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

2. 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

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

4. 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

5. 13.3
Past values
Prediction
Predicted
Present value
Prediction error
Actual
Present value

6. 13.3
Example: motion estimation of image compression
Predicted
Prediction error

7. 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

8. 13.3
Time domain: autocorrelation function
Correlation characteristics of a source
Frequency domain: power spectrum
Examine: short-term speech signal

9. 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

10. 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

11. 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

12. 13.3
N-tap predictive DPCM
encoder
encoder
decoder

13. 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

14. 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

15. 13.3.1
One-tap prediction

16. 13.3.1
One-tap prediction
eq c

17. 13.3.1
One-tap prediction

18. 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 MHz sample rate
eq c
Nyquist rate = two times the bandwidth of a band-limited function/channel

19. 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

20. 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

21. 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

22. 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

23. 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

24. 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

25. 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

26. 13.4
Forward adaptation
Side information

27. 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)

28. 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

29. 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