Signals and systems in communications
A rectangular pulse leaves 5 % of its energy outside twice its null-to-null width. A raised-cosine pulse of the same duration leaves 100 times less: the pulse shape decides how much it invades its neighbors.
01Thinking in frequency
In communications almost everything is reasoned in the frequency domain: what part of the spectrum a signal occupies, what a channel lets through, where an interference falls. The tools are those of Signals and Systems —the series and the Fourier transform, their properties, the spectra and Parseval's relation—, applied to concrete questions: how much bandwidth a signal needs and what a channel does to it.
One principle runs through the whole subject: the duration and the bandwidth of a signal are inversely related. A pulse of duration \( T \) occupies a bandwidth on the order of \( 1/T \). Transmitting faster requires more spectrum.
02How wide is a bandwidth?
Except in ideal cases, the spectrum of a real signal does not end at any frequency: it decays, but never reaches zero. That is why there are several definitions of bandwidth, and one has to say which one is being used:
| Definition | What it measures |
|---|---|
| −3 dB | Up to where the power spectral density falls to half |
| Null-to-null | Up to the first zero of the spectrum; for a rectangular pulse, \( 1/T \) |
| Fractional power (99 %) | The band that contains 99 % of the energy: the one used by regulatory bodies |
| Noise equivalent | That of an ideal rectangular filter that passes the same noise power |
Three pulses of the same duration \( T \): rectangular, triangular and raised cosine. The plot shows their energy spectral density in dB, normalized; the shaded strip is the chosen band. Below, the fraction of the energy of each pulse that falls inside it.
03Distortionless transmission
A channel does not distort a signal if the output is an attenuated and delayed copy of the input, \( y(t) = K\,x(t - t_0) \). In frequency, this requires two things over the whole band of the signal:
The ear is not very sensitive to phase distortion, but digital data is: if the components of a pulse arrive with different delays, the pulse spreads and invades its neighbors (intersymbol interference). That is why modems and digital receivers have equalizers, filters that compensate for the magnitude and phase of the channel.
04Nonlinear systems
A saturated amplifier or a mixer is not linear, and generates frequencies that were not at the input. With two tones \( f_1 \) and \( f_2 \) at the input, harmonics and intermodulation products appear at \( m f_1 \pm n f_2 \). The third-order ones, \( 2f_1 - f_2 \) and \( 2f_2 - f_1 \), are the most harmful: they fall right next to the signals, inside the same channel, and no filter separates them. The linearity of the amplifiers of a transmitter or of the front end of a receiver is specified with the third-order intercept point.
05In the lab
With a pulse generator and the oscilloscope FFT, view the spectrum of rectangular pulses of 1 µs and 0.1 µs, and measure their first zeros.
Pass a square wave through an all-pass filter, which does not change the magnitude, and observe how its shape changes.
Inject two close tones into an amplifier and use a spectrum analyzer to measure the third-order products as the level is increased.
06Common mistakes
- Giving a bandwidth without saying which one. At −3 dB and at 99 % they can differ by several times.
- Believing that a flat-magnitude filter does not distort. If its phase is not linear, it deforms the pulses.
- Using rectangular pulses in a band-limited channel. Their spectrum decays slowly and spills over into the neighboring channels.
- Forgetting intermodulation when raising the power. It grows three times faster than the signal, in decibels.
07Self-assessment
What null-to-null bandwidth does a 1 µs rectangular pulse have?
1 MHz.
What fraction of the energy of a rectangular pulse falls inside the first null?
About 90 %.
Where do the third-order intermodulation products of 100 and 101 MHz fall?
At 99 and 102 MHz.
What phase condition does distortionless transmission require?
Phase linear with frequency: constant group delay.
08Further reading
- B. P. Lathi and Zhi Ding. Modern Digital and Analog Communication Systems. 5th ed., Oxford University Press, 2019. Signal and system analysis oriented to communications, with bandwidths and distortion.
- Leon W. Couch. Digital and Analog Communication Systems. 8th ed., Pearson, 2013. The different definitions of bandwidth compared with examples.
- Simon Haykin and Michael Moher. Communication Systems. 5th ed., Wiley, 2009. Distortionless transmission and nonlinear systems.