1. PULSE CODE MODULATION & SOURCE CODING Sampling Theory

2. LEARNING OBJECTS
Sampling Theory
Signal Reconstruction
Aliasing

3. Basic elements of a PCM system

4. Sampling Theory
In many applications, e.g. PCM, it is useful to represent a signal in terms of sample values taken at appropriately spaced intervals.
The signal can be reconstructed from the sampled waveform by passing it through an ideal low pass filter.
In order to ensure a faithful reconstruction, the original signal must be sampled at an appropriate rate as described in the Nyquist’s sampling theorem.
A real-valued band-limited signal having no spectral components above a frequency of B Hz is determined uniquely by its values at uniform intervals spaced no greater than seconds apart.

5. Sampling Theory

6. Impulse Sampling

7. Impulse Sampling

8. Sampling Visualized in Frequency Domain

9. Interpolation
From the spectrum of the sampled signal, we can see that the original signal can be recovered by passing its samples through a LPF

10. Ideal Interpolation

11. Ideal Interpolation

12. Ideal Interpolation

13. Practical Considerations in Nyquist Sampling

14. Gradual Roll-Off Low Pass Filter

15. Gradual Roll-Off Low Pass Filter

16. Aliasing
Resultantly, they will be not band limited.

17. Aliasing

18. A Solution: The Antialiasing Filter
The anti-aliasing, being an ideal filter, is unrealizable. In practice we use a steep cutoff which leaves a sharply attenuated residual spectrum beyond the folding frequencies.

19. Practical Sampling

20. Practical Sampling

21. Some Applications of Sampling Theorem
Sampling theorem is very important in signal analysis, processing and transmission because it allows to replace a continuous time signal by a discrete sequence of numbers. This leads into the area of digital filtering.
In communication, the transmission of continuous-time message reduces to the transmission of a sequence of numbers. This opens the doors to many new techniques of communicating continuous-time signals by pulse trains.
The continuous-time signal g(t) is sampled, and sampled values are used to modify certain parameters of a periodic pulse train.

22. Some Applications of Sampling Theorem
The sampled value can be used to vary amplitude, width or position of the pulse in proportion to the sample values of the signal g(t). Accordingly we get

23. Pulse Modulated Signals

24. Some Applications of Sampling Theorem
Pulse modulation permits simultaneous transmission of several signals on a time-sharing basis: Time Division Multiplexing. Because a pulse modulated signal occupies only a part of the channel time, therefore several pulse modulated signals can be transmitted on the same channel by interweaving.
Similarly several baseband signals can be transmitted simultaneously by frequency division multiplexing where spectrum of each message is shifted to a specific band not occupied by any other signal.

25. Time Division Multiplexing

26. Pulse Code Modulation sampling
Most useful and widely used of all the pulse modulations.
PCM is a method of converting an analog signal into a digital signal (A/D conversion).
An analog signal’s amplitude can take on any value over a continuous range while digital signal amplitude can take on only a finite number of values.
An analog signal can be converted into a digital signal by means of three steps:
sampling
quantizing, that is, rounding off its value to one of the closest permissible numbers (or quantized levels)
Binary coding, that is conversion of quantized samples to 0s and 1s.

27. Amplitude Quantization

28. Scalar Quantizer
(a) Mid-tread
(b) Mid-rise

29. Quantization

30. Quantization

31. Quantization Error

32. Quantization Error

33. Quantization Noise

34. Quantization Noise

35. Quantization Noise

36. Quantization SNR
, 6dB per bit

37. Non-uniform Quantization
Motivation
Speech signals have the characteristic that small-amplitude samples occur more frequently than large-amplitude ones
Human auditory system exhibits a logarithmic sensitivity
More sensitive at small-amplitude range (e.g., 0 might sound different from 0.1)
Less sensitive at large-amplitude range (e.g., 0.7 might not sound different much from 0.8)
histogram of typical
speech signals

38. Non-uniform Quantization

39. Non-uniform Quantization

40. Non-uniform Quantization
Non-uniform Quantization = Compression + Uniform quantization

41. Non-uniform Quantization

42.  Law / A Law
The μ-law algorithm (μ-law) is a companding algorithm, primarily used in the digital telecommunication systems of North America and Japan.
Its purpose is to reduce the dynamic range of an audio signal.
In the analog domain, this can increase the SNR achieved during transmission, and in the digital domain, it can reduce the quantization error (hence increasing signal to quantization noise ratio).
A-law algorithm used in the rest of worlds.
A-law algorithm provides a slightly larger dynamic range than the μ-law at the cost of worse proportional distortion for small signals.
By convention, A-law is used for an international connection if at least one country uses it.

43.  Law Compression

44. A-Law Compression

45. Binary Coding
From practical viewpoint, a binary digital signal (a signal that can take on only two values) is very desirable because of its simplicity, economy, and ease of engineering. We can convert an L-ary signal into a binary signal by using pulse coding.
This code, formed by binary representation of the 16 decimal digits from 0 to 15, is known as the natural binary code (NBC).
Each of the 16 levels to be transmitted is assigned one binary code of four digits. The analog signal m(t) is now converted to a (binary) digital signal. A binary digit is called a bit for convenience.

46. Binary Coding Now each sample is encoded by four bits.
To transmit this binary data, we need to assign a distinct pulse shape to each of the two bits.
One possible way is to assign a negative pulse to a binary 0 and a positive pulse to a binary 1 so that each sample is now transmitted by a group of four binary pulses (pulse code). The resulting signal is a binary signal.

47. Sigma-Delta ADC

48. Pulse Code Modulation Examples

49. Transmission Bandwidth and SNR
For a binary PCM, we assign a distinct group of n binary digits (bits) to each of the L quantization levels. Because a sequence of n binary digits can be arranged in distinct 2n patterns, L=2n or n=log2L Each quantized sample is, thus, encoded into n bits. Because a signal m (t) band-limited to B Hz requires a minimum of 2B samples per second, we require a total of 2nB bits per second (bps), that is, 2nB pieces of information per second. Because a unit bandwidth (1 Hz) can transmit a maximum of two pieces of information per second, we require a minimum channel of bandwidth Hz, given by BT=nB Hz This is the theoretical minimum transmission bandwidth required to transmit the PCM signal.

50. Transmission Bandwidth and SNR
We know that L2 = 22n, and the output SNR can be expressed as
where
Lathi book

51. Transmission Bandwidth and SNR
We observe that the SNR increases exponentially with the transmission bandwidth BT. This trade of SNR with bandwidth is attractive. A small increase in bandwidth yields a large benefit in terms of SNR. This relationship is clearly seen by rewriting using the decibel scale as

52. Transmission Bandwidth and SNR
This shows that increasing n by 1 (increasing one bit in the code word) quadruples the output SNR (6-dB increase).
Thus, if we increase n from 8 to 9, the SNR quadruples, but the transmission bandwidth increases only from 32 to 36 kHz (an increase of only 12.5%).
This shows that in PCM, SNR can be controlled by transmission bandwidth.
Frequency and phase modulation also do this. But it requires a doubling of the bandwidth to quadruple the SNR. In this respect, PCM is strikingly superior to FM or PM.

53. Differential PCM

54. Differential Pulse Code Modulation (DPCM)

55. Differential Pulse Code Modulation (DPCM)

56. DPCM
If the estimate is , then the difference
is transmitted.

57. DPCM

58. How Does the Predictor Works ?

59. The Predictor .

60. The Linear Predictor
A tapped delay-line (transversal) filter used as a linear predictor with tap gains equal to prediction coefficients

61. Linear Prediction Coding (LPC)
Consider a finite-duration impulse response (FIR)
discrete-time filter which consists of three blocks :
1. Set of p ( p: prediction order) unit-delay elements (z-1)
2. Set of multipliers with coefficients w1,w2,…wp
3. Set of adders (  )

62. DPCM
The DPCM transmitter

63. SNR Improvement

64. Coded Excited Linear Prediction (CELP)
By exploiting redundancies from the speech signal, prediction can be improved
Predictor coefficients are derived from the sampled signal and transmitted along with the signal
Prediction can be so good that after some time only the predictor coefficients are sent.
We get transmission at 8-16 kbps with the same quality of PCM

65. Delta Modulation

66. In DM, we use a first-order predictor which is just a delay.
Delta Modulation (DM)
In DM, we use a first-order predictor which is just a delay.

67. Delta Modulation (DM)

68. Delta Modulation (DM)

69. DM System: Transmitter and Receiver

70. DM System: Transmitter and Receiver
( differentiator )
( Integrator)

71. Delta Modulation (DM)

72. Slope Overload Distortion
and Granular Noise

73. Slope Overload Distortion
and Granular Noise

74. Adaptive Delta Modulation
Slope overload and granular noise reduce the dynamic range of DM
Adaptive DM adjusts the step size according to frequency
Output SNR is proportional to
(For single integration case) (BT/B)^3
(For double integration case) (BT/B)^5
Comparison with PCM: at low BT/B, DM is superior; at high BT/B, the advantage is reversed

75. Performance Comparison:
PCM Vs DPCM/DM
Single Integration
Double Integration
Comparison with PCM

76. Line Coding

77. Digital Data Transmission
Source
Input to a digital system is in the form of sequence of digits. It could be from a data set, computer, digitized voice signal (PCM or DM), digital camera, fax machine, television, telemetry equipment etc.

78. Line Coding and Decoding

79. Data Rate Vs. Signal Rate
Data rate: the number of data elements (bits) sent in 1sec (bps). It’s also called the bit rate or transmission rate.
Signal rate: the number of signal elements sent in 1sec. It’s also called the pulse rate, the modulation rate, symbol rate or the baud rate.
Transmission bandwidth is related to baud rate.
We wish to:
increase the data rate (increase the speed of transmission)
decrease the signal rate (decrease the bandwidth requirement)

80. Line Codes
Output of the transmitter is coded into electrical pulses or waveforms for the purposes of transmission over the channel or to modulate a carrier.
This process is called line coding or transmission coding.
There are many possible ways to assign a waveform (pulse) to a digital data based of various desirables.

81. Line coding schemes

82. On-Off Return to Zero (RZ)
1 is encoded with p(t) and 0 is encoded with no pulse.
Pulse returns to zero level after every 1.

83. Polar Return to Zero (RZ)
1 is encoded with p(t) and 0 is encoded with –p(t).
Pulses returns to zero level after every 1 and 0.

84. Bipolar Return to Zero (RZ)
1 is encoded with p(t) or –p(t) depending on
whether previous 1 is encoded p(t) or –p(t)
while 0 is encoded with no pulse.
Pulses returns to zero level after every 1 and 0.
Also known as Pseudoternary or Alternate Mark Inversion (AMI)

85. On-Off Non Return to Zero (NRZ)
1 is encoded with p(t)
while 0 is encoded with no pulse.
Pulses do not return to zero level after every 1 and 0.

86. Polar Non Return to Zero (NRZ)
1 is encoded with p(t)
while 0 is encoded with –p(t).
Pulses do not returns to zero level after every 1 and 0.

87. Desirable Properties of Line Codes
Transmission bandwidth
Power efficiency
Error detection and correction capability
Favorable power spectral density
Adequate timing content
Transparency

88. Desirable Properties of Line Codes
Transmission bandwidth
It should be as small as possible.
Power efficiency
For a given bandwidth and specified detection error probability, transmitted power should be as small as possible.
Error detection and correction capability
It should be possible to detect and if possible to correct detected errors.

89. Desirable Properties of Line Codes
Favorable power spectral density
It is desirable to have zero PSD at w=0 (dc) as ac coupling and transformers are used at the repeaters.
Adequate timing content
It should be possible to extract timing or clock information from the signal.
Transparency
It should be possible to transmit a digital signal correctly regardless of the pattern of 1’s and 0’s.

90. PSD of Various Line Codes: Assumptions
Pulses are spaced Tb seconds apart. Consequently, the transmission rate is Rb=1/ Tb pulses per second.
The basic pulse used is denoted by p(t) and its Fourier transform is P(w).
The PSD of the line code depends upon that of the pulse shape p(t). We assume p(t) to be a rectangular pulse of width Tb/2 i.e.

91. PSD of Polar Signaling

92. Polar Signaling Essential bandwidth of the signal is 2Rb Hz.
This is four times the theoretical BW (Nyquist)
Polar signaling has no error detection capability.
It has non-zero PSD at w=0.
Polar signaling is the most power-efficient scheme.
Transparent

94. PSD On-Off Signaling

95. On-Off Signaling Excessive transmission bandwidth
For a given transmitted power, it is less immune to noise interference than polar scheme.
Made up of a polar signal plus periodic signal; hence, BW is similar to polar signaling (Fig 7.2. Page 296, Lathi).
Contains a discrete component of clock frequency (Eq 7.19, Lathi).
PSD of On-Off signaling is ¼ of that of polar signaling (Eq 7.19, Lathi).
Non-transparent.
All the disadvantages of polar schemes such as:
Excessive transmission bandwidth
Non-zero power spectrum at w=0
No error detection capability.

96. PSD of Bipolar Signaling

97. Advantages of Bipolar (Pseudoternary or AMI) Signaling
Spectrum has DC null.
Bandwidth is not excessive
Has single error detection capability (If error then violation of AMI rule).
If rectified, an off-on signal is formed that has a discrete component at clock frequency.

98. Disadvantages of Bipolar (Pseudoternary or AMI) Signaling
Required twice (3db) as much power as polar signal.
Not transparent (long strings of zeros problematic)
Various substitution scheme are used to prevent long strings of zeros