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I T B L I T F Intersymbol Interference

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I T B L I T F Intersymbol Interference ItIntersym blbol ItfInterference • Any signal can be decomposed as the sum of orthogonal waveforms (basis functions) ()t ()t dt 0 xt() xi i () t and ij for i j i Modulation : mapping constellation symbols x i to waveforms • Signa l transm itted over non-idea l ch annel h(t)(t) rt() xt ()*() ht xi ( ht ()*ii ()) t xhti () i i In general, ht i () are not orthogonal Success ive transm itted symblbols in terf ere with each oth er Transmit Filter (Modulation Basis Function) • Most common choice for basis function i (t) (t iT ) where T is symbol period • Analog transmitted waveform is generated by modulating the transmit (t) filter byyy the symbols xi Throughput this course, unless otherwise stated, transmit filter is lumped with channel filter and receiver front-end continuous matched filter into one filter h(t) 2 Scatter Diagram 3 Time-Domain View t kT t For LTI channel, received noisy & match-filtered analog signal n(t) 0 x j r()t xjht ( jT )nt () h(t) r(t) j rk Note that matched filter t kT t Sampldled at t ime 0 ( t 0 is sampling offset) is absorbed in h (t) def rk r(t0 kT) xkh(t0 T) xjh(t0 (k j)T) n(t0 kT) jk 0 is decision delay Desired ISI Noise Signal Cursor ( j = k- ) pre-cursor post-cursor ISI ISI (j > k- ) (j < k- ) 4 Freqqyuency-Domain View • Ideal (Memory-less or ISI-f)free) ChChlannel • Constant Magnitude Response ht() K (tt 0 ) j 2ft • Linear Phase Response H( f ) Ke 0 • T-spaced sampling replicated spectrum at in teger multip les of 2 (radi an frequency) T • Nyquist’s Criterion for No ISI 2n H ( w ) Constant n T 5 Nyquist Pulses • Satisfyyyq Nyquist’s condition for no ISI • Impulse response is zero for all sampling instants except desired one, hence, hk k • h(t) = sinc (t/T) is only Nyquist pulse w/ minimum bandwidth equal to (sensitive to timing errors and T difficult to realize in practice ) • Most popular choices in practice are raised-cosine and square-root raised cosine pulses with 15-35% excess BW, implemented as FIR digital filters (for 30% excess factor, 37 filter taps needed and the first sidelobe is 40 dB down) 6 Raised-Cosine Filter t t cos( ) ht() sinc ( ) T T 2 t 1 ()2 T T ||w (1 ) T H (w) 0 ()||1 w T T T (1 si(in( (|w| ))) 2 2 T ()||()11 w T T 7 Causes of ISI • Rece ive Filter ing (t(out-of-bdband noise rejection, desired channel selection) • Transm it pulse shap ing (e.g. to reduce side- lobes, narrow main-lobe) • MliMultipat h propagation -chlhannel frequency selectivity • Higher transmission rates (using wider transmission bandwidth) 8 Transmitted Waveform Received Waveform 9 ISI Distortion Criteria Peak Distortion Criterion D p | hk x j k | | x |max | hk | k k • Represents worst-case distance loss between signal points • Worst-case ISI (rare in pract ice) - ifllinput pattern of all xmax Mean Squ are Distortion 2 2 Dms S x | hk | E[( xi hk i ) ] k k i • Assumes zero-mean I.I.D. input sequence • Dms is added to noise v ariance in probability of error Q-function calculations (only accurate if ISI is Gaussian) 10 Graphical Display of ISI • Channel Impulse Response Single impulse for ideal channel. ISI results in scaled & delayed impulses. • Channel Frequency Response Flat magnitude response and linear phase response for ideal channel. Nulls indicate severe ISI • Eye Diagrams Generated using oscilloscope to observe received signal when symbol timing is used as a trigger • Scatter Diagram For ideal channels looks like input signal constellation Eye Diagrams BPSK Constellation 4-PAM Constellation Rece ive d waveforms superimposed and fold ed over durati on of 2 symbol period s 12 Simple Example 2-Tap ISI Channel ykkk x x 1 n k ^ k xk nk yk xk -1 -3 -3 -1 1 3 0 -3 0.35 -5.65 -3 1 -1 0.05 -3.95 -3 232 3 -010.1 191.9 1 4-PAM 3 1 0.25 4.25 3 Symbol-by-symbol detection is sub-optimum in presence of ISI because it does not exploit channel memory 13 ISI Channel Model • Received analoggg signal is p assed throug h an analog matched filter and sampled at the symbol rate • T-spaced samples at matched filter output are sufficient statistics (i.e . no loss of information as far as data detection is concerned) for representing the ISI channel (Forney ‘72) Without loss of generality , the combined effects of transmit filter , channel, and receive filter are modeled as FIR filter w/ memory y km h x km n k m 0 14 15 Examples of ISI Channels • Wire less TiiTransmission ChlChannels • Digital Cellular Radio (2G,3G,4G) • Dig ita l Video Broad cast (DVB-T, DVB-H) • Local Area Network (IEEE802.11x) • Wireline Transmission Channels • TitdTwisted-Pa ir Copper Lines (XDSL) • Coaxial Cable (DOCSIS) • Power Line Communi cati ons (PLC) Mobile Digital Cellular Radio • Frequency band : around 1-2 GHz • Coverage area divided into cells (eac h w ith it s own b ase st ati on) • 2G Standards : IS-136, GSM, IS-95, EDGE,.. f7 • 3G standards : CDMA based f6 f2 f2f1 f6 • 4G standards : OFDM based (LTE) f5 f3 f4 Impairments : • Path loss (proportional to R : 2 . 5 5 ) • Resolvable multipath reflections (in-band nulls), frequency –selective channel • SiSigna l fdifading : fas t small-scale fa ding due to multi pa th , and sl ow large-scale fading (shadowing) due to obstacles in direct path v • Doppler shift (mobility f ), time-selective channel d • Co-channel interference (a.k.a. inter-cell interference) frequency re-use factor) • Thermal Noise (modeled as additive white Gaussian noise (AWGN)) Wireless Channels : Challenges Remote Dominant Reflector Local Scatterers to Base Co-Channel Mobile Base Station Local Scatterers to Mobile Local Scattering Fading Multipath Propagation Intersymbol Interference MbilMtiMobile Motion Time Vary ing Channe l Local Scatterers Cellular Spectrum Reuse Co-channel Interference to Base Remote Dominant 18 Reflector Signal Level in Wireless Channels Short Term Fading Mean Path Loss dB) (( Signal Level Long Term Fading Distance (dB) • Slow fading (shadowing) caused by large obstructions between transmitter and receiver • Fast fading is due to reflection and scattering of the signal by objects near transmitter • Path loss proportional to 1/r <5 19 Signal Fading • Long-term (l(slow ) fdifading (k(a.k.a. shadowing) occurs over long distances and is log-normal distributed (i.e . Gaussian in dB) about the mean path loss (which is inversely proportional to nth power of propagation distance where 2.5<n<5 • Short-term (fast) fading is Rayleigh- distributed relative to local mean P P ( ) P 20 Pr[Preceived P] 1 e Cell Planning • Typical reuse factors are K= 4,7, and 12 • Tradeoffs : for small cells, transmitted signals encounter smaller propagation loss which translates into transmit power savings. Also, smaller cells allow for more frequency re-use which translates into capacity increase (assuming effective interference cancellation). However, more base stations are needed (infrastructure cost) 21 Multipppgath Propagation Multipath Delay Spread of Channel Range of time delays over which an impulse transmitted at time 0 is received with non-zero energy (also called memory of channel) Coherence Bandwidth of Channel Frequency range over which two transmitted sinusoids are affected the same (in magnitude & phase) by the channel Delay Spread = 1 / Coherence Bandwidth Frequency no n-sel ecti ve ch ann el (m em or yl ess, IS I-free, no n-ddspesve)ispersive) Signal Bandwidth << Coherence Bandwidth (negligible delay spread narrow-band signaling) Symbol period >> Delay Spread 22 Typical Numbers • IdIndoor environmen t (e.g. cubic le offices ) 100nano -sec Bc 10MHz • Outdoor environment (e.g. urban cellular) 5 micro -sec Bc 200kHz 23 Multipath Propagation Doppler Spread of Channel Range of frequencies over which a tone transmitted at time 0 is received with non-zero energy Coherence Time of Channel Time range over which two transmitted impulses are affected the same (in magnitude & phase) by the channel Doppler Frequency = 1 / Coherence Time Cond itio n fo r S low l y Tim e Va ry ing C ha nne l Transmission Block Duration << Coherence Time 1/(Block Length * Symbol period) >> Doppler Frequency 1 v fd v 24 NTs NTs Example fc 3GHz 0.1m fd 10 • Pedestrian Speed : 3m / sec fd 30Hz Tc 33msec • Highway Speed : 120Km / hr 33.3m / sec fd 333.3Hz Tc 3msec Guidelines for choosing block length : Doppler, complexity, memory, overhead 25 Narrowband vs. Wideband • If a signal u(t) propagates distance “d” experiencing attenuation of “A”, then the ppggyassband received signal is given by y(t) {A.u(t )e j2fc (t )} 2 j d {A.u(t )e j2fcte } d d where f f c c c Therefore, the transfer function of the 2 j d equivalent baseband channel is Ae j2f e 26 Narrowband vs. Wideband In narrowband transmission,,pp channel appears to have constant gain & delay for all frequencies 2 2 jd jd jf2 1 jf2 2 2-path model 1 2 A12e e A e e Using superposition 2 2 jd1 A jd jf2 1 2 jf2 Ae1 e ()1 ee A 1 wavelength delay spreadd path length diff. Condition for Narrow-band transmission Onlyyq frequency -dependent term in channel magnitude response is 1 e j2f Non-resolvable where is complex const. multipath ||f max 1 1 transmission BW | f | coherence BW max Example 2−path channel, 1 microsec delay spread 2−path channel, 1 microsec delay spread 1 5 10 10 0 10 −5 10 −10 10 −15 10 −20 gnitude Response agnitude Response 10 Mag Magn −25 10 −30 10 −35 0 10 10 0 1 2 3 4 5 6 7 8 9 10 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Frequency (Hz) 6 Frequency (Hz) x 10 jf2 Hf() 1 e 1 sec 2 2 | H ( f ) | 4cos (f ) Coherence BW = 1MHz Indoor/Outdoor Wireless Standards WWAN/WMAN/WRAN (few Km) 2.5/3G/4G Channel is time (GPRS/WCDMA/LTE/DVB-H) and frequency Long delay spreads Large selective, wider (10 micro) # users coverage WLAN (100M) 802.11a/b/g/n/ad Medium delay Medium spreads # users (1 micro) WPAN (10M) Bluetooth, Zigbee 802Short.11 delay spreads Low mobility (0.05-0.1 micro) Shorter Coverage Small # users Range 10m 100m 1km+ 29 A single technology may not be ‘optimal’ for all spheres source : Intel Wireless Comm.
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