CM · Digital modulation

Sampling and PCM

Sample a continuous signal at regular intervals and only a handful of dots remain. Yet if the sampling rate exceeds twice the highest frequency, the original can be reconstructed exactly. Learn why sampling too slowly disguises a high frequency as a fake low one.

Do the samples bring the signal back?

Sweep the sampling rate fs. Above twice the signal frequency the reconstruction (gold) overlaps the original; below it, the samples fold down into a fake low frequency (red).

Sampling rate fsfs = 1.14 × f0

Sweep the rate. fs = 2 f0 is the boundary (Nyquist).

Rate versus the Nyquist bound

fs < 2 f0 → Aliasing

Aliasing

Samples are dots

Sampling reads the continuous signal once every fixed time step and writes down the value. A smooth curve shrinks to a handful of dots. The natural doubt is this: we threw away everything between the dots — how could the original curve ever come back?

Nyquist — faster than twice

The sampling theorem answers it. If the signal’s highest frequency is fmax, sampling at fs > 2 fmax lets the samples reconstruct the original perfectly. That boundary, 2 fmax, is the Nyquist rate. When fs falls short, a component near fmax folds into the band below fs/2 and appears as a slower sinusoid that passes through the very same samples. This disguise is aliasing, and once folded it cannot be undone.

Observefs > 2 fmax

The rate must exceed twice the highest frequency.

Choosef_a = | fs ? f0 |

When undersampled, the apparent frequency folds down.

PCM — quantize and encode

A sample is still a continuous value. PCM rounds that value to the nearest of L levels (quantizing) and writes each level as n = log₂ L bits. Sampling fs times per second, the bit rate is R = n fs. The error from rounding is quantization noise, which shrinks as the number of levels L grows (as you spend more bits). Sampling chops time, quantizing chops value, and together they turn the signal into numbers.

Fill inn = log₂ ?

L levels need log₂ L bits per sample.

On your ownR = ? (bit/s)

The bit rate is bits per sample times the sampling rate.

Back to the first screen

As you raised the rate the dots packed tighter, and the instant fs passed twice the signal frequency the reconstruction lay exactly over the original and turned gold. Below it, a slower sinusoid through the same samples appeared and disguised itself in red. The one thing to hear in sampling is this: discarding everything between the dots loses nothing as long as fs > 2 fmax, and breaking that boundary folds a high frequency into a fake low one for good.

Sampling reads a continuous signal at the rate fs, once per interval. By the sampling theorem, fs > 2 fmax lets the samples reconstruct the original perfectly, and the boundary 2 fmax is the Nyquist rate. If fs is too small, high frequencies fold below fs/2 and aliasing occurs. PCM then quantizes each sample to L = 2ⁿ levels, encodes it as n = log₂ L bits, and the bit rate is R = n fs.

In the next unit

Now the message is a stream of bits. Next is how those bits shake the carrier. Switch the carrier’s amplitude on and off with the bits and you get ASK; flip between two frequencies, FSK; reverse the phase, PSK. We compare the three branches of digital modulation side by side.