Fourier Analysis and Signal Representations

Introduction

Fourier analysis provides a mathematical language for representing signals as sums or continua of sinusoids. A time-domain waveform describes how a signal varies with time, while its frequency-domain representation shows the amplitudes and phases of its frequency components. This dual viewpoint is fundamental in electronics and communication: filters are specified by frequency response, modulation shifts spectra, sampling creates spectral replicas, and Parseval’s theorem connects signal behavior in time and frequency.

A signal may be deterministic or random, continuous-time or discrete-time, periodic or aperiodic, and an energy or power signal. The appropriate Fourier representation depends on these properties. This chapter develops Fourier series for periodic signals, Fourier transforms for aperiodic signals, convolution and filtering, sampling and aliasing, and energy and power relationships.

Learning Objectives

By the end of this chapter, a student should be able to:

1. Derive trigonometric and complex Fourier series coefficients.
2. Determine amplitude, phase, and power spectra.
3. Apply the continuous-time Fourier transform and its properties.
4. Explain convolution, filtering, bandwidth, and Parseval’s theorem.
5. Analyze ideal sampling, spectral replication, and aliasing.
6. Distinguish energy signals from power signals.
7. Apply Fourier methods to communication and electronic systems.

1. Periodic Signals and Fourier Series

A continuous-time signal x(t) is periodic if there is a positive number T₀ such that

The smallest positive T₀ is the fundamental period, and f₀ = 1/T₀ is the fundamental frequency in hertz. The corresponding angular frequency is

A sufficiently well-behaved periodic signal can be represented as a sum of harmonically related sinusoids. The trigonometric Fourier series is

where

Here n is a positive integer and t₀ is any interval beginning; integration over one complete period gives the same coefficients. The coefficient a₀ is the average, or dc, value of the signal.

The series may also be written in amplitude-phase form:

where

The function atan2 preserves the correct quadrant of the phase. The complex exponential form is usually more convenient for systems analysis:

with

For a real signal, C₋ₙ = Cₙ, where denotes complex conjugation. The relationships between forms are C₀ = a₀ and, for n > 0,

Symmetry shortcuts

If x(t) is even, x(−t) = x(t), all bₙ are zero. If x(t) is odd, x(−t) = −x(t), a₀ and all aₙ are zero. Half-wave symmetry, x(t + T₀/2) = −x(t), eliminates the dc term and all even harmonics. These properties reduce calculation and explain spectral patterns.

Convergence and discontinuities

At a point where x(t) is continuous, a convergent Fourier series equals x(t). At a jump discontinuity, it converges to the midpoint of the left and right limits. Near a jump, finite partial sums may overshoot; this is the Gibbs phenomenon. The overshoot does not disappear in height as more terms are added, although its width becomes narrower.

2. Worked Example: Square-Wave Spectrum

Consider the odd, unit-amplitude square wave of period T₀ defined over one period by x(t) = 1 for 0 < t < T₀/2 and x(t) = −1 for −T₀/2 < t < 0. Because the signal is odd, a₀ = aₙ = 0. Thus

Using odd symmetry and integrating over the positive half,

Therefore bₙ = 4/(nπ) for odd n and bₙ = 0 for even n. The series is

The waveform therefore contains only odd harmonics. Its spectrum consists of discrete lines at odd multiples of f₀, with amplitudes decreasing as 1/n. This explains why sharp digital transitions require substantial high-frequency bandwidth.

3. Spectra and Bandwidth

For a periodic signal, the line spectrum consists of coefficients Cₙ at frequencies n f₀. The magnitude spectrum plots |Cₙ| and the phase spectrum plots ∠Cₙ. For the trigonometric form, the nth sinusoidal amplitude is Aₙ.

Bandwidth is not always uniquely defined. For a strictly band-limited signal, the highest frequency containing nonzero spectral content defines an absolute bandwidth. In practical systems, bandwidth may mean the frequency range containing a specified percentage of power or satisfying a transmission mask. A square wave has theoretically infinite bandwidth, even though most useful power may lie in its first few harmonics.

The spectrum of a real signal has conjugate symmetry:

Consequently, its magnitude spectrum is even and its phase spectrum is odd, apart from phase discontinuities caused by zeros.

4. Continuous-Time Fourier Transform

The Fourier series describes periodic signals. For an aperiodic signal x(t), the continuous-time Fourier transform (CTFT) is

and the inverse transform is

Here t is in seconds, f is in hertz, and X(f) has units equal to the units of x(t) multiplied by seconds. An angular-frequency convention uses ω in radians per second:

The two conventions are equivalent when ω = 2πf. A transform exists under several mathematical conditions; absolute integrability, ∫|x(t)|dt < ∞, is a sufficient condition but not necessary. Impulses and other generalized signals are handled using distributions.

Important transform pairs

With the hertz convention, useful pairs include:

Here δ(t) is the Dirac impulse and u(t) is the unit-step function. These pairs are interpreted in the generalized-function sense where necessary.

Transform properties

Linearity, time shift, frequency shift, scaling, differentiation, and convolution are central properties. If x(t) ↔ X(f), then

The scaling factor 1/|a| is essential for preservation of area and dimensional consistency.

5. Convolution and Linear Time-Invariant Systems

For a linear time-invariant (LTI) system with impulse response h(t), the output is the convolution

The convolution theorem states

where H(f) is the frequency response. Thus an LTI system multiplies each frequency component by a complex gain. |H(f)| determines amplitude modification and ∠H(f) determines phase shift.

The inverse operation is not generally valid without care: deconvolution requires division by H(f), and frequencies at which H(f) is zero cannot be recovered. In the presence of noise, very small values of |H(f)| can cause severe noise amplification.

For a periodic input, each harmonic is independently scaled:

This result is the basis of Fourier-series analysis of filters and steady-state circuits.

6. Energy, Power, and Parseval’s Theorem

The energy of a signal is

A finite-energy signal has E < ∞ and usually has zero average power. Average power is

A periodic signal generally has infinite total energy but finite average power. For a periodic signal, power over one period is

Parseval’s theorem for the CTFT is

For the complex Fourier series,

These equations require consistent transform conventions. The equality shows that signal energy can be calculated in either domain. In a resistor R, if x(t) is voltage, instantaneous power is |x(t)|²/R; if x(t) is current, it is R|x(t)|². Thus the mathematical signal power must be converted using the physical impedance.

Example: power in a square wave

For a square wave with values +V and −V across a resistance R, |x(t)|² = V² at every time except discontinuities, which do not affect integration. Therefore

Using the Fourier series, the same result follows from the sum of the powers of all odd harmonics. This is a useful check on harmonic calculations.

7. Sampling and Aliasing

Uniform sampling converts a continuous-time signal x(t) into samples x[n] = x(nTₛ), where Tₛ is the sampling interval and fₛ = 1/Tₛ is the sampling frequency. Ideal sampling can be modeled as multiplication by an impulse train:

The spectrum is a repeated set of shifted copies of X(f):

If x(t) is band-limited to |f| ≤ B, exact reconstruction is possible in the ideal model when

This is the Nyquist sampling condition. The frequency 2B is the Nyquist rate; fₛ/2 is the Nyquist frequency of the sampler. The strict inequality avoids spectral contact in practical designs, where transition bands and nonideal filters exist.

An ideal reconstruction filter selects the central spectral replica and has gain Tₛ in the above convention. The corresponding interpolation formula is

where sinc(z) = sin(πz)/(πz). This formula assumes ideal, infinite-duration sinc interpolation and a sufficiently band-limited signal.

Aliasing

Aliasing occurs when shifted spectral replicas overlap. The resulting sampled data cannot distinguish certain continuous-time frequencies. For a sinusoid at fᵢ, frequencies separated by integer multiples of fₛ produce the same samples:

where k is chosen so that f_alias lies in [0, fₛ/2]. Aliasing is irreversible after sampling. An analog anti-aliasing low-pass filter must therefore suppress frequencies above the usable band before the analog-to-digital converter.

8. Applied Case: Digital Acquisition of a Sensor Signal

Background

Consider an illustrative industrial vibration sensor whose useful vibration content extends to 18 kHz. The sensor output also contains interference near 31 kHz and broadband noise.

Problem

An engineer proposes a sampling frequency of 30 kHz because it is greater than 18 kHz. This is insufficient: the Nyquist frequency is only 15 kHz, so the desired 18 kHz content aliases into the baseband.

Analysis and solution

The minimum theoretical sampling rate for a 18 kHz band limit is greater than 36 kHz. A practical design may choose fₛ = 50 kHz, followed by an analog low-pass anti-aliasing filter. The filter should pass the required band with acceptable amplitude and phase distortion, while attenuating the 31 kHz interference and frequencies near or above 25 kHz, the new Nyquist frequency. A digital filter may then provide additional noise reduction after conversion.

If the 31 kHz interference were sampled at 50 kHz, its first-fold alias would be |31 − 50| = 19 kHz, outside the desired band but potentially troublesome if later processing assumes a wider band. At 30 kHz, it would alias to |31 − 30| = 1 kHz, creating a false low-frequency vibration component.

Outcome and learning points

The corrected architecture prevents high-frequency components from masquerading as low-frequency signals. The case demonstrates that sampling-rate selection, analog filtering, spectral interpretation, and system specifications must be designed together. Increasing the rate alone does not correct aliasing already introduced by inadequate analog filtering.

9. Practical Design Considerations and Limitations

Real signals are finite in duration and measurements are discrete. Multiplication by a finite observation window convolves the true spectrum with the window spectrum, causing leakage. A rectangular window gives narrow main-lobe width but comparatively large sidelobes; tapered windows reduce sidelobes at the cost of frequency resolution. The discrete Fourier transform (DFT) is

and the inverse DFT is

For sampling rate fₛ and record length N, the bin spacing is ∆f = fₛ/N. Zero-padding produces a more finely sampled plotted spectrum but does not increase the underlying resolution determined by record duration and window properties. The fast Fourier transform (FFT) is an efficient algorithm for computing the DFT; it does not define a different transform.

Common best practices include specifying transform conventions, preserving units, checking conjugate symmetry for real signals, distinguishing one-sided from two-sided spectra, applying anti-aliasing filters, documenting window and normalization choices, and validating energy or power using Parseval’s theorem.

Fourier analysis also has limitations. Nonstationary signals may require short-time Fourier transforms or wavelets. A spectrum shows which frequencies are present but not, by itself, when they occur. Noise, finite records, quantization, clock jitter, and nonlinear distortion can all alter measured spectra. In safety-critical or regulated instrumentation, sampling and filtering assumptions must be documented and verified through calibration and testing.

Chapter Summary

Fourier series represent periodic signals as dc components and harmonically related sinusoids. Trigonometric, amplitude-phase, and complex forms provide equivalent descriptions, while symmetry often simplifies coefficient calculations. The Fourier transform extends the representation to aperiodic signals and relates time-domain behavior to a continuous spectrum. Convolution in time becomes multiplication in frequency, making Fourier analysis central to LTI filter design. Energy and average power are evaluated through time-domain integrals or Parseval’s theorem in the frequency domain. Sampling creates periodically repeated spectra; the Nyquist condition permits ideal reconstruction only when the signal is adequately band-limited, and aliasing must be prevented before sampling. Practical spectral analysis must account for finite records, leakage, windowing, DFT normalization, noise, and hardware limitations.

Key Terms

Fourier series: Expansion of a periodic signal into harmonically related sinusoids.

Fourier transform: Integral representation of an aperiodic signal in frequency.

Fundamental period: Smallest positive period of a periodic signal.

Harmonic: Frequency component at an integer multiple of the fundamental frequency.

Complex coefficient: Coefficient Cₙ in the complex Fourier series.

Magnitude spectrum: Frequency plot of spectral magnitudes.

Phase spectrum: Frequency plot of spectral phases.

Bandwidth: Frequency range required to represent or transmit a signal under a stated criterion.

Convolution: Integral operation describing the output of an LTI system.

Impulse response: Output of an LTI system to a unit impulse.

Frequency response: Fourier transform H(f) of an LTI system’s impulse response.

Energy signal: Signal with finite total squared magnitude.

Power signal: Signal with finite nonzero average power.

Parseval’s theorem: Equality between time-domain and frequency-domain energy or power measures.

Sampling: Conversion of a continuous-time signal into values at discrete times.

Nyquist rate: Twice the highest frequency of a strictly band-limited signal.

Nyquist frequency: Half the sampling frequency.

Aliasing: Irreversible frequency ambiguity caused by spectral overlap during sampling.

Spectral leakage: Spreading of measured spectral energy caused by finite observation.

Anti-aliasing filter: Analog filter that suppresses frequencies before sampling.

Review Questions

Short Answer Questions

1. Define the fundamental frequency and fundamental angular frequency of a periodic signal.
2. State the Fourier-series symmetry conditions for an even and an odd signal.
3. What physical meaning is assigned to |H(f)| and ∠H(f) in an LTI system?
4. Distinguish an energy signal from a power signal.
5. Why is an anti-aliasing filter placed before an analog-to-digital converter?

Long Answer Questions

1. Derive the complex Fourier-series coefficients from the trigonometric Fourier series.
2. Explain the Fourier-transform convolution theorem and its significance in filter analysis.
3. Derive the spectrum of an ideal impulse-train sampled signal.
4. Explain Parseval’s theorem for both aperiodic and periodic signals.
5. Discuss spectral leakage, windowing, DFT bin spacing, and the limitations of zero-padding.

Analytical / Application Questions

1. A 2 kHz sinusoid is sampled at 3 kHz. Determine its observed baseband alias frequency and explain the result.
2. A periodic waveform has only odd harmonics and zero average value. Identify two likely symmetry properties and justify them.
3. Design the principal stages of an acquisition system for a sensor whose useful band is 0–12 kHz, including a defensible sampling rate and the role of analog and digital filters.

Multiple Choice Questions with Answers

Question 1

For a periodic signal with period T₀, what is its fundamental angular frequency?

A. 2πT₀

B. T₀/(2π)

C. 2π/T₀

D. 1/(2πT₀)

Correct Answer: C. 2π/T₀

Question 2

Which Fourier-series coefficients vanish for an even signal?

A. Only a₀

B. All bₙ

C. All aₙ

D. Only the odd harmonics

Correct Answer: B. All bₙ

Question 3

If x(t) ↔ X(f), the transform of x(t − t₀) is:

A. X(f − t₀)

B. X(f)e^{−j2πft₀}

C. X(f)e^{j2πt₀}

D. t₀X(f)

Correct Answer: B. X(f)e^{−j2πft₀}

Question 4

For an LTI system, the frequency-domain output is:

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

B. Y(f)=X(f)/H(f) always

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

D. Y(f)=X(f)−H(f)

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

Question 5

A signal band-limited to 10 kHz requires, in the ideal sampling model, a sampling frequency:

A. Less than 5 kHz

B. Exactly 10 kHz

C. Greater than 20 kHz

D. Greater than 5 kHz but less than 10 kHz

Correct Answer: C. Greater than 20 kHz

Question 6

A 31 kHz sinusoid sampled at 30 kHz aliases to:

A. 1 kHz

B. 15 kHz

C. 30 kHz

D. 61 kHz

Correct Answer: A. 1 kHz

Question 7

Which theorem equates the total time-domain energy with frequency-domain energy?

A. Superposition theorem

B. Parseval’s theorem

C. Thevenin’s theorem

D. Green’s theorem

Correct Answer: B. Parseval’s theorem

Question 8

For an N-point DFT sampled at fₛ, the frequency-bin spacing is:

A. Nfₛ

B. fₛ/N

C. N/fₛ

D. 1/(Nfₛ)

Correct Answer: B. fₛ/N

Question 9

What is the main purpose of an anti-aliasing filter?

A. To create spectral replicas

B. To increase sampling jitter

C. To suppress out-of-band components before sampling

D. To remove all phase response

Correct Answer: C. To suppress out-of-band components before sampling

Question 10

A unit-amplitude odd square wave contains, in its ideal Fourier series:

A. Only even harmonics

B. Only odd harmonics

C. Only a dc component

D. No harmonics

Correct Answer: B. Only odd harmonics