DDPM noise schedule: alpha-bar and signal-to-noise decay

What's actually being asked

Work out α_t, the cumulative products ᾱ_t, and the signal-to-noise term √ᾱ_t / √(1−ᾱ_t) for a 3-step DDPM schedule, then explain the trend across steps.

Answer

Note on the numbers: the source text for this question lost the actual β_1, β_2, β_3 values during copy-paste — they never made it into what's given here. So below is the method worked fully and correctly on an illustrative 3-step schedule (β_1 = 0.1, β_2 = 0.2, β_3 = 0.3) — swap in the real numbers and redo these same three steps if you get them.

Step 1 — α_t

α_t = 1 − β_t

t	β_t	α_t = 1 − β_t
1	0.1	0.9
2	0.2	0.8
3	0.3	0.7

Step 2 — cumulative product ᾱ_t

ᾱ_t = Π_{s=1}^{t} α_s — multiply every α up to and including step t.

• ᾱ_1 = α_1 = 0.9
• ᾱ_2 = α_1·α_2 = 0.9 × 0.8 = 0.72
• ᾱ_3 = α_1·α_2·α_3 = 0.72 × 0.7 = 0.504

ᾱ_t only ever shrinks as t grows — every α_s < 1, so multiplying in one more always pulls it down further.

Step 3 — √ᾱ_t and √(1−ᾱ_t)

From the closed form x_t = √ᾱ_t · x_0 + √(1−ᾱ_t) · ε, √ᾱ_t is the weight on the original signal x_0 and √(1−ᾱ_t) is the weight on the injected noise ε.

t	ᾱ_t	√ᾱ_t	1−ᾱ_t	√(1−ᾱ_t)	√ᾱ_t / √(1−ᾱ_t)
1	0.900	0.9487	0.100	0.3162	3.000
2	0.720	0.8485	0.280	0.5292	1.603
3	0.504	0.7099	0.496	0.7043	1.008

(Check at t=1: 0.9/0.1 = 9, and √9 = 3 exactly, so no rounding slop there.)

Analysis — signal vs noise across steps

The ratio √ᾱ_t / √(1−ᾱ_t) is effectively the signal-to-noise ratio (SNR) of x_t. In this toy schedule it falls from 3.0 at t=1 to about 1.0 by t=3 — by the third step the signal term and the noise term are contributing almost equally to x_t. Matlab shabdo mein, har step ke saath original image ka contribution kam hota jaa raha hai aur random noise ka contribution badhta jaa raha hai.

Two things worth carrying forward:

• ᾱ_t is monotonically decreasing in t, always — it's a product of numbers each less than 1, so it can never go back up.
• A real DDPM schedule (e.g. linear β from 1e-4 to 0.02 over T = 1000 steps) decays far more gently per step than this fast 3-step toy example — SNR only approaches 0 near the final timestep, by which point x_T is, by design, indistinguishable from pure Gaussian noise N(0, I).

Exam tip

If the paper gives actual β values, repeat the exact three steps in order — α_t → ᾱ_t (cumulative product, not sum) → the two square roots. The most common mistake is summing β_t instead of multiplying (1−β_t) terms; always convert to α_t first, then take the running product.