1. 4.2 Digital Transmission Pulse Modulation (Part 2.1)
Outlines
Pulse Modulation (Part 2.1)
Pulse Code Modulation (Part 2.2)
Delta Modulation (Part 2.3)
Line Codes (Part 2.4)

2. Using Nyquist’s sampling theorem ,we have
Example 4.2 
For a PCM system with a maximum audio input frequency 4 kHz determine the minimum sample rate and the alias frequency produced if a 5 kHz audio signal were allowed to enter the sample-and-hold circuit.
Solution:
Using Nyquist’s sampling theorem ,we have
fs ≥ 2fm therefore, fs ≥ 8 kHz 
If a 5 kHz audio frequency entered the sample-and-hold circuit, the output spectrum shown in Figure 4.4 is produced. It can be seen that the 5 kHz signal produces an alias frequency of 3 kHz that has been introduced into the original audio spectrum.
Figure 4.2.1: Output spectrum for Example 4.2.

3. PULSE CODE MODULATION (PCM)
Basic scheme of PCM system
Quantization
Quantization Error
Companding
Block diagram & function of TDM-PCM communication system

4. Basic scheme of PCM system
The most common technique for using digital signals to encode analog data is PCM.
Example: To transfer analog voice signals off a local loop to digital end office within the phone system, one uses a codec.

5. Cont’d...
Because voice data limited to frequencies below Hz, a codec makes 8000 samples/sec. (i.e., 125 microsecond/sample).
If a signal is sampled at regular intervals at a rate higher than twice the highest signal frequency, the samples contain all the information of the original signal.

6. PCM Block Diagram Four step process
Most common form of analog to digital modulation
Four step process
Signal is sampled using PAM (Sample)
Integer values assigned to signal (PAM)
Values converted to binary (Quantized)
Signal is digitally encoded for transmission (Encoded)

7. 4 Steps Process

8. Cont’d… Analog signal is sampled.
Converted to discrete-time continuous-amplitude signal
(Pulse Amplitude Modulation)
Pulses are quantized and assigned a digital value.
A 7-bit sample allows 128 quantizing levels.
PCM uses non-linear encoding, i.e., amplitude spacing of levels is non-linear
There is a greater number of quantizing steps for low amplitude
This reduces overall signal distortion.
This introduces quantizing error (or noise).
PCM pulses are then encoded into a digital bit stream.
8000 samples/sec x 7 bits/sample = 56 Kbps for a single voice channel.

9. PCM Example

10. Quantization
A process of converting an infinite number of possibilities to a finite number of conditions (rounding off the amplitudes of flat-top samples to a manageable number of levels).
For example, a sine wave with a peak amplitude of 5V varies between +5V and -5V passing through every possible amplitudes in between. A PCM code could have only eight bits, or 256 combinations. To convert samples of a sine wave to PCM requires some rounding off.
With quantization, the total voltage range is subdivided into a smaller number of subranges, as shown in Table 4.1

11. Cont’d... Analog input signal Figure 4.2.1 Sample pulse PAM signal
PCM code
Figure 4.2.1

12. Cont’d 1st sample (occurs at t1) : Input voltage=+2V. PCM code = 110
No quantization error
2nd sample (occurs at t2) :
input voltage=-1V.
PCM code = 001
3rd sample (occurs at t3):
Input voltage=+2.6V
Since no PCM code for +2.6V, magnitude of sample is rounded off to the nearest amplitude,+3V
PCM code = 111
Quantization error = 0.4V

13. Example 4.2.2
For the PCM coding scheme in Figure 4.2.1, determine the quantized voltage, quantization error(Qe), and PCM code for the analog sample voltage of +1.07V.
Solution:
Quantized level=sample voltage/resolution
= +1.07/1
= 1.07
= 1V (rounded off to the nearest quantization level)
Quantization error = original sample voltage – quantized level
= 1.07 – 1=0.07
PCM code for +1=101

14. QUANTIZATION ERROR
A difference between the exact value of the analog signal & the nearest quantization level.

15. Types of Quantization
Midtread
Midrise

16. Types of Quantizer
1. Uniform type : The levels of the quantized amplitude are uniformly spaced. 2. Non-uniform type : The levels are not uniform.

17. Dynamic Range (DR)
Ratio of the largest possible magnitude/smallest possible magnitude (other than 0V) that can be decoded by DAC in receiver.
In dB:
Where
DR = absolute value of dynamic range
Vmax = the maximum voltage magnitude
Vmin = the quantum value (resolution)
n = number of bits in the PCM code

18. The number of bits used for a PCM code depends on dynamic range.
For a minimum number of bits:
where n=number of bits in a PCM code(excluding sign bit)
DR=absolute value of dynamic range

19. Example 4.2.3
For a PCM system with following parameters :
Maximum analog input frequency = 4kHz
Maximum decoded voltage at receiver =±2.55V
Minimum dynamic range = 46 dB
Determine:
Minimum sample rate
Minimum number of bits used in PCM code
Resolution
Quantization error
Solution :
Minimum sample rate, fs

20. Dynamic range (absolute value):
Minimum no. of bits, n:
Since amplitude range is ±2.55V, one additional bit (sign bit is required. Total number of bits =9 bits.
Resolution:

21. Maximum quantization error:

22. Example 1
Calculate the dynamic range for a linear PCM system using 16-bit quantizing.
Calculate the number of bits in PCM code if the DR = dB

24. Coding Efficiency
A numerical indication of how efficiently a PCM code is utilized.
The ratio of the minimum number of bits required to achieve a certain dynamic range to the actual number of PCM bits used.

25. Signal to Quantization Noise Ratio (SQR)
The worst possible SQR occurs when the input signal is at its minimum amplitude (101 or 001).
The worst-case voltage SQR
For PCM code shown in Figure :
SQR for a maximum input signal

26. For a maximum amplitude input signal of 3V;
The signal power-to-quantizing noise power ratio
Where R =resistance (ohm)
v = rms signal voltage
q = quantization interval
= average signal power (W)

27. Example 2
Calculate the SQR (dB) if the input signal = 2 Vrms and the quantization noise magnitudes = 0.02 V.
Determine the voltage of the input signals if the SQR = dB and q =0.2 V.

28. Effect of Non-Linear Coding

29. Nonlinear Encoding
Quantization levels not evenly spaced. The step size increases with the amplitude of the input signal.
Reduces overall signal distortion
Can also be done by companding

30. Companding The process of compressing and then expanding.
The higher amplitude analog signals are compressed (amplified less than lower amplitude signals) prior to transmission and then expanded(amplified more than lower amplitude signals in receiver).
Improving the DR of a communication system.

31. Companding Functions At lower Vin, gain is high, Vout high
Vin increase, gain lower, curve begin to flatten.

32. Method of Companding
For the compression, two laws are adopted: the -law in US and Japan and the A-law in Europe.
-law
A-law
The typical values used in practice are: =255 and A=87.6.
After quantization the different quantized levels have to be represented in a form suitable for transmission. This is done via an encoding process.
Vmax= Max uncompressed analog input voltage
Vin= amplitude of the input signal at a particular of instant time
Vout= compressed output amplitude
A, = parameter define the amount of compression

33. Example 3
A companding system with µ = 255 used to compand from 0V to 15 V sinusoid signal. Draw the characteristic of the typical system.
Draw an 8 level non-uniform quantizer characteristic that corresponds to the mentioned µ.

34. Cont’d...
A-law
μ-law

35. PCM Line Speed
The data rate at which serial PCM bits are clocked out of the PCM encoder onto the transmission line.
Where
Line speed = the transmission rate in bits per second
Sample/second = sample rate, fs
Bits/sample = no of bits in the compressed PCM code

36. Example 4
For a single PCM system with a sample rate fs = 6000 samples per second and a 7 bits compressed PCM code, calculate the line speed.

37. Virtues & Limitation of PCM
The most important advantages of PCM are:
Robustness to channel noise and interference.
Efficient regeneration of the coded signal along the channel path.
Efficient exchange between BT and SNR.
Uniform format for different kind of base- band signals.
Flexible TDM.

38. Cont’d…
Secure communication through the use of special modulation schemes of encryption.
These advantages are obtained at the cost of more complexity and increased BT.
With cost-effective implementations, the cost issue no longer a problem of concern.
With the availability of wide-band communication channels and the use of sophisticated data compression techniques, the large bandwidth is not a serious problem.

39. Time-Division Multiplexing
This technique combines time-domain samples from different message signals (sampled at the same rate) and transmits them together across the same channel.
The multiplexing is performed using a commutator (switch). At the receiver a decommutator (switch) is used in synchronism with the commutator to demultiplex the data.

40. Cont’d…
TDM system is very sensitive to symbol dispersion, that is, to variation of amplitude with frequency or lack of proportionality of phase with frequency. This problem may be solved through equalization of both magnitude and phase.
One of the methods used to synchronize the operations of multiplexing and demultiplexing is to organize the multiplexed stream of data as frames with a special pattern. The pattern is known to the receiver and can be detected very easily.

41. Block diagram of TDM-PCM communication system

42. An audio signal with a frequency from 4 kHz to 8 kHz and a peak to peak voltage of 25.5 V is being sampled at a rate of 10% higher than minimum sampling frequency of Nyquist rate. A sampled signal is then being quantized with minimum step size of 0.1 V. Determine the :
i) sampling frequency used,
ii) quantization level used,
iii) no. of bits used to encode the signal,and
iv) quantization noise

43. END OF PART 2.2