Chapter 6m

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  • In the previous chapters, we have considered applications of continuous wave (CW) modulation where we deal with analog signal such as the AM and FM signals. In this chapter, we will focus on digital transmission of analog signal. There is an increase use of digital communication systems To transmit analog messages such as voice and video signals by digital means, the signal has to be converted to a digital signal. This process is known as analog to digital conversion . It I also referred to digital pulse modulation .
  • The (bandlimited) signal is first sampled, thus converting the analog signal into a discrete-time continuous-amplitude signal. The amplitude of each signal sample is quantized into one of 2B levels, where B is the number of bits used to represent a sample. The discrete amplitude levels are represented or encoded into distinct binary words each of length B bits.
  • A single-channel PCM transmission system.
  • f s : sampling frequency or sampling rate x(n)= x(t)  (t-nT s )
  • Note s(t) is periodic with period T s and its Fourier series representation is c k = 1/T s , Thus
  • From figure in previous slide, The signal can be recovered by passing the x(n) through a LPF. The LPF has to be ideal to recover completely the signal In practice, the LPF is not ideal.
  • What conditions should be considered in order to avoid any distortion in the recovered signal?
  • Effects of changing the sampling rate If T s decreases, f s increases and all replicas of X(f) moves farther apart. . If T increases, f s decreases and all replicas of X(f) moves closer. . Soon a point is reached beyond which a reduction in the sampling rate will result in overlap between spectral densities. This point is reached when . f s =1/T s = 2B
  • Let B = f max When f s < 2f max , there will be aliasing or folding over of image frequencies into the desired frequency band.
  • A signal m ( t ) is called a band-limited signal if M ( f ) = 0 for | f | > f max where f max is the highest-frequency spectral component of m ( t ). The sampling theorem specifies the rate at which an analog signal should be sampled to ensure that all the relevant information contained in the signal is retained by sampling . Sampling at less than the rate specified by the sampling theorem leads to a folding over or aliasing of image frequencies into the desired frequency band.
  • In proving the sampling theorem, we assumed an ideal samples obtained by multiplying a signal x(t) by an impulses train which is physically non-existent. These signals are illustrated in the next slides
  • If x(t) is an analog band-limited signal, the PAM signal that uses natural sampling is x s (t)= x(t)*s(t), Figure c) shows the resulting PAM signal for a duty cycle d=  /T s =1/3 The Pam waveform with natural sampling is relatively easy to generate, since it only requires the use of an analog switch that is readily available in CMOS hardware. See next slide for an example.
  • Baseband analog waveform Impulse train sampling waveform Resulting PAM signal duty cycle d=  /T s =1/3 The sampled waveform, produced by practical sampling devices that are of sample and hold types, has the form
  • Let’s discuss each case in detail The sampled PAM is x s (t)= s(t) x(t), then X s (f)= X(f)*S(f) Note s(t) is periodic with period T s and its Fourier series representation is We can show that c k = d sinc (nd) where d=  /T s , then Thus
  • The spectrum of the PAM signal is given as a function of the spectrum of x(t) as illustrated in next slide. For the Figure, we use f s = 4B and d=1/3.
  • For X(f) The sampling theorem still applies f s ≥ 2 B. For signal reconstruction, we recover x(t) from the x s (t) by passing the signal through a LPF. For unit gain LPF, we can use an amplifier to compensate for the gain factor d . Compare the bandwidth of X s (f) and X(f)? Compare the bandwidth of X s (f) and X s,ideal (f)? Note, we can control B Xs by varying f s and 
  • The sampled waveform, produced by practical sampling devices that are the sample and hold types, has the form shown above. This type of sampling is known as flat-top sampling or sample and Hold .
  • Let h(t)=rect(t/  ), Note that Then Thus, x s (t)=h(t)* x s (t) ideal Hence, X s (f)=H(f)  X s (f) ideal
  • Example of a sampled signal spectrum X s (f). we see that by using flat-top sampling we have introduced amplitude distortion, and the primary effect is an attenuation of high-frequency components. This effect is known as the aperture effect . If the pulse duration  is chosen such that  <<T s , then H ( f) is essentially constant over the baseband and and it is just a LPF. Else, we can use a LPF such that H eq ( f)= 1/ H (f). The LPF is called an equalization filter .
  • Pulse-amplitude modulation is now rarely used, having been largely superseded by pulse-code modulation , and, more recently, by pulse-position modulation . In particular, all telephone modems faster than 300 bit/s use quadrature amplitude modulation (QAM). (QAM uses a two-dimensional constellation ). It shall be noted, however, that widely popular Ethernet communication standard is a good example of PAM usage. In particular, 100BASE-T2 (running at 100Mb/s) Ethernet medium is utilizing 5 level PAM modulation running at 25 megapulses/sec over two wire pairs. Special technique is used to reduce inter-symbol interference between the unshielded pairs. Later, 1000BASE-T medium raised the bar to use 4 pairs of wire running each at 125 megapulses/sec to achieve 1000Mb/s data rates, still utilizing PAM5 for each pair. PBX : Short for p rivate b ranch e x change, a private telephone network used within an enterprise. Users of the PBX share a certain number of outside lines for making telephone calls external to the PBX.
  • A PWM waveform consists of a sequence of pulses, each of which have a width that is proportional to the values of a message signal at the sampling instants. Since the width of a pulse cannot be negative, a dc bias is added to the message signal prior to modulation. Pulse-width modulation of a signal or power source involves the modulation of its duty cycle to either convey information over a communications channel or control the amount of power sent to a load . PWM is unsuitable for TDM transmission due to the varying pulse width .
  • A PPM signal consists of pulses in which the pulse displacement from a specific time reference is proportional to the sample values of the information bearing signal. Also known as Pulse Time Modulation, Pulse-position modulation is a form of signal modulation in which M message bits are encoded by transmitting a single pulse in one of 2 M possible time-shifts. This is repeated every T seconds, such that the transmitted bit rate is M/T bits per second. It is primarily useful for optical communications systems, where there tends to be little or no multipath interference A dc bias must be added to n th message signal prior to modulation so that the input to the PPM modulator is non-negative for all values of time. Again, the random arrival rate of pulses makes this unsuitable for transmission using TDM techniques.
  • After sampling, the analogue amplitude value of each sampled ( PAM) signal is quantized into one of a number of L discrete levels. The result is a quantized PAM signal. A codeword can then be used to designate each level at each sample time. This procedure is referred to as “Pulse Code Modulation”
  • Pulse code modulation is very popular because of the many advantages it offers. These include: Inexpensive digital circuitry may be used in the system. All-digital transmission. PCM signals derived from analogue signals may be timedivision multiplexed with data from digital computers and transmitted over a common high-speed channel. Further digital signal processing such as encryption is possible. Errors may be minimised by appropriate coding of the signals. Signals may be regularly reshaped or regenerated using repeaters at appropriate intervals.
  • An analogue message m ( t ) is first sampled at or above the Nyquist sampling rate. These sampled signals are then converted into a finite number of discrete amplitude levels. The conversion process is called quantisation . Figure 14.2 shows how an analogue message is converted into 8 amplitude levels with equal spacing by an 8-level quantiser.
  • Pulse code modulation is very popular because of the many advantages it offers. These include: Inexpensive digital circuitry may be used in the system. All-digital transmission. PCM signals derived from analogue signals may be timedivision multiplexed with data from digital computers and transmitted over a common high-speed channel. Further digital signal processing such as encryption is possible. Errors may be minimised by appropriate coding of the signals. Signals may be regularly reshaped or regenerated using repeaters at appropriate intervals.
  • A PCM system contains three main blocks: PCM transmitter Transmission path Receiver In the receiver, the coded signal is decoded and reconstructed to obtain the recovered signal. In the PCM transmitter, there are 3 main stages: Sampling (PAM, Sample/Hold is commonly used) Quantization Encoding
  • Sampler converts the analog signal into a discrete one that is discrete in time but continuous in amplitude Quantizer converts the discrete time signal into a sampled and quantized signal that is discrete in both time and amplitude. The sample level is rounded off to the closest allowed level (only a fixed finite number of levels are allowed) Due to sampling and quantizing the analog signal is mapped to a finite discrete set of sample values or quantized samples
  • Consider the analog signal m(t)= 7*sin(2*pi*f m *t); where f m =200Hz. Let the sampling frequency be f s = 10 f m The amplitude of m s (t) can be confined to the range [-m q , m q ] (example [-8,8]), This range can be divided in L zones, each of step  . In this case, if L=8, then  =2.  = 2 m q /L The sample amplitude value is approximated by the midpoint of the interval in which it lies. The input-output characteristics of this quantizer is shown in next slide.
  • The input-output characteristics of this quantizer is shown in next slide. The input is m s (t) and the output is m q (t) This quantization discussed so far is said to be uniform since all of the steps  are of equal size
  • The quantizing error, q(t)=m s (t)- m q (t) From the diagram, note that |q(t)|   /2.
  • Since we are approximating the analog sample values by using a finite number of levels ( L =8 in this illustration), error is introduced into the recovered output analog signal because of the quantizing effect. The error waveform is illustrated in Figure
  • Those who are familiar with the theory of probability Where p(q) is the probability density. Since  =2 m q /L, then we get the result above. S 0 = <m 2 (t)> and N o =<q 2 (t)>, thus S o /N o is expressed as shown above.
  • Solution: In this case, m q =A , Thus (S/N) o = 3 L 2 (0.5 A 2) /A = 3 L 2 / 2
  • Uniform PCM is good for uniform signal distribution, but it is bad for nonuniform distributions. For example, in speech communication it is found (statistically) that smaller amplitudes predominate in speech and large amplitudes are relatively rare. The uniform quantizing is thus wasteful for speech signals. See next slide for Figure
  • Among several choices, two compression laws have been accepted as desirable standards by the CCITT: the  -law compand used in North America and Japan, and the A-law compand used in Europe .
  • The  -law is given by the above formula. For positive amplitude m >0 In the United States, Canada, and Japan, the telephone companies use a compression parameter  =255 in their PCM systems for L=128 levels.
  • The compression characteristic of the  -law is shown in Figure for several values of the compression parameter  , The horizontal axis is the normalized input signal |m(kT s )/m q |, and the vertical axis is the output compressed signal. The parameter  control the compression factor It is noted that  = 0 corresponds to linear amplification (uniform quantization overall). In the United States, Canada, and Japan, the telephone companies use a  =255 compression characteristic in their PCM systems.
  • The A-law is given by the above formula. In Europe, the telephone companies use a A=87.6 compression characteristic in their PCM systems for L=128 levels.
  • The compression characteristic of the A-law is shown in Figure for several values of the compression parameter A, The horizontal axis is the normalized input signal |m(kT s )/m q |, and the vertical axis is the output compressed signal. The parameter A control the compression factor It is noted that A = 1 corresponds to linear amplification (uniform quantization overall). In Europe, the telephone companies use A=87.6 compression characteristic in their PCM systems.
  • The expander characteristic is the inverse of the compression characteristic. Practical application: A logarithmic compressor can be realized by a combination of semiconductor diodes and resistors.
  • Output S/N of 8-bits PCM systems with and without companding.
  • The decimal-to-binary conversion can be done in various ways. Next slide shows two possible coding rules (binary and gray coding) for converting a 16-level sample into 4 binary digits.
  • SNR is controlled by the PCM bandwidth since it is controlled by the number of bits.
  • Output S/N of 8-bits PCM systems with and without companding.
  • For the second system: Gain= 9 dB= 10 0.9 = 8 If P in = 6 W, What is P out ?
  • A good question to ask is: What is the spectrum of a PCM signal? The PCM signal is a nonlinear function of the input signal. Consequently, the spectrum of the PCM signal is not directly related to the spectrum of the input analog signal. It can be shown that the spectrum of the PCM signal depends on the bit rate, the correlation of the PCM data, and on the PCM waveform pulse shape (usually rectangular) used to describe the bits The equality is obtained if a ( sin x)/x type of pulse shape is used to generate the PCM waveform. The exact spectrum for the PCM waveform will depend on the pulse shape that is used as well as on the type of line encoding. [2] Couch, L.W., Digital and Analog Communication Systems, 5th ed., Prentice Hall, Upper Saddle River, NJ, 1997. [3] Couch, L.W., Modern Communication Systems: Principles and Applications, Macmillan Publishing, New York, NY, 1995.
  • Each quantized sample is, thus encoded into n bits. Note, a signal m(t) band-limited to B Hz requires a minimum of 2B samples per second for the sampling stage. For each sample we will assign n bits. Thus, we require a total 2nB bits per second (bps). Thus the minimum transmission bandwidth required to transmit the PCM signal B T = B ch = n B Hz.
  • Peak quantization SNR: SQNR=3 L 2 = 53 dB Another example: CD player (see the text by Proakis and Salehi
  • Samples of a band-limited signal are correlated  previous sample gives information about the next one. Example: if previous samples are small, the next one will be small with high probability. For the same SNR as the PCM, the DPCM can reduce the number of bits used (hence transmission bandwidth). For the same number of bits as the PCM, the DPCM can reduce the quantizer SNR.
  • Thus m q (k)= m(k)+ q(k) is the quantized version of m(k)
  • Since only samples of a message signal are transmitted, the channel is occupied only for a short time slot in pulse modulation systems. Consequently, samples of N message signals may be transmitted over the same channel. Message signals 1 ; 2 ; … ;N are separated in the time domain. The commutator is usually implemented by using electronic switching circuitry. Note, the multiplexed signal is the input to the pulse modulator
  • Figures shown in this slide and next one, show a complete 10-channel PCM system and its associated signal shapes at various transmitting points. Clearly, the bandwidth required at the output of the binary encoder is three times the bandwidth required at the input and the output of the quantizer. Thus, a binary PCM system requires more transmission bandwidth than the PAM and the quantized PAM systems.
  • Figure shows signal shapes at various transmitting points shown in the pervious slide.
  • f s : sampling frequency for each signal and T s : sampling period Minimum sampling frequency for each signal f s =2B T TDM : time spacing between adjacent samples= T s /N Minimum sampling frequency for the multiplexed signal f TDM = N f s = 2 N B If n bits are used, then the data rate is R= 2 n N B Thus transmission bandwidth is B T = n N B
  • Consider a multiplexed PAM wave generated by the commutator. The time interval T F containing one sample from each message signal is called a frame. Synchronization must be established and maintained between the commutator and decommutator . Generally, an extra pulse (called marker ) or a special sequence of pulses are transmitted at the beginning of each frame to help the clock recovery circuit to establish the synchronization
  • Jumbled: mixed
  • Solution: We require f s ≥ 2B=6400 and R b ≥ n f s Thus n  R b / f s  R b / 6400 since 1/ f s  1/ 6400 n  36000/ 6400=5.6 So, we can have n=5, L= 2 5 =32, and f s =R b /n=7.2 kHz
  • Solution: Let 2 m q be peak-to-peak value, the peak error is then Max (error)= 0.005*2*m q =0.01 m q The peak-to-peak error is the maximum step size  =0.02 m q = 2 m q / L Thus L=2 m q /  =2/0.02= 100 Hence the number of bits is n= log 2 (L)=6.69 So n=7
  • Solution: In this case, m q =A , Thus (S/N) o = 3 L 2 (0.5 A 2) /A = 3 L 2 / 2
  • Chapter 6m

    1. 1. Chapter 6 Sampling and Pulse Code Modulation
    2. 2. Outline • introduction • Sampling and sampling theorem • Practical sampling and pulse amplitude modulation (PAM) • Pulse code modulation (PCM) • Differential pulse code modulation (DPCM) • Delta modulation.
    3. 3. Introduction • There is an increase use of digital communication systems • Digital communications offer several important advantages compared to analog communications such as higher performance, higher security and greater flexibility. • digital transmission of analog signals require Analog to digital conversion (AD). • digital pulse modulation
    4. 4. Analog to Digital converter
    5. 5. PCM system
    6. 6. Sampling • A typical method for obtaining a discrete- time sequence x(n) from a continuous- time signal x(t) is through periodic sampling. x(n)= x(nTs), for -∞ < n < ∞ • Ts : sampling period. • fs: sampling frequency or sampling rate s s f T = 1
    7. 7. x(t) s(t) xs (t)
    8. 8. Spectrum of Xs(f) X(f) ∑ ∞ −∞= −= k s s s fkfX T fX )( 1 )(
    9. 9. • Is it possible to reconstruct the analog signal from the sampled valued? Sampler x(t) x(nTs) LPF x(t)
    10. 10. • Given any analog signal, how should we select the sampling period Ts (or the sampling frequency fs) without losing the important information contained in the signal.
    11. 11. Spectrum of Xs(f)
    12. 12. Sampling Theorem • Let m(t) be a real valued band-limited signal having a bandwidth B, and m(nTs) be the sample values of m(t) where n is an integer. • The sampling theorem states that the signal m(t) can be reconstructed from m(nTs) with no distortion if the sampling frequency fs ≥2B • The minimum sampling rate 2B is called the Nyquist sampling rate.
    13. 13. Typical sampling rates for some common applications Application B fs Speech 4 kHz 8 kHz Audio 20 kHz 40 kHz Video 4 MHz 8 MHz
    14. 14. Example Determine the Nyquist rate of the following analog signal and plot the spectrum of the sampled signal for : 1. fs=15oHz 2. fs=300Hz 3. fs=500 Hz x(t) = 3cos(50πt) + 10sin(300πt) - cos(100πt)
    15. 15. To Avoiding aliasing • Band-limiting signals (by filtering) before sampling. • Sampling at a rate that is greater than the Nyquist rate. Anti- Aliasing Filter Sampler fs≥ 2B x(t) xs(t)
    16. 16. Practical Sampling • In practice, we multiply a signal x(t) by a train of pulses of finite width. • There are two types of practical sampling –Natural Sampling (Gating) –Instantaneous Sampling. Also known as flat- top PAM or sample-and-hold.
    17. 17. Natural Sampling s(t)
    18. 18. Generation of PAM with natural sampling X(t)
    19. 19. Another example of natural sampling
    20. 20. Sample-and-Hold τ xs(t)
    21. 21. Sample-and-hold (S/H) circuit.
    22. 22. Natural Sampling (Gating) )()()( tstxtxs = ∑ ∞ −∞=       − = k skTt rectts τ )(
    23. 23. Natural Sampling (Gating): Spectrum • The spectrum (FT) of the sampled (PAM) signal is ( ) sinc( ) ( )s s k X f d kd X f kf ∞ =−∞ = −∑ sT d τ = Duty cylce of s(t)
    24. 24. Natural Sampling (Gating): Spectrum d X( f ) d sinc(kd)
    25. 25. Sample-and-Hold( flat-top sampling) τ xs(t) ∑ ∞ −∞=       − = k s ss kTt rectTkxtx τ )()(
    26. 26. Sample and hold: Spectrum ∑ ∑ ∞ −∞= ∞ −∞= −      =       − = k ss k s ss kTtTkx t rect kTt rectTkxtx )()(* )()( δ τ τ ( ) sin ( ) ( )s s k X f c f X f Kfτ τ ∞ =−∞ = −∑
    27. 27. Sample and hold: Spectrum sinc(τf) X(f)τ
    28. 28. • we see that by using flat-top sampling we have introduced amplitude distortion, and the primary effect is an attenuation of high- frequency components. This effect is known as the aperture effect. • If τ <<Ts, then H( f) represents a LPF. • Else, we can use a LPF such that Heq( f)= 1/H(f) The LPF is called an equalization filter.
    29. 29. Reasons for intentionally lengthening the duration of each sample are: • Reduce the required transmission bandwidth: B is inversely proportional to pulse duration • To get the exact signal value, the transient must fade away
    30. 30. Pulse Modulation • Pulse modulation results when some characteristic of a pulse is made to vary in one- to-one correspondence with the message signal. • A pulse is characterized by three qualities: – Amplitude – Width – Position • Pulse amplitude modulation, Pulse width modulation, and Pulse position modulation
    31. 31. Pulse amplitude modulation (PAM) • In Pulse Amplitude Modulation, a pulse is generated with an amplitude corresponding to that of the modulating waveform. • There are two types of PAM sampling –Natural Sampling (Gating) –Flat-top or sample-and-hold.
    32. 32. PAM System • A system transmitting sample values of the analog signal is called a pulse-amplitude modulation (PAM) system.
    33. 33. • Like AM, PAM is very sensitive to noise. • While PAM was deployed in early AT&T Dimension PBXs, there are no practical implementations in use today. However, PAM is an important first step in a modulation scheme known as Pulse Code Modulation.
    34. 34. Note • PBX: Short for private branch exchange, a private telephone network used within an enterprise. • Users of the PBX share a certain number of outside lines for making telephone calls external to the PBX.
    35. 35. Pulse Width Modulation (PWM) • In PWM, pulses are generated at a regular rate. The length of the pulse is controlled by the modulating signal's amplitude.
    36. 36. Pulse Position Modulation (PPM) • PPM is a scheme where the pulses of equal amplitude are generated at a rate controlled by the modulating signal's amplitude.
    37. 37. Pulse Code Modulation
    38. 38. Advantages of PCM • Inexpensive digital circuitry may be used in the system. • All-digital transmission. • Further digital signal processing such as encryption is possible. • Errors may be minimized by appropriate coding of the signals. • Signals may be regularly reshaped or regenerated using repeaters at appropriate intervals.
    39. 39. A single-channel PCM transmission system
    40. 40. Advantages of PCM • Inexpensive digital circuitry may be used in the system. • All-digital transmission. • Further digital signal processing such as encryption is possible. • Errors may be minimized by appropriate coding of the signals. • Signals may be regularly reshaped or regenerated using repeaters at appropriate intervals.
    41. 41. A single-channel PCM transmission system
    42. 42. Quantization • Quantizer converts the discrete time signal into a sampled and quantized signal that is discrete in both time and amplitude
    43. 43. m(t) and its sampled value m(kTs) 0 0.002 0.004 0.006 0.008 0.01 -8 -6 -4 -2 0 2 4 6 8 ∆
    44. 44. Input-output characteristics of the quantizer Output Input L=8 mq=8-mq=-8
    45. 45. • Quantization can be uniform and nonuniform • The quantization discussed so far is said to be uniform since all of the steps ∆ are of equal size. • Nonuniform quantization uses unequal steps
    46. 46. Uniform Quantization • The amplitude of ms(t) can be confined to the range [-mq, mq] • This range can be divided in L zones, each of step ∆ such that ∆= 2 mq / L • The sample amplitude value is approximated by the midpoint of the interval in which it lies.
    47. 47. Quantization Noise • The difference between the input and output signals of the quantizer becomes the quantizing error or quantizing noise mq(t) mmqq(t)+(t)+ ∆∆/2/2mmqq(t)-(t)- ∆∆/2/2 ms(t)ms(t) 2 )( 2 ∆ ≤≤ ∆ − tq
    48. 48. Quantization Error or Noise
    49. 49. • Assuming that the error is equally likely to lie anywhere in the range (-∆/2, ∆/2), the mean- square quantizing error is given by 12 1 22/ 2/ 22 ∆ = ∆ = ∫ ∆ ∆− dqqq 2 2 2 3L m q q = 2 2 2 )( 3 qo o m tm L N S =
    50. 50. Example • For a full-scale sinusoidal modulating signal m(t)= A cos(ωmt), show that • or 2 3 2 L N S o o = )()(log2076.1 10 dBL N S dBo o +=      2 2 3 q o o o m S L N S =
    51. 51. Nonuniform Quantization • For many classes of signals the uniform quantizing is not efficient. • Example: speech signal has large probability of small values and small probability of large ones. • Solution: allocate more levels for small amplitudes and less for large. Thus, total quantizing noise is greatly reduced
    52. 52. Example of Nonuniform quantization 0 0.005 0.01 0.015 0.02 -6 -4 -2 0 2 4 6
    53. 53. Nonuniform Quantization
    54. 54. • The effect of nonuniform quantizing can be obtained by first passing the analog signal through a compression (nonlinear) amplifier and then into the PCM circuit that uses a uniform quantizer. • At the receiver end, demodulate uniform PCM and expand it. • The technique is called companding. • Two common techniques 1. µ-law companding 2. A-law companding
    55. 55. µ-law Compression Characteristic • where         + + = qm mm y µ µ 1ln )1ln( )sgn( 1≤ qm m
    56. 56. µ-law Compression Characteristic |m/mq| y
    57. 57. Α-law Compression Characteristic • where 1≤ qm m         ≤≤                 + + ≤         + = 1 1 ,ln1 ln1 )sgn( 1 , ln1 1 qq qq m m Am m A A m Am m m m A y
    58. 58. A-law Compression Characteristic |m/mq| y
    59. 59. • The compressed samples must be restored to their original values at the receiver by using an expander with a characteristics complementary to that of the compressor. • The combination of compression and expansion is called companding
    60. 60. • It can be shown that when a µ-law compander is used, the output SNR is • where [ ]2 2 )1ln( 3 µ+ ≈ L N S o o )(2 2 2 tm mq >>µ
    61. 61. Coding of Quantized Samples • The coding process in an A/D converter assigns a unique binary number to each quantization level. For example, we can use binary and gray coding. • A word length of n bits can create L= 2n different binary numbers. • The higher the number of bits, the finer the quantization and the more expensive the device becomes.
    62. 62. Binary and Gray coding of samples.
    63. 63. Output SNR • SNR is controlled by the PCM bandwidth )(6 dBn N S dBo o +=      α c10log10=α [ ]        + = caseeduncompress m tm casecompressed c q , )(3 , )1(ln 3 2 2 2 µ
    64. 64. Comments About dB Scale • The decibel can be a measure of power ratio • It can also be used for measuring power       =      N S N S dB 10log10 mW P P PP W dBm WdBW 1 log10 log10 10 10 = =
    65. 65. Examples Gain= (Pout/Pin) = 2 = +3 dB Pout= 48 mW
    66. 66. Bandwidth of PCM • What is the spectrum of a PCM signal? • The spectrum of the PCM signal depends on the bit rate, the correlation of the PCM data, and on the PCM waveform pulse shape (usually rectangular) used to describe the bits. • The dimensionality theorem [2] shows that the bandwidth of the PCM waveform is bounded by BPCM ≥ R/2 = n fs/2 where R: bit rate
    67. 67. Transmission Bandwidth • For L quantization levels and n bits L= 2n or n= log2L • The bandwidth of the PCM waveform BPCM ≥ n B Hz • Minimum channel bandwidth or transmission bandwidth B = n B Hz
    68. 68. Example: PCM for Telephone System • Telephone spectrum: [300 Hz, 3400 Hz] • Min. sampling frequency: fs,min = 2 Fmax=6.8 kHz • Some guard band is required: fs= 2 Fmax + ∆fg = 8 kHz • n=8-bit codewords are used  L=256. • The transmission rate: R=n* fs =64 kbits/s • Minimum PCM bandwidth: ΒPCM = R/2=32 kHz
    69. 69. Example 6.3 • A signal m(t) of bandwidth B= 4 kHz is transmitted using a binary companded PCM with µ=100. Compare the case of L=64 with the case of L=256 from the point of view of transmission bandwidth and the output SNR.
    70. 70. Differential PCM (DPCM) • Samples of a band-limited signal are correlated. • This can be used to improve PCM performance: to decrease the number of bits used (and, hence, the bandwidth) or to increase the quantization SNR for a given bandwidth. • Main idea: quantize and transmit the difference between two adjacent samples rather than sample values. • Since two adjacent samples are correlated,
    71. 71. DPCM System-Modulator mq(k) d(k) m(k) Quantizer Predictor dq(k) )(ˆ kmq )(ˆ)()( kmkmkd q−= )()()( kqkdkdq += )()(ˆ)( kdkmkm qqq += mq(k)= m(k)+ q(k) is the quantized version of m(k)
    72. 72. DPCM System-Demodulator Predictor Output mq(k) )(ˆ kmq Input dq(k) )(ˆ)()( kmkmkd qqq −= )(ˆ)()( kmkdkm qqq +=
    73. 73. Time Division Multiplexing (TDM)
    74. 74. Ten-Channel PCM System (a) Transmitter (b) Receiver
    75. 75. Signal Shapes
    76. 76. Bandwidth Requirements for TDM • If N band-limited signals are multiplexed each with bandwidth B • The minimum TDM sampling rate is fTDM= 2 N B • If each sample is coded with n bits, then the minimum transmitted data rate is R= 2 n N B • The minimum transmission bandwidth is BT = n N B
    77. 77. TDM: Concept of Framing and Synchronization • The time interval TF containing one sample from each message signal is called a frame. • an extra pulse (called marker) is transmitted for synchronization
    78. 78. Comparison of Time and Frequency Division Multiplexing • Time division multiplexing: Individual TDM channels are assigned to distinct time slots but jumbled together in the frequency domain. Channels are separated in the time domain • Frequency division multiplexing: Individual FDM channels are assigned to distinct frequency regions but jumbled together in the time domain. Channels are separated in the frequency domain
    79. 79. Comparison of Time and Frequency Division Multiplexing • Many of the TDM advantages are technology driven. The digital circuits are much cheaper and easier to implement • In FDM, imperfect bandpass filtering and nonlinear cross-modulation cause cross talk. TDM is not sensitive to these problems.
    80. 80. Example • A binary channel with bit rate Rb=36000 bits/s is available for PCM transmission. Find appropriate values of the sampling rate fs, the quantizing level, and the binary digits n, assuming the signal bandwidth is B=3.2 kHz.
    81. 81. Example • An analog signal is quantized and transmitted by using a PCM system. If each sample at the receiving end of the system must be known to within ±0.5% of the peak- to-peak full-scale value, how many binary digits must each sample contain ?
    82. 82. Example • For a full-scale sinusoidal modulating signal m(t)= A cos(ωmt), show that • or 2 3 2 L N S o o = )()(log2076.1 10 dBL N S dBo o +=      2 2 3 q o o o m S L N S =
    83. 83. Suggested problems • 6.1-1, 6.2-3 and 6.2-5.

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