Spectrum (signal)

The spectrum of a signal $x(t)$ is its Fourier transform $X(f)$. It’s a complex-valued function of frequency that captures how the signal is distributed across frequencies.

Two views of the spectrum:

• Amplitude spectrum $|X(f)|$: how much of frequency $f$ is in $x(t)$.
• Phase spectrum $\angle X(f)$: where the cosine-sine combination sits in time at that frequency.

For real $x(t)$, the amplitude spectrum is even in $f$ and the phase spectrum is odd (conjugate symmetry). The spectrum is generally complex even when $x(t)$ is real.

Spectral classification

By where the energy in $|X(f)|$ is concentrated, signals are classified:

• Lowpass: most energy near $f=0$. Smooth in time, slow oscillations.
• Highpass: most energy at high frequencies. Spikes and rapid transitions in time.
• Bandpass: energy concentrated in a narrow band around some center frequency $f_c$ (away from DC).
• Narrowband: small bandwidth (sharp spectrum). Long time-domain extent.
• Wideband: broad bandwidth. Short time-domain extent. See Time-bandwidth tradeoff.

This vocabulary is used constantly when designing filters in Chapter 9.

Line spectrum vs continuous spectrum

A periodic signal has a line spectrum — discrete impulses at the harmonic frequencies $k{f}_0$, with strengths equal to the Fourier series coefficients. An aperiodic signal has a continuous spectrum — a smooth function $X(f)$.

Both cases are unified within the CTFT framework: the transform of a periodic signal is a train of impulses at the harmonic locations, $X(f)={\sum }_k{c}_x[k]\delta (f-k{f}_0)$.

Units

If $x(t)$ has units (volts, amperes, pascals…), $X(f)$ has those units times time. For voltage in $\text{V}$ and time in seconds, $X(f)$ has units $\text{V}\cdot \text{s}=\text{V/Hz}$. This is why $X(f)$ is sometimes called a spectral density — it’s “how much voltage per Hz of bandwidth.”

Physical meaning at a point

The value $X(f_0)$ at a single frequency is the complex amplitude of the (infinitesimal) sinusoidal component at $f_0$. Writing $X(f_0)=|X(f_0)|e^{j\varphi }$, the contribution to $x(t)$ from a small frequency band $df$ around $f_0$ is $2|X(f_0)|\cos (2\pi f_{0}t+\varphi )df$ (the factor of 2 comes from the conjugate-pair of $\pm f_0$ contributions).

So the spectrum encodes, at every frequency, both how strong a component there is and its phase. The reconstruction integral

$x(t)={\int }_{-\infty }^{\infty }X(f)e^{j2\pi ft}df$

is a continuous superposition of these sinusoidal pieces, weighted by $X(f)$.

In LTI systems

For an LTI system, the output spectrum is the input spectrum multiplied by the frequency response:

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

Drawing $X(f)$, $H(f)$, and their product $Y(f)$ is the standard way to visualize what a system does to a signal. Filters with sharp passbands and stopbands shape the output spectrum cleanly; broad-tailed filters produce broader output spectra.