Signal Spectrum

Beyond the time domain

Time-domain analysis shows how a signal's amplitude changes over time. In telecommunications, this analysis is complemented by the frequency domain. The analysis reveals which frequency components, represented by simple sine waves, make up the signal. Its graphical representation is the signal's spectrum.

Jean-Baptiste Fourier's work shows how a complex periodic signal can be represented as a sum of simple sine waves. A musical chord provides an analogy, since its sound combines components associated with the notes played together.

Spectrum of a simple signal

Consider a signal that is the sum of two sine waves with different frequencies and amplitudes. In the time domain it has a complex waveform, while the frequency-domain representation shows its components separately.

The spectrum shows two distinct lines, one at frequency $f_1$ with amplitude $A_1$, and another at frequency $f_2$ with amplitude $A_2$. The horizontal axis represents frequency, and the vertical axis represents the amplitude of each component. This representation makes analysis easier than the complex time-domain waveform.

Spectrum of a square wave: infinite harmonics

A square wave is a useful digital-signal example with a complex frequency-domain structure. A perfect square wave symmetric about zero with a 50% duty cycle is composed of a fundamental frequency ($f_0$) and an infinite series of odd harmonics ($3f_0, 5f_0, 7f_0, ...$), with amplitudes that decrease in inverse proportion to the harmonic number.

Practical implications

• Infinite Bandwidth requirement: To transmit a perfect square wave without distortion, a communication channel would need infinite bandwidth because the wave contains all harmonics.
• Signal distortion: Channels have limited bandwidth. When a square wave passes through one, its higher harmonics are attenuated or removed. The signal edges become less sharp, which distorts the waveform.

Spectrum in telecommunications

Spectrum analysis is used in the design of telecommunication systems. It allows engineers to:

• Determine bandwidth requirements: The spectrum indicates how much bandwidth a signal needs. A 1 kHz square wave requires more bandwidth than a 1 kHz sine wave because of its harmonics.
• Design channels and filters: Knowledge of the spectrum supports channels and filters that preserve essential frequency components and attenuate unwanted noise and interference.
• Sharing a medium through multiplexing: Technologies like Frequency Division Multiplexing (FDM) shift the spectra of different signals into separate frequency ranges within one channel. This permits simultaneous transmission with separated bands.