-j2\pi f_0 \tau}\,d\tau}_{H(f_0)} = H(f_0)\,e^{j2\pi f_0 t}$ . $\blacksquare$

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LTI System Block Diagram — An LTI system characterised by impulse response $h(t)$ and transfer function $H(f)$ . The output $y(t) = x(t) * h(t)$ in time is equivalent to $Y(f) = X(f) H(f)$ in frequency. Cascading two LTI systems multiplies their transfer functions.

Common Mistake: The Wireless Channel Is NOT Exactly LTI

Mistake:

Treating the wireless channel as a time-invariant system in all analyses.

Correction:

The wireless channel is a linear time-variant (LTV) system (Section 4.5). The LTI model is a useful approximation when the channel changes slowly compared to the symbol duration (quasi-static or block-fading model), but fast fading requires the full LTV framework.

Quick Check

What is $x(t) * \delta(t - t_0)$ ?

$x(t)$

$x(t - t_0)$

$x(t_0)$

$x(t) \cdot \delta(t - t_0)$

Correction:

x(t - t_0)

Correct. Convolution with a shifted delta simply delays the signal: $x(t) * \delta(t - t_0) = x(t - t_0)$ .

Impulse Response

The output of an LTI system when the input is a Dirac delta: $h(t) = \mathcal{H}\{\delta(t)\}$ . Completely characterises the system.

Convolution

$(x * h)(t) = \int_{-\infty}^{\infty} x(\tau) h(t-\tau)\,d\tau$ . The output of an LTI system with impulse response $h$ and input $x$ .

BIBO Stability

An LTI system is BIBO (bounded-input bounded-output) stable iff $\int |h(t)|\,dt < \infty$ .

Continuous-Time Signals and LTI Systems