Wireless Fading Channels Dr. Syed Junaid Nawaz Assistant Professor Department of Electrical Engineering COMSATS Institute of Information Technology Islamabad, Paistan. Email: junaidnawaz@ieee.org Courtesy of, Dr. Noor M. Khan (MA Jinnah University, Paistan) Dr. M. N. Patwary(Staffordshire University, UK) Dr Muhammad Ali Imran (University of Surrey, UK) Typical Cellular Mobile Environment Remote Dominant Reflectors/Scatterers (Influential) Medium-Distance Dominant Reflectors/Scatterers (Influential) Elevated BS Antenna MS Scatterers local to BS (Non-influential) Scatterers local to MS (Influential)
Shannon s Wireless Communication System Message Signal Channel code word Modulated Transmitted Signal Source Source Encoder Channel Encoder Modulator Estimate of Message signal Estimate of channel code word Wireless Channel User Source Decoder Channel Decoder Demodulator Received Signal Source Coding
Source Coding Source Coding [Analog and Digital Communication B.P. Lathi]
Our first model y=hx+n Channel Source Destination Baseband? Modulation? Communications taes place at [f c -W/, f c +W/] Processing occurs at baseband [-W/, W/]
Modulation/Sampling in baseband Wireless Channel Why wireless is different from wired lin/networ: No privacy no dedicated wire channel High interference due to the lac of privacy Variations in the channel strength (why? will see shortly!) Recall original model: y=hx+n The variations in channel strength are modelled by h A multiplicative factor with the input, models the attenuation/amplification caused by the channel We have to revert to our original model: y=hx+n 10/30
Variations in the channel strength 11/30 Variations in the received signal with the mobility Fading (Random variations in received signal strength) Factor1: Path Loss Factor : Shadowing Factor 3: Multipath Echoes
Source Close Factor 1: Path Loss (very large scale) Physical phenomena, longer distance implies the radiation is spread over larger surface area z Power attenuated proportional to inverse of a power of distance ( in free space) => Power- Law Path Loss Far y x 13/30 Factor : Shadowing (Large Scale) Power attenuated due to absorption by several obstructions (just lie light is obstructed by opaque objects, causing shadows) Several independent random multiplicative factors: modelled by lognormal distribution will see details later 1 Object Source 3 Objects Objects 14/30
Factor 3: Multipath Echoes (Small scale) Each wireless lin is composed of several reflected paths and (optionally) the line-of-sight (specular) path How does this change the channel strength? 15/41 Scales of variation Increasing separation between the receiver and transmitter Slow Medium Fast An illustration from Goldsmith: Not realistic data 16/30
Wireless Multipath Channel 16 db 3sec Real Measurements Courtesy Prof D Tse and Qualcom Large Scale Fading Path-loss + Shadowing T-R separation distances are large Main propagation mechanism: reflections Attenuation of signal strength due to power loss along distance traveled: shadowing Distribution of power loss in dbs: Log-Normal dx Log-Normal shadowing model n (t) = β n (γ n X n (t))dt + δ n dw t 6 5 Fluctuations around a slowly 4 3 varying mean Large Scale Fading X n (t) [db s] 1 0 0 50 100 150 00 50 300 6 dχ n (t) = α n (γ n χ n (t))dt + σ n sqrt(χ n ) dw t 5 4 χ n (t) 3 1 Small Scale Fading 0 0 50 100 150 00 50 300 time [us]
Small Scale Fading Small scale propagation: T-R separation distances are small Heavily populated, urban areas Main propagation mechanism: scattering Multiple copies of transmitted signal arriving at the transmitted via different paths and at different time-delays, add vectotrially at the receiver: fading dx n (t) = β n (γ n X n (t))dt + δ n dw t 6 Distribution of signal attenuation coefficient: Rayleigh, Ricean. Short-term fading model Rapid and severe signal fluctuations around a slowly varying mean X n (t) [db s] χ n (t) 5 4 3 1 0 0 50 100 150 00 50 300 6 5 4 3 1 dχ n (t) = α n (γ n χ n (t))dt + σ n sqrt(χ n ) dw t Large Scale Fading Small Scale Fading 0 0 50 100 150 00 50 300 time [us] Log-Distance Path Loss Model d Pd ( db) Pd 10n log 0 d 0 Path loss exponent n= n=3 n= n=3
Log-Distance Path Loss Model Path loss exponent Environment Free space Urban area cellular.7 to 3.5 Shadowed urban cell 3 to 5 In building LOS 1.6 to 1.8 Obstructed in building 4 to 6 Obstructed in factories to 3 n Log-Distance Path Loss Model
Log-Normal Shadowing Model d P ( db) P 10nlog Xˆ d d0 d 0 Random variable X has the log-normal distribution, if ln(x) has the normal distribution. n=3 n= Empirical Models for Propagation Losses to Environment Oumura: Empirical study of path loss in Japanese cities. Useful for planning urban cellular systems. Includes correction factors for rural and suburban areas. Urban macrocells 1-100m, frequencies 0.15-1.5GHz, BS antenna 30-100m high; Hata: similar to Oumura, but simplified GHz band COST 31: Hata model extended by European study to GHz
Empirical Models for Propagation Losses to Environment HATA-OKUMURA Model Large cell coverage (distances up to 100 m), Frequency up to the GHz band. PL = 69.55 + 6.16 log (f ) - 13.8 log (ht) - a (hm) + [44.9-6.55 log (ht)] log (d) db a (hm) = [1.1 log(f ) - 0.7] hm - [1.56 log(f ) - 0.8] db for midsize city a(hm) - correction factor for mobile unit antenna height (db) f = 1500 MHz, ht = 40 m, hm = 1.5 m
Geometry of rooftop diffraction and shadowing Multi-Path Fading Channel Base Station Mobile Station
Multipath Channel s Impulse response Multipath Channel s Impulse response Time delays in multipath signals causes ISI. Severe ISI requires complex equalizers.
Multi-path Propagation Multi-path smears or spreads out the signal delay spread Causes inter-symbol interference limits the maximum symbol rate Base Station Mobile Station Transmitted Symbol Received Symbol t Delay Spread Base Station Space Mobile Station Transmitted Symbol Time Received Symbol t
Inter-symbol Interference Transmitted Symbol of Interest Received Symbol of Interest t Transmitted Symbols Received Symbols t Average Delay Spread Average delay spread τ τ a a τ P (τ P (τ )τ ) τ 0 t P ( t ) dt 0 P ( t ) dt ) P (τ 1 ) P (τ Multi-path Profile (Discrete) P (τ 0 ) P (τ ) ) P (τ t τ 0 0 τ τ 1 τ
RMS Delay Spread (Discrete) RMS delay spread σ τ τ τ where, τ a a τ P(τ )τ P(τ ) Measurements Type of Delay Spread d Environment (s) Open area <0. Suburban area 0.5 Urban area 3
Coherence Bandwidth Coherence bandwidth B c is a range of frequencies over which the channel can be considered flat passes all spectral components with approximately equal gain and liner phase Bandwidth where the correlation function R T () for signal envelopes is high, or in other words the approximate maximum bandwidth over which two frequencies of a signal are liely to experience correlated amplitude fading. Therefore two sinusoidal signals with frequencies that are farther apart than the coherence bandwidth will fade independently. Coherence Bandwidth If R T () is approximately 90% 1 B C 50 If R T () is approximately 50% B C 5 An exact relationship between coherence bandwidth & delay spread does not exist 1 R T () correlation between responses at different frequencies.
Correlation function. What is the correlation between received signals that are spaced in frequency Δf=f1-f?? How can be measured? 1) Transmitting a pair of sinusoids separated in frequencies by Δ f=f-f1 ) Cross correlating the complex spectra of the two separately received signals 3) Repeat the process by increasing the separation Δ f. Coherence bandwidth represents a frequency range over which a signal's frequency components have a strong potential for amplitude correlation. Example (Power delay profile) P r () 0 db -10 db -0 db -30 db 4.38 µs 1.37 µs 0 1 5 (µs) _ (1)(5) (0.1)(1) (0.1)() (0.01)(0) 4. 38s [0.010.1 0.11] _ (1)(5) (0.1)(1) (0.1)() (0.01)(0) 1.07s [0.010.1 0.11] 1.07(4.38) 1. 37s
Frequency Flat Fading If the mobile radio channel has a constant gain and linear phase over a bandwidth greater than the bandwidth of the transmitted signal - the received signal will undergo flat fading Please, observe that the fading is flat (or frequency selective) depending on the signal bandwidth relative to the channel coherence bandwidth. Frequency Selective Fading If the mobile radio channel has a constant gain and linear phase over a coherence bandwidth, smaller than the bandwidth of the transmitted signal - the received signal will undergo frequency selective fading Again, the signal bandwidth is wider then the channel coherence bandwidth, causing one or more areas of attenuation of the signal within the signal bandwidth
Small-Scale Fading (Frequency selectivity) Spread in delay, causes frequency selectivity Small-Scale Fading (Based on time delay spread) Frequency Flat Fading 1. BW of signal < Coherence BW of channel. Delay Spread < Symbol period Frequency Selective Fading 1. BW of signal > Coherence BW of channel. Delay Spread > Symbol period Example 1. B S << B C & T S >> Classify the frequency selective behavior of the channel???
Example. B S > B C & T S < Classify the frequency selective behavior of the channel??? Need of Equalization If a transmitted signal s bandwidth is greater than the 50% coherence bandwidth, then the channel is frequency selective An equalizer (adaptive tapped delay filter) will be needed at the receiver Flat-fading channels do not require equalization
Inter-symbol Interference For no Inter-symbol Interference the transmission rate R for a digital transmission is limited by delay spread and is represented by: R < 1/ ; If R >1/ Inter-symbol Interference (ISI) occurs Need for ISI removal measures (Equalizers) Doppler Shift f c broadening from f c to (f c + f m ) f d f c v c v v BS1
Relativistic Doppler Frequency The observed frequency is where the relative velocity v is positive if the source is approaching and negative if receding. f c - carrier freq., c-speed of light, f d -Doppler shift c v 1 c v 1 f f c c v f f f f c c d Doppler Shift f c broadening from f c to (f c + f m ) v c v f f c m cos m D f f
Doppler Spread & Coherence Time Describes the time varying nature of the channel in a local area Doppler Spread B D, is a measure of the spectral broadening caused by the time rate of change f c broadening from (f c -f m ) to (f c + f m ) If the base-band signal bandwidth is much greater than B D, the effects of Doppler spread are negligible at the receiver Coherence Time Coherence Time is the time domain dual of Doppler spread Doppler spread and coherence time are inversely proportional T C = 1/f m Statistical measure of the time duration over which the channel impulse response is invariant
Coherence Time If the coherence time is defined as the time over which the correlation function is above 0.5, then T C 9 16 Rule of thumb for modern digital communication defines TC as the geometric mean of two expressions for TC for different correlation functions T C f m 9 16 f m Fading in Brief What is the correlation between received signals sent at different time, with difference Δt=t1-t?? How can be measured? 1) Transmitting a sinusoid at time t1 and send the signal again at time t, Δt=t-t1 ) Cross correlating the channel responses of the two signals received at different times 3) Repeat the process by increasing the difference Δt. Coherence time is a measure of the expected time duration over which the channel's response is essentially invariant.
Fading in Brief What is the correlation between received signals that are spaced in frequency Δf=f1-f?? Bc Tc What is the correlation between received signals that are spaced in delay Δt=t1-t?? Small-Scale Fading (Time Variability) Small-Scale Fading (Based on Doppler spread) Fast Fading 1. High Doppler spread. Coherence time < Symbol period 3. Channel variations faster then base-band signal variations Slow Fading 1. Low Doppler spread. Coherence time > Symbol period 3. Channel variations slower then base-band signal variations
Time Fast Fading The channel impulse response changes rapidly within the symbol duration - coherence time < symbol period T S > T c and B S < B D Channel specifies as a fast or slow fading channel does not specify whether the channel is flat fading or frequency selective fading Fast The channel impulse response changes rapidly within the symbol period of the transmitted signal. Time Slow Fading The channel impulse response changes at a rate much slower than the transmitted base-band signal. Doppler spread is much less than the bandwidth of the base-band signal T S << T c and B S >> B D Velocity of the MS and the base-band signaling determines whether a signal undergoes fast or slow fading
Mobile Channel Parameters Time delay spread Coherence Bandwidth -> ISI Doppler Spread Coherence Time -> Unstable channel Flat fading Frequency selective fading Fast fading Slow fading Types of Small-Scale Fading
Fading (Continued) Bandwidth B c Flat in Time and Selective in Frequency Selective in both Time and Frequency Flat in Time and Frequency T c Flat in Frequency and Selective in Time Time Fading (Continued) Bandwidth B c Flat in Time and Selective in Frequency Selective in both Time and Frequency Flat in Time and Frequency T c Flat in Frequency and Selective in Time Time
Fading in Brief Flat fading B S B C Multi path time delay Frequency selective fading B S B C Small Scale Fading Doppler spread Fast fading T S T C Slow fading T S T C Fading in Brief
Fading in Brief RMS Delay Spread: Typical values Delay spread is a good measure of Multipath Manhattan San Francisco Suburban Office building Office building 1 10ns 50ns 150ns 500ns 1µs µs 5µs 10µs 5µs 3m 15m 45m 150m 300m 600m 3Km 7.5Km
Example For the power delay profile shown in the figure below, a) Find the rms delay spread of the channel. b) Calculate the 55 % correlation bandwidths. c) For a mobile travelling with a speed of 40m/s, receiving the signal at the carrier frequency of 800 MHz through the channel, calculate the time over which the channel appears stationary. d) Classify behavior of the channel as slow or fast fading? Example τ a a τ P (τ )τ P (τ ) _ (0.1)(1) (1.0)() (0.001)(3) 1.9101s [0.11.00.001] τ a a τ P(τ )τ P(τ ) _ (0.1)(1) (1.0)() (0.001)(3) 3.731 s [0.11.0 0.001] σ τ τ τ 3.731 (1.9101) 0.89s
Example B C 1 5 B C 1 5 0.89 1 B C 5 0.89 10-6 BC 691.56 KHz Example m v 40 6 f f 80010 106.667 Hz m c 8 c 310 1 9 9 0.43 T c Tc 0.004 sec f 16 fm 16 (106.667) 106.667
Example T s 1 1 1 0.065 sec 6 F f 810 s c Tc 0.004 sec T c T s Slow Fading Channel Statistical Modeling and Simulation of Multipath Channels Transmitter Receiver
Statistical Modeling and Simulation of Multipath Channels x(t) Statistical Modeling and Simulation of Multipath Channels y(t)
Doppler Spread (Continued) h is the channel impulse response h has a complex normal distribution with zero mean h is Raleigh distributed Phase φ is uniformly distributed between 0 and π is Chi-square distributed Rayleigh Fading 1 The received envelope (amplitude) of a flat fading signal is described as a Rayleigh distribution r Square root sum r, of two quadrature Gaussian noise signals x I and y Q has a Rayleigh distribution (Papoulis65) p( r) exp ; (0 r ) x I y Q r r
Rayleigh Fading PDF Rayleigh Fading PDF and CDF
Rayleigh Fading PDF Ricean Fading 1 Rayleigh fading, when there is No dominant stationary signal component (i.e., Non LoS scenario) r r - s pr ( ) = e ; for r³ 0 s When there is a dominant stationary signal component At the output of an envelope detector - adding a DC component at the random multi-path ( r + a ) ar s 0 ç r - æ ö pr ( ) = e Iç ; for ( a³ 0, r³ 0) s çès ø
Ricean Fading a - pea amplitude of the dominant signal I 0 (.) - modified Bessel function of the first ind and zero order, (available in Matlab with, besseli) Described in terms of a Ricean factor, K a = s a K ( db) = 10log ( db) s Ricean PDF
Rayleigh/Ricean Schematic Rayleigh(a) and Ricean(b) Schematic
Rayleigh Fading One second of Rayleigh fading with a maximum Doppler shift of 10 Hz. One second of Rayleigh fading with a maximum Doppler shift of 100 Hz. Fading in Brief
Channel Equalization Equalizer The goal of equalizers is to eliminate inter symbol interference (ISI) and the additive noise as much as possible. The inter symbol interference of received symbols (bits) must be removed before decision maing (the case is illustrated below for a binary signal, where symbol = bit): Equalizer
Equalizer Equalizer
Equalization Techniques or Structures Three Basic Equalization Structures Linear Transversal Filter Simple implementation using Tap Delay Line or FIR filters FIR filter has guaranteed stability (although adaptive algorithm which determines coefficients may still be unstable) Decision Feedbac Equalizer Extra step in subtracting estimated residual error from signal Maximal Lielihood Sequence Estimator (Viterbi) Optimal performance High complexity and implementation problem (not heavily used) Linear Transversal Equalizer This is simply a linear filter with adjustable parameters The parameters are adjusted on the basis of the measurement of the channel characteristics A common choice for implementation is the transversal filter (Tap Delay Line) or the FIR filter with adjustable tap coefficient Fig. 3.6 Total number of taps = N+1 Total delay = N
Linear Equalizers Linear Equalizer:
LMS: Initialize:, Repeat for X W * y e X X W W e End W 1 y 0 1 max Mapping
LMS Results: H=[0.9], SNR=18 db, µ = 0.08, α=0.85, γ=0.1, E-taps= LMS Results: H=[0.9], SNR=18 db, µ = 0.08, α=0.85, γ=0.1, E-taps=
LMS Results: H=[0.9], SNR=18 db, µ = 0.08, α=0.85, γ=0.1, E-taps= Least Squares Channel Estimation training data Tx Rx
Least Squares Channel Estimation training Tx data Rx Least Squares Channel Estimation Representation 1: training Tx data Rx Representation :
Summary Channel Estimation: training data Tx Rx Channel Equalization: Example
Example Moore Penrose pseudoinverse Example pseudoinverse BER=0/4=0
Decision feedbac equalization (DFE): Performances better than LE, due to ISI cancellation of tails of previously received symbols. Decision feedbac equalization (DFE):
Wiener solution R = correlation matrix (M x M) of received (sampled) signal values p = vector (of length M) indicating cross-correlation between received signal values and estimate of received symbol copt = vector (of length M) consisting of the optimal equalizer coefficient values LMS converges to Decision feedbac equalization (DFE):
Basic Idea: Linear, Zero-Forcing Linear, Zero-Forcing
ISI Baseband Communication System Model where h ( t) Impulse response of the transmitte r T h ( t) Impulse response of the channel C h ( t) Impulse response of the receiver R y( T) h 0 a anhe ( T nt) ne ( T) n, n where he ( t) ht ( t)* hc ( t)* hr ( t), n ( t) n( t ) * h ( t) e R Nyquist condition for zero ISI A pulse will produce zero ISI at sampling instants if it satisfies h e 1, ( nt) 0, n 0 n 0 provided that its Fourier Transform satisfy n n He f T T
A sinc(.) function will satisfy this but we see a pulse shape that Has a more gradual transition in the frequency domain Is more robust to timing errors Yet still satisfies Nyquist s condition for zero ISI Consider a pulse shape that satisfies: In this case, pulse touch and almost begin to overlap There are many H e (f) for which H(f) = T B T or B T 1 1 Raised Cosine Pulse W f W f W W W W W W f W W f f H 0,, ) ( 4 cos 1, ) ( 0 0 0 0 0 0 0 0 ) ) (4( 1 ] ) ( cos[ ) ( ) ( t W W t W W t W Sa W t h e Frequency domain Time domain where W= absolute bandwidth, W 0 =R s /=1/T is the minimum theoretical bandwidth and W-W 0 = excess bandwidth, r = (W-W 0 )/W0 is the rolloff factor and is in the range 0 r 1.
If we denote A RC rolloff pulse shape is defined in this case by the rolloff factor f W W 0 r f 0 W 0 where f o is the 6 db bandwidth of the pulse f1 and f are related to the pulse bandwidth B (or W) as shown in the figure: Solving for bandwidth in terms of the roll off factor and symbol rate, we have: 1 W (1 r ) R s The DSB bandwidth can be written as: W ( 1 r ) DSB R s Example 3.3: Find the minimum bandwidth for the baseband transmission of a 4 level PAM having a R=400bits/sec and r=1.
Root RC rolloff Pulse Shaping We saw earlier that the noise is minimized at the receiver by using a matched filter If the transmit filter is H(f), then the receive filter should be H*(f) The combination of transmit and receive filters must satisfy Nyquist s first method for zero ISI H ( f ) H( f ) H e * ( f ) H( f ) H ( f ) e Transmit filter with the above response is called the root raised cosine-rolloff filter Root RC rolloff pulse shapes are used in many applications such as IS- 54 and IS-136 0.1 R R C P ulse S hape 0. 1 0.08 0.06 0.04 0.0 0-0.0-0.0 4-5 -4-3 - -1 0 1 3 4 5 0.1 R C P ulse S hape (E xcess B W = 0.5) 0.08 0.06 0.04 0.0 0-0.0-5 -4-3 - -1 0 1 3 4 5
Eye Patterns An eye pattern is obtained by superimposing the actual waveforms for large numbers of transmitted or received symbols Perfect eye pattern for noise-free, bandwidth-limited transmission of an alphabet of two digital waveforms encoding a binary signal (1 s and 0 s) Actual eye patterns are used to estimate the bit error rate and the signal to- noise ratio Concept of the eye pattern
1 0.8 0.6 0.4 0. 0-0. -0.4-0.6-0.8 EYE DIAGRAM -1 0 0. 0.4 0.6 0.8 1 1. 1.4 1.6 1.8 Time (sec) 1.5 EYE DIAGRAM WITH NOISE (Variance =0.1) 1 0.5 0-0.5-1 -1.5 0 0. 0.4 0.6 0.8 1 1. 1.4 1.6 1.8 Time (sec)
3 EYE DIAGRAM WITH NOISE (Variance =0.5) 1 0-1 - -3 0 0. 0.4 0.6 0.8 1 1. 1.4 1.6 1.8 Time (sec) 3.4 Equalization Nyquist filtering and pulse shaping schemes assumes that the channel is precisely nown and its characteristics do not change with time However, in practice we encounter channels whose frequency response are either unnown or change with time For example, each time we dial a telephone number, the communication channel will be different because the communication route will be different However, when we mae a connection, the channel becomes time-invariant The characteristics of such channels are not nown a priori Examples of time-varying channels are radio channels These channels are characterized by time-varying frequency response characteristics
To compensate for channel induced ISI we use a process nown as Equalization: a technique of correcting the frequency response of the channel The filter used to perform such a process is called an equalizer Since H R (f) is matched to H T (f), we usually worry about H C (f) The goal is to pic the frequency response H EQ (f) of the equalizer such that 1 jc ( f ) H c ( f ) H EQ ( f ) 1 H EQ ( f ) e where H C ( f ) 1 H and the phase characteristics EQ( f ) EQ ( f ) C ( f ) H ( f ) C Problems with Equalization It can be difficult to determine the inverse of the channel response If the channel response is zero at any frequency, then the inverse is not defined at that frequency The receiver generally does not now what the channel response is. Channel changes in real time so equalization must be adaptive The equalizer can have an infinite impulse response even if the channel has a finite impulse response The impulse response of the equalizer must usually be truncated
N is chosen sufficiently large so that equalizer spans length of the ISI. Normally the ISI is assumed to be limited to a finite number of samples The output y of the Tap Delay Line equalizer in response to the input sequence {x } is N y c x, N,... N n n N n where c n is the weight of the n th tap Ideally, we would lie the equalizer to eliminate ISI resulting in y 1, 0, 0 0 But this cannot be achieved in practice. However, the tap gains can be chosen such that y 1, 0 0, 1,,... N There are two types of such equalizer (i.e., linear equalizers) Preset Equalizer: Transmits a training sequence that is compared at the receiver with a locally generated sequence Requires an initial training sequence Differences between sequences are used to update the coefficient c n Time varying channel can change the sequence, since the coefficients are fixed Adaptive Equalizer: Equalizer adjust itself periodically during transmission of data
The tap weights constitute the adaptive filter coefficient The two techniques can be combined into a robust equalizer In this case, there are two modes of operation: Training Mode For the training mode, a nown sequence is transmitted and a synchronized version is generated at the receiver Decision-directed mode When training mode is complete, the adaptive algorithm is switched on The tap weights are then adjusted with info from training mode The impulse response of the transversal filter is h eq ( t ) H n N eq ( N f c ( t ) n N n N n ) c n e j fn If x(t) is the signal pulse corresponding to X ( f ) H ( f ) H ( f ) H ( f T C R ) then the equalized output signal is N y ( t ) c x ( t n ) Nyquist zero ISI condition implies that n N n y N 1, y( T ) cn x( T n ) n N 0, 0 1,,..., N
Since there are N+1 equalizer coefficients, we may express in matrix form as: y=xc where: x = (N+1) x(n+1) matrix with elements x(t - n) c = (N+1) column coefficient vector y = (N+1) column vector Since this design forces the ISI to be zero at sampling instants t = T, the equalizer is called zero-forcing equalizer (ZFE) Thus we obtain a set of (N+1) linear equations for the ZFE In Figure 3.6 is chosen as high as T = T Symbol-spaced equalizer; < T Fractional-spaced equalizer Example 3.5: (Page 155) 1 Received Pulse 0.8 0.6 0.4 0. 0-0. -0.4-5 -4-3 - -1 0 1 3 4 5
Decision Feedbac Equalizer A decision-feedbac equalizer (DFE) is a nonlinear equalizer that employs previous decisions to eliminate the ISI caused by previously detected symbol It consists of a feedforward section a feedbac section and a detector connected together as shown The filters are usually fractionally spaced FIR with adjustable tap coefficients The detector is a symbol-by-symbol detector