Lesson 2: Signals & Systems (Core Telecom Brain)

Mastering system behaviors for signal processing.

Key Concepts:

• Continuous-time vs discrete-time systems
• LTI systems
• Impulse response
• Sampling theorem (Nyquist)
• Aliasing
• Filtering (LPF, HPF, BPF)
• Analog-to-digital & digital-to-analog conversion

Why It Matters: This forms the “brain” of telecom engineering, enabling design of efficient systems.

Labs/Practice: Implemented filters and sampling in labs; avoided aliasing in digital signal experiments.

Tools Used: MATLAB, Python (SciPy), Simulink.

This is the bridge from pure math (Lesson 1) to actual telecom engineering.
Signals & Systems is where you start thinking like a telecom engineer: how do we process, sample, filter, and digitize signals without losing (or corrupting) the information?

This topic is foundational for digital communications, RF/wireless, cellular networks, and almost every tool & project later (MATLAB, GNU Radio, srsRAN, etc.).

Why Signals & Systems Is the “Core Brain” of Telecom

• Real-world signals are continuous-time analog (voice, radio waves).
• Modern telecom is almost entirely digital after the antenna.
• To go from analog → digital reliably → you need sampling, quantization, filtering, and system analysis.
• Every modulation scheme, equalizer, channel estimator, beamformer, etc., is built on LTI system theory.
• Skip or rush this → you’ll understand what OFDM or QAM does, but not why it survives multipath or noise.

1. Continuous-Time vs Discrete-Time Systems

• Continuous-time (analog): Signals defined for all real t
  Example: s(t) = cos(2π·1000·t)
  Systems process them continuously (analog filters, amplifiers).

• Discrete-time (digital): Signals only at integer multiples of sampling period
  x[n] = x(n·Tₛ), n = …, -2, -1, 0, 1, 2, …
  Processed by DSP chips, FPGAs, software (Python/MATLAB).

Telecom reality
RF front-end is analog/continuous → baseband processing is discrete/digital.
5G NR, Wi-Fi 6, LTE → massive digital signal processing at baseband.

2. LTI Systems (Linear Time-Invariant) — The Golden Assumption

Almost every telecom system we analyze is modeled as LTI:

• Linear: Scaling + superposition
  ax₁(t) + bx₂(t) → a·y₁(t) + b·y₂(t)
• Time-Invariant: Time shift in input → same time shift in output

Why LTI is huge in telecom

• Filters (channel filters, anti-aliasing, matched filters)
• Channels themselves (in many models, especially AWGN)
• Equalizers, interpolators, beamformers
• Allows powerful math: convolution, frequency response, z-transform/DFT

3. Impulse Response — The DNA of an LTI System

The impulse response h(t) or h[n] completely characterizes an LTI system.

• Input: δ(t) (Dirac delta) → Output: h(t)

• For any input x(t):
  y(t) = x(t) * h(t) (convolution)

• Discrete case:
  y[n] = ∑ₖ x[k] · h[n-k]

Intuition
h(t) tells you “how the system rings” after a sharp impulse.

Telecom examples

• Matched filter (optimal receiver in AWGN): h(t) = conjugate time-reversed transmitted pulse
• Multipath channel impulse response: multiple delayed copies → causes ISI
  → solved by OFDM or adaptive equalizers

Example: Impulse response of a simple RC low-pass filter (exponential decay — classic “memory” of the filter)

(Exponential decay curve showing how the filter responds to an impulse)

Another view — step response of RC low-pass (shows smoothing effect):

4. Sampling Theorem (Nyquist-Shannon) — The Most Important Rule in Digital Telecom

Theorem
If a signal is bandlimited to maximum frequency fₘₐₓ (no energy above fₘₐₓ),
then sampling at
fₛ ≥ 2·fₘₐₓ
perfectly reconstructs the original signal (using ideal sinc interpolation).

• Nyquist rate = 2·fₘₐₓ
• Nyquist frequency = fₛ / 2

Telecom examples

• Audio: human hearing ~20 kHz → CD uses 44.1 kHz
• 5G baseband: channels up to 100–400 MHz → sampling rates in Gsamples/s in high-end radios

Visual: Proper sampling vs aliasing (sine wave example)

Left: Proper sampling (fₛ > 2×f_max) reconstructs perfectly. Right: Undersampling causes aliasing.

Another clear aliasing demo:

5. Aliasing — The Enemy You Must Understand

When fₛ < 2·fₘₐₓ, frequencies above fₛ/2 fold back into the baseband:

Consequence
High-frequency components masquerade as low-frequency ones → irreversible distortion.

Prevention in real systems

• Always apply anti-aliasing low-pass filter before ADC
  Cutoff frequency ≤ fₛ / 2
• In software-defined radio: digital down-conversion + careful decimation

6. Filtering: LPF, HPF, BPF

Filters remove unwanted frequencies.

Example frequency responses (magnitude and phase for Butterworth-style filters)

(Typical LPF roll-off at -3 dB cutoff)

(HPF example — passes high frequencies)

7. Analog-to-Digital & Digital-to-Analog Conversion (ADC / DAC)

ADC chain

1. Anti-aliasing LPF
2. Sample-and-hold
3. Quantization (→ quantization noise)
4. Encoding (binary)

DAC: reverse process + reconstruction filter (smooth stair-steps)

Key impairments in telecom

• Quantization noise (increases with fewer bits)
• Aperture jitter (critical at high sampling rates)
• Dynamic range requirements → 12–16 bit ADCs common in base stations

Block diagram of ADC process (with anti-aliasing filter highlighted)

Anti-aliasing filter → Sample & Hold → Quantizer

Another overview:

Tools to Start Using Right Now

• MATLAB — fft, conv, filter, impz, freqz
• Python — NumPy, SciPy.signal, Matplotlib

```python

Quick aliasing demo

import numpy as np import matplotlib.pyplot as plt

t = np.linspace(0, 1, 10000) fs = 100 # sampling rate (too low for some components) f1, f2 = 5, 55 # 55 Hz will alias when fs=100

x = np.sin(2np.pif1t) + 0.7np.sin(2np.pif2*t)

Sampled version

n = np.arange(0, len(t), int(len(t)/(fs*1))) ts = t[n] xs = x[n]

plt.figure(figsize=(12,5)) plt.subplot(121); plt.plot(t, x, label=’continuous’); plt.title(‘Original’); plt.grid(True) plt.subplot(122); plt.stem(ts, xs, label=’sampled @ 100 Hz’); plt.title(‘Aliasing visible’); plt.grid(True) plt.tight_layout(); plt.show()