Principles of Baseband Digital Data Transmission Prof. Wangrok Oh Dept. of Information Communications Eng. Chungnam National University Prof. Wangrok Oh(CNU) / 3
Overview Baseband Digital Data Transmission Systems 2 Line Codes and Their Power Spectra 3 Effect of Filtering of Digital Data: ISI 4 Pulse Shaping Filter Prof. Wangrok Oh(CNU) 2 / 3
Baseband Digital Data Transmission Systems 26 Chapter 5 Principles of Baseband Digital Data Transmission Message source ADC (if source is analog) Line coding Pulse shaping Channel (f iltering) Receiver f ilter Sampler Thresholder DAC (if source is analog) Figure 5. Block diagram of a baseband digital data transmission system. Analog-to-digital converter (ADC) block Synchronization use of an ADC isatrequired the transmitter, if the thesource inverse operation produces must antake analog placemessage at the receiver output in order ADC to convert consists the digital of two signal operations back to analog form (called digital-to-analog conversion, or DAC). As seen Sampling Chapter 2, after converting from binary format to quantized samples, this can be as simple as a lowpass filter or, as analyzed in Problem 2.6, a zero- or higher-order 2 Quantization hold operation can be used. The Quantization next block, line coding, will be dealt with in the next section. It is sufficient for now to simply Rounding state that thepurposes samples of to linethe coding nearest varied, quantizing and include spectral shaping, synchronization 2 Converting considerations, themand tobandwidtha binary considerations, number representation among other reasons. Pulse To shaping avoid might aliasing be used to further shape the transmitted signal spectrum in order for it to be better accommodated by the transmission channel available. In fact, we will discuss the effects of filtering Source and had how, to if inadequate be bandlimited attention to is W paid Hz to it, severe degradation can result from transmitted Sampling pulses interfering rate had with to satisfy each other. f s > This 2Wissamples termed intersymbol per second interference (sps) (ISI) and Signal can very is not severely bandlimited impact overall or f s system < 2Wperformance Aliasing if steps results are not taken to counteract DAC it. On (digital-to-analog the other hand, we will converter) also see that is careful the inverse selection operation of the combination of ADCof pulse shaping (transmitter filtering) and receiver filtering (it is assumed that any filtering done by the channel is not open to choice) can completely eliminate ISI. Prof. Wangrok Oh(CNU) Baseband Digital Data Transmission Systems 3 / 3
Baseband Digital Data Transmission Systems Line Coding The purposes of line coding Spectral shaping Synchronization considerations Bandwidth considerations Pulse shaping Shaping the transmitted signal spectrum in order for it to be better accommodated by the transmission channel available Severe degradation can result from transmitted pulses interfering with each other: Intersymbol interference (ISI) We will see that careful selection of the combination of pulse shaping (transmitter and receiver filtering can completely eliminate ISI Synchronization At the output of the receiver filter, it is necessary to synchronize the sampling times to coincide with the received pulse epochs The samples of the received pulses are then compared with a threshold in order to make a decision as to whether a or a was sent Prof. Wangrok Oh(CNU) Baseband Digital Data Transmission Systems 4 / 3
Line Codes and Their Power Spectra Description of Line Codes The spectrum of a digitally modulated signal is influenced both by Baseband data format used to represent the digital data 2 Pulse shaping filtering used to prepare the signal for transmission Commonly used baseband data formats Nonreturn-to-zero (NRZ): is represented by a positive level A and is represented by A 2 NRZ mark: is represented by a change in level and is represented by no change in level 3 Unipolar return-to-zero (RZ): is represented by a -width pulse and 2 is represented by no pulse 4 Polar RZ: is represented by a positive RZ pulse and is represented by a negative RZ pulse 5 Bipolar RZ: is represented by a level and s are represented by RZ pulses that alternate in sign 6 Split phase (Manchester): is represented by A switching to A at 2 the symbol period and is represented by A switching to A at 2 the symbol period Prof. Wangrok Oh(CNU) Line Codes and Their Power Spectra 5 / 3
Line Codes and Their Power Spectra 5.2 Line Codes and Their Power Spectra 27 NRZ change NRZ mark Unipolar RZ Polar RZ Bipolar RZ Split phase 2 4 6 8 2 4 6 8 2 4 6 8 2 4 6 8 2 4 6 8 2 4 6 8 2 4 6 8 2 4 6 8 2 4 6 8 2 4 6 8 2 4 6 8 2 Time, seconds Figure 5.2 Abbreviated list of binary data formats. 4 6 8 Prof. Wangrok Oh(CNU) applied to Line certain Codes ofand these. Their Briefly, Power during Spectra each signaling interval, the following descriptions 6 / 3
Line Codes and Their Power Spectra Two of the most commonly used formats are NRZ and split phase Split phase can be obtained from NRZ by multiplying it with a square wave clock waveform with a period equal to the symbol duration Considerations in choosing data formats Self-synchronization 2 Power spectrum 3 Transmission bandwidth 4 Transparency 5 Error detection capability 6 Good bit error probability performance Prof. Wangrok Oh(CNU) Line Codes and Their Power Spectra 7 / 3
Line Codes and Their Power Spectra Power Spectra for Line-Coded Data A pulse-train signal X(t) = k= a k p(t kt ) () where a k is a sequence of r.v.s with the average R m = a k a k+m, m =, ±, p(t) is a deterministic pulse-type waveform is a r.v. that is independent of the value of a k and uniformly distributed in the interval ( T/2, T/2) Autocorrelation function where r(τ) = T R X(τ) = m= p(t + τ)p(t)dt R mr(τ mt ) (2) Prof. Wangrok Oh(CNU) Line Codes and Their Power Spectra 8 / 3
Line Codes and Their Power Spectra Power spectral density where S X(f) = F[R X(τ)] = F = = m= m= = S r(f) m= R mr(τ mt ) (3) R mf[r(τ mt )] (4) R ms r(f)e j2πmt f (5) m= R me j2πmt f (6) S r(f) = F[r(τ)] r(τ) = T p(t + τ)p(t)dt = P (f) 2 p( t) p(t) Sr(f) = T T Prof. Wangrok Oh(CNU) Line Codes and Their Power Spectra 9 / 3
Line Codes and Their Power Spectra Example (PSD of a Split Phase Line Coding) Time average R m { R m = p(t) = ( ) t+t /4 ( ) t T /4 T /2 T /2 Fourier transform of p(t) 2 A2 + 2 ( A)2 = A 2 if m = 4 A2 4 A2 4 A2 + 4 A2 = if m P (f) = T 2 sinc ( T 2 f ) e j2π T 4 f T 2 sinc ( T 2 f ) e j2π T 4 f (8) (7) = T ( ) T 2 sinc 2 f (e j2π T 4 f e j2π T 4 f ) (9) ( ) ( ) T πt = jt sinc 2 f sin 2 f () Prof. Wangrok Oh(CNU) Line Codes and Their Power Spectra / 3
Line Codes and Their Power Spectra Example (Continued) From (): S r(f) = jt sinc T PSD of a split phase line coding ( ) ( T πt 2 f sin 2 f = T sinc 2 ( T 2 f ) sin 2 ( πt 2 f ) S SP = A 2 T sinc 2 ( T 2 f ) sin 2 ( πt 2 f ) ) 2 () (2) (3) Prof. Wangrok Oh(CNU) Line Codes and Their Power Spectra / 3
PSD Line Codes and Their Power Spectra.6.5.4.3.2. -5-4 -3-2 - 2 3 4 5 Frequency Prof. Wangrok Oh(CNU) Line Codes and Their Power Spectra 2 / 3
Effect of Filtering of Digital Data: ISI One source of degradation in a digital data transmission is inter-symbol interference (ISI) ISI is induced when a sequence of signal pulses is passed through a channel with a bandwidth insufficient to pass the signal One of trivial transmit pulse shapes is a rectangular pulse with duration equal to the symbol duration Spectrum of the transmitted signal is H(f) 2 Spectral characteristics of the rectangular pulse shape is undesirable We wish to reduce the bandwidth required to transmit symbols To reduce the required bandwidth Increase the duration of h(t) In order to obtain a bandlimited signal, the duration of h(t) should be unlimited The increase in the duration of h(t) causes interference between transmitted symbols Prof. Wangrok Oh(CNU) Effect of Filtering of Digital Data: ISI 3 / 3
X n Impulse Modulator S (t) Transmit Filter h(t) S B(t) Channel b(t) S C(t) Synchronizer N(t) ˆX n Slicer T s Q(t) Receive Filter f(t) R(t) Prof. Wangrok Oh(CNU) Pulse Shaping Filter 4 / 3
Let us assume for the time being that b(t) = δ(t) and N(t) = Q(t) = S B (t) f(t) (4) = X n δ(t nt s ) h(t) f(t) (5) = = n= n= n= X n h(t nt s ) f(t) (6) X n p(t nt s ) (7) where p(t) = h(t) f(t) is called the overall pulse shape Prof. Wangrok Oh(CNU) Pulse Shaping Filter 5 / 3
Q(t) is sampled at the symbol rate to yield the decision variable Q k Q k = Q(kT s ) = = X n p[(k n)t s ] (8) n= n= X n p k n (9) where p l = p(lt s ) We wish Q k = X k For Q k = X k, we require that p = and p l = for l If the above condition is not satisfied, we will get Q k = X k + n k X n p k n (2) = X k + I k (2) where I k is called inter-symbol interference (ISI) Prof. Wangrok Oh(CNU) Pulse Shaping Filter 6 / 3
The condition p l = δ l is called the condition for zero ISI or the Nyquist condition (for zero ISI) The overall pulse shape p(t) satisfying the zero ISI condition is called Nyquist pulse Cleary, we would like to design p(t) to be a Nyquist pulse Prof. Wangrok Oh(CNU) Pulse Shaping Filter 7 / 3
The interpretation of the Nyquist condition in the frequency domain If p(t) satisfies the Nyquist condition: p(t) δ(t kt s) = δ(t) (22) k= FT of the both side of (22) P (f) T s k= T s k= ) δ (f k Ts ) P (f k Ts = (23) = (24) To satisfy the Nyquist condition, the folded spectrum k= P (f k T s ) must be constant! Prof. Wangrok Oh(CNU) Pulse Shaping Filter 8 / 3
Raised Cosine Filter What is the minimum bandwidth required to transmit symbols at the rate of without ISI? T s P R(f) 2T s 2T s f P R(f) satisfies the Nyquist condition and there can be no pulse shape with a smaller bandwidth that does Corresponding time-domain pulse shape sin p R(t) = ( ) πt T s πt T s ( ) t = sinc T s (25) Prof. Wangrok Oh(CNU) Pulse Shaping Filter 9 / 3
.8.6 pr(t).4.2 -.2-8 -6-4 -2 2 4 6 8 Why isn t p R(t) universally employed? Impossible to obtain symbol synchronization 2 Infinite sensitivity to timing error t Prof. Wangrok Oh(CNU) Pulse Shaping Filter 2 / 3
Consider Q( ) Consider a data pattern X =, X ± =, X ±2 = +, X ±3 =, Q( ) = X p( ) + n X np(nt s + ) (26) = X sinc( ) + n X nsinc(nt s + ) (27) } {{ } =I Note that the envelope of sinc(t) decrease only linearly in t The term I will not converge and result in Q( ) = for any > Prof. Wangrok Oh(CNU) Pulse Shaping Filter 2 / 3
One solution to this problem is to truncate the sinc function to have a finite duration 2 2 2T s 4T s 8 8 6 6 4 4 2 2.5.5.5.5 2 2 8T s 6T s 8 8 6 6 4 4 2 2.5.5.5.5 Prof. Wangrok Oh(CNU) Pulse Shaping Filter 22 / 3
The spectrum of the truncated sinc pulse is simply the convolution of sinc pulse and a rectangular pulse This example shows that we need a better or rather practical design for p(t) The Raised Cosine Filter We observed that the truncated sinc pulse is not a desirable approach α We allow an excess bandwidth of 2T s Hz beyond 2T s Transition band from α 2T s to +α 2T s The transition band is designed to satisfy the Nyquist condition P (f) f + 2Ts 2Ts 2Ts 2Ts The factor α is called the roll-off factor Prof. Wangrok Oh(CNU) Pulse Shaping Filter 23 / 3
2T s The transition band must be symmetric about Nyquist condition is satisfied! This will also result in a bandlimited pulse which is not time limited! We will obtain a pulse with an envelope that decreases faster than the sinc pulse which will allow us to truncate the pulse with a shorter duration window The DeFacto standard for the overall pulse shape is the Raise Cosine (RC) An RC pulse with a roll-off factor α p RC(t) = sin ( ) πt T s πt T s ( t = sinc T s ( απt cos T s ) ( ) 2 2αt T s (28) ) m(t) (29) ( ) t Note that p RC(t) = sinc T s g(t) satisfies the Nyquist condition for any weighting function g(t) Prof. Wangrok Oh(CNU) Pulse Shaping Filter 24 / 3
RC filter is just a special case where g(t) = m(t) The envelope of the RC pulse decreases inverse linearly in t 3 as opposed to t for the sinc pulse Clearly, it is desirable that the transition band be smooth so as to allow practical filter design The spectrum of p RC(t) P RC(f) = T s T s 2 { + cos [ ( ) ]} πt s f α α 2T s if f α 2T s if α 2T s f +α 2T s if f > +α 2T s (3) Prof. Wangrok Oh(CNU) Pulse Shaping Filter 25 / 3
Raised cosine pulses versus roll-off factor α p RC(t) = =.25 = t =.5 Prof. Wangrok Oh(CNU) Pulse Shaping Filter 26 / 3
Spectrum of raised cosine pulses versus roll-off factor α P RC(f) = =.5 =.25 2T s 2T s f Prof. Wangrok Oh(CNU) Pulse Shaping Filter 27 / 3
Eye Diagram Transmitted signal with the RC filter: Roll-off factor=.25 2.5 Transmitted Signal.5 -.5 - -.5-2 2 25 3 35 4 45 5 55 6 Time The above plot gives no idea as to the amount of ISI in the signal Eye diagram Useful in evaluating the performance of a digital communication systems Prof. Wangrok Oh(CNU) Pulse Shaping Filter 28 / 3
2 Especially useful in getting a first-hand idea of the amount of ISI in a given system The amount of ISI can easily be seen by folding this plot every nt s seconds and overlapping the plots If the plot contains data for a large number of symbols, it will contain most of the possible transition paths Such a plot is called the eye diagram Eye diagram: X n {, } and a RC pulse of roll-off factor.25 2.5.5 Amplitude -.5 - -.5-2 - -.8 -.6 -.4 -.2.2.4.6.8 Time Prof. Wangrok Oh(CNU) Pulse Shaping Filter 29 / 3
Eye diagrams versus roll-off factors 2 2 2.5.5.5.5.5.5 Amplitude -.5 Amplitude -.5 Amplitude -.5 - - - -.5 -.5 -.5-2 - -.8 -.6 -.4 -.2.2.4.6.8 Time -2 - -.8 -.6 -.4 -.2.2.4.6.8 Time -2 - -.8 -.6 -.4 -.2.2.4.6.8 Time =.25 =.5 = The Nyquist condition is exactly met and there is no ISI at the sampling points The width of the eye is wider for larger roll-off factors This makes the system less sensitive to timing (sampling time) errors For α =, we see well defined level (e.g. zero) between eyes which can be exploited to achieve timing synchronization Prof. Wangrok Oh(CNU) Pulse Shaping Filter 3 / 3