CHAPTER 4. PULSE MODULATION Part 2

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1 CHAPTER 4 PULSE MODULATION Part 2

2 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 2

3 Digital Pulse Modulation 3

4 Analog input signal Sample at discrete time instants Analog pulse modulation, PAM signal Digital pulse modulation, PCM code 4

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. 5

6 PCM Block Diagram Most common form of analog to digital modulation 6

7 Sampling, Quantizing, and Encoding The PCM signal is generated by carrying out three basic operations: 1. Sampling 2. Quantizing 3. 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. 7

8 Analog Input Signal Sample ADC Quantize Encode Digital Output Signal

9 Analog Input Signal ADC Sample Quantize Encode Sampling Makes the signal discrete in time. the minimum sample frequency such that the signal can be reconstructed without distortion, fs >= 2fmax 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. Digital Output Signal Eeng 360 9

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. n L = 2 = log L n 2 10

11 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 11

12 Quantization Example Analogue signal Sampling TIMING Quantization levels. Quantized to 5-levels Quantization levels Quantized 10-levels 12

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

14 Types of Uniform Quantization 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 14

15 -8 Output sample X Q -6-4 Dynamic Range: (-8, 8) Quantization Characteristic Input sample X Example: Uniform n =3 bit quantizer L=8 and X Q = {±1,±3,±5,±7} 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. 15 Eeng

16 Dynamic Range (DR) Largest possible magnitude/smallest possible magnitude. DR = V V max = min Vmax resolution DR = 2 n 1 DR ( db) = 20 log( DR) 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 DR( db) 6n for n > 4 16

17 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. Coding Efficiency = Minimum number of bits x 100 Actual number of PCM bits 17

18 Example 1 1. Calculate the dynamic range for a linear PCM system using 16-bit quantizing. 2. Calculate the number of bits in PCM code if the DR = db. Determine the coding efficiency in this case. 18

19 The quantization quantum = the magnitude difference between adjacent steps, Δv 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. 19

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. 20

21 Quantization Noise The process of quantization can be interpreted as an additive noise process. Signal X Quantized Signal, X Q Quantization Noise, n Q The signal to quantization noise ratio (SNR) Q=S/N is given as: ( SNR) Q = Average Power{ X} Average Power{ n } Q 21

22 Signal to Quantization Noise Ratio (SQR) The worst possible signal voltage-to-quantization noise voltage ratio (SQR) occurs when input signal is at its minimum amplitude. SQR is directly proportional to resolution. The worst-case voltage SQR SQR = (min) resolution Q e 22

23 SQR for a maximum input signal SQR (max) = V max Q e R =resistance (ohm) v = rms signal voltage q = quantization interval Qe = quantization error The signal power-to-quantizing noise power ratio SQR ( db) = 10log = 10log average signal power average quantization noise power ( q 2 v 2 12 R ) R = 10log v q = log v q 23

24 Example 2 1. Calculate the SQR (db) if the input signal = 2 Vrms and the quantization noise magnitudes = 0.02 V. 2. Determine the voltage of the input signals if the SQR = db and q =0.2 V. 24

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 X Q Example: Nonuniform 3 bit quantizer Input sample X

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.

26 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. 26

27 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. 27

28 Companding Functions 28

29 Method of Companding! For the compression, two laws are adopted: the µ-law in US and Japan and the A-law in Europe.! µ-law!! A-law V V out out = V max ln( 1 + ln(1 + µ Vin ( ) A Vmax Vmax 1+ ln A = 1+ ln( A Vmax 1+ ln A Vin ( ) Vin ( ) µ )! The typical values used in practice are: µ=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. V max Vin 0 Vmax 1 V A V in max 1 A 1 V max = Max uncompressed analog input voltage V in = amplitude of the input signal at a particular of instant time V out = compressed output amplitude A,µ = parameter define the amount of compression 29

30 Cont d... µ-law A-law 30

31 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. 31

32 Example 4 A companding system with µ = 200 is used to compand -4V to 4V signal. Calculate the system output voltage for V in = -4, -2, 0, 2 and 4V. Equation: V out = V max ln( 1 + ln(1 + µ Vin ( ) µ ) V max V in (V) V out (V)

33 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. 33

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/x rms ) for Uniform Quantizer V is the peak signal level and x rms is the rms value α = log[ln(1 + µ)] for µ-law companding α = log[1 + Ln A] for A-law companding 34

35 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. n n = log2 L = 2 L 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. 35 Eeng

PCM encoding example Levels are encoded using this table Table:

Quantization and digitalization of a signal.

36 PCM encoding example Levels are encoded using this table Table: Quantization levels with belonging code words L=8 Chart 1. Quantization and digitalization of a signal. Signal is quantized in 11 time points & 8 quantization segments. Chart 2. Process of restoring a signal. PCM encoded signal in binary form: Total of 33 bits were used to encode a signal 36

37 PCM Example

38 Nonlinear Encoding Quantization levels not evenly spaced Same concept as non-uniform quantization Reduces overall signal distortion Can also be done by companding 38

39 PCM Line Speed The data rate at which serial PCM bits are clocked out of the PCM encoder onto the transmission line. line speed = samples second Where Line speed = the transmission rate in bits per second Sample/second = sample rate, f s Bits/sample = no of bits in the compressed PCM code Line speed also known as bit rate X bits sample 39

40 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. 40

41 Channel Bandwidth The channel bandwidth, B required to transmit a pulse is given by B = κnw Where κ = a constant with a value between 1 to 2 n = number of bits W = signal bandwidth Channel BW = transmission BW 41

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=nf s Where n is the number of bits in the PCM word (M=2 n ) and f s is the sampling rate. 42

43 For no aliasing case (f s 2B), the MINIMUM Bandwidth of PCM B pcm(min) is: B pcm(min) = R/2 = nf s/ /2 The Minimum Bandwidth of nf s/ /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: B pcm = R = nf s 43

44 Example 6 A signal with a bandwidth of 4.2 MHz is transmitted using binary PCM. The number of representation levels is 512. Calculate (a)the code word length (b)the bit rate (c)the transmission bandwidth, assuming that, κ = 2 (d)find the SQR in db for the signal given that peak signal voltage is 5Vp 44

45 PCM transmitter/receiver Analog signal LPF BW=B Bandlimited Analog signal Sampler & Hold Flat-top PAM signal PCM signal Encoder Quantized PAM signal Quantizer No. of levels=m Channel, Telephone lines with regenerative repeater PCM signal Decoder Quantized PAM signal Reconstruction LPF Analog Signal output 45

46 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: 46

47 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. P e 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. 47

48 Effects of Quantizing Noise If P e 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 = 2 n α = 4.77 for peak SNR α = 0 for average SNR 48

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 B T and SNR. Uniform format for different kind of baseband signals. Flexible TDM. 49

50 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 B T. With cost-effective implementations, the cost issue no longer a problem of concern. W i t h t h e a v a i l a b i l i t y o f w i d e - b a n d communication channels and the use of sophisticated data compression techniques, the large bandwidth is not a serious problem. 50

51 Application: PCM in Wired Telephony Voice circuit bandwidth is 3400 Hz. Sampling rate is 8 KHz (samples are 125 µs 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. 51

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 52

53 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. 53

54 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 54

Pulse Code Modulation

Pulse Code Modulation Modulation is the process of varying one or more parameters of a carrier signal in accordance with the instantaneous values of the message signal. The message signal is the signal