Delta modulation and DPCM

Watch a one-bit staircase chase a sine. Shrink the step or the sampling rate and it falls behind (slope overload); grow the step and it hunts on the flat tops (granular noise). Then give each sample n bits and compare DPCM's SNR with plain PCM's.

Message (green), staircase the receiver rebuilds (blue), samples that lost track (red)

Error m − x̂ at each sample; shaded: the ±0.20 band it stays inside while tracking

Transmitted bits (1 = step up, 0 = step down)

111110000000001111111100000000111

A \text{ (message amplitude)}

1.00

f_s / f_m \text{ (samples per message period)}

16×

\Delta \text{ (step size)}

0.200

Staircase speed

\Delta\cdot f_s

3.200

Steepest slope

2\pi A f_m

6.283

Slope overload

Smallest safe Δ = 0.393 · largest safe A = 0.509

Bits per second1 × f_s = 16 f_m

Samples that lost track (after the first period)16

Measured SNR6.5 dB

Noise: measured vs Δ²/31.11e-1 vs 1.33e-2

What to notice

Slope overload. The staircase can climb at most $\Delta f_s = 3.200$ per period, but the sine climbs at up to $2\pi A f_m = 6.283$ . On the steep parts the bits run 111… or 000… and the staircase falls behind. Fix it with a bigger $\Delta$ (at least 0.393), a higher $f_s$ , or a smaller $A$ .
Granular noise is the ±Δ hunting on the flat tops, where the bits alternate 1010…. Its power is about $\Delta^2/3$ (error spread evenly over $\pm\Delta$ ) — measured here 1.11e-1 against 1.33e-2, though overload adds to the measured value. A big $\Delta$ cures overload and worsens this: that trade-off is why adaptive DM exists.
One bit per sample, so the bit rate equals $f_s$ . To match PCM's quality DM needs a much higher $f_s$ — here 16× the message frequency against PCM's minimum of 2×.

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