Differential Quantization and Delta Modulation

February 7, 2020

In the previous lessons, it was established that SQNR (signal-to-quantization noise ratio) can be increased by:

  1. Increasing the number of bits per sample, or
  2. Increasing the sample rate

A third method of increasing SQNR is changing the modulation technique. In earlier lessons, the mathematics assumed that sampling was performed using a technique known as pulse code modulation (PCM). PCM quantizes the signal waveform 𝗑(𝑡) every sample period, 𝜏, and yields the sample sequence 𝗒(𝑛) ≜ 𝖰[𝗑(𝑛𝜏)].

Although widely used, PCM disregards a feature of the band-limited waveform. Waveforms that are band-limited cannot change quickly at high sample rates. If they did change quickly, they would, by definition, have high-frequency content and would not be band-limited.

Differential Quantization

Because band-limited signals cannot change quickly, the difference between consecutive samples should be relatively small. Therefore, the difference between consecutive samples 𝗑(𝑛) − 𝗑(𝑛 − 1) should be quantized rather than quantizing each raw sample 𝗑(𝑛). Sampling by quantizing the difference is called differential quantization. Differential quantization requires fewer bits for an equivalent SQNR value.

So far, this description is on qualitative, intuitive grounds. For quantitative analysis, the variance of the difference between two consecutive samples should be sampled and then compared to the variance using PCM-based sampling.

As shown in the proposition below, the best method depends on how correlated the signal is. If the signal is highly correlated, as might be expected using a high sample rate relative to the bandwidth, then the differential approach tends to yield a lower variance (has less distortion). If the signal is not highly correlated, as might be expected using a low sample rate, then the PCM approach tends to yield less distortion.

Proposition

Let Rxx(m) be the auto-correlation of x(n). Let Yxx (m) ≜ [ Rxx (m)] / (𝜎x ) 2 be the auto-covariance of x(n). Let (𝜎x ) 2 be variance of x(n). Let (𝜎d ) 2 be the variance of x(n) – x(n-1) (using differential quantization).

Then:

(1)   \begin{equation*} (\sigma_{d})^2=2(\sigma_{x})^2[1-y_{xx}(1)] \end{equation*}

Proof

(2)   \begin{equation*} (\sigma_{d})^2\triangleq\text{E}([x(n)-(\sigma_{d})^2]b)\text{ by definition of variance }\sigma^2 \end{equation*}

(3)   \begin{equation*} =\text{E}[x^2(n)]-2\text{E}[x(n)\times(n-1)]+\text{E}[x^2(n-1)]\text{ by the binomial theorem} \end{equation*}

(4)   \begin{equation*} =2\text{E}[x^2(n)]-2\text{E}[x(n)\times(n-1)]\text{ by the wide-sense stationary property} \end{equation*}

(5)   \begin{equation*} =2(\sigma_{x})^2-2\text{E}[x(n)\times(n-1)]\text{ by definition of variance }\sigma^2 \end{equation*}

(6)   \begin{equation*} =2[(\sigma_{x})^2-\text{R}_{xx}(1)]\text{ by the definition of }\text{R}_{xx}(m) \end{equation*}

(7)   \begin{equation*} =2[(\sigma_{x})^2-(\sigma_{x})^2\text{y}_{xx}(1)]\text{ by the definition of }\text{y}_{xx}(m) \end{equation*}

(8)   \begin{equation*} =2[(\sigma_{x})^2(1-\text{y}_{xx}(1))]\text{ by the definition of }\text{y}_{xx}(m) \end{equation*}

 

Small Correlation

 

Large Correlation

 

According to the proposition above, if yxx (1) > 1/2, then the differential encoding introduces less distortion.