Sampling

Sampling as time discretization

Sampling is the first step in converting an analog signal into a digital one. It records the amplitude of a signal that is continuous in time at regularly spaced instants, producing a sequence of samples that is discrete in time.

In telecommunications, the sampled quantity is the amplitude of an analog waveform, such as speech. Each recorded value is a sample; the sample sequence retains amplitude values until a later quantization step.

Sampling discretizes time only. The amplitude remains continuous until quantization assigns it a code. The theorem below assumes uniform sampling, a band-limited signal, and ideal reconstruction.

The Nyquist-Shannon sampling theorem

A band-limited analog signal can be reconstructed from uniformly spaced samples when its highest frequency is strictly below half the sampling frequency. Sampling at or below that limit can cause spectral overlap and aliasing. This condition is expressed by the Nyquist-Shannon sampling theorem.

$f_s > 2 \cdot f_{\text{max}}$

• Sampling Frequency ($f_s$): This tells us how many samples are taken per second.
• Maximum Signal Frequency ($f_{\text{max}}$): This is the highest frequency contained within the signal we want to digitize.

The value $f_s/2$ is the Nyquist frequency. Two samples per cycle mark the boundary of the ideal condition, so their positions alone do not guarantee recovery of a sinusoid. Practical systems sample above this boundary and filter the input before sampling. A reconstruction filter performs the corresponding interpolation after conversion.

Practical example: sampling human voice for telephony

ITU-T G.711 specifies the sampling rate and code length used for narrow-band PCM voice. The following calculation explains why 8 kHz is used for a telephone band ending near 4 kHz.

1. Determine $f_{\text{max}}$: The narrow-band telephone band is specified as 300 Hz to 3400 Hz in ITU-T recommendations. We therefore set $f_{\text{max}} = 3400 \text{ Hz}$. Filter design commonly reserves a band up to approximately 4 kHz.
2. Calculate the lower sampling bound: The theorem gives $f_s > 2 \cdot 3400 \text{ Hz} = 6800 \text{ Hz}$. A practical design therefore chooses a rate above 6800 Hz.
3. Use the PCM rate: ITU-T G.711 specifies 8000 samples/s (8 kHz) for voice PCM. The rate leaves room for the transition of a practical input filter between the telephone band and the first spectral image.

The corresponding sampling period ($T_s$) is the time between consecutive samples:

The 125 microsecond interval is the sample period for the 8 kHz PCM stream and the 125 μs frame period used in basic SDH/SONET framing. With 8-bit G.711 samples, the same stream carries 64 kbit/s per voice channel.

Aliasing from undersampling

If the signal contains frequencies at or above the available limit ($f_s \le 2 \cdot f_{\text{max}}$ for the stated band), the samples contain an irreversible ambiguity known as aliasing.

A sinusoid at frequency $f$ and one at $|f-kf_s|$ produce the same sampled frequency after folding into the interval from 0 to $f_s/2$, for an appropriate integer $k$. The second component is the alias of the first.

The anti-aliasing filter

To limit aliasing, an analog input filter is placed before the sampler. It attenuates components outside the desired band that could alias into the retained band, and it has a passband, transition band, and stopband rather than an ideal vertical cutoff. The filter is a low-pass (anti-aliasing) filter before the sampling circuit. For 8 kHz telephony PCM, the filter preserves the telephone band and attenuates frequencies in the transition band between the telephone band and approximately 4 kHz.

The sampled output: the PAM signal

The direct result of the sampling process is a new type of signal called a Pulse Amplitude Modulation (PAM) signal. In a PAM signal, the amplitude of a train of regularly spaced pulses is varied in proportion to the value of the analog signal at the sampling instants. These amplitudes have not yet been replaced by binary codes.

Ideal impulse PAM is discrete in time, while a sample-and-hold output remains at each sampled amplitude for the hold interval. In both cases the amplitude is still analog and can take any value in the input range. The flat-top waveform is generated by a circuit called a Sample-and-Hold (S&H). Quantization follows sampling in the A/D conversion chain.