BASEBAND DATA TRANSMISSION by Dr. Uri Mahlab Dr. Uri Mahlab

Communication system Dr. Uri Mahlab

Block diagram of an Binary/M-ary signaling scheme + Channel noise + Dr. Uri Mahlab

Block diagram Description {dk}={1,1,1,1,0,0,1,1,0,0,0,1,1,1} For Tb For Tb Dr. Uri Mahlab

Block diagram Description (Continue - 1) {dk}={1,1,1,1,0,0,1,1,0,0,0,1,1,1} Dr. Uri Mahlab

Block diagram Description (Continue - 2) {dk}={1,1,1,1,0,0,1,1,0,0,0,1,1,1} 100110 Dr. Uri Mahlab

Block diagram Description (Continue - 3) {dk}={1,1,1,1,0,0,1,1,0,0,0,1,1,1} Tb Timing 100110 t t Dr. Uri Mahlab

Block diagram Description (Continue - 4) Timing HT(f) HR(f) (X(t Information source Pulse generator Trans filter + Receiver filter Channel noise n(t) Tb 2Tb 5Tb 6Tb t 3Tb 4Tb t t Dr. Uri Mahlab

Block diagram Description (Continue - 5) Tb 2Tb 5Tb 6Tb t 3Tb 4Tb t t Dr. Uri Mahlab

Block diagram of an Binary/M-ary signaling scheme Timing + + Dr. Uri Mahlab

Block diagram Description Tb 2Tb 5Tb 6Tb t 3Tb 4Tb t t t 1 0 0 0 1 0 1 0 0 1 1 0 Dr. Uri Mahlab

Typical waveforms in a binary PAM system Dr. Uri Mahlab

Block diagram of an Binary/M-ary signaling scheme Timing + + Dr. Uri Mahlab

Explanation of Pr(t) Dr. Uri Mahlab

The element of a baseband binary PAM system Dr. Uri Mahlab

Analysis and Design of Binary Signal Dr. Uri Mahlab

For and is the total time delay in the system, we get. The input to the A/D converter is For and is the total time delay in the system, we get. t t t2 t3 t tm t1 Dr. Uri Mahlab

tm =mTb+td t t The output of the A/D converter at the sampling time t2 Dr. Uri Mahlab

ISI - Inter Symbol Interference t2 t3 t tm t1 Dr. Uri Mahlab

Explanation of ISI t t f f Dr. Uri Mahlab

Explanation of ISI - Continue Tb 2Tb 5Tb 6Tb t 3Tb 4Tb Dr. Uri Mahlab

Dr. Uri Mahlab

-The pulse generator output is a pulse waveform If kth input bit is 1 if kth input bit is 0 -The A/D converter input Y(t) Dr. Uri Mahlab

Dr. Uri Mahlab

5.2 BASEBAND BINARY PAM SYSTEMS - minimize the combined effects of inter symbol interference and noise in order to achieve minimum probability of error for given data rate. Dr. Uri Mahlab

5.2.1 Baseband pulse shaping The ISI can be eliminated by proper choice of received pulse shape pr (t). Doe’s not Uniquely Specify Pr(t) for all values of t. Dr. Uri Mahlab

To meet the constraint, Fourier Transform Pr(f) of Pr(t), should satisfy a simple condition given by the following theorem Theorem Proof Dr. Uri Mahlab

Which verify that the Pr(t) with a transform Pr(f) Satisfy ______________ Dr. Uri Mahlab

1 The condition for removal of ISI given in the theorem is called Nyquist (Pulse Shaping) Criterion 1 -2Tb -Tb Tb 2Tb Dr. Uri Mahlab

The Theorem gives a condition for the removal of ISI using a Pr(f) with a bandwidth larger then rb/2/. ISI can’t be removed if the bandwidth of Pr(f) is less then rb/2. Tb 2Tb 5Tb 6Tb t 3Tb 4Tb Dr. Uri Mahlab

Particular choice of Pr(t) for a given application p (t) r Dr. Uri Mahlab

A Pr(f) with a smooth roll - off characteristics is preferable over one with arbitrarily sharp cut off characteristics. Pr(f) Pr(f) Dr. Uri Mahlab

In practical systems where the bandwidth available for transmitting data at a rate of rb bits\sec is between rb\2 to rb Hz, a class of pr(t) with a raised cosine frequency characteristic is most commonly used. A raise Cosine Frequency spectrum consist of a flat amplitude portion and a roll off portion that has a sinusoidal form. Dr. Uri Mahlab

raised cosine frequency characteristic Dr. Uri Mahlab

Summary The BW occupied by the pulse spectrum is B=rb/2+b. The minimum value of B is rb/2 and the maximum value is rb. Larger values of b imply that more bandwidth is required for a given bit rate, however it lead for faster decaying pulses, which means that synchronization will be less critical and will not cause large ISI. b =rb/2 leads to a pulse shape with two convenient properties. The half amplitude pulse width is equal to Tb, and there are zero crossings at t=3/2Tb, 5/2Tb…. In addition to the zero crossing at Tb, 2Tb, 3Tb,…... Dr. Uri Mahlab

Optimum transmitting and receiving filters 5.2.2 Optimum transmitting and receiving filters The transmitting and receiving filters are chosen to provide a proper Dr. Uri Mahlab

Where td, is the time delay Kc normalizing constant. -One of design constraints that we have for selecting the filters is the relationship between the Fourier transform of pr(t) and pg(t). Where td, is the time delay Kc normalizing constant. In order to design optimum filter Ht(f) & Hr(f), we will assume that Pr(f), Hc(f) and Pg(f) are known. Portion of a baseband PAM system Dr. Uri Mahlab

If we choose Pr(t) {Pr(f)} to produce Zero ISI we are left only to be concerned with noise immunity, that is will choose Dr. Uri Mahlab

Noise power Spectral Density - Channel transfer function - Noise Immunity Problem definition: For a given : Data Rate - Transmission power - Noise power Spectral Density - Channel transfer function - Raised cosine pulse - Choose Dr. Uri Mahlab

Error probability Calculations At the m-th sampling time the input to the A/D is: We decide: Dr. Uri Mahlab

The noise is assumed to be zero mean Gaussian at the receiver input A=aKc The noise is assumed to be zero mean Gaussian at the receiver input then the output should also be Zero mean Gaussian with variance No given by: Dr. Uri Mahlab

b Dr. Uri Mahlab

Dr. Uri Mahlab

-A A Dr. Uri Mahlab

Dr. Uri Mahlab

Q(u) dz= U Dr. Uri Mahlab

Hence we need to maximize the signal to noise Ratio Perror decreases as increase Hence we need to maximize the signal to noise Ratio Thus for maximum noise immunity the filter transfer functions HT(f) and HR(f) must be xhosen to maximize the SNR Dr. Uri Mahlab

Optimum filters design calculations We will express the SNR in terms of HT(f) and HR(f) We will start with the signal: The PSD of the transmitted signal is given by:: Dr. Uri Mahlab

And the average transmitted power ST is The average output noise power of n0(t) is given by: Dr. Uri Mahlab

The SNR we need to maximize is Or we need to minimize Dr. Uri Mahlab

Using Schwartz’s inequality The minimum of the left side equaity is reached when V(f)=Const*W(f) If we choose : Dr. Uri Mahlab

The filter should have a linear phase response in a total time delay of td Dr. Uri Mahlab

Finally we obtain the maximum value of the SNR to be: Dr. Uri Mahlab

Rectangular pulse can be used at the input of HT(f). For AWGN with and pg(f) is chosen such that it does not change much over the bandwidth of interest we get. Rectangular pulse can be used at the input of HT(f). Dr. Uri Mahlab

5.2.3 Design procedure and Example The steps involved in the design procedure. Example:Design a binary baseband PAM system to transmit data at a a bit rate of 3600 bits/sec with a bit error probability less than The channel response is given by: The noise spectral density is Dr. Uri Mahlab

If we choose a braised cosine pulse spectrum with Solution: If we choose a braised cosine pulse spectrum with Dr. Uri Mahlab

We choose a pg(t) We choose Dr. Uri Mahlab

Plots of Pg(f),Hc(f),HT(f),HR(f),and Pr(f). Dr. Uri Mahlab

For Pr(f) with raised cosine shape To maintain a For Pr(f) with raised cosine shape And hence Which completes the design. Dr. Uri Mahlab

DUOBINARY BASEBAND PAM SYSTEM 5.3 In order to transmit data at a rate of rb bits/sec with zero ISI PAM data transmission system requires a bandwidth of at least rb /2 HZ. + Binary PAM data transmission at a rate of rb bits/sec with zero ISI 1.such filters are physically unrealizable 2. Any system with this filters will be extremely sensitive to perturbations. Dr. Uri Mahlab 204

Dr. Uri Mahlab

The duobinary signaling schemes use pulse spectra Pr(f): The duobinary scheme utilizes controlled amounts of ISI for transmitting data at a rate of rb /2 HZ. The shaping filters for duobinary are easier to realize than the ideal rectangular filters. The duobinary signaling schemes use pulse spectra Pr(f): The pulse response Pr(t): Dr. Uri Mahlab

use of controlled ISI in duobinary signaling scheme 5.3.1 The output Y(t) of the receive filter can be written as: If the output is sampled at tm= mTb/2+ td than it is obvious that in the absence of noise: The Am' s can assume one of two values +/-A depending on whether the m th input bit is 1 or 0. Since Y(tm) depends on Am & Am-1 assuming no noise : +2A if the m th and (m-1st)bits are both 1's 0 if the m th and (m-1st)bits are different -2A if the m th and (m-1st)bits are both zero Y(tm) = Dr. Uri Mahlab

Transmitting and receiving filters for optimum performance 5.3.2 The receiving levels at the input to the A/D converter are 2A, 0, and -2A with probabilities 1/2, 1/4. The probability of bit error pe is given by: Since no is a zero mean Gaussian random variable with a variance No we can write pe as: For the direct binary PAM case : Dr. Uri Mahlab

The integral can be evaluated as : Where /2=Gn(f) is the noise power spectral density the probability of error is: The integral can be evaluated as : Dr. Uri Mahlab

M-ARY SIGNALING SCHEMES 5.4 M-ARY SIGNALING SCHEMES In baseband binary PAM we use pulses with one of 2 possible amplitude, In M-ary baseband PAM system we allowed M possible levels (M>2) and there M distinct input symbols. During each signaling interval of duration the source is converted to a four-level PAM pulse train by the the pulse generator. The signal pulse noise passes through the receiving filter and is sampled by the A/D converter at an appropriate rate and phase. the M-ary PAM scheme operating with the preceding constraints can transmit data at a bit rate of rs log2M bit/sec and require a minimum bandwidth of rs/2 HZ. Dr. Uri Mahlab

5.7 MISCELLANEOUS TOPICS 5.7.1 Eye Diagram The performance of baseband PAM systems depends on the amount of ISI and channel noise. The received waveform with no noise and no distortion is shown in Figure 5.20a the “open” eye pattern results Figure 5.20b shows a distorted version of the waveform the corresponding eye pattern. Figure 5.20c shows a noise distorted version of the received waveform and the corresponding eye pattern. Dr. Uri Mahlab

Dr. Uri Mahlab

In the figure 5.21 we see typical eye patterns of a duobinary signal Dr. Uri Mahlab

If the signal-to-noise ratio at the receiver is high then the following observations can be made from the eye pattern shown simplified in Figure 5.22: Dr. Uri Mahlab

5. The sampling time is midway between zero crossing. 1. The best time to sample the received waveform is when the eye is opening is largest. 2. The maximum distortion is indicated by the vertical width of the two branches at sampling time. 3. The noise margin or immunity to noise is proportional to the width of the eye opening. 4. The sensitivity of the system to timing errors is revealed by the rate of closing of the eye as sampling time is varied. 5. The sampling time is midway between zero crossing. 6. Asymmetries in the eye pattern indicate nonlinearities in the channel. Dr. Uri Mahlab

synchronization 5.7.2 Three methods in which this synchronization can be obtained are: 1. Derivation of clock information from a primary or secondary standard. 2. Transmitting a synchronizing clock signal. 3. Derivation of the clock signal from the received waveform itself. An example of a system used to derive a clock signal from the received waveform is shown in figure 5.23. To illustrate the operating of the phase comparator network let us look at the timing diagram shown in figure 5.23b Dr. Uri Mahlab

Dr. Uri Mahlab

Scrambler and unscrambler 5.7.3 Scrambler and unscrambler Scrambler: The scrambler shown in figure 5.24a consists a “feedback” shift register. Unscrambler:The matching unscrambler have a ”feed forward” shift register structure. In both the scrambler and unscrambler the outputs of several stages of of shift register are added together modulo-2 and the added to the data stream again in modulo-2 arithmetic Dr. Uri Mahlab

Dr. Uri Mahlab

Scrambler affects the error performance of the communication system in that a signal channel error may cause multiple error at the output of the unscrambler. The error propagation effect lasts over only a finite and small number of bits. In each isolated error bit causes three errors in the final output it must also be pointed out that some random bit patterns might be scrambled to the errors or all ones. Dr. Uri Mahlab