## Example 7.2

#### A telephone system uses filters to band limit the signal from each user to a maximum of 4 kHz. The maximum signal generated by each user is 1 V, and the minimum signal that must be supported is 72 dB below this value. What is the minimum bit rate that must be supported by the telephone system for each user on the network?
If companding is used within the A/D conversion process to achieve equivalent dynamic range, but with only eight bits, what will be the bandwidth required of a binary baseband data link for a single digitized voice channel? If 32 channels are multiplexed into an E1 frame, what is the number of bits per frame and the data rate on the E1 link?

### Solution

#### A 4 kHz maximum input signal requires a minimum sampling rate of 8000 samples per second. In order to maintain a 72 dB dynamic range, the A/D converters must use 72/6 = 12 bits, giving a total bit rate per telephone channel of 8000 x 12 = 96000 bps.
With the use of companding, the bit rate per channel reduces to 8 x 8000 = 64 kbps. For a baseband channel, the maximum bandwidth efficiency is 2 bits/second/Hz for binary signalling, hence 32 kHz of bandwidth is needed per voice band.
For 32 channels per frame, the frame length of the E1 line is 32 x 8 = 256 bits. The frame rate must equal the sampling rate of any one of the channels, that is, 8000 frames per second, hence the E1 data rate = 256 x 8000 = 2048 Mbps.
A description of the E1 and T1 channel types is given in Section 1.3.)