Chapter 3 Baseband Pulse and Digital Signaling Based on the fundamentals learned in Chapters 12, we now consider specific communication issues. 1. Pulse code modulation and delta modulation 2. N-ary digital signals 3. Intersymbol interference 4. Multiplexing 5. Transmission Basic Model of Communication Digital Source Transmitter Channel Receiver Destination Source generates from a finite set of symbols. Transmitter Source Encoder Channel Encoder Baseband Signaling • Source Encoder translates the out put of the source in an efficient manner for communication (e.g., compression). • Channel Encoder transforms the coded source to enable error detection and correction at the receiver (e.g., add redundancy). • Baseband signaling encodes digital information in a sequence of analog pulses. Practical consideration of Channel Modulation Physical Transmission Physical Medium Physical Reception Demodulation Noise Channel model used in this class (simplified) H(f) Receiver Optimal Filter Channel Decoder Source Decoder Optimal filter makes “best guess” of transmitted analog pulses. Channel decoder inverses operation of channel encoder (i.e., error detection and correction). Source decoder inverses operation of source encoder. Pulse Amplitude Modulation (PAM) • Baseband operation • Transforming continuous time analog signals into discrete time analog pulses • Information carried in amplitude of pulses. • First step in the analog to digital conversion (A/D) • Pre-cursor to Pulse Code Modulation (PCM) • Sometimes, PAM signals are used directly for transmission without making it into PCM • Two types of PAM – Gating – Sample and hold Analog Source over Digital Communication Analog Source A/D Conversio n Digital Communication System D/A Conversion Analog Output •PAM can be thought of as - Hybrid analog / digital communication system - Part of analog-to-digital conversion process Gated signal w s (t) of given original signal bandlimite w s ( t ) w ( t ) s ( t ) where s ( t ) t kT s d signal w(t) (to B hertz) 1 , and f 2B s Ts Theorem : Spectrum, W s (f) , of gated PAM W s (f) F w s (t) d sin nd nd Any one of these has the same shape as the original W(f). Thus gated PAM is a linear operation. W ( f nf s ) , where d Ts Demodulation of Gated PAM signal Oscillator multiplies cos(nwst) to the received PAM signal. This operation is equivalent to bringing down the frequency of received signal by nws. Let Sample - and - Hold (S/H) signal of given bandlimite S/H PAM is also called the flot top w s (t) and hold the sampled d signal w(t) (to B hertz) be w s (t) W s (f) . PAM. S/H PAM is created by instantane value for a certain duration ously sampling before the next sample is taken. Theorem. where Ws ( f ) 1 H(f) Ts W ( f kf s ) k sin f f s 2 B and H ( f ) F h(t) f They do not have the same shape as the original W(f). Thus S/H PAM is a non linear operation. . Pulse Code Modulation (PCM) is a special form of A/D conversion. It consists of sampling, quantizing, and encoding steps. It is widely popular because: - Used for long time in telephone systems - Inexpensive electronics exists - Errors can be corrected during long haul transmission - Can use time division multiplexing PCM signal Signals in PCM Process Design Issues for PCM - Analog to Digital Conversion Aliasing Sample timing accuracy Quantization noise D/A accuracy Reconstruction filter - Digital Communication Technique Encoding and decoding Signal format Transmit and receive filters Channel effects Statistical decision making error Bandwidth of PCM Assume w(t) is bandlimited to B hertz. Minimum sampling rate = 2B samples / second A/D output = n bits per sample (quantization level M=2n) Assume a simple PCM without redundancy. Minimum channel bandwidth = bit rate /2 Bandwidth of PCM signals: BPCM nB (with sinc functions as orthogonal basis) BPCM 2nB (with rectangular pulses as orthogonal basis) For any reasonable quantization level M, PCM requires much higher bandwidth than the original w(t). Effects of Noise Types of Noise • Quantizing noise (during A/D conversion) • Environment noise (e.g., EM interference) • Filtering noise (low pass filtering at decoder) Types of Quantization Noise • Overload noise (input too large) • Random noise (input too small) • Granular noise (non uniform error jump) • Hunting noise (too long of quite time) Special quantizers are used (µ-law, A-law quantizers) peak signal to noise ratio : signal power is measured at it peak value . 2 3M S 3M 2 N pk out 1 4 M 1 Pe 2 (when Pe is negligible average signal to noise ratio : signal power is measured .) at it average value. 2 M S M 2 N out 1 4 M 1 Pe 2 M : quantizati on levels Pe : probabilit y of bit error in channel S n Let M 2 M N out 2 2 (when Pe is negligible n 2 2 .) 2n S S 2n 20 n log 2 6 . 02 n 10 log 10 log 2 N N dB out This equation states that for each bit added to the PCM scheme, about 6 - dB is gained in the signal - to - noise ratio. Performance of PCM (Pe 0, uniform quantization steps) Example) PCM used in telepho Voice is considered into an 8 - bit binary number. Bit rate of binary PCM signal, R f s samples/se cond bandwidth null bandwidth S 8 32 N pk out d at 4 k Hz ( B). f s 8 ,000 samples to be bandlimite Each sample is converted minimum ne systems n 8 bits per sample R: n bits/sampl : B min 1 es 64 kbits/seco nd (this is known as DS - 0 signal.) R 32 k Hz (when sinc function 2 52.9dB is used.) 2 : B PCM R 64 k Hz (when rectangula per second r pulse is used) Quantization Quantization is a non linear transformation which maps elements from a continuous set to a finite set. It is also the second step required by A/D conversion. Analog Signal - Continuous time - Continuous value Sample Quantize - Discrete time - Continuous value Digital Signal - Discrete time - Discrete value Uniform Quantization V output w2(t) -V V input w1(t) -V Region of operation For M=2n levels, step size : = 2V /2n = V(2-n+1) Quantization Error, e V output w2(t) -V V input w1(t) -V /2 -/2 Error, e input w1(t) Error is symmetric around zero. 0 Average error power : 3 V 2 2 n 1 1 2 2 2 V 2 2 2 e ( s ) ds x dx 2V V 0 3 12 12 Then the average signal power is 2 3 S 2n 2 N out 2 V 2 2 2n 3 r wave between V and V . Suppose the input signal is a triangula V . Definition. The dynamic range of an input signal is the ratio of the largest to the smallest power levels which the input signal can take on and be reproduced with the acceptable signal distortion. The dynamic range of the quantizer input in the PCM system is 6n dB. Nonuniform Quantizer Used to reduce quantization error and increase the dynamic range when input signal is not uniformly distributed over its allowed range of values. allowed values values for most of time input time “Compressing-and-expanding” is called “companding.” Nonuniform quantizer Compressor Uniform Quantizer digital signals •••• Discrete samples •••• Channel received digital signals Decoder Expander output Compression Techniques A - law compressor w1 ( t ) 1, A0 A w1 ( t ) 1 ln A w 2 (t ) 1 ln A w1 ( t ) 1 ln A 0 w1 ( t ) 1 A - law compressor (very popular internatio nally) w1 ( t ) 1 w 2 (t ) ln 1 w1 ( t ) ln 1 In the U.S., 255 is used. 1 A w1 ( t ) 1 Practical Implementation of µ-law compressor Output SNR of 8-bit PCM systems with and without companding. Baseband Signaling Receiver Transmitter Baseband Signaling w(t) Channel H(f) w#(t) Optimal Filter • Once the sending end prepared digital signals (e.g., PCM) to send, now it is the job of Baseband Signaling to prepare the signals suited for the channel. • What should w(t) be? Orthogonal set of signals {k (t), k=1,2,3, ..., N} N w ( t ) w k k ( t ) , for 0 t T o k 1 Note • For practical implementation, we can only use a finite number, N, of the orthogonal set of signals {k (t), k=1,2,3, ..., N}. • Again, for practical implementation, the time duration must be finite, To < . • The goal is to find a set {k (t), k=1,2,3, ..., N} such that w(t) represents the digital signals prepared (e.g., PCM) and a small amount of distortion in the channel does not affect the recovery of w(t) from the received signal, w#(t). Example. In ASCII character, “X” is 0001101. Then, using a certain {k (t), k=1,2,3, ..., 7}, “X” is represented (for 0 < t < To) as w ( t ) w11 ( t ) w 2 2 ( t ) w 3 3 ( t ) w 4 4 ( t ) w 5 5 ( t ) w 6 6 ( t ) w 7 7 ( t ) where w1 0 , w 2 0 , w 3 0 , w 4 1, w 5 1, w 6 0 , w 7 1 . Or we can view the coeffients w k ' s as components of a vector. Then, w1 , w 2 , w 3 , w 4 , w 5 , w 6 , w 7 0 , 0 , 0 ,1,1, 0 ,1 Generally, we can use a vector space notation. w N w k φ k w1 , w 2 , w 3 ,..., w N k 1 where k ,1, 2 ,3 ,... N is a set of N - dimensiona l orthogonal unit vecto rs. Definition. Baud (symbol) rate D = N / To. Definition. Bit rate R = n / To where n is the number of data bit sent in To seconds. If wk is binary, n = N and w(t) is a binary signal. If wk is not binary, n N and w(t) is a multilevel signal. 1 To At the receiver, w # k w ( t ) k ( t ) dt , for k 1, 2 ,3,..., N . # Kk 0 This process is called the matched basis to get the original * filter (i.e., use the same orthogonal signal back.) Then, the receiver reconstruc ts w ( t ) by N w (t ) w # k k (t ), for 0 t T o . If the channel was clean (i.e., no noise), k 1 w(t) is recovered without error. Example. 3 - bit binary signal in the figure. 3 s (t ) d j p j ( t ) where j 1 1 p j ( t ) p t j T . 2 d d 1 , d 2 , d 3 1 ,0 ,1 p j (t ) Let j ( t ) To 1 T j (t ) 0 p j (t ) 25 T o 2 p j ( t ) dt 0 j 1 T t jT elsewhere s s1 , s 2 , s 3 5 T , 0 , 5 T 3 s (t ) s j j 1 j (t ) ( N 3) with M 256 messages. Binary signaling T o 8 ms. M 2 2 256 . n 8 n 2 Given a codeword 01001110 : w w1 , w 2 , w 3 , w 4 , w 5 , w 6 , w 7 , w 8 0 ,1, 0 , 0 ,1,1,1, 0 Case 1. Retangular pulse as basis set Pulse duration Amplitude T b 1 ms 1 first null bandwidth Case 2. Sinc functions k (t ) B D 1, 000 Hz as basis set sin t kT s Ts Ts t kT s T s is a sampling frequency. For this case, T s Tb . minimum bandwidth B 1 2 D 500 Hz Multilevel signaling ( L 4 levels) with M 256 messages. T o 8 ms. L 2 . Then, we need to encode l- bit binary data into one signal level (out of L levels). l If L 4 , then the following might Binary Input 11 10 00 01 be used Output Voltage +3 +1 -1 -3 Example. 01 00 11 10 -3 -1 +3 +1 w1 = -3, w2 = -1, w3 = +3, w4 = +1 Note that 2ms is allowed for sending each symbol. Line Code • On the channel, we might want to send binary numbers directly. • The resulting bit patterns on the channel might create a static voltage, which is not desired. • Use line code to eliminate the average static voltage. - Save power - Save bandwidth (possibly) 1 5 volt 1 1 1 1 average static voltage 0 volt 0 0 0 0 0 0 Types of Line Code • Unipolar signaling: 1 = +A volt, 0 = 0 volt • Polar signaling: 1 = +A volt, 0 = -A volt • Biopolar signaling: 1 = +A or –A, 0 = 0 volt (Also called the alternate mark inversion – AMI) • Machester signaling: 1 = +A (half duration) followed by –A (half duration) 0 = -A (half duration) followed by +A (half duration) Additional combinations can be made along with RZ (return to zero) and NRZ (non return to zero). Desired Properties of Line Code • Self synchronization • Low probability of bit error • Spectral efficiency • Low transmission speed • Error detection capability • Transparency Power Spectral Density for Line Code (We will not follow the details in the book.) N At the source, w T ( t ) a n f ( t nT b ) n 1 where f ( t ) is a symbol pulse. w T ( t ) is the signal observed for 0 t T (T NT b ). a n is data value for the n P s ( f ) lim 1 T T wT ( t ) th 2 symbol. Eye Pattern Seen in oscilloscope The Cleaner, the better Good indication of transmission quality Regenerative Repeater Suppose for any given bit, Pe probabilit y that thi s bit is incorrectl y regenerate d by a regenerati ve repeater. If this bit were to go through m i m i Pi Pe 1 Pe where Pi is the probabilit i by i regenerati ve repeaters. if an odd number a series of m regenerati ve repeaters, y that thi s bit is incorrecte ly regenerate d After m regenerati ve repeaters, this bit will of errors take place. m Pme probabilit be in error, y a bit in error after m regenerati ve repeaters i 1 i is odd m i m i Pe 1 Pe mP e i Bit Synchronization To accurately detect received signals, synchronization timing is needed. - derived from received data - separate signal sent from source Synchronization - bit level - frame level - carrier level Binary-to-Multilevel Conversion Spectral Efficiency Definition . Spectral efficiency η R bits per second. B By Shannon : η max Line Code Unipolar NRZ Polar NRZ Unipolar RZ Bipolar RZ Manchester NRZ Multilevel polar NRZ S log 2 1 B N C First Null Bandwidth (Hz) R R 2R R 2R R/l Spectral Efficiency =R/B bits/s 1 1 0.5 1 0.5 l Intersymbol Interference • No channel has infinite bandwidth • Most transmission schemes require higher bandwidth than available in the channel. - Square wave requires infinite bandwidth. - Synch function is not possible due to causality violation. - Modified synch function to satisfy the causality requires higher bandwidth. • Each symbol may be smeared into adjacent time slots. • Intersymbol Interference (ISI) is the spreading of symbol pulses from one slot into adjacent slots. Baseband Pulse-Transmission System Let w in ( t ) be a flat top w in ( t ) Suppose n w in ( t ) n L - level signal. ( a n can take on one of L values.) t a n h ( t nT s ) where h ( t ) Π . Ts 1 Symbol rate D T s a n h ( t ) ( t nT s ) a n ( t nT s ) h ( t ) n Then, w out ( t ) a n ( t nT s ) h e ( t ) where h e ( t ) h ( t ) hT ( t ) hC ( t ) h R ( t ). n h e ( t ) : equvalent impulse Choose H e ( f ) to minimize HR( f ) response function. ISI by selecting He( f ) H ( f ) HT ( f ) HC( f ) h e ( t ) H e ( f ) H ( f ) H T ( f ) H C ( f ) H R ( f ). (or tuning ) H R ( f ). H R ( f ) : equalizing filter. Finding He(f)? Nyquist’s First Method (Zero ISI) k 0 C h e ( kT s ) 0 Find h e ( t ) such that k 0 is needed to account for the offset in the receiver sampling clock. If such h e ( t ) can be found, a single input pulse at t 0 creates a non zero value at the output of the receiver at t , but all other sampling If 0 , then one possible This is impractica solution times is t kT s (for k 0 ), the receiver he ( t ) sin f s t f st He( f ) output is zero. zero ISI. f f s f s 1 l because : h e ( t ) is non causal. The sampling times at the receiver output have to be precise to maintain zero ISI. Finding He(f)? Raised Cosine-Rolloff Nyquist Filtering 1 He( f ) 2 Find H e ( f ) such that B : absolute Rolloff bandwidth, factor : r f fo D 2B 1 r 1 f f 1 1 cos 2 f 0 f B fo , f1 f o f , f f1 f1 f B f B f o : 6 - dB bandwidth sin 2 f o t H e ( f ) h e ( t ) 2 f o 2 f o t of filter cos 2 f t 2 1 4 f t Theorem. A filter is said to be a Nyquist if the effective transfer function f Y(f) 2 f o He( f ) 0 Y(f) satisfies filter is f 2 fo Otherwise (1) Y ( f ) Y ( f ) for f 2 f o and (2) Y ( f f o ) Y ( f f o ) for f f o . There is no ISI if D f s 2 f o . Raised Cosine-Rolloff Filter as Nyquist Filter. Other Nyquist Filters H(f) 1 H(f) 1 f -2fo -fo fo 2fo f -2fo -fo fo 2fo Example. Binary PCM system Input : Analog signal (e.g., voice) Quantized Channel fo to 16 levels - bipolar signal H e ( f ) : Raised Cosine - Rolloff B 1 r 4 x10 filter wit 3 1 0.5 2 . 667 x 10 2 . 667 kHz 3 Baud rate D 2 f o 5 .333 kHz 16 quantizati on levels 4 bits per sample signal sample rate f s 4 D 21 . 332 kHz Bandwidth B fs 2 0 . 667 kHz 10 . 666 kHz h r 0 . 5 and B 4 kHz How are Nyquist filters realized? H(f) HT(f) HC(f) HR(f) channel characteristics is unknown. Equalization: HR(f) is an adaptive filter. tune HR(f) till match is achieved H(f) HT(f) HC(f) HR(f) input test sequence Desired He(f) specify desire characteristics + + compare - - OftenDifferential voice and video signalsCode do not Modulation change much from Pulse one (DPCM) sample to next. - Such signals has energy concentrated in lower frequency. - Sampling faster than necessary generates redundant information. Can save bandwidth by not sending all samples. * Send true samples occasionally. * In between, send only change from previous value. * Change values can be sent using a fewer number of bits than true samples. Examples (CCITT standards) * 32 k bits / s (4-bit quantization and 8 k samples /s) for 3.2kHz * 64 k bits / s (4-bit quantization and 16 k samples /s) for 7 kHz For slowly varying signals, a future sample can predicted from past samples. s(t) + Predictor Transmitter Side + - e(t) e(t) + s(t) + + Predictor Receiver Side Transversal filter can perform the prediction process. One Implementation of DPCM Quantization error is accumulated. Another Implementation of DPCM Quantization error is not accumulated. Delta Modulation (DM) - Special type of DPCM with M = 2. Inexpensive and simple to implement. DM Waveform Some notes about DM Bit rate = sampling rate Reconstructed signal where y(iTs) = +1 or n -1 z ( nT s ) y ( iT s ) i 1 and is the step size. Types of noise * Quantization noise: step size takes place of smallest quantization level. * Granular noise: z(nTs) is always different from z((n-1)Ts). * Slope overload noise: maximum slope of output signal is / Ts. too small: slope overload noise too large: quantization noise and granular noise There is an optimum value for in terms of signal bandwidth, signal power, and sampling frequency. Let w(t) A cos 2 πf o t and the sampling Example. where k is an integer, dw ( t ) dt Ts k 2 . What is the minimum 2 Af o π sin 2 πf o t which 1 fs has the maximum 1 kf o For no slope overload, Ts 2 A fo . 2 Aπ k frequency, f s kf o value of δ for no slope overload? value of 2 Af o π . Adaptive Delta Modulation Inappropriate step size creates noise. Make adaptive. number of successive 1’s or 0’s size 1 2 3 4 . step 2 4 . ADM Block Diagram. Speech Coding - Waveform coders: output approximates original voice signal. * PCM, DPCM, DM, CVSD (24 – 64 k bits/s) - Vocoder: parameterize voice signals based on speech models * CELP, VSELP (2 -16 k bits/s) Time Division Multiplexing • Time interleaving of samples from different sources to be transmitted over a single communication channel. Frame Synchronization • Framing is done to delimit the boundaries of data units. (e.g., 24 PCM samples collected from 24 difference sources, each corresponding to a voice sample) • The receiver sees a continuous stream of symbols (for binary signals, 1’s and 0’s). • How does the receiver know, for example, where the different PCM samples are? Certain unique string of bits is used to indicate the boundaries of frames. • The channel data may contain the bit patterns that happen to be identical to the framing bit string. Certain bit stuffing and de-stuffing methods are necessary to avoid such situations. • Probability of any arbitrary bit string matching a K bit framing string: 1 Pf 2 K 2 K Frame Synchronizer North American TDM Hierarchy. T1 TDM Format for One Frame. (8000 samples / s)

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