White noise

Ideal discrete-time white noise is an uncorrelated sequence: the autocorrelation is a scaled impulse,

R_xx[k] = E{ (x[n]−μ)(x[n+k]−μ) } = σ_x² · δ[k] (28)

so the power spectral density is flat. SigGer draws i.i.d. samples from a chosen distribution, then applies a scale and a mean. All three noise windows share the sine-wave layout with renamed controls:

Faceplate control	Role	Range
Scale factor a	Multiplies the zero-mean random sequence	1.0 – 100.0 (linear)
Variance σ²	Variance of the unscaled sequence	0 – 1000 (linear)
Sampling frequency	Stored with the file	1–192 kHz
Sequence mean μ	Output mean (not multiplied by a)	−100 – +100, unitless

x[n] = a · ξ[n] + μ (29)

ξ[n] is zero-mean with Var(ξ) = σ². Therefore

E{x} = μ , Var(x) = a² σ² (30)

If σ² = 0, then ξ ≡ 0 and x[n] = μ (constant). Floating- vs fixed-point still selects the saved dtype, but the three noise controls do not switch to volt/count ranges.

Each Run draws a new independent realization (NumPy Generator). The oscilloscope preview uses a deterministic stand-in so the trace does not flicker while you turn the dials.

1. Gaussian distribution

ξ[n] ∼ 𝒩(0, σ²), i.e. ξ = σ Z with Z standard normal. The density of the unscaled variable is

f_ξ(ξ) = (1 / √(2π σ²)) exp( − ξ² / (2σ²) ) (31)

x[n] = a · √(σ²) · Z[n] + μ , Z[n] ∼ 𝒩(0, 1) i.i.d. (32)

Gaussian white noise is the classical model for thermal noise and for many least-squares estimators (AWGN channel). See Wikipedia: White noise and Gaussian noise.

2. Uniform distribution

A zero-mean uniform random variable on [−w, w] has variance w² / 3. Matching Var(ξ) = σ² gives

w = √(3 σ²) , ξ ∼ Uniform(−w, w) (33)

f_ξ(ξ) = 1 / (2w) for |ξ| ≤ w , 0 otherwise (34)

Then (29) applies. Uniform noise is a common model for quantization error when the step is small compared with the signal (Widrow quantization model).

3. Laplacian distribution

The Laplace (double-exponential) law with scale b > 0 is

f_ξ(ξ) = (1 / (2b)) exp( − |ξ| / b ) (35)

Its variance is 2b², so

b = √(σ² / 2) , ξ ∼ Laplace(0, b) (36)

Laplacian samples have heavier tails than a Gaussian of the same variance (more large spikes). They appear in speech residual models and in some robust estimators. Implementation: numpy.random.Generator.laplace (NumPy documentation).

4. Comparison

Family	Support of ξ	Tails	Generator
Gaussian	(−∞, ∞)	exp(−ξ²)	`standard_normal` × √σ²
Uniform	[−√(3σ²), +√(3σ²)]	None (hard bounds)	`uniform(−w, w)`
Laplacian	(−∞, ∞)	exp(−\|ξ\|)	`laplace(0, √(σ²/2))`

Kurtosis (excess) is 0 for the Gaussian, −6/5 for the uniform, and 3 for the Laplace — a useful check if you inspect a saved record in SigScope or a notebook.