## Sampling

The cornerstone of an explanation of how PCM works is the Nyquist sampling theorem (Ref. 1), which states:

If a band-limited signal is sampled at regular intervals of time and at a rate equal to or higher than twice the highest significant signal frequency, then the sample contains all the information of the original signal. The original signal may then be reconstructed by use of a low-pass filter.

Consider some examples of the Nyquist sampling theorem from which we derive the sampling rate.

1. The nominal 4-kHz voice channel: sampling rate is 8000 times per second (i.e., 2 x 4000).

2. A 15-kHz program channel:1 Sampling rate is 30,000 times per second (i.e., 2 x 15,000).

'A program channel is a communication channel that carries radio broadcast material such as music and commentary. It is a facility offered by the PSTN to radio and television broadcasters.

3. An analog radar product channel 56 kHz wide: Sampling rate is 112,000 times per second (i.e., 2 x 56,000).

Our interest here, of course, is the nominal 4-kHz voice channel sampled 8000 times per second. By simple division, a sample is taken every 125 ^sec, or 1 sec/8000.

6.2.1.1 The Pulse Amplitude Modulation (PAM) Wave. With several exceptions, practical PCM systems involve time-division multiplexing. Sampling in these cases does not just involve one voice channel but several. In practice, one system (T1) samples 24 voice channels in sequence and another (E1) samples 30 voice channels. The result of the multiple sampling is a pulse amplitude modulation (PAM) wave. A simplified PAM wave is shown in Figure 6.1. In this case, it is a single sinusoid (sine wave). A simplified diagram of the processing involved to derive a multiplex PAM wave is shown in Figure 6.2. For simplicity, only three voice channels are sampled sequentially.

The sampling is done by gating, which is just what the term means: A "gate" is opened for a very short period of time, just enough time to obtain a voltage sample. With the North American DS1 (T1) system, 24 voice channels are sampled sequentially and are interleaved to form a PAM multiplexed wave. The gate is open about 5.2 ^sec (125/24) for each voice channel. The full sequence of 24 channels is sampled successively from channel 1 through channel 24 in a 125-^sec period. We call this 125-^sec period a frame, and inside the frame all 24 channels are successively sampled just once.

Another system widely used outside of the United States and Canada is E1, which is a 30-voice channel system plus an additional two service channels, for a total of 32 channels. By definition, this system must sample 8000 times per second because it is also optimized for voice operation; thus its frame period is 125 ^sec. To accommodate the 32 channels, the gate is open 125/32 ^sec, or 3.906 ^sec.