Baseand Data Transmission III Reerences Ideal yquist Channel and Raised Cosine Spectrum Chapter 4.5, 4., S. Haykin, Communication Systems, iley. Equalization Chapter 9., F. G. Stremler, Communication Systems, Addision esley. F.
Ideal yquist Channel The simplest way o satisying n= P ( n / T ) = T is a rectangular unction: < < p( ) = 0 > Equation or zero ISI =/ T F.
Ideal yquist Channel The special value o the it rate R = is called the yquist rate, and is called the yquist andwidth. This ideal aseand pulse system is called the ideal yquist channel p( t) = sin(πt) πt F.3
Ideal yquist Channel F.4
Ideal yquist Channel In practical situation, it is not easy to achieve it due to. The system characteristics o P() e lat rom - /T up to /T and zero elsewhere. This is physically unrealizale ecause o the transitions at the edges.. The unction p (t) decreases as / t or large t, resulting in a slow rate o decay. Thereore, there is practically no margin o error in sampling times in the receiver. F.5
F.6 Raised Cosine Spectrum e may overcome the practical diiculties encountered y increasing the andwidth o the ilter. Instead o using > < < = p 0 ) ( we use > < = + + + P p p 0 ) ( ) ( ) (
Raised Cosine Spectrum A particular orm is a raised cosine ilter. F.7
F.8 Raised Cosine Spectrum The requency characteristic consists o a lat amplitude portion and a roll-o portion that has a sinusoidal orm. The pulse spectrum p() is speciied in terms o a roll o actor α as ollows: > < < = 0 ( sin 4 0 ) ( p π The requency parameter and andwidth are related y / = α
Raised Cosine Spectrum where α is the rollo actor. It indicates the ecess andwidth over the ideal solution (yquist channel) where =/T. The transmission andwidth is ( + α) F.9
Raised Cosine Spectrum The requency response o α at 0, 0.5 and are shown in graph elow. e oserved that α at and 0.5, the unction P() cuto gradually as compared with the ideal yquist channel and is thereore easier to implement in practice. F.0
Raised Cosine Spectrum The time response p(t) is otained as ( cos(παt) p t) = (sin c(t))( 6α t ) The unction p(t) consists o two parts. The irst part is a sinc unction that is eactly as yquist condition ut the second part is depended on α. The tails is reduced i α is approaching. Thus, it is insensitive to sampling time errors. F.
Raised Cosine Spectrum F.
F.3 Raised Cosine Spectrum Eample: For α =, ( = 0) the system is known as the ull-cosine rollo characteristic. > < < + = p 0 0 cos 4 ) ( π and 6 ) sinc( ) ( t t t p =
Raised Cosine Spectrum This time response ehiits two interesting properties: At t = ± T / = ± /4 we have p(t) = 0.5; that is, the pulse width measured at hal amplitude is eactly equal to the it duration T. There are zero crossings at t = ± 3T /, ± 5T /,... in addition to the usual crossings at the sampling times t= ± T, ± T, These two properties are etremely useul in etracting a timing signal rom the received signal or the purpose o synchronization. However, the price paid or this desirale property is the use o a channel andwidth doule that required or the ideal yquist channel corresponding to α = 0. F.4
Raised Cosine Spectrum Eample: A sequence o data transmitting at a rate o 33.6 Kit/s hat is the minimum B at yquist rate. B==33.6/=6.8KHz I a 00% rollo characteristic B=(+α)=33.6 KHz F.5
Raised Cosine Spectrum Eample Bandwidth requirement o the T system T system: multiple 4 voice inputs, ased on an 8- it PCM word. Bandwidth o each voice input (B) = 3. khz yquist sampling rate yquist = B = 6. khz Sampling rate used in telephone system s = 8 khz F.6
Raised Cosine Spectrum ith a sampling rate o 8 khz, each rame o the multipleed signal occupies a period o 5µs. In particular, it consists o twenty-our 8-it words, plus a single it that is added at the end o the rame or the purpose o synchronization. Hence, each rame consists o a total o 93 its. Correspondingly, the it duration is 0.647 µs. F.7
Raised Cosine Spectrum The minimum transmission andwidth is / T = 77kHz (ideal yquist channel) The transmission andwidth using ull-cosine rollo characteristics is ( +α) = 77kHz =. 544MHz F.8
Eye Patterns This is a simple way to give a measure o how severe the ISI (as well as noise) is. This pattern is generated y overlapping each signal-element. Eample: Binary system 0 0 0 T F.9
Eye Patterns Eye pattern is oten used to monitoring the perormance o aseand signal. I the S/ ratio is high, then the ollowing oservations can e made rom the eye pattern. The est time to sample the received waveorm is when the eye opening is largest. The maimum distortion and ISI are indicated y the vertical width o the two ranches at sampling time. The noise margin or immunity to noise is proportional to the width o the eye opening. The sensitivity o the system to timing errors is determined y the rate o closure o the eye as the sampling time is varied. F.0
Eye Patterns F.
Equalization In preceding sections, raised-cosine ilters were used to eliminate ISI. In many systems, however, either the channel characteristics are not known or they vary. Eample The characteristics o a telephone channel may vary as a unction o a particular connection and line used. It is advantageous in such systems to include a ilter that can e adjusted to compensate or imperect channel transmission characteristics, these ilters are called equalizers. F.
Eample Equalization F.3
Equalization Transversal ilter (zero-orcing equalizer) k F.4
Equalization The prolem o minimizing ISI is simpliied y considering only those signals at correct sample times. The sampled input to the transversal equalizer is ( kt) = k The output is ( yt) = y k For zero ISI, we require that k = 0 y k = 0 k 0 (*) F.5
Equalization The output can e epressed as y k = n= a n k n There are + independent equations in terms o a n. This limits us to + constraints, and thereore (*) must e modiied to y k = 0 = k 0 k = ±, ±,..., ± F.6
F.7 Equalization The + equations ecomes = + + + 0 0 0 0 0 0 0 0 M M L L L L M M L L M M L L L L o a a a a a
Equalization Eample Determine the tap weights o a three-tap, zero-orcing equalizer or the input where = 0.0, = 0., 0 =.0, = 0.3, = 0., = 0 or k > k The three equations are a + 0.a 0.3a 0.a + a 0 0.3a 0 0 + 0.a + a = 0 = = 0 Solving, we otain a = 0.779, a = 0.847, a0 = 0.8897 F.8
Equalization The values o the equalized pulse are y = 0.0, y = 0.0356, y y 3 = 0.0, y 0 = 0.0036, y =.0, y 3 = 0.0, = 0.085 This pulse has the desired zeros to either side o the peak, ut ISI has een introduced at sample points arther rom the peak. F.9
Equalization F.30
Duoinary Signaling Intersymol intererence is an undesirale phenomenon that produces a degradation in system perormance. However, y adding intersymol intererence to the transmitted signal in a controlled manner, it is possile to achieve a signaling rate equal to the yquist rate o symols per second in a channel o andwidth Hz. F.3
Coding and decoding Consider a inary input sequence { k } consisting o uncorrelated inary symols and 0, each having duration T. This sequence is applied to a pulseamplitude modulator producing a two-level sequence o short pulses (approimating a unit impulse), whose amplitude is i symol k is ak = i symol k is 0 This sequence is applied to a duoinary encoder as shown elow: { } a k { } c k yquist channel c k = ak + ak Delay T F.3
Coding and decoding One o the eects o the duoinary encoding is to change the input sequence { a k } o uncorrelated twolevel pulses into a sequence { c k } o correlated threelevel pulses. This correlation etween the adjacent pulses may e viewed as introducing intersymol intererence into the transmitted signal in an artiicial manner. F.33
Coding and decoding Eample: Consider { } = 0000 is a startup it. where the irst it k Encoding: { k }: 0 0 0 0 { a k }: - - + - + + - c : - 0 0 0 + 0 { } k Decoding: Using the equation a k = ck ak, i.e., I c k = +, decide that a k = +. I c k =, decide that a k =. I c k = 0, decide opposite o the previous decision. F.34
Duoinary Signaling: Impulse response and requency spectrum Let us now eamine an equivalent model o the duoinary encoder. The Fourier transer o a delay can e descried as e πt, thereore, the transer unction o the encoder is H I ( ) is jπt H ( ) = + e I The transer unction o the yquist channel is < / T H ( ) = 0 otherwise F.35
Duoinary Signaling: Impulse response and requency spectrum The overall equivalent transer unction H ( ) o the is then given y H ( ) = H ( ) H ( ) or < / T I = ( + e = ( e = e jπt jπt jπt + e ) jπt cosπt ) e jπt H() has a gradual roll-o to the and edge which can e easily implemented F.36
F.37 Duoinary Signaling: Impulse response and requency spectrum The corresponding impulse response h(t) is ound y taking the inverse Fourier transorm o H() ) ( ) / sin( ) / ( ) / sin( / ) / sin( ) / ( ) ) / ( sin( / ) / sin( ) ( T t t T t T T T t T t T t T t T T t T T t T t T t t h = + = + = π π π π π π π π π π
Duoinary Signaling: Impulse response and requency spectrum otice that there are only two nonzero samples, at T-second intervals, give rise to controlled ISI rom the adjacent it. The introduced ISI is eliminated y use o the decoding procedure. F.38