7. Baseband Digital Transmission

1 7. Baseband Digital Transmission Baseband Transmission is the simplest form for the communication of information is communicated with specific symbols selected from a finite set ofsymbols. In Baseband Transmission , symbols are simply communicated as a pulse with a discretevoltage level and, for binary Transmission , only two voltages are used. A series of pulses forms apulse train that carries the full message. Prior to Transmission , especially in radio systems, thesepulses are shaped to limit their high frequency content so as to minimize crosstalk with adjacentcommunication channels. During Transmission through a bandlimited channel, pulses aredispersed (spread) in time and can overlap with each other giving rise to intersymbol interference(ISI). When pulses reach the receiver, dispersion and other distortions can be partiallycompensated with an IMPULSE RESPONSE IN A BANDLIMITED CHANNELWe first consider a series of narrow symbol pulses.

2 Restricted channel bandwidth,disperses (or spreads) the pulse in time and necessitates an increased interval between maximum rate at which symbols can be sent is proportional to channel bandwidth. Pulsedispersion relates directly to channel impulse response which can be determined through theFourier transform of H(f), the channel frequency response. We now consider a single rectangular pulse with amplitude h and interval . Theamplitude spectrum of the pulse is determined using the Fourier transform. As illustrated below,the spectrum has magnitude h at zero frequency and a sin f / f variation with addition to amplitude spectrum, there is a phase spectrum (not shown) that has values 0 and when the pulse is centered on the t = 0 axis. Note the spectral nulls at f = 1/ , 2/ , 3/ , .. EnergyWaveformv(volts)hx(t)t (sec)-/2 0/2-3/-2/-1/0hAmplitude Spectral Density (V/Hz)f (Hz)1/2/3/X (f) = hSinffFigure 7-1 Fourier transform of a single pulse that if the weight ( area) of the pulse, h , is held constant while the width isdecreased, the spectral peak remains constant at h and the spectrum spreads out in to the limit, the pulse approaches an impulse and the spectrum becomes "flat".

3 Next we consider a narrow rectangular pulse passing through a bandlimited transmissionsystem modeled by an ideal lowpass filter (LPF). If the input pulse spectrum is approximatelyChapter 7: Baseband Digital transmission136 flat , and the Transmission system has a rectangular brickwall frequency response, the receiverwill see a rectangular spectrum centered about 0 Hz. Applying the inverse Fourier transform tothis rectangular spectrum, we determine that the received voltage response has peak amplitude2fch and sin fct / fct variation with time. VImpulsehp(t) t /2 /20P(f) = h Sin f f AmplitudeSpectral Density(V/Hz)fh 01/ Band LimitedChannelReceivedSignal-1/ ffc-fc0h R(f)ImpulseResponseV2fch r(t)t01/2fc-1/2fc-1/fc1/fc -1r(t) 2fch Sin 2fct 2fct Figure 7-2 Impulse response of a bandlimited channelNote that the impulse response tails oscillate at the LPF cutoff frequency.

4 This can berelated to a practical low pass filter where transfer function poles nearest the cutoff frequencyhave the lowest damping ratio ( high Q) and, following an impulse, the filter continues tooscillate at the cutoff frequency because of the high Q poles. Example - An impulse of amplitude 5 kV and duration 5 ns is input to a 50 ohm cable which includes aninline LPF with unity gain and 5 MHz cutoff frequency. The cable is terminated with a 50 ohm load. Determinethe amplitude and zero crossing interval of the impulse response at the load. Show that the weight (or area) of theoutput response is independent of the LPF cutoff :The impulse weight, h = (5 kV)(5 ns) = 25 V-sPeak voltage of impulse response = 2fch = 2(5 MHz)(25 V-s) = 250 VZero crossing interval (in the tails) = 1/2fc = 1/10 MHz = 100 nsArea (or weight) = 222fchfctfctdt sin = 22fchfcxxdx sin where dxfcdt= 2= 2220fchfcxxdx sin = h 22 = h (which is independent of fc )Note that there is conservation of charge Impulse ampere-seconds = output response ampere-secondsThe integrals sin xxdx02 = and sinxxdx = 202 are useful for analysis of the above pulsespectra and impulse to Communication Systems A Multimedia Workbook137Ch7-Eye-02m5 Dodds, Univ.

5 Of Saskatchewan, Canada3/25 MAXIMUM SIGNALING RATEM essage symbols occur in a sequence at rate Rs called the baud rate of the Transmission ;one baud equals one symbol per second. With limited channel bandwidth (as in a lowpass filter),symbol pulses are spread out in time (dispersed) and, if interference between successive pulses isto be avoided, there must be a minimum interval between pulses. The maximum symbol rate, Rmax, is therefore limited by the channel bandwidth. We shall see that by setting the symbol rateequal to twice the lowpass filter bandwidth and by sampling the received signal at appropriatetimes, we have the maximum symbol rate that can be attained without intersymbol Impulse signaling with an ideal filterAssume a first impulse (not a rectangular pulse) transmitted through an ideal lowpasschannel with unity gain from 0 Hz to the cutoff frequency fc.

6 The received signal (the channelimpulse response) will be of the form sin fct / fct with a main lobe having peak voltage at time t1and with zero crossings at time intervals 1/2fc about time t1 . If the first impulse is followed by asecond impulse, transmitted after delay 1/2fc, the second impulse response will have a main lobewith voltage peak at time t2 occurring at a zero crossing of the first impulse response. The secondsymbol may then be received without interference provided that the received signal is sampledprecisely at the zero crossing of the first impulse response. Further zero crossings of bothimpulse responses will be coincident at intervals of 1/2fc and, if additional impulse symbols aretransmitted at time intervals of 1/2fc, these too can be received without ISI. Symbols may be communicated without inter-symbol interference at a rate of fsy = 2fc.

7 This symbol rate, fsy, is known as the Nyquist signaling rate. In Figure 7-3 , five impulses with weights +3 and +1 are shown after Transmission through alowpass channel with cutoff frequency at 500 kHz. The received voltage is the superposition ofall impulse responses from the transmitted symbol sequence. A sample taken at time t = 8 us,for example, contains no output from pulses p1, p2 p4 and p5 and is therefore responsive only tothe amplitude of p3. This condition of zero ISI is independent of the other pulse (volts) fc = 500 kHz+30t(us)+3+3+1+1 p1 p2 p3 p4 p52468101214 Figure 7-3 Intersymbol InterferenceFor zero ISI we must exactly match the channel bandwidth to the reciprocal of thesignaling rate. In this case, the channel includes the transmit filter, the Transmission link and thereceive filter. In practice, we use an interconnecting Transmission link with somewhat largerbandwidth and then control the channel bandwidth with accurate transmitter and receiver filterswith cutoff at fc.

8 Precise cut-off frequencies can be implemented with clocked Digital 7: Baseband Digital Impulse signaling with practical filtersThe sharp cut-off, rectangular brickwall filters illustrated in the previous section cannotbe implemented in practice; the transition from passband to stopband must occur over somefrequency range. Filters are implemented with a roll-off that is symmetric about fc extending upto (1+r) fc where r is the channel roll-off factor. The transition region characteristic usuallyapproximates the first 180 of a raised cosine leading to the moniker raised cosine filter . Withroll-off factor r , the transmitted spectrum is 30% in excess of what would be transmittedwith a brickwall filter ( 30% excess bandwidth ).The gradual filter transition results shortening the ripple tails in the channel impulseresponse.

9 This reduction of tail amplitude and duration significantly reduces ISI and relaxes theneed for precise matching of signaling rate to the zero crossing rate. For small timing offsets, theprevious impulse has a zero crossing near the desired sampling time. On the other hand,interfering impulses from the distant past may have displaced zero crossings however theiramplitude has now become insignificant. Figure 7-4 illustrates excess bandwidth and reduced impulse response duration. The filtercharacteristic follows a raised cosine function in the transition region and has a gain of (-6dB)at the frequency fc. The total Transmission bandwidth required is fb = (1+r) fc. =12 = 0h1(t) =sin t/T() t/Tr = ( f )r = = 0fH2( f )h2(t) =sin t/T() t/Tcosr t/T()1 4r2t2/T2 Figure 7-4 raised cosine frequency response and impulse response (r = ).

10 It is not possible to create a filter with perfectly sharp cut off in the frequency domain -all practical filters must have some excess bandwidth. The raised cosine pulse is defined infrequency byHf21()=012 frTHfTrfrT22212()cos= 1212 <<+rTfrTHf20()=12+ rTfAdditionally, an ideal filter must have an impulse response that extends to infinite timebefore and after the pulse peak and, if the filter is to be causal (output response occurs after theinput is applied), the peak output would occur an infinite time after the input. This wouldIntroduction to Communication Systems A Multimedia Workbook139Ch7-Eye-02m5 Dodds, Univ. of Saskatchewan, Canada3/25/02certainly not make for a useful communication system since one would need to wait more than alifetime for the message to reach the this point we have considered filter structures (transversal filters, for example) withconstant delay and therefore phase shift that increases linearly with frequency.