CHAPTER 3 PULSE MODULATION

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Description: CHAPTER 3 PULSE MODULATION

CHAPTER 3 PULSE MODULATION
Digital Communication Systems 2012 R.Sokullu

Digital Communication Systems 2012
Outline 3.1 Introduction 3.2 Sampling Process 3.3 Pulse Amplitude Modulation 3.4 Other Forms of Pulse Modulation 3.5 Bandwidth-Noise Trade-off 3.6 The Quantization Process Digital Communication Systems 2012 R.Sokullu

3.1 Introduction This chapter is a transitional chapter between analog and digital modulation techniques. In CW modulation, as we have studied in chapter 2, one parameter of the sinusoidal carrier wave is continuously varied in accordance with a given message signal. In the case of pulse modulation we have a pulse train and some parameter of the pulse train is varied in accordance with the message signal. Digital Communication Systems 2012 R.Sokullu

CommSystems 1 – Analog Communication Techniques
In the first part of Communication Systems we studied transmission techniques of analog waveforms (analog sources) over analog signals (lines). Why is modulation necessary? What types of modulation did we study? When we studied a specific modulation type what were the specific subjects we discussed? Digital Communication Systems 2012 R.Sokullu

CommSystems 2 – Digital Communication Techniques
In the second part we have two major topics analog waveforms (analog sources) transmission using baseband signals digital waveforms (digital sources) transmission using band-pass signals Digital Communication Systems 2012 R.Sokullu

Why digital? Digital approximation of analog signals can be made as precise as required Low cost of digital circuits Flexibility of digital approach – possibility of combining analog and digital sources for transmission over digital lines Increased efficiency – source coding/channel coding separation Digital Communication Systems 2012 R.Sokullu

Goals of this course: To study digital communication systems and their conceptual basis in information theory To study how analog waveforms can be converted to digital signals (PCM) Compute spectrum of digital signals Examine effects of filtering – how does filtering affect the ability to recover digital information at the receiver. filtering produces ISI in the recovered signal Study how to multiplex data from several digital bit streams into one high speed digital stream for transmission over a digital system (TDM) Digital Communication Systems 2012 R.Sokullu

Motivation and Development
Digital transmission –1960s Real application – after 1970s developments in solid state electronics, micro-electronics, large scale integration all common information sources are inherently analog Historical steps Sampled analog sources transmitted using analog pulse modulation (PAM, PPM) Samples are quantized to discrete levels (PCM, DM) Conversion from analog and transmission were implemented as a single step Today Layered approach – different steps are distinguished and separately optimized (source coding and channel coding) Digital Communication Systems 2012 R.Sokullu

We distinguish between: analog pulse modulation a periodic pulse train is used as a carrier wave; a parameter of that train (amplitude, duration, position) is varied in a continuous manner in accordance with the corresponding sample value of the message signal; information is transmitted basically in analog form, but at discrete times. digital pulse modulation message represented in a discrete way in both time and amplitude; sequence of coded pulses is transmitted. Digital Communication Systems 2012 R.Sokullu

R.Sokullu

Source Coding Problem of coding: efficient representation of source signals (speech waveforms, image waveforms, text files) as a sequence of bits for transmission over a digital network Paired problem of source decoding – conversion of received bit sequence (possibly corrupted) into a more-or-less faithful replica of the original Digital Communication Systems 2012 R.Sokullu

Channel Coding Problem of the efficient transmission of a sequence of bits through a lower layer channel 4 KHz telephone channel, wireless channel Recovery at the channel output in the remote receiver despite distortions Digital Communication Systems 2012 R.Sokullu

Why separate source and channel coding?
Basic theorem of information theory: If a source signal can be communicated through a given point-to-point channel within some level of distortion (by any means) then the separate source and channel coding can also be designed to stay within the same limits of distortion. WHY then…(delay, complexity…) Pros and cons? Does it always hold true? Digital Communication Systems 2012 R.Sokullu

Shannon and the Channel Coding Theorem
Channel coding can help reduce the error probability without reducing the data rate Date rate depends on the channel itself – channel capacity Channel bandwidth W, input power P, noise power then the channel capacity in bits is: Digital Communication Systems 2012 R.Sokullu

Digital Interface Interface between source coding/channel coding – issues continuity, rate etc. continuous sources packet sources complex combinations Here: min number of bits from source and max transmission speed over channel source coder rate = channel encoder rate (source-channel coding theorem) protocols discussed in details in Data Communications course Digital Communication Systems 2012 R.Sokullu

3.2 Sampling Process Sampling converts an analog signal into a corresponding sequence of samples that are uniformly distributed in time. Proper selection of the sampling rate is very important because it determines how uniquely the samples would represent the original signal. It is determined according to the so called sampling theorem. Digital Communication Systems 2012 R.Sokullu

The model: we consider an arbitrary signal g(t) with finite energy, specified for all time we sample the signal instantaneously and at an uniform rate, once every Ts seconds we obtain an infinite sequence of samples spaced Ts seconds apart; they are denoted by [g(nTs)], where n can take all possible integer values Ts is referred to as the sampling period, and fs=1/Ts is the sampling rate. let gδ(t) denote the signal obtained by individually weighting the elements of a periodic sequence of delta functions spaced Ts seconds apart by the numbers [g(nTs)]. Digital Communication Systems 2012 R.Sokullu

The sampling process. (a) Analog signal. (b) Instantaneously sampled version of the analog signal. Figure 3.1 Digital Communication Systems 2012 R.Sokullu

For the signal gδ(t), called the ideal sampled signal, we have the following expression : As the idealized delta function has unit area, the multiplication factor g(nTs) can be considered as “mass” assigned to it (samples are “weighted”); A delta function weighted in this manner is approximated by a rectangular pulse of duration Δt and amplitude g(nTs)/Δt. Digital Communication Systems 2012 R.Sokullu

Knowing that the uniform sampling of a continuous-time signal of finite energy results into a periodic spectrum with a period equal to the sampling rate using the FT gδ(t) can be expressed as: So if we take FT on both sides of (3.1) we get: discrete time Fourier transform Digital Communication Systems 2012 R.Sokullu

The relations derived up to here apply to any continuous time signal g(t) of finite energy and infinite duration. If the signal g(t) is strictly band-limited, with no components above W Hz, then the FT G(f) of g(t) will be zero for |f| ≥ W. Digital Communication Systems 2012 R.Sokullu

(a) Spectrum of a strictly band-limited signal g(t). (b) Spectrum of the sampled version of g(t) for a sampling period Ts = 1/2 W. Figure 3.2 Digital Communication Systems 2012 R.Sokullu

For a sampling period Ts=1/2 W after substitution in 3.3 we get the following expression: and using 3.2 for the FT of gδ(t) we can also write: m=0 Digital Communication Systems 2012 R.Sokullu

and for the conditions specified about f we get: and when we substitute (3.4) and (3.6) we get: Digital Communication Systems 2012 R.Sokullu

Conclusion: 1. If the sample values g(n/2W) of an analog signal g(t) are specified for all n, then the FT G(f) of the signal is uniquely determined by using the discrete-time FT of equation (3.7). 2. Because g(t) is related to G(f) by the inverse FT, the signal g(t) is itself uniquely determined by the sample values g(n/2W) for -∞ < n <+∞. Digital Communication Systems 2012 R.Sokullu

Second part: reconstructing the signal from the samples
We substitute equation (3.7) in the inverse FT formula and after some reorganizing we get: which after integration ends to be: Digital Communication Systems 2012 R.Sokullu

This is an interpolation formula for reconstructing the original signal g(t) from a sequence of sample values [g(n/2W)]. The sinc function sinc(2Wt) is playing the role of interpolation function. Each sample is multiplied by a suitably delayed version of the interpolation function and all the resulting waveforms are summed up to obtain g(t). Digital Communication Systems 2012 R.Sokullu

Sampling Theorem 1. A band-limited signal of finite energy, which has no frequency components higher than W Hz, is completely described by specifying the values of the signal at instants of time separated by 1/2W (means that sampling has to be done at a rate twice the highest frequency of the original signal). 2. A band-limited signal of finite energy, which has no frequency components higher than W Hz, may be completely recovered from a knowledge of its samples taken at the rate of 2W samples per second. Digital Communication Systems 2012 R.Sokullu

Note: The sampling rate of 2W for a signal of bandwidth W Hz, is called the Nyquist rate; Its reciprocal 1/2W (seconds) is called the Nyquist interval; Digital Communication Systems 2012 R.Sokullu

The derivations of the sampling theorem so far were based on the assumption that the signal g(t) is strictly band limited. Practically – not strictly band-limited; the result is under sampling so some aliasing is produced by the sampling process. Aliasing is the phenomenon of a high-frequency component in the spectrum of the signal taking on the identity of a lower frequency in the spectrum of its sampled version. Digital Communication Systems 2012 R.Sokullu

(a) Spectrum of a signal. (b) Spectrum of an undersampled version of the signal exhibiting the aliasing phenomenon. Figure 3.3 Digital Communication Systems 2012 R.Sokullu

Practically there are two possible engineering solutions: prior to sampling, a low-pass anti-aliasing filter is used to attenuate the high-frequency components that are not essential to the information baring signal. the filtered signal is sampled at a rate slightly higher than the Nyquist rate. Note: This also makes the design of the reconstructing filter easier. Digital Communication Systems 2012 R.Sokullu

Anti-alias filtered spectrum of an information-bearing signal. (b) Spectrum of instantaneously sampled version of the signal, assuming the use of a sampling rate greater than the Nyquist rate. (c) Magnitude response of reconstruction filter. Figure 3.4 Digital Communication Systems 2012 R.Sokullu

The reconstruction filter is low-pass, pass-band –W to +W. The transition band of the filter is fs- W where fs is the sampling rate. Digital Communication Systems 2012 R.Sokullu

3.3 Pulse Amplitude Modulation
Definition: In Pulse Amplitude Modulation (PAM) the amplitudes of regularly spaced pulses are varied in accordance with the corresponding sample values of the continuous message signal; Note: Pulses can be rectangular or some other form. Digital Communication Systems 2012 R.Sokullu

Flat-top samples, representing an analog signal.
Figure 3.5 Digital Communication Systems 2012 R.Sokullu

PAM Steps in realizing PAM: Instantaneous sampling of the message signal every Ts seconds, with sampling rate fs chosen according to the sampling theorem. Lengthening the duration of each sample to obtain a constant value of T (duration of pulses). These two are known as “sample and hold”. Question is: how long should be the pulses (T)? Digital Communication Systems 2012 R.Sokullu

Assume: s(t) sequence of flat-top pulses generated as described. where Ts is the sampling period, m(nTs) is the sample value at time t=nTs standard rectangular pulse is represented by: by definition the instantaneously sampled version of m(t) is: time-shifted delta function Digital Communication Systems 2012 R.Sokullu

after convolution and applying the sifting property of the delta function we get: Digital Communication Systems 2012 R.Sokullu

The result in the previous slide means that (compare 3.10 and 3.14) the PAM signal s(t) is mathematically represented by 3.15: Digital Communication Systems 2012 R.Sokullu

After taking FT on both sides we get: Using formula 3.2 for the relation between Mδ(f) and M(f), the FT of the original message m(t) we can write: Finally, after substitution of 3.16 into 3.17 we get which represents the FT of the PAM signal s(t). Digital Communication Systems 2012 R.Sokullu

Second part: recovery procedure assume that the message is limited to bandwidth W and the sampling rate is fs which is higher than the Nyquist rate. pass s(t) through a low-pass filter whose frequency response is defined in 3.4c the result, according to 3.18 is M(f)H(f), which is equal to passing the original signal m(t) through another low-pass filter with frequency response H(f). Fig. 3.4 Digital Communication Systems 2012 R.Sokullu

To determine H(f) we use the FT of a rectangular pulse, plotted on fig. 3.6a and 3.6b: By using flat-top samples to generate a PAM signal we introduce amplitude distortion and delay of T/2 This distortion is known as the aperture effect. This distortion is corrected by the use of an equalizer after the low-pass filters to compensate for the aperture effect. The magnitude response of the equalizer is ideally: Digital Communication Systems 2012 R.Sokullu

Rectangular pulse h(t). (b) Spectrum H(f), made up of the magnitude |H(f)|, and phase arg[H(f)] Figure 3.6 Digital Communication Systems 2012 R.Sokullu

Conclusion on PAM: 1. Transmission of a PAM signal imposes strict requirements on the magnitude and phase responses of the channel, because of the relatively short duration of the transmitted pulses. 2. Noise performance can never be better than a base-band signal transmission. 3. PAM is used only for time division multiplexing. Later on for long distance transmission another subsequent pulse modulation is used. Digital Communication Systems 2012 R.Sokullu

3.4 Other Forms of Pulse Modulation
Rough comparison between CW modulation and pulse modulation shows that latter inherently needs more bandwidth. This bandwidth can be used for improving noise performance. Such additional improvement is achieved by representing the sample values of the message signal by some other parameter of the pulse (different than amplitude): pulse duration (width) modulation (PDM) – samples are used to vary the duration of the individual pulses. pulse-position modulation (PPM) – position of the pulse, relative to its un-modulated time of occurrence in accordance with the message signal. Digital Communication Systems 2012 R.Sokullu

Illustrating two different forms of pulse-time modulation for the case of a sinusoidal modulating wave. (a) Modulating wave. (b) Pulse carrier. (c) PDM wave. (d) PPM wave. Figure 3.8 Digital Communication Systems 2012 R.Sokullu

Comparison: 1. In PDM long pulses require more power, so PPM is more effective. 2. Additive noise has no effect on the position of the pulse if it is perfectly rectangular (ideal) but in reality pulses are not so PPM is affected by channel noise. 3. As in CW systems the noise performance and comparison can be done using the output signal-to-noise ratio or the figure of merit. 4. Assuming the average power of the channel noise is small compared to the peak pulse power, the figure of merit for a PPM system is proportional to the square of the transmission bandwidth BT, normalized with respect to the message bandwidth W. 5. In bad noise conditions the PPM systems suffer a threshold of its own – loss of wanted message signal. Digital Communication Systems 2012 R.Sokullu

3.5 Bandwidth-Noise Trade-Off
As far as the analog pulse modulation schemes are concerned the pulse position modulation exhibits optimum noise performance. In comparison with CW modulation schemes it is close to the FM systems. both systems have a figure of merit proportional to the square of the transmission bandwidth BT normalized with respect to the message bandwidth. both systems exhibit a threshold effect as the signal-to-noise ratio is reduced. can we do better – yes but not with analog methods…. Digital Communication Systems 2012 R.Sokullu

what is required is discrete representation in both time and amplitude. discrete in time – sampling discrete in amplitude – quantization Digital Communication Systems 2012 R.Sokullu

3.6 Quantization Process For a continuous signal (voice, music) the samples have a continuous amplitude range. But humans can detect only finite intensity differences So an original signal can be approximated, without loss of perception, by a signal constructed of discrete amplitudes selected on a min error basis. This is the basic condition for the existence of pulse code modulation. Digital Communication Systems 2012 R.Sokullu

Definition: Amplitude quantization is defined as the process of transforming the sample amplitude m(nTs) of a message signal m(t) at time t=nTs into a discrete amplitude of v(nTs) taken from a finite set of possible amplitudes. We assume that the quantization process is memoryless and instantaneous. (This means that the transformation at time t is not affected by earlier or later sample values.) Digital Communication Systems 2012 R.Sokullu

Description of a memoryless quantizer.

Types of quantizers based on the way representation values are distributed and positioned around the origin: unifrom – equally spaced representation levels non-uniform – non-equally; considered later mid-read – origin lies in the middle of a read; mid-rise – origin lies in the middle of the rising part of the staircase graph symmetric about the origin Digital Communication Systems 2012 R.Sokullu

Two types of quantization: (a) midtread and (b) midrise.

Quantization Noise Definition: The error caused by the difference between the input signal m and the output signal v is referred to as quantization noise. Digital Communication Systems 2012 R.Sokullu

Illustration of the quantization process.

The model Assume: input value m, which is the sample value of a zero-mean RV M; output value v which is the sample value of a RV V; quantizer g(*) that maps the continuous RV M into a discrete RV V; respective samples of m and v are connected with the following relation: or Digital Communication Systems 2012 R.Sokullu

We are trying to evaluate the quantization error Q. zero mean because the input is zero mean for the output signal-to-noise (quantization) ratio we need the mean square value of the quantization error Q. the amplitude of m varies (-mmax, mmax); then for uniform quantizer the step size is given by: with L being the total number of representation levels; for uniform quantizer the error is bounded by –Δ/2≤q≤Δ/2 if step size is small Q is uniformly distributed (L large) Digital Communication Systems 2012 R.Sokullu

as mean is zero, variance is: Digital Communication Systems 2012 R.Sokullu

hidden in Δ is the number of levels used, which directly influences the error. typically an L-ary number k, denoting the kth representation level of the quantizer is transmitted to the receiver in binary form. Let R denote the number of bits per sample used in the binary code. Digital Communication Systems 2012 R.Sokullu

so for the step size we get and for the variance If P denotes the average power of the message signal m(t) we can find the output signal-to-noise ratio as: Digital Communication Systems 2012 R.Sokullu

Conclusion: The output SNR of the quantizer increases exponentially with increasing the number of bits per sample, R. Increasing R means increase in BT. So, using binary code for the representation of a message signal provides a more efficient method for the trade-off of increased bandwidth for improved noise performance than either FM or PPM. Note: FM and PPM are limited by receiver noise, while quantization is limited by quantization noise. Digital Communication Systems 2012 R.Sokullu

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