1. EKT343-Principles of Communication Engineering
PULSE
MODULATION
CHAPTER 4
Part 2
EKT343-Principles of Communication Engineering

2. Digital Pulse Modulation
Pulse Code Modulation (PCM)
Sample
Quantize:
Types of quantization : Uniform, non-uniform
Uniform quantization: midtread, midrise
Quantization error and SQR
Non-uniform quantization-> Companding
Encode
PCM Transmission
Line speed
Bandwidth
Noise in PCM
Advantages & Application
EKT343-Principles of Communication Engineering

3. EKT343-Principles of Communication Engineering
Pulse Modulation
Analog pulse modulation: Sampling, i.e., information is transmitted only at discrete time instants. e.g. PAM, PPM and PDM
Digital pulse modulation: Sampling and quantization, i.e., information is discretized in both time and amplitude. e.g. PCM
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Analog input signal
Sample at discrete time instants
Analog pulse modulation, PAM signal
Digital pulse modulation, PCM code
EKT343-Principles of Communication Engineering

5. PCM- PULSE CODE MODULATION
DEFINITION: Pulse code modulation (PCM) is essentially analog-to-digital (A/D) conversion where the information contained in the instantaneous samples of an analog signal is represented by digital words in a serial bit stream.
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6. EKT343-Principles of Communication Engineering
PCM Block Diagram
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)
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4 Steps Process
EKT343-Principles of Communication Engineering

8. PCM-Sampling, Quantizing, and Encoding
The PCM signal is generated by carrying out three basic operations:
Sampling
Quantizing
Encoding
Sampling operation generates a flat-top PAM signal.
Quantizing operation approximates the analog values by using a finite number of levels, L.
PCM signal is obtained from the quantized PAM signal by encoding each quantized sample value into a digital word.
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9. EKT343-Principles of Communication Engineering
PCM as ADC
Sampling
Makes the signal discrete in time.
If the analog input has a bandwidth of B Hz, then the minimum sample frequency such that the signal can be reconstructed without distortion, fs >= 2B
Quantization
Makes the signal discrete in amplitude.
Round off to one of q discrete levels.
Encode
Maps the quantized values to digital words that are n bits long.
Analog
Input
Signal
Sample
ADC
Quantize
111
110
101
100
011
010
001
000
Encode
Digital Output
Signal
EKT343-Principles of Communication Engineering
Eeng

10. Definition of 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).
In other words, quantization is a process of assigning the analog signal samples to a pre-determined discrete levels. The number of quantization levels, L determine the number of bits per sample, n.
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Quantization
The output of a sampler is still continuous in amplitude.
Each sample can take on any amplitude value e.g V, V, etc.
The number of possible values is infinite.
To transmit as a digital signal we must restrict the number of possible values.
Quantization is the process of “rounding off” a sample according to some rule.
E.g. suppose we must round to the nearest discrete value, then:
> > 0
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Quantization Example
Analogue signal
Sampling TIMING
Quantization levels.
Quantized to 5-levels
Quantization levels
Quantized 10-levels
EKT343-Principles of Communication Engineering

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

14. Types of Uniform Quantization
Origin lies in the middle of a tread of the staircase like graph in (a)
Origin lies in the middle of a rising part of the staircase like graph (b)
Midtread: Origin lies in the middle of a tread of the staircase like graph in (a), utilized for odd levels
Midrise: Origin lies in the middle of a rising part of the staircase like graph (b), utilized for even levels

15. EKT343-Principles of Communication Engineering
Uniform Quantization
Most ADC’s use uniform quantizers.
The quantization levels of a uniform quantizer are equally spaced apart.
Uniform quantizers are optimal when the input distribution is uniform. When all values within the Dynamic Range of the quantizer are equally likely. 2 4 6 8 1 5 3
Output sample XQ -2 -4 -6 -8
Dynamic Range:
(-8, 8) 7 -7 -3 -5 -1
Quantization Characteristic
Input sample X
Example: Uniform n =3 bit quantizer
L=8 and XQ = {1,3,5,7}
EKT343-Principles of Communication Engineering
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16. EKT343-Principles of Communication Engineering
Dynamic Range (DR)
Largest possible magnitude/smallest possible magnitude.
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
for n > 4
EKT343-Principles of Communication Engineering

17. Coding Efficiency = Minimum number of bits x 100
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.
Coding Efficiency = Minimum number of bits x 100
Actual number of bits
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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. Determine the coding efficiency in this case.
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Cont’d…
The quantization quantum
= the magnitude difference between adjacent steps,
The resolution
= the magnitude of a quantum
= the voltage of the minimum step size.
The quantization error
= the quantization noise
= ½ quantum
= (orig. sample voltage – quantize level)
The quantization range: is the range of input voltages that will be converted to a particular code.
Quantum – Level 2-Level1
EKT343-Principles of Communication Engineering

20. Quantization Error
A difference between the exact value of the analog signal & the nearest quantization level.
Quantization error is a round-off error in the transmitted signal that is reproduced when the code is converted back to analog in the receiver.

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Quantization Noise
The process of quantization can be interpreted as an additive noise process.
The signal to quantization noise ratio (SNR)Q=S/N is given as:
Signal X
Quantized Signal, XQ
Quantization Noise, nQ
EKT343-Principles of Communication Engineering

22. Signal to Quantization Noise Ratio (SQR)
The worst possible signal voltage-to-quantization noise voltage ratio (SQR) occurs when the input signal occurs when input signal is at its minimum amplitude. SQR is directly proportional to resolution.
The worst-case voltage SQR
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Cont'd
R =resistance (ohm)
v = rms signal voltage
q = quantization interval
Qe = quantization error
SQR for a maximum input signal
The signal power-to-quantizing noise power ratio
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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.
EKT343-Principles of Communication Engineering

25. Nonuniform Quantization
Many signals such as speech have a nonuniform distribution.
The amplitude is more likely to be close to zero than to be at higher levels.
Nonuniform quantizers have unequally spaced levels
The spacing can be chosen to optimize the SNR for a particular type of signal.
Output sample XQ 6 4 2
Example: Nonuniform 3 bit quantizer -8 -6 -4 -2 2 4 6 8
Input sample X -2 -4 -6
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Companding
Nonuniform quantizers are difficult to make and expensive.
An alternative is to first pass the speech signal through a nonlinearity before quantizing with a uniform quantizer.
The nonlinearity causes the signal amplitude to be Compressed.
The input to the quantizer will have a more uniform distribution.
At the receiver, the signal is Expanded by an inverse to the nonlinearity.
The process of compressing and expanding is called Companding.
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Cont'd
The process of compressing and then expanding.
The higher amplitude analog signals are compressed
prior to transmission and then expanded in receiver.
Improving the DR of a communication system.
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Companding Functions
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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
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Cont’d...
μ-law
A-law
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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.
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Example 4
A companding system with µ = 200 is used to compand -4V to 4V signal. Calculate the system output voltage for Vin = -4, -2, 0, 2 and 4V.
Equation:
Vin (V) -4 -2 2 4
Vout (V)
-3.48
3.48
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Plot the compression characteristic that will handle input voltage in the given range and draw an 8 level non-uniform quantizer characteristic that corresponds to the given µ.
Couldn’t see non-uniform here because the plot is Vout vs Vin. U can see the difference if u plot Vin vs time
EKT343-Principles of Communication Engineering

34. SNR Performance of Compander
The output SNR is a function of input signal level for uniform quantizing.
But it is relatively insensitive for input level for a compander.
α = Log ( V/xrms) for Uniform Quantizer
V is the peak signal level and xrms is the rms value
α = log[Ln(1 + μ)] for μ-law companding
α = log[1 + Ln A] for A-law companding
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Encoding
The output of the quantizer is one of L possible signal levels.
If we want to use a binary transmission system, then we need to map each quantized sample into an n bit binary word.
Encoding is the process of representing each quantized sample by n bit code word.
The mapping is one-to-one so there is no distortion introduced by encoding.
EKT343-Principles of Communication Engineering
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36. EKT343-Principles of Communication Engineering
PCM encoding example
Levels are encoded using this table
Table: Quantization levels with belonging code words
L=8
Chart 2. Process of restoring a signal.
PCM encoded signal in binary form:
Total of 33 bits were used to encode a signal
Chart 1. Quantization and digitalization of a signal. Signal is quantized in 11 time points & 8 quantization segments.
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PCM Example
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Nonlinear Encoding
Quantization levels not evenly spaced
Same concept as non-uniform quantization
Reduces overall signal distortion
Can also be done by companding
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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
Line speed also known as bit rate
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Example 5
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.
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Channel Bandwidth
The channel bandwidth, B required to transmit a pulse is given by
Where
κ = a constant with a value between 1 to 2
n = number of bits
W = signal bandwidth
Channel BW = transmission BW
EKT343-Principles of Communication Engineering

42. Bandwidth of PCM Signals
The spectrum of the PCM signal is not directly related to the spectrum of the input signal.
The bandwidth of (serial) binary PCM waveforms depends on the bit rate R and the waveform pulse shape used to represent the data.
The Bit Rate R is
R=nfs
Where n is the number of bits in the PCM word (M=2n) and fs is the sampling rate.
restudyyyyyyyy
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For no aliasing case (fs≥ 2B), the MINIMUM Bandwidth of PCM Bpcm(Min) is:
Bpcm(Min) = R/2 = nfs//2
The Minimum Bandwidth of nfs//2 is obtained only when sin(x)/x pulse is used to generate the PCM waveform.
For PCM waveform generated by rectangular pulses, the First-null Bandwidth is:
Bpcm = R = nfs
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Example 6
A signal with a bandwidth of 4.2 MHz is transmitted using binary PCM. The number of representation levels is 512. Calculate
The code word length
The bit rate
The transmission bandwidth, assuming that, κ = 2
Find the SQR in dB for the signal given that peak signal voltage is 5Vp
EKT343-Principles of Communication Engineering

45. PCM transmitter/receiver
LPF
BW=B
Sampler
& Hold
Quantizer
No. of levels=M
Encoder
Analog
signal
Bandlimited
Analog signal
Flat-top
PAM signal
Quantized
PCM
Channel, Telephone lines with regenerative repeater
Decoder
Reconstruction
Signal
output
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Noise in PCM Systems
Two main effects produce the noise or distortion in the PCM output:
Quantizing noise that is caused by the M-step quantizer at the PCM transmitter.
Bit errors in the recovered PCM signal, caused by channel noise and improper filtering.
If the input analog signal is band limited and sampled fast enough so that the aliasing noise on the recovered signal is negligible, the ratio of the recovered analog peak signal power to the total average noise power is:
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Cont’d
The ratio of the average signal power to the average noise power is
M is the number of quantized levels used in the PCM system.
Pe is the probability of bit error in the recovered binary PCM signal at the receiver DAC before it is converted back into an analog signal.
EKT343-Principles of Communication Engineering

48. Effects of Quantizing Noise
If Pe is negligible, there are no bit errors resulting from channel noise and no ISI, the Peak SNR resulting from only quantizing error is:
The Average SNR due to quantizing errors is:
Above equations can be expresses in decibels as,
Where, M = 2n
α = 4.77 for peak SNR
α = 0 for average SNR
EKT343-Principles of Communication Engineering

49. 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.
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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.
EKT343-Principles of Communication Engineering

51. Application: PCM in Wired Telephony
Voice circuit bandwidth is 3400 Hz.
Sampling rate is 8 KHz (samples are 125 s apart) above Nyquist rate, 6.8KHz to avoid unrealizable filters required for signal reconstruction.
Each sample is quantized to one of 256 levels (n=8).
The 8-bit words are transmitted serially (one bit at a time) over a digital transmission channel. The bit rate is 8x8,000 = 64 Kb/s.
The bits are regenerated at digital repeaters.The received words are decoded back to quantized samples, and filtered to reconstruct the analog signal.

52. PCM in Compact Disk (CD)
High definition Audio signal bandwidth is band limited to 15kHz.
Although the Nyquist rate is only 30kHz, the actual sampling of 44.1kHz is used to avoid unrealizable filters required for signal construction
The signal is quantized into a rather large number of levels, L=65,536 (n=16) to reduce quantization noise
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53. EKT343-Principles of Communication Engineering
Exercise 1
A compact disc(CD) records audio signals digitally by using PCM. Assume the audio signal bandwidth to be 15 kHz.
(a) What is the Nyquist rate?
(b) If the Nyquist samples are quantized into L= 65, 536 levels and then binary coded, determine the number of binary digits required to encode the sample.
(c) Determine the number of binary digits per second(bits/s) required to encode the audio signals.
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Exercise 2
This problem addresses the digitization of a television signal using pulse code modulation. The signal bandwidth is 4.5 MHz. Specifications of the modulator include the following:
Sampling : 15% in excess of Nyquist rate
Quantization: uniform with 1024 levels
Encoding : binary
Determine (a) sampling rate and (b) minimum permissible bit rate
EKT343-Principles of Communication Engineering