Digital Communications Chapter 13. Source Coding Subchapter 13.3 & 13.4 Shelly Salim May 13th 2013

Outline 13.1 Sources 13.2 Amplitude Quantizing 13.3 Differential Pulse-Code Modulation 13.4 Adaptive Prediction 13.5 Block Coding 13.6 Transform Coding 13.7 Source Coding for Digital Data 13.8 Examples of Source Coding

Pulse-code modulation Review Digital communication Baseband Bandpass ~pulse Pulse-code modulation

Differential pulse-code modulation 13.3 Differential pulse-code modulation DPCM Prediction of the next sample value is formed from past values Using the redundancy in the signal to form a prediction, the quantization can be performed with a reduced number of decisions for a given quantization level, or with reduced quantization levels for a given number of decisions Subtract the prediction from the next sample value, the difference is called the prediction error

13.3 Past values Prediction Predicted Present value Prediction error Actual Present value

13.3 Example: motion estimation of image compression Predicted Prediction error

Instantaneous/memoryless quantizers 13.3 Quantizers Instantaneous/memoryless quantizers Non-instantaneous quantizers Non-instantaneous quantizers The quantizing methods that take account of sample-to-sample correlation These quantizers reduce source redundancy by converting the input into a new sequence with reduced correlation, reduced variance, or reduced bandwidth The new sequence is then quantized with fewer bits

13.3 Time domain: autocorrelation function Correlation characteristics of a source Frequency domain: power spectrum Examine: short-term speech signal

Examine: short-term speech signal 13.3 Examine: short-term speech signal Power spectrum Gx(f) Observation: Large changes in the signal occur slowly at the low frequency Rapid changes in the signal at the high frequency must be of low amplitude Global maxima: 300 Hz – 800 Hz

Examine: short-term speech signal 13.3 Examine: short-term speech signal Autocorrelation function Rx(T) Observation: A broad and slowly changing function means that there will be only slight change on a sample-to-sample basis

13.3 The difference between adjacent samples for speech is small Source coding techniques have evolved based on transmitting sample-to-sample differences rather than actual sample values N-tap predictive DPCM = predict the next input sample value based on the previous input sample values Predict and correct loop

13.3 N-tap predictive DPCM   encoder encoder decoder

13.3 N-tap predictive DPCM The communication task is that of transmitting the difference (error) between the predicted and the actual data sample If the prediction model forms predictions that are close to the actual sample values, the residues (error) will exhibit reduced variance, relative to the original signal  the residues can be moved through the channel with a reduced data rate From Section 13.2: we know that the number of bits required to move data through the channel with a given fidelity is related to the signal variance

Prediction One-tap prediction N-tap prediction 13.3.1 13.3.2 Predicts the next input sample value based on the previous input sample value Predicts the next sample value based on the linear combination of the previous N sample values

13.3.1 One-tap prediction      

13.3.1 One-tap prediction     eq. 13.50c  

13.3.1 One-tap prediction      

Sigma-Delta modulation 13.3.3 Δ ΣΔ Delta modulation Sigma-Delta modulation The prediction gain could be large if the normalized correlation coefficient, Cx(1), is close to unity To obtain high sample-to-sample correlation, the sampling is operated at a rate that far exceeds the Nyquist rate. For example, the sample rate might be chosen to be 64 times the Nyquist rate, then for a 20 kHz bandwidth with a nominal sample rate of 48 kHz, the sampling would operate at a 3.072 MHz sample rate eq. 13.50c Nyquist rate = two times the bandwidth of a band-limited function/channel

Remove quantizing noise 13.3.3 Δ ΣΔ Delta modulation Sigma-Delta modulation High sample rate is to ensure that the sampled data is highly correlated so DPCM will result in a small prediction error  quantizer will operate with a very small number of bits The simplest form of the quantizer: one-bit quantizer integrator Remove quantizing noise

A modified one-tap DPCM converter 13.3.4 Δ ΣΔ Delta modulation Sigma-Delta modulation A modified one-tap DPCM converter An error-feedback converter/ noise feedback loop We can enhance the correlation of the sampled data by pre-filtering the data with an integrator and compensating for the pre-filter with a post-filtering differentiator The z-transform is a generalization of the Fourier transform of a sampled signal

A modified one-tap DPCM converter 13.3.4 Δ ΣΔ Delta modulation Sigma-Delta modulation A modified one-tap DPCM converter An error-feedback converter/ noise feedback loop We can modify the encoder so that the post-filter differentiator can be moved to the decoder, which then cancels the digital integrator at the input to the decoder Σ Δ

An error-feedback converter/ noise feedback loop 13.3.4 Δ ΣΔ Delta modulation Sigma-Delta modulation A modified one-tap DPCM converter An error-feedback converter/ noise feedback loop When the signal is highly oversampled, not only are the samples highly correlated, the errors are highly correlated as well When errors are highly correlated, they are predictable The previous quantization error can be used as a good estimate of the current error The previous error is stored in a delay register for use as the estimate of the next quantization error

An error-feedback converter/ noise feedback loop 13.3.4 Δ ΣΔ Delta modulation Sigma-Delta modulation A modified one-tap DPCM converter An error-feedback converter/ noise feedback loop Can be used for ADC and DAC The signal flow of previous graph can be redrawn to emphasize two inputs: signal and quantization noise two loops: one including the quantizer and one not including it

Adaptive Prediction 13.4 Non-adaptive prediction, Prediction gain is proportional to the ratio of the signal variance to prediction-error variance Adaptive encoders has additional loops to estimate the parameters to obtain optimal performance These auxiliary loops periodically schedule modifications to the prediction loop parameters and avoid prediction mismatch The International Telegraph and Telephone Consultative Committee (CCITT) has selected a 32-kbits/s adaptive differential pulse-code modulation (ADPCM) coder as a standard for toll-quality speech

13.4 Forward adaptation The input data are buffered and processed to estimate the local statistics, such as the number of samples of the autocorrelation function The correlation sample estimates of the variance of the data The estimation is used to: Adjust the automatic gain control (AGC) to obtain an optimal match of the input signal to the quantizer range  adaptive quantization forward (AQF) Form new filter coefficients for the prediction filter  adaptive prediction forward (APF) The predictor coefficients are derived from the input data, called side information

13.4 Forward adaptation Side information

Synthesis/Analysis Coding 13.4 Synthesis/Analysis Coding Synthesis/analysis coders are very signal specific: designed primarily for speech signals These encoders take advantage of The hearing mechanism responds to the amplitude content of a signal, but it is fairly insensitive to the phase structure Synthesis/analysis coders form a reconstructed signal that approximates the magnitudes and time-varying characteristic of a sequence of the signal, but make no attempt to preserve its relative phase The voice signal is not required to “look” like the original signal, but rather to “sound” like it This type of encoder is best represented as the linear predictive coder (LPC)

Conclusion Differential pulse-code modulation Adaptive Prediction 13.3 Predict the next sample, transmit the prediction error  reduced data rate One-tap prediction and N-tap prediction Delta modulation: 1-bit quantizer Sigma-delta modulation: Integrator at the predict-and-correct loop Predict the next quantization error Adaptive Prediction Modify prediction loop parameters 13.4

Question A signal with correlation coefficient Cx(1) equal to 0.7 is to be quantized with a one-tap LPC filter. Determine the prediction gain when the predictive coefficient is optimized with respect to the minimum prediction error Q