Communication Systems

Chapter 2
Signals, Spectra and System Analysis

Bachelors Program in Electronics and Communication
Communication Systems   •   Bachelor

Signals, Spectra and System Analysis

Introduction

Communication engineering is fundamentally concerned with representing, modifying, transmitting, and recovering information. A microphone converts acoustic pressure into an electrical waveform; a transmitter shapes that waveform for a channel; the receiver extracts the desired information from noise and interference. To analyze such systems, engineers require a language that connects time-domain behavior with frequency-domain behavior. That language is provided by signals, spectra, convolution, filtering, bandwidth, and linear time-invariant (LTI) system theory.

A signal is a mathematical function that represents the variation of a physical quantity with an independent variable, usually time. Signals may be continuous-time or discrete-time, deterministic or random, periodic or aperiodic, and analog or digital. These classifications are not interchangeable. For example, a sampled speech waveform is discrete-time but may still have continuous amplitude, whereas a binary sequence is discrete-time and discrete-amplitude.

This chapter develops the tools required to analyze communication signals and systems. Particular attention is given to energy and power, Fourier series and transforms, convolution, frequency response, bandwidth, and distortion.

Learning Objectives

After studying this chapter, a student should be able to:

  1. Classify continuous-time and discrete-time signals.
  2. Determine whether a signal is an energy signal, a power signal, or neither.
  3. Represent periodic signals using Fourier series and aperiodic signals using Fourier transforms.
  4. Explain the relationship between time-domain convolution and frequency-domain multiplication.
  5. Analyze LTI systems using impulse response and frequency response.
  6. Calculate bandwidth and distinguish amplitude, phase, and delay distortion.
  7. Apply system-analysis methods to practical communication problems.

1. Signal Descriptions and Classifications

A continuous-time signal is written as x(t), where time t varies continuously. A discrete-time signal is written as x[n], where n is an integer sample index. Discrete-time notation does not by itself imply quantized amplitude.

A signal is deterministic if its values can be specified exactly by a mathematical rule. A sinusoid such as x(t)=A cos(2πf0t+φ) is deterministic. A noise waveform is more appropriately modeled as a random signal because its exact future values cannot be predicted.

A periodic continuous-time signal satisfies

x(t + T₀) = x(t)

where T0 is the fundamental period, the smallest positive period when one exists. For a discrete-time signal, periodicity requires

x[n + N₀] = x[n]

where N0 is a positive integer. A discrete-time sinusoid cos(ω0n) is periodic only if ω0/(2π) is rational. This condition differs from the continuous-time case, in which every sinusoid with nonzero frequency is periodic.

A signal is even if x(t)=x(-t) and odd if x(t)=-x(-t). Any signal can be decomposed into even and odd parts:

xe(t) = [x(t) + x(-t)]/2
xo(t) = [x(t) - x(-t)]/2

The same expressions apply to discrete-time signals after replacing t with n. This decomposition is useful because Fourier transforms of even and odd signals have predictable symmetry properties.

Common elementary signals include the unit impulse δ(t), unit step u(t), ramp r(t)=tu(t), exponential, and sinusoid. The continuous-time impulse is defined through its sampling property:

∫₋∞^∞ x(t)δ(t−t₀)dt = x(t₀)

The discrete-time impulse δ[n] equals 1 at n=0 and 0 elsewhere, and therefore

∑ₙ₌₋∞^∞ x[n]δ[n−n₀] = x[n₀]

2. Energy and Power of Signals

Signal energy measures the accumulated squared magnitude. For a continuous-time signal,

E = ∫₋∞^∞ |x(t)|² dt

For a discrete-time signal,

E = ∑ₙ₌₋∞^∞ |x[n]|²

The expressions assume a normalized resistance or impedance. In a physical circuit, instantaneous power may include a factor such as 1/R, depending on the voltage or current representation.

Average power for a continuous-time signal is defined as

P = limT→∞ (1/2T) ∫₋TT |x(t)|² dt

For a discrete-time signal,

P = limN→∞ (1/(2N+1)) ∑ₙ₌₋NN |x[n]|²

An energy signal has finite, nonzero energy and zero average power. A power signal has finite, nonzero average power and infinite energy. A nonzero periodic signal is generally a power signal, because its energy accumulates indefinitely while its average power remains finite. Some signals are neither, such as a continuously growing waveform.

For a periodic signal with period T0, average power can be calculated over one period:

P = (1/T₀) ∫t₀t₀+T₀ |x(t)|²dt

The result is independent of the selected starting time t0. For a discrete periodic signal of period N0,

P = (1/N₀) ∑n=0N₀−1|x[n]|²

Worked Example 1: Energy and Power Classification

Consider x(t)=e-2tu(t). Its energy is

E = ∫₀^∞ e-4tdt = 1/4

Thus it is an energy signal. Its average power is zero because the finite energy is spread over an infinitely long interval.

Now consider x(t)=3cos(100πt). It is periodic and therefore has infinite energy. Its average power is

P = 3²/2 = 4.5

because the average of cos²(·) over a complete cycle is 1/2. It is therefore a power signal.

3. Fourier Series

Periodic signals can be represented as sums of harmonically related sinusoids. The complex exponential Fourier series is

x(t) = ∑k=−∞∞ Ck ejkω₀t

where ω0=2π/T0 is the fundamental angular frequency in radians per second and

Ck = (1/T₀)∫t₀t₀+T₀x(t)e−jkω₀tdt

The coefficients Ck describe the amplitude and phase of the harmonic at kω0. The representation is valid under standard convergence conditions, such as piecewise continuity and a finite number of maxima, minima, and discontinuities in one period. At a jump discontinuity, the Fourier series converges to the midpoint of the left and right limits.

The trigonometric form is

x(t)=a₀ + ∑k=1∞[ak cos(kω₀t)+bk sin(kω₀t)]

with

a₀=(1/T₀)∫x(t)dt
ak=(2/T₀)∫x(t)cos(kω₀t)dt
bk=(2/T₀)∫x(t)sin(kω₀t)dt

where each integral is taken over any interval of length T0. If a waveform is even, all bk coefficients are zero. If it is odd, a0 and all ak coefficients are zero.

Fourier series provide the first important connection between a waveform and its spectrum. The line spectrum consists of spectral components located at integer multiples of the fundamental frequency. A square wave, for example, contains a fundamental and odd harmonics; its sharp transitions require substantial high-frequency content.

4. Fourier Transform and Spectral Interpretation

The Fourier transform represents an aperiodic signal as a continuous distribution of frequency components:

X(f)=∫₋∞^∞x(t)e−j2πftdt

The inverse transform is

x(t)=∫₋∞^∞X(f)ej2πftdf

Here f is frequency in hertz, t is in seconds, and X(f) has units determined by the signal and the transform convention. An angular-frequency form uses ω=2πf:

X(ω)=∫₋∞^∞x(t)e−jωtdt

The inverse then includes the factor 1/(2π). A consistent convention must be used throughout an analysis.

Important transform properties include linearity, time shifting, frequency shifting, scaling, differentiation, and convolution. For example,

x(t−t₀) ⇌ X(f)e−j2πft₀

A delay changes phase but not magnitude. Modulation by a cosine shifts the spectrum:

x(t)cos(2πfct) ⇌ [X(f−fc)+X(f+fc)]/2

This property is the basis of amplitude modulation and frequency translation.

For real signals, X(−f)=X*(f), where the asterisk denotes complex conjugation. Consequently, the magnitude spectrum is even and the phase spectrum has odd symmetry, apart from phase discontinuities caused by zeros.

Parseval’s relation connects time-domain energy with spectral energy:

∫₋∞^∞|x(t)|²dt = ∫₋∞^∞|X(f)|²df

This relation assumes the Fourier-transform pair stated above and is useful for checking calculations and interpreting energy distribution across frequency.

5. Sampling and Discrete-Time Spectra

Although sampling is treated fully in later communication and signal-processing courses, its basic spectral consequence is essential here. Sampling a continuous signal at rate fs produces samples x[n]=x(nTs), where Ts=1/fs. If the signal is band-limited to B hertz, perfect reconstruction requires

fs > 2B

This is the Nyquist sampling condition for a strictly band-limited signal. Sampling at or below twice the highest frequency can cause spectral replicas to overlap, producing aliasing. Anti-aliasing low-pass filters are therefore placed before an analog-to-digital converter. The condition is idealized: real filters have transition bands, and practical systems use a safety margin above the theoretical minimum.

6. Linear Time-Invariant Systems

A system maps an input signal to an output signal. A system is linear if it obeys superposition:

T{a x₁(t)+b x₂(t)} = aT{x₁(t)}+bT{x₂(t)}

for arbitrary constants a and b. It is time invariant if a time shift in the input produces the same time shift in the output:

x(t)→y(t) implies x(t−t₀)→y(t−t₀)

An LTI system is completely characterized by its impulse response h(t). The output is the convolution

y(t)=x(t)*h(t)=∫₋∞^∞x(τ)h(t−τ)dτ

For discrete time,

y[n]=∑k=−∞∞x[k]h[n−k]

Convolution is commutative, associative, and distributive under conditions that ensure the relevant sums or integrals converge. Its physical interpretation is weighted accumulation: each input value generates a shifted and scaled copy of the impulse response, and the output is their sum.

In the frequency domain, convolution becomes multiplication:

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

where H(f) is the frequency response. Therefore, an LTI system modifies each frequency component independently in magnitude and phase.

For stability, the bounded-input bounded-output (BIBO) condition for a continuous-time LTI system is

∫₋∞^∞|h(t)|dt < ∞

For discrete time it is

∑ₙ₌₋∞^∞|h[n]| < ∞

Causality requires h(t)=0 for t<0 in continuous time, or h[n]=0 for n<0 in discrete time. A physically realizable causal system cannot respond before an input is applied.

Worked Example 2: First-Order Low-Pass System

Consider the causal system

H(f)=1/(1+jf/fc)

where fc is the cutoff frequency. Its magnitude and phase are

|H(f)|=1/√[1+(f/fc)²]
∠H(f)=−tan⁻¹(f/fc)

At f=fc, the magnitude is 1/√2≈0.707, or −3.01 dB, and the phase is −45°. Thus, a sinusoid at the cutoff frequency is attenuated and delayed in phase.

7. Filtering, Bandwidth, and Distortion

A filter selectively passes some frequencies and attenuates others. A low-pass filter passes frequencies near zero; a high-pass filter suppresses low frequencies; a band-pass filter passes a selected interval; and a band-stop filter rejects a selected interval.

The bandwidth depends on the definition being used. For a low-pass filter, the 3-dB bandwidth is commonly the frequency at which power falls to half its passband value, corresponding to a voltage or amplitude ratio of 1/√2. For a band-pass filter with lower and upper 3-dB frequencies fL and fH,

B=fH−fL

The center frequency is sometimes defined as the geometric mean,

f₀=√(fL fH)

particularly for filters with logarithmic or resonant behavior. Bandwidth must always be stated with its criterion because null-to-null, occupied, noise-equivalent, and 3-dB bandwidths are not identical.

An ideal filter has constant magnitude in its passband and a linear phase response. A system introduces no waveform distortion if

H(f)=K e−j2πftd

through the signal bandwidth, where K is a constant gain and td is a constant delay. Amplitude distortion occurs when |H(f)| varies across the occupied band. Phase distortion occurs when phase is not a linear function of frequency. The group delay is

τg(f)=−(1/2π)dφ(f)/df

where φ(f) is the unwrapped phase in radians and f is in hertz. Constant group delay across the band is desirable for preserving waveform shape.

In analog communication, distortion can cause intersymbol interference, altered pulse shape, and errors in symbol detection. Equalizers compensate for predictable channel magnitude and phase variation, although equalization generally increases sensitivity to noise outside the desired band.

8. Practical Case: Baseband Data Link over a Limited-Bandwidth Channel

Background

Consider an illustrative wired digital link transmitting binary pulses over a copper cable. The transmitter produces rectangular pulses with significant spectral content extending well above the nominal symbol rate. The cable and receiver input circuit together behave approximately as a low-pass LTI system.

Problem

At a symbol rate of 1 Mbaud, the received waveform shows rounded transitions and occasional errors. Measurements indicate that the channel has a 3-dB bandwidth of 0.6 MHz, while the pulse spectrum contains important components above 1 MHz.

Analysis

The received spectrum is

Y(f)=X(f)H(f)+N(f)

where X(f) is the transmitted spectrum, H(f) is the channel response, and N(f) is additive noise. Since the channel attenuates high-frequency components, abrupt pulse transitions are removed. If phase delay also varies substantially across the signal band, different spectral components arrive with different delays, causing additional pulse spreading.

Approach and Solution

The engineering team first measures the channel magnitude and phase response rather than relying only on a nominal bandwidth number. It then applies a transmit pulse-shaping filter and a receiver equalizer. A raised-cosine or root-raised-cosine design can control occupied bandwidth and reduce intersymbol interference when the transmitter and receiver responses are coordinated. The equalizer is constrained so that noise enhancement remains acceptable.

A practical design must also include margin for component tolerances, temperature variation, cable aging, and electromagnetic interference. If the required data rate cannot be achieved with acceptable error probability, the alternatives include a lower symbol rate, a better cable, adaptive equalization, or a modulation scheme that uses bandwidth more efficiently.

Outcome

After equalization and pulse shaping, the eye opening at the sampling instant increases and the bit-error rate falls. The improvement does not result from creating bandwidth that the cable does not possess; it results from using the available bandwidth more effectively and compensating predictable distortion.

Learning Points

  1. Bandwidth is a property of a specified response and criterion, not merely a label attached to a device.
  2. Time-domain pulse spreading is explained by frequency-domain attenuation and phase variation.
  3. Equalization trades distortion reduction against noise enhancement and implementation complexity.
  4. System design must consider the complete cascade of transmitter, channel, receiver, and measurement equipment.

9. Engineering Practice, Limitations, and Contemporary Relevance

Models are useful because they simplify physical systems, but their assumptions must be checked. LTI analysis is appropriate when system parameters remain approximately constant and superposition is valid. Amplifier saturation, mixers operated outside their intended range, time-varying wireless channels, and nonlinear power amplifiers require extensions such as nonlinear or time-varying system models.

Measured spectra depend on instrument resolution bandwidth, windowing, observation time, and noise floor. A finite record can produce spectral leakage. Window functions reduce leakage but alter amplitude and resolution. Engineers should document sampling rate, record length, window, calibration, and bandwidth definition when reporting spectral measurements.

Modern communication systems use digital signal processors to implement filtering, Fourier transforms, channel estimation, and adaptive equalization. Artificial-intelligence methods may assist in channel prediction or signal classification, but they do not remove the need for sampling theory, stable system design, calibration, and interpretable performance measures. In regulated radio systems, spectral masks, maximum transmitted power, occupied bandwidth, and interference limits are established by national and international regulatory frameworks. Engineers must ensure that signal processing does not create harmful interference or violate privacy and security requirements.

Best practice is to analyze a system in both domains: use the time domain to inspect delays, transients, and pulse shape, and the frequency domain to inspect bandwidth, filtering, interference, and noise allocation. Independent checks using energy, units, limiting cases, and numerical simulation are valuable safeguards against analytical error.

Chapter Summary

Signals may be continuous-time or discrete-time, deterministic or random, periodic or aperiodic, and analog or digital. Energy is the integral or sum of squared magnitude, whereas average power is the long-term average of squared magnitude. Finite-energy signals have zero average power, while nonzero periodic signals generally have finite average power and infinite energy.

Fourier series represent periodic signals as harmonically related components. The Fourier transform represents aperiodic signals through a continuous spectrum. Time shifts alter phase, modulation shifts spectra, and Parseval’s relation connects time-domain energy with spectral energy.

An LTI system is characterized by its impulse response. Convolution gives the output in the time domain, while multiplication by the frequency response gives the output spectrum. Causality and BIBO stability impose important physical and mathematical conditions.

Filters determine which spectral components are passed or attenuated. Bandwidth must be defined using a stated criterion. Amplitude variation causes amplitude distortion, while nonlinear phase causes phase distortion and frequency-dependent delay. Communication-system analysis therefore requires coordinated attention to signal representation, system response, bandwidth, noise, and implementation constraints.

Key Terms

Signal: A mathematical representation of a physical quantity varying with an independent variable.

Continuous-time signal: A signal defined for every value of time.

Discrete-time signal: A signal defined only at discrete sample indices.

Periodic signal: A signal that repeats after a fixed period.

Energy signal: A signal with finite, nonzero total energy.

Power signal: A signal with finite, nonzero long-term average power.

Impulse: An ideal signal used to characterize sampling and system response.

Fourier series: A harmonic representation of a periodic signal.

Fourier transform: A frequency-domain representation of an aperiodic signal.

Spectrum: The distribution of signal magnitude and phase over frequency.

Convolution: The operation that determines the output of an LTI system from its input and impulse response.

Impulse response: The output of a system when the input is an impulse.

LTI system: A system possessing linearity and time invariance.

Frequency response: The complex gain of an LTI system as a function of frequency.

Bandwidth: A specified measure of the frequency range occupied or passed by a signal or system.

Amplitude distortion: Waveform distortion caused by nonuniform magnitude response.

Phase distortion: Waveform distortion caused by nonlinearity of phase response.

Group delay: The negative derivative of phase with respect to angular frequency.

Aliasing: Irreversible spectral overlap caused by inadequate sampling.

Equalizer: A circuit or algorithm that compensates for predictable channel distortion.

Review Questions

Short Answer Questions

  1. Distinguish between a continuous-time signal and a discrete-time signal.
  2. Why is a nonzero periodic signal generally classified as a power signal?
  3. State the Fourier-transform convolution property.
  4. What is meant by the frequency response of an LTI system?
  5. Distinguish amplitude distortion from phase distortion.

Long Answer Questions

  1. Explain the classification of signals using periodicity, symmetry, determinism, and time representation.
  2. Derive the Fourier-series coefficient formula and explain the significance of harmonic components.
  3. Explain convolution and its physical interpretation for an LTI communication channel.
  4. Discuss the conditions for causality and BIBO stability of continuous-time LTI systems.
  5. Explain bandwidth, group delay, and waveform distortion in a practical communication link.

Analytical / Application Questions

  1. Determine the energy and average power classification of x(t)=4e-3tu(t), showing the relevant integral.
  2. A channel has lower and upper 3-dB frequencies of 20 kHz and 80 kHz. Calculate its 3-dB bandwidth and geometric center frequency, and explain why the two quantities are different.
  3. A sampled signal is strictly band-limited to 15 kHz. Determine the theoretical minimum sampling-rate condition and explain the practical role of an anti-aliasing filter.

Multiple Choice Questions with Answers

Question 1

Which condition defines a periodic continuous-time signal?

A. x(t)=x(−t)

B. x(t+T₀)=x(t) for a positive T₀

C. x(t)=0 for t<0

D. ∫|x(t)|dt<∞

Correct Answer: B. x(t+T₀)=x(t) for a positive T₀

Question 2

A nonzero sinusoid extending for all time is generally a:

A. Finite-energy signal

B. Finite-duration signal

C. Power signal

D. Zero signal

Correct Answer: C. Power signal

Question 3

For an LTI system, the output in the time domain is obtained by:

A. Differentiating the input only

B. Convolving the input with the impulse response

C. Adding the input and impulse response

D. Sampling the input at twice its bandwidth

Correct Answer: B. Convolving the input with the impulse response

Question 4

If a signal is delayed by t₀, its Fourier transform is multiplied by:

A. e^{j2πft₀}

B. e^{−j2πft₀}

C. t₀f

D. 1/t₀

Correct Answer: B. e^{−j2πft₀}

Question 5

At the 3-dB cutoff frequency of a first-order low-pass filter, the magnitude ratio is approximately:

A. 0.5

B. 0.707

C. 1.414

D. 3

Correct Answer: B. 0.707

Question 6

The Fourier transform of a time-domain convolution is:

A. X(f)+H(f)

B. X(f)−H(f)

C. X(f)H(f)

D. X(f)/H(f) always

Correct Answer: C. X(f)H(f)

Question 7

For a signal band-limited to 10 kHz, ideal sampling requires a sampling frequency:

A. Less than 5 kHz

B. Exactly 10 kHz

C. Greater than 20 kHz

D. Greater than 100 Hz only

Correct Answer: C. Greater than 20 kHz

Question 8

A causal continuous-time LTI system must have an impulse response that is zero for:

A. t>0

B. t=0

C. t<0

D. All t

Correct Answer: C. t<0

Question 9

Which quantity measures the negative phase slope with respect to frequency?

A. Energy density

B. Group delay

C. Power factor only

D. Sampling rate

Correct Answer: B. Group delay

Question 10

For a band-pass response with 3-dB frequencies 5 kHz and 15 kHz, the 3-dB bandwidth is:

A. 3 kHz

B. 10 kHz

C. 20 kHz

D. 75 kHz

Correct Answer: B. 10 kHz
Communication Systems
Chapter 2 of 15