Chapter 9

Digital Communication Through Band-Limited Channels

Muris Sarajlic

Band – limited channels (9.1)

● Analysis in previous chapters considered the channel bandwidth to be unbounded

● All physical channels are bandlimited, with C(f) = 0 for |f| > W

● Nondistorting (ideal) channel: |C(f)| = const. for | f | < W and is linear

● All other channels are nonideal (distort the signal in amplitude, phase or both)

Signal design for band-limited channels (9.2)

● System model:

x(t) = transmit pulse * channel * receive filter

Signal design for band-limited channels (9.2)

● The equivalent discrete-time model:

● This can be expressed as:

● With :

Signal design for band-limited channels (9.2)

● The equivalent discrete-time model:

● This can be expressed as:

● With :

ISI!

Signal design for band-limited channels – ideal channel (9.2-1)

● Channel is assumed to be ideal band-limited

● Task: to find pulse shapes that will

– Result in zero ISI at the receiver

– Allow us to achieve maximum possible transmission rate with zero ISI (optimum utilization of the given bandwidth)

● The condition for zero ISI:

Signal design for band-limited channels – ideal channel (9.2-1)

● Theorem (Nyquist): The necessary and sufficient condition for x(t) to satisfy

is that its Fourier transform X(f) satisfies

● The replicas of X(f) obtained by sampling x(t) should add up to form a flat spectrum (a delta in time domain)

Signal design for band-limited channels – ideal channel (9.2-1)

● Three different cases of signal design are observed (with respect to signalling rate)

● Case 1: T < 1/2W, or 1/T > 2W

● At this rate, there exists no pulse whose spectrum replicas add to form a flat spectrum → ISI is inevitable at this rate.

Signal design for band-limited channels – ideal channel (9.2-1)

● Case 2: T = 1/2W, or 1/T = 2W

● The illustrated pulse obviously doesn't satisfy the Nyquist criterion for zero ISI

● The only pulse which satisfies the Nyquist criterion is the sinc pulse (rectangular spectrum)

Signal design for band-limited channels – ideal channel (9.2-1)

● Difficulties with sinc pulse:

– It is noncausal and therefore nonrealizable

● A possible solution: delaying it until t0 so that is is approximately 0 for t<0

– It decays as 1/t; if there is a slightest sampling offset, the resulting ISI is infinite (the series is not absolutely summable)

Signal design for band-limited channels – ideal channel (9.2-1)

● Case 3: T > 1/2W, or 1/T < 2W

● At this rate, there exist numerous pulses which satisfy the zero-ISI criterion.

Signal design for band-limited channels – ideal channel (9.2-1)

● A popular choice for x(t) at 1/T < 2W: the raised-cosine pulse. β is the rolloff factor (0 < β < 1)

Signal design for band-limited channels – ideal channel (9.2-1)

● Benefit of raised-cosine pulse: it decays as 1/t3, so any sampling offset results in finite ISI

● With larger β, the pulse decays faster but the bandwidth utilization is poorer

● It is possible to design practical filters that implement the raised cosine pulse

Signal design for band-limited channels – ideal channel (9.2-1)

● Recall:

x(t) = transmit pulse * channel * receive filter● The overall impulse response of the system is then raised-

cosine. For the ideal channel:

where GT(f) and GR(f) are the transmit pulse spectrum and the receive filter spectrum. If the receiver filter is matched to the transmit pulse:

The transmit pulse is a root-raised-cosine pulse. Delay of t0 ensures realizability.

Signal design for band-limited channel with distortion (9.2-4)

● Channel C(f) is not ideal

● The overall frequency response of the system:

where Xd(f) is selected to yield controlled ISI or zero ISI. For zero ISI, Xd(f) = Xrc(f).

Signal design for band-limited channel with distortion (9.2-4)

● Case 1: the channel distortion is precompensated at the transmitter:

● Case 2: compensation of channel distortion is equally split between the transmitter and receiver:

Signal design for band-limited channel with distortion (9.2-4)

● Although ISI is cancelled, the effect of the channel causes SNR degradation

● It can be shown that SNR degradation is lower when the distortion compensation is equally split between TX and RX (case 2)

Optimum receiver for channels with ISI and AWGN (9.3)

● The received signal:

where z(t) is white.

● The received signal is then passed through a filter matched to h(t), so combination of transmit pulse and channel.

● The output of the matched filter is sampled at nT.

● The received signal:

Optimum receiver for channels with ISI and AWGN (9.3)

● In discrete time:

● The matched filter output yk is affected by ISI.

● If we consider that a finite number of symbols ”mixes” to form yk (channel xk has a finite number of taps), then the system has a finite number of states between which it passes.

Optimum receiver for channels with ISI and AWGN (9.3)

● The time evolution of the system can then be described by a trellis

● An optimum decision on a sent information sequence after observing a received sequence can then be made using the Viterbi algorithm.

Discrete-time White Noise Filter Model (9.3-2)

● Observations about the model:

– xk is the autocorrelation of hk

– The noise vk at the matched filter output is colored (correlated), with autocorrelation function xk

Discrete-time White Noise Filter Model (9.3-2)

● In order to perform comparisons of different detectors for ISI channels, it is beneficial to whiten the noise.

● At the output of the whitening filter:

Discrete-time White Noise Filter Model (9.3-2)

● wk is selected such that:

– is causal and minimum-phase

– The noise part is white

● The resulting model (equivalent discrete-time white noise filter model):

Discrete-time White Noise Filter Model (9.3-2)

● This model is used in the comparison of all detection techniques for the ISI channel

● A joint name for the detection techniques for the ISI channel is equalization techniques (or equalization algorithms)

MLSE for the discrete-time white noise filter model (9.3 - 3)

● Since the current detected symbol is affected by L previous symbols, the system has ML states (M is the size of the alphabet)

● Time evolution of the system described by a trellis

● Viterbi algorithm used to give the maximum-likelihood estimate of the transmitted sequence (MLSE)

MLSE for the discrete-time white noise filter model – performance (9.3 - 4)

● There exists a certain degradation in SNR due to ISI

● It depends on the channel, and there is an analytical model for finding the worst-case SNR degradation and the corresponding channel tap coefficients fk

● For MLSE and a channel with L = 1 – no SNR degradation

Linear equalization (9.4)

● MLSE-based equalization is prohibitively complex – the complexity grows exponentially with the channel length (ML)

● Instead of the Viterbi algorithm, a simple digital filter can be employed to perform the equalization

● It has suboptimum performance but the complexity (length of the equalization filter) is now linear with channel length.

● Cj are the coefficients (taps) of the equalization filter

● Determined by using different criteria

Peak distortion criterion (Zero forcing, ZF) (9.4-1)

● An infinite-length equalizer is analyzed

● Peak distortion: worst-case ISI

● With an infinite-length equalizer, it can be made zero

● The condition: impulse response of the channel-equalizer cascade is a delta (we force the effect of all interfering symbols to zero and extract only the current one)

Peak distortion criterion (Zero forcing, ZF) (9.4-1)

● The cascade of whitening filter and equalizer:

Zero forcing equalizer – performance (9.4-1)

● Although ISI can be completely eliminated, there is a problem with ZF

● Equalizer processes the signal and noise equally

● If there are nulls in the spectrum of the channel, the equalizer ”divides” the noise by zero – noise explodes!

● The ”noise enhancement” effect

Mean-Square Error (MSE) Criterion (also MMSE) (9.4-2)

● Instead of forcing the ISI to zero, the mean square error of symbol estimates:

is minimized.

● It can be shown that the transfer function of the MMSE equalizer is

● The joint transfer function of the whitening filter and MMSE equalizer:

MMSE equalizer (9.4-2)

● Compared with ZF equalizer, the N0 term appears in the denominator of the equivalent equalizer transfer function

● The effect of noise enhancement is reduced

● With N0 ≠ 0 there is both residual ISI and noise at the equalizer output

● However, the performance of the MMSE equalizer in general is better than ZF

● For N0 → 0 (high SNR) they perform similar

MMSE equalizer – performance (9.4 - 3)

● Three channels (frequently used to test equalization algorithms)

MMSE equalizer – performance (9.4 - 3)

Decision-feedback equalization (DFE) (9.5)

● Nonlinear equalizer structure

● Suboptimum but in general performs better than linear structures

● The idea: estimate symbols and use the estimates to recreate ISI affecting future symbols. This ISI is then removed from future symbols

Decision-feedback equalization (DFE) (9.5)

● Coefficients of DFE can be determined both using the zero-forcing or MMSE criterion (MMSE is more common)

DFE performance (9.5-2)

● The problem with DFE: if the first symbol estimate is wrong, the subsequent ISI cancellation is wrong as well, producing further symbol estimation errors...

● The error propagates

DFE performance (9.5-2)

● Comparison with optimum MLSE:

Iterative equalization and decoding, turbo equalization (9.7)

● A transmitter using a convolutional encoder + interleaver + modulator

● Signal transmitted over a linear time-dispersive channel causing ISI

● Channel can be viewed as an inner encoder

● Iterative equalization and decoding based on turbo principle can be performed

Iterative equalization and decoding, turbo equalization (9.7)

Iterative equalization and decoding, turbo equalization (9.7)

Iterative equalization and decoding, turbo equalization (9.7)

● If parallel concatenated convolutional code is used in the TX (instead of ordinary convolutional code), and a turbo channel decoder in the RX instead of MAP channel decoder, the resulting RX structure is a turbo equalizer