Time and Frequency Domain Equalization

Time and Frequency Domain Time and Frequency Domain

Equalization Equalization

Presented By:

Khaled Shawky Hassan Under Supervision of:

Prof. Werner Henkel

Introduction to Equalization

Non-ideal analog-media such as telephone cables and radio channels typically distort the transmitted signal.

Due to the non-ideal situation and high data rate, Inter Symbol Interference (ISI) is introduced to the received signal.

ISI arises in all pulse-modulation systems, including Pulse Amplitude Modulation (PAM) frequency-shift keying (FSK), phase-shift keying (PSK) and

quadrature amplitude modulation (QAM).

Data Transmission System

Transmission system:

Encoder &

Filter

Telephone Circuit or Channel

Modulator De-

Modulator

Filter &

Equalizer

Decision Device Decoder Input Bit T

Stream Output Bit

Stream

Baseband system model:

Channel impulse response:

The received signal is the superposition of the channel impulse response to each transmitted symbol and AWGN

The received signal:

If we sample the received signal at (kT+t

), where t

accounts for the channel delay and sampler phase, we obtain:

( )

( - ) ( )

j

r t = ∑ x h t jT + n t

0 0 0

( )

( 0)

( - ) ( )

j k

r t kT x h t Int x h t kT jT n t kT

≠

+ = + ∑ + + +

Conclusion:

Interference term is proportional to a sample of the channel impulse response, h(t

- jT).

i.e.: ISI = 0, iff h(t

- jT) = 0 (Zero crossing at the T - spaced interval)

Nyquist’s first criterion

When the impulse response has such uniformly-spaced zero crossings, it is said to satisfy Nyquist’s first criterion.

In frequency domain terms, this condition is equivalent to:

'( ) constant for | | 1/ 2 and

'( ) ( ) ( 1/ ), 0 1/

H f f T

H f H f H f T f T

= ≤

= + − ≤ ≤

In practice, the effect of IS1 can be seen In practice, the effect of IS1 can be seen from a trace of the received signal on an from a trace of the received signal on an oscilloscope with its time base

oscilloscope with its time base synchronized to the symbol rate.

synchronized to the symbol rate.

Time Domain Equalization

In time domain equalization schemes a (digital) filter (FIR) is inserted in the signal path after the channel in the receiver part, to compact ISI.

Types of time domain equalizers:

Linear Equalizers Example

Transversal – Zero-forcing Equalizer: (neglects the effect of noise)

2

1

^ *

estim ated symbols

from Peak D istortion

|

: ( ) 1

( ) ,

N

k n k n

n N

d C y

w here C z

F z

−

= −

=

=

∑

Finite-length ZF equalizer is guaranteed to minimize the peak distortion or worst-case IS1 only if the peak distortion before equalization is less than 100 percent; i.e., if a binary eye is initially open or Minimum phase channel

The least mean-square (LMS) equalizer is more robust, as the

equalizer coefficients are chosen to minimize the MSE

Linear Adaptive Equalizers (LMS/RLS)

LMS Coefficient Update:

*

1 , 0,1,

k k k k

C ₊ =C + ∆e X k = L

Performance Comparison

' * 2 0

| |

where weighting factor w is selected 0<w<1

k k n

n k n

n

w I C X

ξ

=

=

∑

RLS Cost function

Non-linear Equalizers Example

Decision Feedback equalizer: Is useful for channels with severe amplitude distortion, uses decision feedback to cancel the

interference from symbols which have already been detected.

Low Complexity Equalization by Guard Insertion

DMT consists of a large number of QAM modulated carriers that are orthogonal.

Obviously, independent transmission channels (assumed N parallel channels) are obtained only if appropriate symbol synchronization is performed at the receiver to compensate for the delay introduced by the channel

Mathematical description of DMT modulation, transmission,

and demodulation:

The Inter-carrier and Inter-symbols Interference

When the cyclic prefix is shorter than the memory of the channel, inter-carrier interference (ICI) as well as ISI are unavoidable.

Interference can be written as a weighted sum of the QAM

symbols transmitted during the previous and next DMT symbol.

In addition, because of the loss of orthogonality between the carriers in the current symbol, extra ICI is generated.

The goal:

Searching for low-complexity equalization schemes to

obtain insight in the different interference mechanisms.

Interference Type in DMT Signal

ICI

is the interference from QAM symbols transmitted over the other carriers of the m

symbol.

ISI is the interference from a

and a

, the QAM symbols modulating the k

carrier during the previous and next DMT symbols.

ICI

is the interference caused by the QAM symbols in the

previous and next DMT symbols, transmitted over carriers

other than the k

carrier.

Inter-carrier Interference

Two segments of the impulse response contribute to the interference: the head and tail.

ICI can be expressed as a function of the FFT of the head and tail.

ICI

is calculated from carrier i and j, and weighted by W(k-j).

ICI is reciprocal, i.e.: the interference power from carrier k on

j equals the interference power from carrier j on k.

.

Y = H X n +

A (digital) filter is inserted in the signal path before the FFT function.

Time domain equalizer (TEQ) coefficient initialization for true capacity optimization leads to a highly nonlinear optimization problem.

The performance of the TEQ is not predictable, as there is no

direct relation between this time domain mean-square-error

(MSE)-optimal channel shortening and the transmission capacity

TEQ Optimization Problem

The cascading of the TEQ and channel impulse response CIR ({hk}) approximately forms an FIR target impulse response (TIR) ({bk}), with an impulse response length shorter than the cyclic prefix.

The relative delay between the equalizer and the TIR is denoted by ∆

The unknown parameters ∆, {wk}, and {bk} are computed based

on a mean squared error (MSE) criterion (Channel Shortening

Problem), and can be found from the next figure.

In the per-tone equalization, the MSE optimization is

performed for each carrier separately, leading to improved as well as more predictable and reproducible performance.

Every single tone is given its own distinct and optimal equalizer.

The computational requirement is roughly kept at the same level, or even smaller, as the computational requirement in a TEQ-based modem.

An appropriate initialization scheme (based on equalizer training) is included.

The memory requirement obviously increases

significantly, but this is not considered a major

implementation.

Frequency (Per-tone) Domain Equalization

Our approach is based on transferring the TEQ- operations to the

frequency domain (i.e., after the FFT-

demodulation).

A (complex) –tap FEQ for tone.

Then allow each tone to have its own tap FEQ for tone i.

instead of one FFT-operation per symbol, we now apparently need

FFT-operations (one FFT for each column of ) !!!! (Dangerous)

Frequency (Per-tone) Domain Equalization

But:

Fortunately, because of the Toeplitz structure of Y, these FFTs can be calculated efficiently by means of a sliding FFT.

Only one “full” FFT has to be calculated (for the first column

of ) and the remaining FFTs can be deduced as follows:

Complexity of FEQ vs TEQ

We can conclude that the resulting complexities are

comparable and both of order F

T . The complexity of the per tone method can be reduced even further by varying the

equalizer length per tone and setting the length to zero for

non-used tones.

For each of the used tones, we find the MMSE-FEQ (for a particular choice of ) by minimizing the following cost function:

Equalization Initialization

For each of the used tones, we find the MMSE-FEQ (for a particular choice of ) by minimizing the following cost function:

NOTE:

The modified T-tap FEQ Per-tone reduces the complexity by

incorporating the liner combination of the FEQ coefficients, such that

the global FEQ for each tone i has as its inputs the i

output of the FFT

and T-1 (real) difference terms.

Simulation Results

ADSL simulation results are presented for downstream to compare the different equalizers. Standard channel T1.601-

#9 with NEXT from 12 DSL and 12 HDSL disturbances is

considered. To compute capacity, we take the SNR gap

Γ=9.8 dB, noise margin γ

= 6 dB, coding gain γ

= 3 dB. The

number of bits assigned to tone i is:

Per-tone Equalization Application

MULTIPLE-INPUT/MULTIPLE-OUTPUT EQUALIZATION

• Other multicarrier systems must solve equalization differently. Receiver structures have been investigated that are valid not only for IFFT-based DMT transmitters without cyclic extension, but also for any other filter- bankbased multicarrier system.

• Is a cyclic extension is still useful? when the cyclic prefix length is shorter than the MIMO order, then cyclic prefix has no use anymore.

• What receiver structure could be derived if no cyclic extension was

used at all (as in case of the high Order MIMO)? Swapping the filtering operations of the MIMO channel and the FFT, we can show that shown that each tone of a MIMO OFDM system can be viewed as a MIMO SC system. As a result, the existing equalization approach for MIMO SC systems can be applied to each tone of a MIMO OFDM system. This so- called Per tone TEQ (PTEQ) approach for MIMO OFDM systems.

wireless doubly selective channel

1-Tap TEQ, to convert doubly selective channel into pure frequency

selective channel.

1-Tap FEQ, that is optimized for each tone, and the

performance is maximized

by summing up the N tones.

Per Tone Equalization for OFDM over wireless doubly selective channel

Important Notes:

Important Notes:

The proposed ICI mitigation technique is simply achieved by taking P FFTs of different modulated versions y

(n) of the received sequence y(n).

Note that this actually corresponds to over-sampling the received sequence in the frequency-domain by a factor of P.

To detect a symbol on the k

subcarrier of the i

OFDM

block, neighboring subcarriers are combined at the output of the pth FFT.

The resulting outputs are subsequently combined to obtain the symbol transmitted on that subcarrier

For each subcarrier, we can find the MMSE equalizer

coefficients by minimizing the cost function.

Conclusion

For the same equalizer order, the PTEQ approach always has a better performance than the TEQ approach. Moreover, it can be shown that the equalization complexity for both approaches is comparable.

The performance of the PTEQ approach is a much smoother function of the synchronization delay than the performance of the TEQ approach. Hence, for the PTEQ approach the

synchronization delay setting is less critical than for the TEQ approach.

Since a per-tone equalizer works on the symbol level, whereas a

time-domain equalizer works on the sample level, a per-tone

equalizer can much more easily be designed in practice than a

time-domain equalizer.