Signals and Systems: Fourier Analysis

Fourier analysis is the backbone of frequency-domain thinking in signals and systems. It provides a precise way to describe how a signal is built from sinusoids and how a system responds to each sinusoidal component. Once you can move between time and frequency, problems that look complicated in one domain often become straightforward in the other, especially when filtering, characterizing frequency response, or analyzing bandwidth.

This article covers the core tools: Fourier series, the Fourier transform, frequency response, and filtering, with practical interpretations throughout.

Why the Frequency Domain Matters

A time-domain waveform can hide structure. A speech signal, an ECG trace, or a vibration measurement may look irregular, yet each contains periodicities and dominant frequency content that matter for perception, diagnosis, or mechanical integrity.

The frequency domain answers questions like:

• Which frequencies are present, and with what strength?
• How does a system amplify or attenuate each frequency?
• How can we remove noise without destroying the information we care about?

The key idea is sinusoidal eigenfunctions for linear time-invariant (LTI) systems: if you input a sinusoid at frequency $\omega$, the output is a sinusoid at the same frequency, scaled and phase-shifted. Fourier methods exploit that property by representing general signals as combinations of sinusoids.

Fourier Series: Periodic Signals as Harmonic Sums

For a periodic signal with period $T$, the Fourier series represents it as a sum of harmonics at integer multiples of the fundamental frequency ${\omega }_0=\frac{2\pi }{T}$.

A common complex-exponential form is:

$$x(t)=\sum _{k=-\infty }^{\infty }{c}_k{e}^{jk{\omega }_{0}t}$$

where the coefficients $c_k$ quantify the contribution of the $k$th harmonic. Intuitively:

• Large $|c_k|$ for small $|k|$ means the signal is dominated by low-frequency content and changes slowly within a period.
• Significant energy at high $|k|$ indicates sharp transitions or fine detail. A square wave, for example, requires many harmonics to approximate its edges.

Practical insight: why harmonics matter

In audio, a musical note is not just its fundamental frequency. The harmonic spectrum shapes timbre. In power systems, the ideal sinusoidal voltage at 50 or 60 Hz can pick up harmonics due to nonlinear loads, and those harmonics can cause heating, distortion, or interference. Fourier series coefficients give a direct way to quantify and manage that distortion.

Convergence and real-world signals

Fourier series is mathematically clean for periodic signals, but many real signals are not perfectly periodic. Still, it remains useful when a signal is approximately periodic over an interval, or when systems are driven by periodic excitations.

Fourier Transform: A Continuous Spectrum for Aperiodic Signals

For nonperiodic signals, frequency content is not confined to discrete harmonics. The Fourier transform generalizes the idea to a continuous range of frequencies:

$$X(\omega )={\int }_{-\infty }^{\infty }x(t)e^{-j\omega t}dt$$

and the inverse transform reconstructs the signal:

$$x(t)=\frac{1}{2\pi }{\int }_{-\infty }^{\infty }X(\omega )e^{j\omega t}d\omega$$

Here, $X(\omega )$ is the spectrum. Its magnitude $|X(\omega )|$ tells you how much of each angular frequency $\omega$ is present; its phase $\angle X(\omega )$ encodes time alignment and waveform shape.

Time-frequency tradeoff in plain terms

Shorter events in time tend to spread out in frequency. A sharp pulse contains many frequencies; a long, smooth waveform concentrates energy near low frequencies. This is not just a rule of thumb. It reflects a fundamental tradeoff between localization in time and localization in frequency.

Key properties used constantly in systems work

Fourier analysis is powerful partly because of its algebraic properties:

• Linearity: transforms of sums are sums of transforms.
• Time shift: delaying $x(t)$ multiplies $X(\omega )$ by a phase term $e^{-j\omega t_0}$.
• Differentiation: differentiation in time corresponds to multiplication by $j\omega$ in frequency, which explains why high frequencies are emphasized by derivatives.

Most importantly for filtering and LTI systems:

• Convolution in time becomes multiplication in frequency.

If $y(t)=x(t)\ast h(t)$, then:

$$Y(\omega )=X(\omega )H(\omega )$$

This single relationship is the workhorse of frequency-domain system analysis.

Frequency Response: How Systems Shape Spectra

An LTI system is characterized by its impulse response $h(t)$. Its frequency response is the Fourier transform of the impulse response:

$$H(\omega )={\int }_{-\infty }^{\infty }h(t)e^{-j\omega t}dt$$

When a signal passes through the system, each frequency component is multiplied by $H(\omega )$. Two parts matter:

• Magnitude response $|H(\omega )|$: gain or attenuation versus frequency
• Phase response $\angle H(\omega )$: phase shift versus frequency

Why phase is not optional

Magnitude response tells you what frequencies survive, but phase response affects waveform shape. In audio, phase distortion can smear transients. In communications, phase characteristics influence symbol timing and intersymbol interference. Two systems can have identical magnitude responses yet produce noticeably different outputs if their phase responses differ.

Filtering: Designing and Understanding Spectral Modification

Filtering is the practical application of frequency response shaping. A filter aims to pass certain frequency bands and suppress others.

Common filter types

• Low-pass: passes low frequencies, attenuates high frequencies. Used for smoothing, anti-aliasing, and removing high-frequency noise.
• High-pass: passes high frequencies, attenuates low frequencies. Used for removing drift or DC offsets.
• Band-pass: passes a middle band. Used in radio receivers to isolate channels.
• Band-stop (notch): rejects a narrow band. Used to remove power-line interference in biomedical signals.

Filtering in the frequency domain

Because $Y(\omega )=X(\omega )H(\omega )$, filtering can be interpreted as sculpting the spectrum. If noise occupies frequency regions separate from the desired signal, filtering can be highly effective. If they overlap, there is an unavoidable tradeoff: suppressing noise also suppresses signal content in that band.

Real-world example: removing 60 Hz interference

An ECG recording might include a strong 60 Hz component from mains coupling. If the physiological content of interest is mostly below that, a notch filter near 60 Hz can reduce interference. The choice of notch width matters: too narrow may fail to remove drifting interference; too wide can distort legitimate signal components.

Reading Spectra Correctly: Amplitude, Power, and Units

In practice, you may encounter spectra displayed as magnitude, magnitude-squared, or in decibels. Each serves a purpose:

• Magnitude highlights component strength directly.
• Power spectral representations relate more naturally to energy and noise analysis.
• Decibels compress dynamic range and are standard in audio and RF.

Interpreting a spectrum also requires awareness of scaling conventions and whether the analysis is in angular frequency $\omega$ (rad/s) or ordinary frequency $f$ (Hz), connected by $\omega =2\pi f$.

Putting It Together: A Frequency-Domain Workflow

A typical signals-and-systems workflow using Fourier analysis looks like this:

1. Represent the signal in frequency using Fourier series (periodic) or Fourier transform (aperiodic).
2. Characterize the system by its frequency response $H(\omega )$.
3. Predict the output spectrum via multiplication: $Y(\omega )=X(\omega )H(\omega )$.
4. Interpret results in terms of attenuation, bandwidth, and phase effects.
5. Design or select filters that meet performance goals while respecting tradeoffs.

Fourier analysis is not just a mathematical transformation. It is a lens that reveals structure, clarifies system behavior, and makes filtering and spectral reasoning precise. Once you learn to think in both time and frequency, many core problems in signals and systems become easier to model, analyze, and solve.