Improvement in Signal-to-Quantization Noise Ratio (SQNR) in PCM System

Introduction:
In Pulse Code Modulation (PCM) system, analog signals are sampled and quantized to be represented by digital code words. The code word length determines the number of quantization levels and ultimately affects the Signal-to-Quantization Noise Ratio (SQNR).

Signal-to-Quantization Noise Ratio (SQNR):
SQNR is a measure of the quality of a PCM system. It represents the ratio of the power of the signal to the power of the quantization noise. A higher SQNR indicates a better quality system with less quantization noise.

Effect of Code Word Length on SQNR:
The code word length directly affects the number of quantization levels that can be represented. As the code word length increases, the number of quantization levels also increases. This leads to a finer representation of the analog signal and reduces the quantization noise.

Comparison of 6-bit and 8-bit Code Word Lengths:
When the code word length is increased from 6 bits to 8 bits, the number of quantization levels is increased from 2^6 to 2^8. This means that the number of quantization levels is increased by a factor of 2^(8-6) = 2^2 = 4.

Calculation of Improvement in SQNR:
The SQNR improvement can be calculated by taking the ratio of the number of quantization levels for the 8-bit code word length to the number of quantization levels for the 6-bit code word length.

Number of quantization levels for 6-bit code word length = 2^6
Number of quantization levels for 8-bit code word length = 2^8

SQNR improvement = (Number of quantization levels for 8-bit code word length) / (Number of quantization levels for 6-bit code word length)
= (2^8) / (2^6)
= 2^(8-6)
= 2^2
= 4

Therefore, the SQNR improves by a factor of 4, which is equivalent to an improvement of 16.

Conclusion:
When the code word length is increased from 6 bits to 8 bits in a PCM system, the SQNR improves by a factor of 16. This improvement is due to the increased number of quantization levels, resulting in a finer representation of the analog signal and reduced quantization noise.

When word length is 6