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How To Understand Quantization And Bit Rate

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How To Understand Quantization And Bit Rate

Question

How should the quantization slide be understood, especially the relations M = 2^n and R = n/T = n f_s?

Short Answer

Quantization maps each sampled analog amplitude to one of a finite number of discrete amplitude levels.

If each sample is represented by n bits, then the number of possible quantization levels is:

[ M = 2^n ]

If the ADC produces f_s samples per second, and each sample takes n bits, then the bit rate is:

[ R = n f_s ]

Equivalently, if T_s = 1/f_s is the sampling period:

[ R = \frac{n}{T_s} ]

How To Read The Figure

  • The left waveform is a flat-top sampled analog signal: each sampled value is held briefly.
  • The horizontal voltage lines are quantization levels.
  • Each sampled amplitude is rounded or assigned to the nearest allowed level.
  • The binary labels on the right are the digital codes assigned to those levels.
  • The red dashed vertical lines indicate sample instants, separated by the sampling period T_s.

Meaning Of The Symbols

  • M: number of quantization levels
  • n: number of bits per sample
  • f_s: sampling frequency, in samples per second
  • T_s: sampling period, so T_s = 1/f_s
  • R: bit rate, in bits per second
  • T_b: bit period, so T_b = 1/R

Important Distinction

The T in the slide formula R = n/T is best read as the sampling period T_s, not the bit period.

One sample arrives every T_s seconds, and that one sample must be encoded using n bits. Therefore:

[ R = \frac{\text{bits per sample}}{\text{seconds per sample}} = \frac{n}{T_s} ]

The bit period is instead:

[ T_b = \frac{1}{R} = \frac{T_s}{n} ]

Answer To The Slide Question

If the bitrate goes up while n stays the same, the bit period gets shorter.

Claim status: EXTRACTED for the formula relationships shown on the slide; INFERRED for the notation clarification between T_s and T_b.

Example

For a 3-bit quantizer:

[ M = 2^3 = 8 ]

If the sampling frequency is 1000 samples/s, then:

[ R = 3 \cdot 1000 = 3000 \text{ bits/s} ]

So each bit lasts:

[ T_b = \frac{1}{3000} \text{ s} ]

If the bitrate doubles and n stays the same, T_b halves.

Graph View

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