Quantization, SNR, Bits, and Sample Rate

1. Power and SNR

We start from a discrete-time signal $x[n]$ (after sampling, possibly before or after quantization).

For SNR, we think in terms of power:

• Average signal power (discrete-time):

  $$P_{\text{signal}}=\mathbb{E}[x[n{]}^2]\approx \frac{1}{N}\sum _{n=0}^{N-1}x[n{]}^2$$

  Intuition: if $x[n]$ behaves like a voltage, then power is proportional to $x[n{]}^2$. Any constant factors (like $1/R$) cancel when we form ratios.

Quantization introduces an error:

$$e[n]=x[n]-x_q[n]$$

where $x_q[n]$ is the quantized value.

• Noise power:

  $$P_{\text{noise}}=\mathbb{E}[e[n{]}^2]$$

Then the signal-to-noise ratio (SNR) is

$$\text{SNR}=\frac{P_{\text{signal}}}{P_{\text{noise}}}$$

and in decibels (for power ratios):

$${\text{SNR}}_{\text{dB}}=10{\log }_{10}\left(\frac{P_{\text{signal}}}{P_{\text{noise}}}\right)$$

Some useful rules of thumb:

• $10$ dB $\Rightarrow$ power ratio of $10$
• $20$ dB $\Rightarrow$ power ratio of $100$
• $3$ dB $\Rightarrow$ power ratio of about $2$
• $6$ dB $\Rightarrow$ power ratio of about $4$

2. How bit depth controls SNR

Assume a uniform mid-tread quantizer:

• Bit depth: $B$ bits $\Rightarrow L=2^B$ quantization levels.
• Range: [-X_\max, X_\max], symmetric.
• Step size: \Delta = \frac{2 X_\max}{L} = \frac{2 X_\max}{2^B}

Quantization error model

Textbook assumptions:

• The input is “busy enough” and not strongly correlated with quantization levels.
• Quantization error $e[n]$ is modeled as uniform on $[-\Delta /2,\Delta /2]$.

Then:

• Mean: $\mathbb{E}[e[n]]=0$
• Variance (and thus noise power): $${\sigma }_e^2=\mathbb{E}[e[n{]}^2]=\frac{{\Delta }^2}{12}$$

So:

$$P_{\text{noise}}=\frac{{\Delta }^2}{12}$$

Signal power for a full-scale sine

Take a sinusoid:

$$x[n]=A\sin (\omega n)$$

The average power is

$$P_{\text{signal}}=\mathbb{E}[x[n{]}^2]=\frac{A^2}{2}$$

If the quantizer is used at full scale:

• Set A = X_\max
• Then P_\text{signal} = \dfrac{X_\max^2}{2}

Putting signal and noise together

We already have

\Delta = \frac{2 X_\max}{2^B} \Rightarrow \Delta^2 = \frac{4 X_\max^2}{2^{2B}}

So

P_\text{noise} = \frac{\Delta^2}{12} = \frac{4 X_\max^2}{12 \cdot 2^{2B}} = \frac{X_\max^2}{3 \cdot 2^{2B}}

Thus

\text{SNR} = \frac{P_\text{signal}}{P_\text{noise}} = \frac{X_\max^2 / 2}{X_\max^2 / (3 \cdot 2^{2B})} = \frac{1/2}{1/(3 \cdot 2^{2B})} = \frac{3 \cdot 2^{2B}}{2}

Now convert to dB:

$${\text{SNR}}_{\text{dB}}=10{\log }_{10}\left(\frac{3\cdot 2^{2B}}{2}\right)=10\left[{\log }_{10}\left(\frac{3}{2}\right)+{\log }_{10}\left(2^{2B}\right)\right]$$

But

$${\log }_{10}\left(2^{2B}\right)=2B{\log }_{10}(2)\approx 2B\cdot 0.3010=0.602B$$

and ${\log }_{10}\left(\frac{3}{2}\right)\approx 0.1761$

So

$${\text{SNR}}_{\text{dB}}\approx 10(0.1761+0.602B)=1.76+6.02B$$

Key conclusions:

• Each extra bit increases SNR by about $6.02$ dB (for a full-scale sine in an ideal uniform quantizer).
• Example:
  • $B=16$ bits: $\text{SNR}\approx 6.02\cdot 16+1.76\approx 98$ dB
  • $B=8$ bits: $\text{SNR}\approx 6.02\cdot 8+1.76\approx 50$ dB

The uniform error model and full-scale sine are idealizations, but this formula is a very common rule of thumb.

3. How sample rate affects the representation

Bit depth and sample rate affect different aspects of the signal representation:

• Bit depth $B$

  • Controls how finely amplitude is quantized.
  • Determines quantization step size $\Delta$.
  • Directly affects SNR / dynamic range via the $6.02B+1.76$ dB relation.

• Sample rate $f_s$

  • Controls how often we sample in time.
  • Determines the maximum representable (non-aliased) frequency: roughly the Nyquist frequency $f_s/2$.
  • Controls time resolution (samples per second).
  • Does not directly change the per-sample quantization noise power (for fixed $\Delta$).

Formally:

• Sampling period: $T_s=1/f_s$
• If the analog input is band-limited to f_\max and f_s > 2 f_\max, the signal is ideally reconstructible (Nyquist–Shannon sampling theorem).
• Increasing $f_s$ increases the bandwidth of the discrete-time representation (up to $f_s/2$).

In practice:

• Unlike speech, music requires the use of the full frequency spectrum. That means sampling the signal at a higher rate, i.e., the standard sampling rates of music recordings are 44.1kHz or 48 kHz vs. 16 kHz for speech.
• This also means longer sequences.

Noise spectral density intuition

With a fixed bit depth $B$ and quantization step $\Delta$:

• Total quantization noise power over the full Nyquist band $[0,f_s/2]$ is still $P_{\text{noise}}=\frac{{\Delta }^2}{12}$

• If you increase $f_s$, the same total noise power is spread over a wider frequency range

• Therefore, the noise power per Hz (noise spectral density) decreases.

• If you later low-pass to a fixed audio band (e.g. $0$–$20$ kHz), oversampling plus filtering can improve in-band SNR.

This is the idea behind oversampling converters.

Summary: bit depth vs sample rate

• Bit depth $B$:

  • Controls dynamic range and quantization SNR.
  • More bits $\Rightarrow$ smaller $\Delta$ $\Rightarrow$ lower $P_{\text{noise}}$ $\Rightarrow$ higher SNR.

• Sample rate $f_s$:

  • Controls bandwidth and time resolution.
  • Higher $f_s$ $\Rightarrow$ can represent higher frequencies (up to $f_s/2$).
  • For fixed $B$ and $\Delta$, it does not change the basic $6.02B+1.76$ dB relationship, but it changes how that noise is distributed over frequency.