# OpenStax-CNX

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# Transmission Bandwidth

Module by: Don Johnson. E-mail the author

Summary: To determine the transmission bandwidth of certain signal sets, consider the baseband version.

What is the transmission bandwidth of these signal sets? We need only consider the baseband version as the second is an amplitude-modulated version of the first. The bandwidth is determined by the bit sequence. If the bit sequence is constant — always 0 or always 1 — the transmitted signal is a constant, which has zero bandwidth. The worst-case—bandwidth consuming—bit sequence is the alternating one shown in Figure 1. In this case, the transmitted signal is a square wave having a period of 2T 2 T .

From our work in Fourier series, we know that this signal's spectrum contains odd-harmonics of the fundamental, which here equals 12T 1 2 T . Thus, strictly speaking, the signal's bandwidth is infinite. In practical terms, we use the 90%-power bandwidth to assess the effective range of frequencies consumed by the signal. The first and third harmonics contain that fraction of the total power, meaning that the effective bandwidth of our baseband signal is 32T 3 2 T or, expressing this quantity in terms of the datarate, 3R2 3 R 2 . Thus, a digital communications signal requires more bandwidth than the datarate: a 1 Mbps baseband system requires a bandwidth of at least 1.5 MHz. Listen carefully when someone describes the transmission bandwidth of digital communication systems: Did they say "megabits" or "megahertz"?

## Exercise 1

Show that indeed the first and third harmonics contain 90% of the transmitted power. If the receiver uses a front-end filter of bandwidth 32T 3 2 T , what is the total harmonic distortion of the received signal?

### Solution

The harmonic distortion is 10%.

## Exercise 2

What is the 90% transmission bandwidth of the modulated signal set?

### Solution

Twice the baseband bandwidth because both positive and negative frequencies are shifted to the carrier by the modulation: 3R 3 R .

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