Communications Engineering Fundamentals

date2026-07-14document_iddoc_607729d375d0265c1e70bcb1f9031bdddescription信号の周波数表現、線形時不変システム、変調、信号対雑音比、帯域の基礎を説明する。prerequisitesフーリエ変換の基本 / 複素数 / 確率の基本type講義statusactive

informationcommunicationsundergraduatelecture

data/lecture/information/communications/communications-engineering-portal.lecture.n.md

1Introduction

This lecture explains how to represent signals in both time and frequency domains and how to analyze channel effects, modulation, noise, and bandwidth constraints.

data/lecture/math/analysis/introduction-to-fourier-transform.lecture.n.md

2Signals and Frequency

A signalSignal is a function of a variable such as time or space that conveys or represents information. This lecture considers a continuous-time signal $x (t)$ .

FrequencyFrequency is the number of repetitions of a periodic variation per unit time. Frequency $f$ is measured in hertz, and its relation to angular frequency is $ω = 2 π f$ .

Communications engineering conventionally denotes the imaginary unit by $j$ , where $j^{2} = - 1$ . Assume that $x$ and its Fourier transform are both integrable and that $x$ is continuous. Define the Fourier transform by

X (f) = \int_{- \infty}^{\infty} x (t) e^{- j 2 π f t} d t .

The inverse transform, which holds at every time under these assumptions, is

x (t) = \int_{- \infty}^{\infty} X (f) e^{j 2 π f t} d f .

$X (f)$ represents the amplitude and phase of each frequency component in $x (t)$ . It is related to the angular-frequency convention $X_{ω} (ω)$ used in the mathematical reference by $X (f) = X_{ω} (2 π f)$ . For more general signals, the Fourier transform can be extended to square-integrable functions and distributions.

3Linear Time-Invariant Systems

A linear time-invariant systemLinear time-invariant system; LTI system satisfies linearity, which preserves superposition, and time invariance, under which a time shift of the input causes the same time shift of the output.

If $h (t)$ is the impulse response of an LTI system, its input $x (t)$ and output $y (t)$ satisfy

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

Under conditions for the Fourier transforms to exist, the frequency responseFrequency response $H (f)$ gives

Y (f) = H (f) X (f) .

Complex exponentials are eigenfunctions of LTI systems: the system preserves the input complex exponential's frequency and multiplies its complex amplitude by $H (f)$ . This property permits frequency-by-frequency analysis of filters and channels.

4Modulation and Spectral Translation

ModulationModulation is the operation of varying a carrier's amplitude, frequency, phase, or another property according to a baseband information signal. To exhibit spectral translation explicitly, consider double-sideband suppressed-carrier amplitude modulation:

s (t) = m (t) \cos (2 π f_{c} t) .

The frequency-shift property of the Fourier transform gives

S (f) = \frac{1}{2} {M (f - f_{c}) + M (f + f_{c})} .

The baseband spectrum is therefore copied around $\pm f_{c}$ . If $M (f) = 0$ for $| f | > B$ and $f_{c} > B$ , the positive-frequency sideband extends from $f_{c} - B$ to $f_{c} + B$ and has occupied bandwidth $2 B$ . Modulation permits use of a frequency band suited to an antenna or channel and allocation of distinct bands to multiple signals. Demodulation requires receiver processing corresponding to the selected modulation scheme.

5Noise and SNR

Under an additive-noise model, a received signal is

r (t) = s (t) + n (t),

where $n (t)$ is noise. The signal-to-noise ratioSignal-to-noise ratio; SNR is the ratio of signal power $P_{s}$ to noise power $P_{n}$ , evaluated at the same measurement point and over the same measurement bandwidth:

S N R = \frac{P_{s}}{P_{n}}, P_{n} > 0.

For a deterministic signal $u (t)$ , define average power, when the limit exists, by

P_{u} = \lim_{T \to \infty} \frac{1}{2 T} \int_{- T}^{T} | u (t) |^{2} d t .

If random noise $n (t)$ is assumed to be wide-sense stationary, $P_{n} = E [| n (t) |^{2}]$ is independent of time and represents its average power. This value equals the variance only when the noise has zero mean.

In decibels,

{S N R}_{d B} = 10 \log_{10} \frac{P_{s}}{P_{n}} .

SNR is one measure of reception quality. The error rate can nevertheless differ at the same SNR because it also depends on modulation, coding, the noise distribution, and the decision rule.

6Bandwidth

BandwidthBandwidth denotes a frequency range occupied by a signal or passed by a system, or the width of that range. Its precise definition depends on context. A strictly band-limited baseband signal satisfies $X (f) = 0$ for $| f | > B$ for some $B > 0$ ; $B$ is then called its one-sided bandwidth or highest frequency, while the two-sided support interval $[- B, B]$ has width $2 B$ . Practical signals and filters may not become exactly zero, so conventions such as occupied bandwidth or 3 dB bandwidth are used according to purpose.

A wider bandwidth permits more rapid waveform changes and provides scope for a higher information rate, but bandwidth alone does not determine an achievable communication rate. SNR, modulation, coding, and the acceptable error rate also matter.

7Key Points

The Fourier transform decomposes a signal into frequency components.
An LTI system is described by convolution, which becomes multiplication by its frequency response in the frequency domain.
Modulation can translate an information signal's spectrum into a frequency band suitable for transmission.
SNR is the ratio of signal power to noise power at the same measurement point and over the same bandwidth.
A bandwidth statement must specify whether it refers to strict band limitation, occupied bandwidth, 3 dB bandwidth, or another convention.

8Next Lecture

data/lecture/information/communications/sampling-and-sampling-theorem-basics.lecture.n.md