te 4 pulse_modulation

TELECOMMUNICATIONS ENGINEERING

INSTRUCTOR

Md Hasib Noor

Faculty of EngineeringDepartment of Electrical and Electronic Engineering

Pulse Modulation

Voice is analog in character moves in the form of waves.

3-important wave-characteristics: Amplitude Frequency Phase

Why Voice Digitization?? Ensures better quality (than analog) Provides higher capacity (than analog) Deals with longer distance (than analog)

Digitization is just a discrete electrical voltage. The amplitude of Electrical pulses can be varied to represent characteristics

of an analog voice signal.

Basic Concepts

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PAM is the first step in digitizing an analog waveform. Establishes a set of discrete times at which the input signal waveform is sampled. The sampling process is equivalent to amplitude modulation of a constant

amplitude pulse train, thus, PAM.

Nyquist Sampling Rate : The minimum sampling frequency required to extract allinformation in a continuous, time-varying waveform. Nyquist Criterion: fs > 2*(BW)

where fs = sampling rate, BW = bandwidth of the input signal.

Figure 1: PAM

Pulse Amplitude Modulation (PAM)

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The Nyquist frequency, named after electronic engineer Harry Nyquist, is halfof the sampling rate of a discrete signal processing system.

It is sometimes known as the folding frequency of a sampling system.

The “Nyquist frequency” should not be confused with the “Nyquist rate”,which is the minimum sampling rate that satisfies the Nyquist sampling criterionfor a given signal or family of signals.

The Nyquist rate is twice the maximum component frequency of the functionbeing sampled.

For example, the Nyquist rate for the sinusoid at 0.6 fs is 1.2 fs, which meansthat at the fs rate, it is being under sampled.

Thus, Nyquist rate is a property of a continuous-time signal, whereas Nyquistfrequency is a property of a discrete-time system.

Nyquist rate & Nyquist frequency)

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Spectrum of PAM Signal:

The PAM spectrum can be derived by observing that a continuous train of impulseshas a frequency spectrum consisting of discrete terms at multiples of the samplingfrequency.

The input signal amplitude modulates these terms individually. Thus a double-sideband spectrum is produced about each of the discrete frequency terms in thespectrum of the pulse train.


5Figure 2: Spectrum of PAM Signal

The original signal waveform is recovered by a low-pass filter designed toremove all but the original signal spectrum.

As shown in the figure 2, the reconstructive low-pass filter must have a cut-offfrequency that lies between BW and (fs – BW).

Hence, separation is only possible if (fs – BW) is greater than BW (i.e., (fs > 2BW).


6Figure 2: Spectrum of PAM Signal

Foldover Distortion: If the input is under sampled (i.e. fs < 2BW), the original waveform cannot be

recovered without distortion. As indicated in figure 3, this output portion arises because the frequency

spectrum centered about the sampling frequency overlaps the original spectrumand cannot be separated from the original spectrum by filtering.

Since it is a duplicate of the input spectrum “folded” back on top of the desiredspectrum that causes the distortion, this type of sampling impairment is called“foldover distortion.” Another term for this impairment is “aliasing”.


7Figure 3: Foldover spectrum produced by under sampling an input

PAM System:

Complete PAM system includes a band-limiting filter (or anti-aliasing filter)before sampling to ensure that no source related signals get folded back into thedesired signal bandwidth.

End-to-End PAM system:


8Figure 5

Pulse Code Modulation (PCM):

PCM is an extension of PAM wherein each analog sample is quantized into adiscrete value for representation as a digital code-word.

PAM system can be converted to PCM if we add ADC* at the source and DAC**at the destination.

Pulse Code Modulation (PCM)

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Figure 6: PCM

*ADC: analog to digital converter**DAC: digital to analog converter

Pulse amplitude modulation systems are not useful over long distance, for thevulnerability of individual pulse amplitudes to noise, distortion and crosstalk.

The susceptibility of amplitude may be eliminated by converting the PAMsamples into a digital format. (Using regenerative repeaters)

A finite number of bits are used for coding PAM samples.

n bit number can represent 2n samples.

PAM samples amplitude can take on an infinite range of values.

The PAM sample amplitude is quantized to the nearest of a range of discreteamplitude levels.

Quantization and Binary Coding

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Signal V is confined to a rangeof VL and VH. This range is dividedinto M (M=8) equal steps. The step size S is given by:

S = (VH - VL) / M The center of each steps locate thequantization levels V0 , V1…V8. Quantized signal Vq takes any ofthe quantized level value. A signal V is quantized to its nearestquantization level. The convention followed to quantize the signal is Figure 7: The Process of Quantization

Vq = V3 (if (V3 - S/2) ≤ V < (V3 + S/2) Vq = V4 (if (V4 - S/2) ≤ V < (V4 + S/2)

Thus, the signal Vq makes quantum jump of step size S and at any instant of time the quantization error (V - Vq) has magnitude which is equal or less than S/2. The quantization in which the step size is uniform is called linear or uniform

quantization.

Quantization Process

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Quantization brings about a certain amount of noise in immunity to the signal. Repeaters with quantizers are used after certain distance to control the

variation in instantaneous amplitude for attenuation and channel noise within ±S/2.

If instantaneous noise level is larger than S/2, error occurs in the quantizationlevel.

The quantized signal is an approximate of the original signal. Quality can be increased by increasing the number of quantization levels. Sometimes increased levels introduces noise in the repeaters. The susceptibility to noise can be greatly minimized by resorting the digital

coding of the PAM sample amplitude. Each quantized level is represented by a code number and transmitted instead of

the level value. If binary arithmetic is used the number will be transmitted as a series of pulses. Such a system is called PCM System.

Quantization

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Let’s assume: The analog signal is limitedin

its excursions to the range -4Vto +4V. The step size is 1 volt. Eight quantization levels areused and are located at -3.5V,-2.5V …., +3.5V. Code number000 is assigned to -3.5V and so on. If the analog samples aretransmitted the 1.3, 2.7, 0.5 etcwill be transmitted. If the quantized values are transmittedvoltages 1.5, 2.5, 0.5 etc will be transmitted. In binary PCM the binary code patterns

101, 110,100 are transmitted.

Binary PCM

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Figure 8: Binary PCM - Features

The functional diagram for PCM isshown in the figure 9. The analog input V is band-limited to3.4 KHz to prevent aliasing and sampledat 8 KHZ. Samples are quantized to producePAM signals, and applied to encoder.

Encoder generates a unique pulsepattern for each quantized sample level. The quantizer and encoder together Figure 9: A PCM system for speech communication

work as Analog to Digital Converter (ADC). Receiver first separates the noise from the signals. A quantizer does it by determining the two voltage levels of the pulse. Then it regenerates the appropriate pulse depending on the decision.

PCM System

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The regenerated pulse train is now fed to a decoder which assembles the pulsepattern and generates a corresponding quantized voltage level.

Quantizer and decoder work together as a Digital to Analog converter (DAC). The quantized PAM is now passed through a filter which rejects the frequency

components lying outside the baseband signal.

Figure 9: A PCM system for speech communication

PCM System

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After sampling, the analogue amplitude value of each sampled (PAM) signal isquantized into one of a number of L discrete levels. The result is a quantizedPAM signal.

A code-word can then be used to designate each level at each sample time. Thisprocedure is referred to as “Pulse Code Modulation”.

Figure 10: Analogue to Digital Conversion

Analogue to Digital Conversion

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After quantization, a digit is assigned to each of the quantized signal levels in sucha way that each level has a one-to-one correspondence with the set of realintegers. This is called digitization of the waveform.

Each integer is then expressed as an n-bit binary number, called code-word, orPCM word.

The number of code-words, M , is related to n by: 2n= M.

Encoding

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Quantization followed by digitization maps input amplitudes into PCM words.

There are M integers, PCM words, or codewords to correspond to the M allowedoutput amplitudes of the quantizer.

Codebook is the set of all these M codewords.

Codeword

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Analog to Digital:

Figure 11: Process of digitization

Analog to Digital

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The quantized signal is an approximation to the original signal and some error. The instantaneous error e = V-Vq is randomly distributed within the range S/2 and

is called quantization error or noise. The mean square quantization error is S2. For linear quantization the probability distribution of the error is constant

within the ± (S/2).

Figure 12: Probability distribution of error due to linear quantization

Quantization Noise

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The average quantization noise output power is given by the variance.

Where µ = mean, which is zero for quantization noise.

The range of quantization error ±(S/2) determines the limits of integration.

Quantization Noise

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de

Signal to quantization noise ratio (SQR) is a good measure of performance ofa PCM system transmitting speech.

If Vr is the r.m.s value of the input signal and the resistance level is 1 ohm, thenSQR is given by

Quantization Noise

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If the input signal is a sinusoidal wave and Vm as the maximum amplitude, SQRmay be calculated from the full range sine wave as:

Expressing S in terms of Vm and the number of steps, M, we have

Quantization Noise

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Quantity 1.225M represents the signal to quantization noise voltage ratio for afull range sinusoidal input voltage.

M = 2n, where n is the number of bits used to code a quantization level.Therefore:

The table 1 is showing the values of SQR for different binary code word sizes forsinusoidal input systems.

Every additional code bits gives an increment of 6 dB in SQR.

Quantization Noise

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Example: A sine wave with a 1-V maximum amplitude is to be digitized with aminimum SQR of 30 dB. How many uniformly spaced quantization intervals areneeded, and how many bits are needed to encode each sample?

Solution: Using Equation:SQR = 7.78 + 20 log10 (Vm / S) Given,

The maximum size of a quantization interval is SQR = 30 dBdetermined as: Vm = 1-V

S = (1) 10 –(30-7.78)/20

= 0.078 V

Thus 13 quantization intervals are needed for each polarity for a total of 26 intervals inall. The number of bits required to encode each sample is determined as:

N = log2 (26) = 4.7 = 5 bits per sample

Quantization

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Uniform vs Non-Uniform Quantization

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An alternative is to first pass the speech signal through a nonlinearity beforequantizing 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 to avoidsignal distortion. .

The process of compressing and expanding is called Companding.

Companding

Compression + Expansion Companding

)(ty)(tx )(ˆ ty )(ˆ tx

x

)(xCy = x

yCompress Uniform Qauntize

ChannelExpand

Transmitter Receiver27

Companding

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Companding

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Various compression–expansion characteristics can be chosen to implement acompandor

by increasing the amount of compression, we increase the dynamic range at theexpense of the signal-to-noise ration for large amplitude signals.

There are two types of companding characteristics:

µ-law Companding: used in North America and Japan

A-law Companding: recommended by CCITT for Europe and most of the rest of theworld

Comparison between A-law and mu law

• µ-Law has a larger dynamic range compared to A-law

• µ-Law has worse distortion with small signals compared to A-law

• µ-Law is used in North-America and Japan while A-law is commonly used in Europe

• A-law takes precedence over µ-law with international calls

*** Please go through “Digital Communication” by John Bellamy for further understanding for different Companding techniques

(Differential PCM):

A special kind of PCM technique that codes the difference between sample points to compress the digital data.

Because audio waves propagate in predictable patterns, DPCM predicts the next sample and codes the difference between the prediction and the actual point.

The differences are smaller numbers than the numerical value of each sample on the full scale and thereby reduce the resulting bit-stream.

DPCM

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This is a special kind of DPCM technique that requires much simpler circuitrythan PCM

This technique is widely used for transmission of voice information where quality isnot of primary importance

In this method, an analog waveform is tracked, using a binary 1 to represent a risein voltage, and a 0 to represent a drop.

Transmits only one bit per sample.

The Present sample value is compared with the previous sample value and thisresult, whether the value is increased or decreased, is transmitted.

Delta Modulation

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Delta Modulation

Delta modulation components (transmitter)

integrator

converts the difference between the input signal and the average of the previous steps.

+-

Previous comparator output

comparator referenced to 0 (two levels quantizer), whose output is 1 or 0 if the input signal is positive or negative.

+Δ

-Δ

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Delta demodulation components (receiver)

The demodulator is simply an integrator (like the one inthe feedback loop) whose output rises or falls with each1 or 0 received. The integrator itself constitutes a low-pass filter

Delta Modulation34

Delta Modulation

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• Slope overload distortion/noise - is caused by use of step size delta which is too smallto follow portions of waveform that has a steep slope. …Can be reduced by increasing the step size.

• Granular noise - is caused by too large step size in signal parts with small slope. It can be reduced by decreasing the step size.

Adaptive DM:

A better performance can be achieved if the value of δ is not fixed. In adaptive deltamodulation, the value of δ changes according to the amplitude of the analog signal.

Quantization Error:

It is obvious that DM is not perfect. Quantization error is always introduced in theprocess. The quantization error of DM, however, is much less than that for PCM.

Adaptive Delta Modulation

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END

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