Dr. Uri Mahlab1

2 Informationsource TransmitterTransmitter Channel ReceiverReceiver Decision Communication system

Dr. Uri Mahlab3 InformationsourcePulsegeneratorTransfilterchannel (X(t (X(t (X T (t (X T (t Timing Receiverfilter Clockrecoverynetwork A/D + Channel noise Channel noisen(t) Output Block diagram of an Binary/M-ary signaling scheme+ H T (f) H R (f) Y(t) H c (f)

Dr. Uri Mahlab4 InformationsourcePulsegeneratorTransfilter (X(t (X(t (X T (t Timing Block diagram Description H T (f) {d k }={1,1,1,1,0,0,1,1,0,0,0,1,1,1} TbTbTbTb TbTbTbTb TbTbTbTb TbTbTbTb For b k =1 For b k =0

Dr. Uri Mahlab5 InformationsourcePulsegeneratorTransfilter (X(t (X(t (X T (t Timing Block diagram Description (Continue - 1) Block diagram Description (Continue - 1) H T (f) {d k }={1,1,1,1,0,0,1,1,0,0,0,1,1,1} TbTbTbTb TbTbTbTb TbTbTbTb For b k =1 For b k =0 Transmitterfilter TbTbTbTb TbTbTbTb

Dr. Uri Mahlab6 InformationsourcePulsegeneratorTransfilter (X(t (X(t (X T (t Timing Block diagram Description (Continue - 2) Block diagram Description (Continue - 2) H T (f) {d k }={1,1,1,1,0,0,1,1,0,0,0,1,1,1} TbTbTbTb TbTbTbTb T b 3T b 4T b 5T b t 6T b

Dr. Uri Mahlab7 InformationsourcePulsegeneratorTransfilter (X(t (X(t (X T (t Timing Block diagram Description (Continue - 3) H T (f) {d k }={1,1,1,1,0,0,1,1,0,0,0,1,1,1} TbTbTbTb TbTbTbTb T b 3T b 4T b 5T b t 6T b TbTbTbTb 2T b 3T b 4T b 5T b t 6T b

Dr. Uri Mahlab8 Informationsource PulsegeneratorTransfilter (X(t (X(t Timing Block diagram Description (Continue - 4) H T (f) TbTbTbTb 2T b 3T b 4T b 5T b t 6T b TbTbTbTb 2T b 3T b 4T b 5T b t 6T b Channel noise n(t) Channel noise n(t) + t Receiverfilter H R (f)

Dr. Uri Mahlab9 Block diagram Description (Continue - 5) TbTbTbTb 2T b 3T b 4T b 5T b t 6T b TbTbTbTb 2T b 3T b 4T b 5T b t 6T b t

Dr. Uri Mahlab10 InformationsourcePulsegeneratorTransfilterchannel (X(T (X(T (X t (T (X t (T Timing Receiverfilter Clockrecoverynetwork A/D + Channel noise Channel noisen(t) Output Block diagram of an Binary/M-ary signaling scheme+ H T (f) H R (f) Y(t) H c (f)

Dr. Uri Mahlab11 Block diagram Description TbTbTbTb 2T b 3T b 4T b 5T b t 6T b TbTbTbTb 2T b 3T b 4T b 5T b t 6T b t t

Dr. Uri Mahlab12 InformationsourcePulsegeneratorTransfilterchannel (X(t (X(t (X T (t (X T (t Timing Receiverfilter Clockrecoverynetwork A/D + Channel noise Channel noisen(t) Output Block diagram of an Binary/M-ary signaling scheme+ H T (f) H R (f) Y(t) H c (f)

Dr. Uri Mahlab13 Transfilterchannel P g (t) P g (t) Receiverfilter Explanation of P r (t) H T (f) H R (f) H c (f) P r (t) P r (t) H T (f) H c (f) H R (f) P g (f) P g (f) P r (f) P r (f)

Dr. Uri Mahlab14 The output of the pulse generator X(t),is given by P g (t) is the basic pulse whose amplitude a k depends on.the k th input bit

Dr. Uri Mahlab15 For t m =mT b +t d and t d is the total time delay in the system, we get. t t t t1t1t1t1 t2t2t2t2 t3t3t3t3 tmtmtmtm The input to the A/D converter is

Dr. Uri Mahlab16 t t1t1t1t1 t2t2t2t2 t3t3t3t3 tmtmtmtm The output of the A/D converter at the sampling time t m =mT b +t d TbTbTbTb 2T b 3T b 4T b 5T b t 6T b

Dr. Uri Mahlab17 t t1t1t1t1 t2t2t2t2 t3t3t3t3 tmtmtmtm ISI - Inter Symbol Interference

Dr. Uri Mahlab18 Transfilterchannel P g (t) P g (t) Receiverfilter Explanation of ISI H T (f) H R (f) H c (f) P r (t) P r (t) P g (f) P g (f) P r (f) P r (f) TransfilterchannelReceiverfilter H T (f) H R (f) H c (f) t f FourierTransform BandPassFilter f FourierTransform t

Dr. Uri Mahlab19 Explanation of ISI - Continue t f FourierTransform BandPassFilter f FourierTransform t TbTbTbTb 2T b 3T b 4T b 5T b t 6T b

Dr. Uri Mahlab20 -The pulse generator output is a pulse waveform If k th input bit is 1 if k th input bit is 0 -The A/D converter input Y(t)

Dr. Uri Mahlab21

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

Dr. Uri Mahlab24 Theorem Proof To meet the constraint, Fourier Transform Pr(f) of Pr(t), should satisfy a simple condition given by the following theorem

Dr. Uri Mahlab25 Which verify that the Pr(t) with a transform Pr(f) Satisfy ZERO ISI

Dr. Uri Mahlab26 The condition for removal of ISI given in the theorem is called Nyquist (Pulse Shaping) Criterion TbTbTbTb 2Tb2Tb2Tb2Tb -Tb-Tb-Tb-Tb -2T b 1

Dr. Uri Mahlab27 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. H T (f) H c (f) H R (f) P g (f) P g (f) P r (f) P r (f) TbTbTbTb 2T b 3T b 4T b 5T b t 6T b

Dr. Uri Mahlab28 Particular choice of Pr(t) for a given application The smallest values near Tb, 2Tb, … In such that timing error (Jitter) will not Cause large ISI will not Cause large ISI Shape of Pr(f) determines the ease with which shaping filters can be realized.

Dr. Uri Mahlab29 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 Mahlab30 In practical systems where the bandwidth available for transmitting data at a rate of r b bits\sec is between r b \2 to r b Hz, a class of p r (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 Mahlab31 raised cosine frequency characteristic

Dr. Uri Mahlab32 The BW occupied by the pulse spectrum is B=r b /2+ . The minimum value of B is rb/2 and the maximum value is r b. Larger values of  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.  =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,…... at Tb, 2Tb, 3Tb,…... Summary

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

Dr. Uri Mahlab34 -One of design constraints that we have for selecting the filters is the relationship between the Fourier transform of p r (t) and p g (t). In order to design optimum filter Ht(f) & Hr(f), we will assume that Pr(f), Hc(f) and Pg(f) are known. Where t d, is the time delay Kc normalizing constant. Portion of a baseband PAM system

Dr. Uri Mahlab35 If we choose Pr(t) {Pr(f)} to produce Zero ISI we are left 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 H T (f) H c (f) H R (f) P g (f) P g (f) P r (f) P r (f)

Dr. Uri Mahlab36 Noise Immunity Problem definition: For a given : Data Rate - r bData Rate - r b Transmission power - S TTransmission power - S T Noise power Spectral Density - Gn(f)Noise power Spectral Density - Gn(f) Channel transfer function - Hc(f)Channel transfer function - Hc(f) Raised cosine pulse - Pr(f)Raised cosine pulse - Pr(f) Choose

Dr. Uri Mahlab37 Error probability Calculations At the m-th sampling time the input to the A/D is: We decide:

Dr. Uri Mahlab38 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 Mahlab39 0A y(t m ) b

Dr. Uri Mahlab40 -AA y(t m ) 0

Dr. Uri Mahlab41 -AA y(t m ) V received V Transmit

Dr. Uri Mahlab42

Dr. Uri Mahlab43 UQ(u)dz=

Dr. Uri Mahlab44 P error decreases as increase Hence we need to maximize the signal to noise Ratio Thus for maximum noise immunity the filter transfer functions H T (f) and H R (f) must be xhosen to maximize the SNR

Dr. Uri Mahlab45 Optimum filters design calculations We will express the SNR in terms of H T (f) and H R (f) We will start with the signal: The psd of the transmitted signal is given by::

Dr. Uri Mahlab46 And the average transmitted power ST is The average output noise power of n 0 (t) is given by:

Dr. Uri Mahlab47 The SNR we need to maximize is Or we need to minimize

Dr. Uri Mahlab48 Using Schwartz’s inequality The minimum of the left side equaity is reached when V(f)=const*W(f) V(f)=const*W(f) If we choose :

Dr. Uri Mahlab49 The filter should have alinear phase response in a total time delay of td

Dr. Uri Mahlab50 Finally we obtain the maximum value of the SNR to be:

Dr. Uri Mahlab51 For AWGN with and p g (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 H T (f).

Dr. Uri Mahlab 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 Mahlab53 Solution: If we choose a braised cosine pulse spectrum with

Dr. Uri Mahlab54 We choose a p g (t) We choose

Dr. Uri Mahlab55 Plots of P g (f),H c (f),H T (f),H R (f),and P r (f).

Dr. Uri Mahlab56 To maintain a For P r (f) with raised cosine shape And hence Which completes the design.