Chapter 3 Pulse Modulation

Size: px

Start display at page:

Download "Chapter 3 Pulse Modulation"

Juliana Moody
5 years ago
Views:

1 Chapter 3 Pulse Modulation Outline Sampling Process: Sampling Theory, Anti-Aliasing Pulse Modulation Analog Pulse Modulation: PAM, PDM, PWM, PPM Digital Pulse Modulation: PCM, DM, DPCM Quantization Process: Uniform; Nonuniform Pulse-code Modulation (PCM): sampling, Quantization, encoding Noise in PCM: Channel noise, Quantization noise Time-Division Multiplexing: T1, Digital Multiplexers: Delta-Modulation: Slope-overload, Granular noise, Delta-Sigma mod. Linear Predictor: Linear Adaptive prediction Differential Pulse-code Modulation (DPCM): Adaptive DPCM 3-1

2 Two families of pulse modulation Pulse Modulation Analog pulse modulation: Carrier wave, CWPulse Train > Discrete in time; Analog in amplitude, duration, position Digital pulse modulation: Pulse Traincoded Pulse > Discrete in both time and amplitude Analog Pulse Modulation 3-2 Digital Pulse Modulation Sampling Process Sampling Process: An operation that is basic to all pulse modulation system, signal processing and digital communication g gnt s t nt s g t t nt s n n ideal sampling T s : Sampling period Analog signal. Instantaneously sampled version of the analog signal. 3-3

3 Sampling Spectrum G f G f f s f nf s f s Gf mf s n G f gnt s exp j2nt s n n : discrete-time Fourier Transform f s : Sampling rate Spectrum of a strictly Spectrum of the sampled version for band-limited signal a sampling period Ts 1/2 W. 3-4 Reconstructing Signal g t can be reconstructed g t : bandlimited; G f 0 for f W. sampling interval 12W. T s g t g t h t 2W n g sin c2wt n 2W n H f 1, f W h t 2Wsinc2Wt Interpolation Formula 3-5

4 Sampling Theory g t can be described by specifying the value of the signal at instants of time separated by 1/2W seconds: Interpolation Formula [ g t : bandlimited; G f 0 for f W ] g t can be completely recovered from a knowledge of its samples taken at the rate of 2W samples per seconds. Aliasing: overlapped region contains that part of the spectrum which is aliased due to undersampling. 3-6 Anti-Aliasing Two ways of eliminating aliasing - Anti-aliasing filter: Signal is prefiltered new maximum frequency Wf s 2. - Sampling Frequency: f s 2W ; sampled at a rate slightly higher than Nyquest rate. Anti-alias filtered spectrum of an information-bearing signal. assuming the use of a sampling rate greater than the Nyquist rate. Spectrum of instantaneously sampled version of the signal Magnitude response of reconstruction filter 3-7

5 PAM (Natural Sampling) a p t t nt s n a p t a t p t v t m t a p t 3-8 Pulse-Amplitude Modulation (PAM) Sample and Hold: Flat-tapped Pulse m t mnt s t nt s n s t mnt s h t nt s n m t h t S f H fm f f s Mf kf s H f k Tsincf Texp j2t f s Mf kf s H k f 3-9

6 PAM Characteristics t T -- TsincfT t T 2 TsincfT T e jft (a) Rectangular pulse h(t). t mt s f s t nf s m n f s 1T s (b) Spectrum H(f), made up of the magnitude H(f), and phase arg[h(f)]. 1 1 f H f Tsincf T sinft Amplitude response of the equalizer: H f Tsincf T System for recovering message signal m(t) from PAM signal s(t) Sample and Hold PAM Why sample and hold Circuit is more practical High-frequency spectral replicates is attenuated significantly > Additional analog post filtering is usually required to finish the filtering process by further attenuating the residual spectral components > Amplitude distortion and a delay of T/2 is introduced Aperture effect: Distortion caused by the use of PAM can be corrected by connecting an equalizer in cascade with the low-pass reconstruction filter > Amplitude response of the equalizer is In comparison to baseband signal transmission more stringent requirements on amplitude and phase response of channel noise performance is worse only time-division multiplexing can be made for long-distance transmission 3-11

7 Other forms of pulse Modulation Pulse-duration Modulation (PDM): - Pulse-width Mod. (PWM); Pulse-length Mod. - Samples of message signal are used to vart duration of individual pulses - Long pulses expend considerable power Pulse-Position Modulation (PPM): - Position of a pulse is varied with message signal - unused power is subtracted from PDM; time transitions are preserved - more efficient (a) Modulating wave. (b) Pulse carrier. (c) PDM wave. (d) PPM wave Generation and Detection of PPM s t gt nt s k p mnt s n s t : strictly nonoverlapping k p mnt s T max s 2 - k p : sensitivity - g t : standard pulse, smaller Thigher BW Detection of PPM - PPMPDM (Integrate) PAMm(t) Slicing: A practical PPM receiver includes a slicer 3-13

8 Noise in PPM Slicer - A noise cleaning device - half peak pulse amp. - Random variations in pulse amp. are removed - random variation in pulse position due to noise will remain Noise in PPM modulation - Noise has no effect on PPM with perfectly rectangular pulse; Perfectly rectangular pulse BWimpractical on finite BW channel - Standard Pulse: A raised pulse SNR - Figure of merit: o B SNR T W 2 c 1 as B T 4.41W B T : Pulse bandwidth, W : message bandwidt - Bandwidth-Noise trade-off: Higher, Lower Noise B T 3-14 Digital Pulse Modulation Pulse code Modulation (PCM) - Sampling - Quantization - Encoding Delta Modulation (DM) 3-15

9 Quantization Process Why Quantization Original continuous signal is approximated by a signal with discrete amplitudes Human sense can detect only finite intensity differences A basic condition of pulse-code modulation (PCM) A process transforming sample amplitude m(nt s )of a message signal m(t) at time tnt s into a discrete amplitude v(nt s ) Description of a memoryless quantizer Types of Quantization Uniform: representation levels are uniformly spaced Nonuniform: A variable separation between the levels is preferable in certain applications - Fewer steps are needed - Step-size increases as the separation from the origin - Nonuniform quantizer compressor (-law) + uniform qualizer - Optimality of scalar quantizers Midtread: original lies in the middle of a tread Midrise: original lies in the middle of a rising part Two types of quantization: (a) midtread and (b) midrise. 3-17

10 Uniform Quantization Quantization noise (error): Difference between the input signal m and output v. Quantization Error q m - v quan. error dist. f Q step size 2m max 2 Quant. error v 2 12 S v P v 2 q P 2 2 2R m max L m 2 max 2 2R R: # of bits Illustration of the quantization process Examples A full-load sinusoidal modulating signal of amplitude A m : PA 2 m ; m max A m ; SNR 3P m 2 max 2 2R R(dB); If R7 bits, SNR (db); 2W 8,000 samples; Transmission rates 56k bits/sec. Signal is Gaussian distributed ~ N(0,1); Signal power P 2 1; Quantization noise 0.5 N q 2 x e x2 2 x e 1 x x e x2 2 x e 1 x x e 1 x

11 Optimality of Scalar Quantizer Minimize average quantization power - Fixed number of representation levels - Select the representation levels Average distortion D L k 1 m k d m k f M mdm, where dm k m k 2 : mean-square distortion f M m : Probability density function of M Two components - Encoder in TX: - Decoder in RX: k L k 1 k L k 1 Two Conditions - Condition I: Given k L k 1, Find k L k 1 - Condition II: Given k L k 1, Find k L k 1 Find k and k to minimize D 1 2 k : m k m m k + 1 for k 1 2 L Input output Condition I Optimality of Encoder for a given Decoder A decoder means that we have a certain codebook in mind Codebook is defined by : k k 1 ; Given k L k 1, Find k L k 1 that minimizes average distortion D A d mg m f M mdm min dm k f M mdm A Find dm k d m j L k 1 m g m k, k 1 2 L holds for all j k k k - input m are closer to k than any other element - A specified codebook is recognized as nearest neighbor condition 1 m k -- 2 k + k m 1 m 2 m 3 m 4 4 nearest neighbor 5 6 m 5 m 6 m 7 m

12 Condition II Optimality of Decoder for a given Encoder Optimize codebook k L k 1, given k L k 1 ; Encoder is fixed L D m 2 k f M D k kopt k 1 m k mdm 2 m k f M mdm 0 m m k k mf M mdm f M mdm m k p k Pm k M m k + 1 Conditional mean of random variable M Optimality algorithm Optimze the encoder m k in accordance with condition I Optimze the encoder k in accordance with condition II Continue in this manner until average distortion D reaches a minimum EM m k M m k + 1 k opt m k m k Example (4-level uniform &Optimum) opt for signal m with Gaussian R.V. ( 0, 2 1) 1 k -- 2k 1 k k k + 1 2, Uniform Quantizer Optimum Quantizer for signal m with Gaussian R.V. (, ) - Find m k and k. - D min dB Optimum Four-level Quantizer L opt D 10logD min L m k k

13 Example (8-level Optimum) Optimum eight-level Quantizer L m k k m 1 m 2 m 3 m m 6 m 7 m 8 m 5 m Pulse-Code Modulation (PCM) A message signal is represented by a sequence of coded pulses The basic elements of a PCM system 3-25

14 Elements of PCM (I) Sampling: message is sampled with a train of narrow rectangular pulses sampling rate > twice the highest frequency A low-pass anti-aliasing filter is used at the front end of the sampler Quantization: provide a new representation of the signal A nonuniform quantizer: step-size increases as separation from the origin of input-output amplitude is increased A nonuniform quantizer a compressor + uniform quantizer Encoding: exploit the advantage of sampling and quantizing more robust to noise, interference or other channel impairments Code element or symbol: one of discrete events in a code Code word or character: a particular arrange of symbols A binary code: two distinct values for each symbol > 0 and 1; maximum advantage over effects of noise; bitacronym for binary; code wordr bits A ternary code: three distinct values for each symbol 3-26 Elements of PCM (II) Regeneration: control the effects of distortion and noise Three basic functions are performed > Equalization: compensate for effects of amplitude and phse distortions due to nonideal transmission of channel > Timing: sample equalized pulses at the instants of time (SNR is a maximum) > Decision making: sample is compared to a threshold Accumulation of distortion and noise is completely removed if disturbance is not too large; Two main reasons for regenerated signal departing from origin > unavoidable presence of channel noise and interferencebit errors > spacing between received pulses deviates from assign valuejitter Decoding: pulses regrouped into codewords a quantized PAM signal pulse amplitue is linear sum of all pulses (weighted by 2 0, 2 1,..) in code word Filtering: A low-pass reconstruction filter (cutoff frequency W) 3-27

15 Quantization (PCM) A nonuniform quantizer a compressor + uniform quantizer Compression Law law law A piece linear approximation to desired curve Expander resotore signal Inversion compression of Compander Compressor + Expander Compression laws. (a) -law. (b) A-law Compression Law d m d law log1 + m log1 + log1 + m m m «1 linear m» 1 logarithmic In US, Canada, Japan m 255 d m d law A m , 0m 1 A 1 + log A A m , 1 A m log A 1 + log A , 0m 1 A A 1 + Am, 1 A m 1 practical values of A ~ 100 uniform quantization used in Europe A Companding circuitry does not produce an exact replica of nonlinear compression curves A piecewise linear approximation to the desired curve using a large enough number of linear segments 3-29

16 Line code (waveform) Line codes for the electrical representations of binary data. (a) Unipolar NRZ signaling. waste of power due to DC; large near zero f (b) Polar NRZ signaling. power spectrum is large near zero freq. (c) Unpolar RZ signaling. 3dB more power than Polar NRZ (d) Bipolar RZ signaling. No DC component, insignificant Low-freq. (e) Split-phase or Manchester code No DC component, insignificant Low-freq Line code (power spectra) Assumptions 0 and 1 are equiprobable average power ~1 frequency f ~ 1/T b bit-timing recovery 3-31

17 Differential Encoding Encode information in terms of signal transitions 0 symbola transition is used 1 symbolno transition is used Original binary information is recovered simply by comparing polarity of adjacent binary symbols A reference bit is required Waveform of differentially encoded data using unipolar NRZ signaling Two major sources of noise Noise in PCM Channel noise: anywhere between transmitter output and receiver input > introduce bit errors into received signalaverage probability of symbol error; bit error rate (BER) for binary Quantization noise: introduced in transmitter and is carried to receiver output; signal-dependent Bit error rate ~ E b /N o (E b : bit energy, N o :noise spectral density) PCM is robust to channel noise and interference (ruggedness to interference) Influence of E b /N o on error probability E b /N o P e bit rate of 10 5 b/s 4.3 db second 8.4dB second 10.6dB seconds 12.0dB seconds 13.0dB day Noise analysis of PCM system. (a) Probability density function of random variable Y at matched filter output when 0 is transmitted. (b) Probability density function of Y when 1 is transmitted. 14.0dB months 3-33

18 Time-Division Multiplexing (TDM) Some of interval between adjacent samples is cleared for use by other independent message sources on a time-shared basis a common communication channel without mutual interference among them Commutator: Electronic switching circuit f s > 2W; interleave N samples Synchronization is essential 3-34 Time-Division Multiplexing (TDM) TDM system is highly sensitive to dispersion in the common channel variations of amplitude with frequency; nonlinearity of phase with frequency. Accurate equalization of amplitude and phase of channel is necessary TDM system is immune to amplitude nonlinearities S 1, S 2,..., S N are not simultaneously impressed on the channel Total power is not too high s 1 s 2 high total Power Interference 3-35

19 T1 System T1: 24 voice channels over separate pairs of wires with regenerative repeaters spaced at ~ 2-km intervals voice signal: 300 ~3100 Hz; a low-pass filter with cutoff freq, 3.1kHz; W3.1kHz, Nyquest band 6.2KHz Quantization: Logarithmic-law with255 > piecewise-linear approximation (Table 3.4): 15 segment:0, 1a1b,2a2b,..., 7a7b; representation levels: bits st bit segment representation inside the swgment b 31 steps 16 steps 0 1a 2a 15 segments 7a 1: voice sample + 0: voice sample An Example of T1 2W 8kHz f s T s 125s voice signal: 24 voice channels bits in one frame synchrinization T b 125/ s f b 1/T b 1.544Mbits signal information: dial pulse; off-hood/on-hood signals Average bits for each voice bits CH1 synchrinization ch 1 frame 2 frame 3 frame 8kHz ch 6 frame least significnant bit signaling bit 125s 1 frame sampling rate (signaling) 8k/61.333kb/s 3-37

Delta Modulation (DM) A simple quantizing strategy for constructing the encoder message is oversampled; f s >> 2Wincrease correlation between samples ent s mnt s m q nt s T s ent s 0 e q nt s mnt s m

20 Delta Modulation (DM) A simple quantizing strategy for constructing the encoder message is oversampled; f s >> 2Wincrease correlation between samples ent s mnt s m q nt s T s ent s 0 e q nt s mnt s m q nt s T s ent s 0 e q nt s mnt s m q nt s T s m q nt s m q nt s T s + e q nt s 3-38 Quantization error of DM Two types of quantization error Slope overload distortion:is too small; m q (t) falls behind m(t) granular noise: is too large; analogous to quantization error ---- max dm t m q nt s increases as fast as the input m q nt s. T s dt Step-size varies with input signal using adaptive DM Adaptive DM e nt s quantization error adaptive DM 3-39

21 Delta-Sigma Modulation Drawback of delta modulation: transmission disturbance (such as noise)accumulative error Accumulative error can be overcome by integrating the message - pre-emphasize low-frequency content - increase correlation to reduce variance of quantization error due to noise - Design of receiver is simplified: only low pass filter is needed Fig, 3.25b is simpler than Fig, 3.25a - smoothness refers to the fact that comparator output is integrated prior to quantization disturbance Error Error propagation add filter to smooth 3-40 Linear Prediction Why Linear Prediction - Delta modulation: sampling rate >> 2W for PCMincrease channel BW - Trade increased system complexity for a reduced channel bandwidth A finite-duration impulse response (FIR) discrete-time filter - Wiener-Hopf equations - Linear Adaptive prediction Linear prediction filter Linear adaptive prediction process DPCM system 3-41

22 Linear Prediction Filter A finite-duration impulse response (FIR) discrete-time filter - Set of p-delay elements, each of which is represented by z -1 - Set of multipliers involving filter coefficients w 1, w 2,..., w p - set of adders used to sum scaled versions of delayed inputs to produce output output prediction error p xˆn w k xn k k 1 e n x n xˆn J Ee 2 n R X kt s Ex nx n k w o : p-by-1 optimum coefficient vector optimum solution for w 1, w 2,..., w p 1 R X w o r X w o R r X : p-by-1 autocorrelation vector X r X mean-square error R X : p-by-p autocorrelation matrix Block diagram of a linear prediction filter of order p 3-42 Linear Adaptive Prediction R X [k] for varying k is not available; we may resort to adaptive predictor - Computation of tap weights w 1, w 2,..., w p ; starting from arbitrary initial values of tap weights; proceed in a recursive manner. - Algorithm used to adjust tap weights is self-designed on basis of available data. gradient vector g k updated value of weight method of steep descent J k 1 2 p w k w k n + 1 w k n g k g k E 2 x nx n k + 2 w j Ex n jx n k step-size parameter p j 1 instantaneous values as estimates of Ex nx n k Least-mean-square (LMS) w k n + 1 w k n x n k e n e n x n w i xn j J p j 1 bowl-shaped error surface initial point predictor error gradient vector Linear adaptive prediction process J min 3-43

Differential Pulse-Code Modulation (DPCM) A more efficient coded signal by removing redundancy - In PCM, resulting sampled signal exhibits a high degree of correlation between adjacent samples.

23 Differential Pulse-Code Modulation (DPCM) A more efficient coded signal by removing redundancy - In PCM, resulting sampled signal exhibits a high degree of correlation between adjacent samples. - Difference between adjacent samples has a smaller variance than variance of signal itself Input to quantizer e n m n mˆn f 1 T s s quantizer output e q n e n + q n prediction-filter input m q n mˆn + e q n m q n mˆn + e n + q n quantization error m q n m n + q n prediction filter DPCM like DM is subject to slope-overload distortion - signal changes too fast for prediction filter DPCM like PCM suffer from quantization noise 3-44

EEE 309 Communication Theory

EEE 309 Communication Theory Semester: January 2017 Dr. Md. Farhad Hossain Associate Professor Department of EEE, BUET Email: mfarhadhossain@eee.buet.ac.bd Office: ECE 331, ECE Building Types of Modulation