Quantization

Quantization

After sampling, the continuous amplitude values must be represented by a finite set of discrete levels — this is quantization. The number of bits $n$ determines how many levels are available:

$\text{levels} = 2^n$

Uniform Quantization

For a signal normalized to $[-1, 1]$, the step size is:

$\Delta = \frac{2}{2^n}$

The quantized value maps input $x$ to the midpoint of its quantization bin:

$x_q = \left(\lfloor (x+1) / \Delta \rfloor + 0.5\right) \cdot \Delta - 1$

Values outside $[-1, 1]$ are clamped.

Quantization SNR

The theoretical signal-to-noise ratio for $n$-bit quantization is:

$\text{SNR}_{\text{dB}} = 6.02 n + 1.76 \quad \text{dB}$

Each additional bit adds roughly 6 dB of dynamic range. A 16-bit audio CD achieves $\approx 98$ dB SNR.

Example

With $n = 2$ bits ($4$ levels, $\Delta = 0.5$):

• Input $0.0 \to$ bin index $\lfloor 2.0 / 0.5 \rfloor = 4$, clamped to $3$... Actually bin $\lfloor 1.0 / 0.5 \rfloor = 2$, midpoint $(2.5)\cdot 0.5 - 1 = 0.25$

Your Task

Implement:

• quantize(x, n_bits) — clamp $x$ to $[-1,1]$, quantize to $2^{n\_bits}$ levels, return midpoint of bin
• quantization_snr_db(n_bits) — returns $6.02 n + 1.76$

import math

def quantize(x, n_bits):
    x = max(-1.0, min(1.0, x))
    levels = 2 ** n_bits
    step = 2.0 / levels
    q = math.floor((x + 1.0) / step)
    q = min(q, levels - 1)
    return (q + 0.5) * step - 1.0

def quantization_snr_db(n_bits):
    return 6.02 * n_bits + 1.76

print(quantize(0.0, 2))                    # 0.25
print(round(quantization_snr_db(8), 4))    # 49.92