Quantization Signal to Noise Ratio (Q-SNR)

Quantization:

Quantization in DSP is the process of mapping a large set of input values to a smaller set. It’s essential for converting analog signals into digital form.

Quantization Explanation

For a signal varies from -8 V to +8 V, giving a total quantization range of 16 V. If the number of quantization levels is 4, the step size will be:

\[ v_{\min} = -8, \quad v_{\max} = 8, \quad L = 4 \]

Quantization step size:

\[ \Delta = \frac{v_{\max} - v_{\min}}{L} = \frac{8 - (-8)}{4} = \frac{16}{4} = 4 \]

Partition boundaries (decision levels):

\[ p_0 = -8, \quad p_1 = -8 + 4 = -4, \quad p_2 = 0, \quad p_3 = 4, \quad p_4 = 8 \]

Quantization codebook (reconstruction levels):

\[ c_i = v_{\min} + \left(i + \frac{1}{2}\right) \Delta, \quad i = 0, 1, 2, 3 \]

Calculate each codeword:

• \[ c_0 = -8 + \left(0 + \frac{1}{2}\right) \times 4 = -8 + 2 = -6 \]
• \[ c_1 = -8 + \left(1 + \frac{1}{2}\right) \times 4 = -8 + 6 = -2 \]
• \[ c_2 = -8 + \left(2 + \frac{1}{2}\right) \times 4 = -8 + 10 = 2 \]
• \[ c_3 = -8 + \left(3 + \frac{1}{2}\right) \times 4 = -8 + 14 = 6 \]

Quantization rule:

For an input \( x \), find \( i \) such that:

\[ p_i < x \leq p_{i+1} \]

then output quantized value:

\[ Q(x) = c_i \]

Summary:

Interval	Output quantized value \( c_i \)
\(-8 < x \leq -4\)	\(-6\)
\(-4 < x \leq 0\)	\(-2\)
\(0 < x \leq 4\)	\(2\)
\(4 < x \leq 8\)	\(6\)

Explore the concept of Quantization Signal-to-Noise Ratio (SNR), a critical parameter in Pulse Code Modulation (PCM) that determines the fidelity of quantized signals in digital communication systems.

Core Concepts of Quantization SNR

1. Definition of Quantization SNR

   Quantization SNR measures the ratio of the power of the quantized signal to the power of the quantization noise introduced during the quantization process.

   Psnr = Ps / Pq, Or, Psnr = Ps / (Δ² / 12) 

   Where Psnr is the quantization SNR, Ps is the average power of the signal, Pq is the quantization noise power, and Δ is the quantization step size.

2. Importance in PCM

   In PCM systems, high quantization SNR ensures better signal reconstruction at the receiver, leading to improved quality and performance.

3. Factors Affecting Quantization SNR
   • Step Size: Smaller step sizes lead to higher quantization SNR.
   • Signal Power: Higher average signal power results in better SNR.

Example of Quantization SNR Calculation

Consider a sine signal with an amplitude of 1. So, average power of the sine signal Ps = (1)^2 = 0.5  and a quantization step size of Δ = 0.25. 

The quantization noise power

Pq = (0.25² / 12) = 0.00520833 

 The quantization SNR can be calculated as follows:

Psnr = Ps / Pq  = 0.5 / 0.00520833 =  96 (Approx.) = 19.82 dB

This indicates that the quantization noise is significantly lower than the signal power, resulting in good signal quality.

Simulation of a typical PCM system using quantization for a signal varying from -8 V to 8 V

In the table above, the signal varies from -8 V to +8 V, giving a total quantization range of 16 V. If the number of quantization levels is 4, the step size will be:

Δ = 16 V / 4 = 4 V

The resulting signal-to-quantization-noise ratio (SQNR) is calculated as:

SQNR_linear = 4 / (((16 / inputSignalAmplitude)²) / 12) = 48

SQNR_dB = 10 · log₁₀(48) ≈ 16.80 dB

and so on.

Quantization Levels and Their Impact

The number of quantization levels directly influences the quantization SNR:

• Increasing quantization levels improves the approximation of the original signal, enhancing SNR.
• However, higher levels also require more bits for representation, leading to potential trade-offs in bandwidth.

Conclusion

Understanding Quantization SNR is essential for designing efficient digital communication systems. By optimizing quantization levels and step sizes, engineers can significantly enhance signal quality.

Further Reading

[1] Understanding Quantization in PCM