Chapter 3 Pulse Modulation CHAPTER 3 PULSE MODULATION

Chapter 3: Pulse Modulation CHAPTER 3 PULSE MODULATION Digital Communication Systems 2012 R. Sokullu 1/68

Chapter 3: Pulse Modulation 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 2/68

Chapter 3: Pulse Modulation 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 3/68

Chapter 3: Pulse Modulation Comm. Systems 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 4/68

Chapter 3: Pulse Modulation Comm. Systems 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 5/68

Chapter 3: Pulse Modulation 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 6/68

Chapter 3: Pulse Modulation 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 7/68

Chapter 3: Pulse Modulation Motivation and Development • Digital transmission – 1960 s • Real application – after 1970 s – 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 8/68

Chapter 3: Pulse Modulation • 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 9/68

Chapter 3: Pulse Modulation Digital Communication Systems 2012 R. Sokullu 10/68

Chapter 3: Pulse Modulation 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 11/68

Chapter 3: Pulse Modulation 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 12/68

Chapter 3: Pulse Modulation 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 13/68

Chapter 3: Pulse Modulation 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 14/68

Chapter 3: Pulse Modulation Digital Interface • Interface between source coding/channel coding – issues continuity, rate etc. – continuous sources – packet sources – complex combinations protocols discussed in details in Data Communications course Here: min number of bits from source and max transmission speed over channel source coder rate = channel encoder rate (source-channel coding theorem) Digital Communication Systems 2012 R. Sokullu 15/68

Chapter 3: Pulse Modulation 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 16/68

Chapter 3: Pulse Modulation 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 17/68

Chapter 3: Pulse Modulation • 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(n. Ts)], 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(n. Ts)]. Digital Communication Systems 2012 R. Sokullu 18/68

Chapter 3: Pulse Modulation The sampling process. (a) Analog signal. (b) Instantaneously sampled version of the analog signal. Figure 3. 1 Digital Communication Systems 2012 R. Sokullu 19/68

Chapter 3: Pulse Modulation • 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(n. Ts) 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(n. Ts)/Δt. Digital Communication Systems 2012 R. Sokullu 20/68

Chapter 3: Pulse Modulation • Knowing that the uniform sampling of a continuoustime 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 21/68

Chapter 3: Pulse Modulation • 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 22/68

Chapter 3: Pulse Modulation (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 23/68

Chapter 3: Pulse Modulation • 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 24/68

Chapter 3: Pulse Modulation • 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 25/68

Chapter 3: Pulse Modulation Conclusion: • 1. If the sample values g(n/2 W) 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/2 W) for -∞ < n <+∞. Digital Communication Systems 2012 R. Sokullu 26/68

Chapter 3: Pulse Modulation 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 27/68

Chapter 3: Pulse Modulation • This is an interpolation formula for reconstructing the original signal g(t) from a sequence of sample values [g(n/2 W)]. • The sinc function sinc(2 Wt) 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 28/68

Chapter 3: Pulse Modulation 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 1/2 W (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 2 W samples per second. Digital Communication Systems 2012 R. Sokullu 29/68

Chapter 3: Pulse Modulation Note: • The sampling rate of 2 W for a signal of bandwidth W Hz, is called the Nyquist rate; • Its reciprocal 1/2 W (seconds) is called the Nyquist interval; Digital Communication Systems 2012 R. Sokullu 30/68

Chapter 3: Pulse Modulation • 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 highfrequency 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 31/68

Chapter 3: Pulse Modulation (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 32/68

Chapter 3: Pulse Modulation • 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 33/68

Chapter 3: Pulse Modulation (a) 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. Digital Communication Systems 2012 R. Sokullu Figure 3. 4 34/68

Chapter 3: Pulse Modulation • 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 35/68

Chapter 3: Pulse Modulation 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 36/68

Chapter 3: Pulse Modulation 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 37/68

Chapter 3: Pulse Modulation Flat-top samples, representing an analog signal. Figure 3. 5 Digital Communication Systems 2012 R. Sokullu 38/68

Chapter 3: Pulse Modulation PAM • Steps in realizing PAM: 1. Instantaneous sampling of the message signal every Ts seconds, with sampling rate fs chosen according to the sampling theorem. 2. Lengthening the duration of each sample to obtain a constant value of T (duration of pulses). 3. These two are known as “sample and hold”. 4. Question is: how long should be the pulses (T)? Digital Communication Systems 2012 R. Sokullu 39/68

• Assume: Chapter 3: Pulse Modulation – s(t) sequence of flat-top pulses generated as described. – where Ts is the sampling period, m(n. Ts) is the sample value at time t=n. Ts – 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 40/68

Chapter 3: Pulse Modulation • after convolution and applying the sifting property of the delta function we get: Digital Communication Systems 2012 R. Sokullu 41/68

Chapter 3: Pulse Modulation • 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 42/68

Chapter 3: Pulse Modulation • 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 43/68

Chapter 3: Pulse Modulation • 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. 4 c • 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 44/68

Chapter 3: Pulse Modulation • To determine H(f) we use the FT of a rectangular pulse, plotted on fig. 3. 6 a and 3. 6 b: • 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 45/68

Chapter 3: Pulse Modulation (a) 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 46/68

Chapter 3: Pulse Modulation 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 baseband 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 47/68

Chapter 3: Pulse Modulation 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 48/68

Chapter 3: Pulse Modulation 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 49/68

Chapter 3: Pulse Modulation 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 50/68

Chapter 3: Pulse Modulation 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 51/68

Chapter 3: Pulse Modulation 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 52/68

Chapter 3: Pulse Modulation 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-tonoise ratio is reduced. – can we do better – yes but not with analog methods…. Digital Communication Systems 2012 R. Sokullu 53/68

Chapter 3: Pulse Modulation • 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 54/68

Chapter 3: Pulse Modulation 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 55/68

Chapter 3: Pulse Modulation 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 56/68

Chapter 3: Pulse Modulation • Definition: Amplitude quantization is defined as the process of transforming the sample amplitude m(n. Ts) of a message signal m(t) at time t=n. Ts into a discrete amplitude of v(n. Ts) 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 57/68

Chapter 3: Pulse Modulation Description of a memoryless quantizer. Figure 3. 9 Digital Communication Systems 2012 R. Sokullu 58/68

Chapter 3: Pulse Modulation 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 59/68

Chapter 3: Pulse Modulation Two types of quantization: (a) midtread and (b) midrise. Figure 3. 10 Digital Communication Systems 2012 R. Sokullu 60/68

Chapter 3: Pulse Modulation 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 61/68

Chapter 3: Pulse Modulation Illustration of the quantization process. Figure 3. 11 Digital Communication Systems 2012 R. Sokullu 62/68

Chapter 3: Pulse Modulation 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 63/68

Chapter 3: Pulse Modulation • 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 64/68

Chapter 3: Pulse Modulation • as mean is zero, variance is: Digital Communication Systems 2012 R. Sokullu 65/68

Chapter 3: Pulse Modulation • 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 66/68

Chapter 3: Pulse Modulation • 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 67/68

Chapter 3: Pulse Modulation 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 68/68