1 © 2006 cisco systems, inc. all rights reserved. adaptive equalization cisco public adaptive...

1© 2006 Cisco Systems, Inc. All rights reserved.AdaptiveEqualization Cisco Public

Adaptive Equalization

Ron Hranac


What is Equalization?

• In a coaxial cable distribution network, higher frequencies are attenuated more than lower frequencies as RF signals travel through the coax.

This results in a tilted frequency response at the input to each amplifier.

• It is necessary to install a fixed-value plug-in equalizer at each amplifier. The equalizer has the opposite amplitude-versus-frequency response of the coaxial cable preceding the amplifier.

• The equalizer “cancels” the tilted frequency response, resulting in a flat amplitude-versus-frequency spectrum at the input to the amplifier’s internal gain stages.

50 MHz 870 MHz 50 MHz 870 MHz

50 MHz 870 MHz

Equalizer’s frequency response

Spectrum’s frequency responseafter equalization


What is an Adaptive Equalizer?

• Adaptive equalization performs a function similar to that of a cable amplifier’s equalizer, but rather than equalizing the entire 50-860 MHz downstream or 5-42 MHz upstream RF spectrum, it deals with just a single channel. Adaptive means the equalizer can change its characteristics as channel conditions change.

Graphic courtesy of Sunrise Telecom



• An adaptive equalizer is a digital circuit that compensates for a digitally modulated signal’s in-channel complex frequency response impairments.

Complex frequency response includes amplitude (or magnitude)-versus-frequency, and phase-versus-frequency.

• The adaptive equalizer uses sophisticated algorithms to derive coefficients for an equalizer solution “on the fly”—in effect, creating a digital filter with essentially the opposite complex frequency response of the impaired channel.

At high SNR (ES/N0) the equalizer will synthesize the opposite response of the channel. At lower SNR doing so would cause noise enhancement, so a compromise solution is derived.



• Ideal equalizer coefficients yield maximum modulation error ratio (MER) by minimizing total impairments including intersymbol interference (ISI), within the limits of the equalizer’s capabilities (number of taps, etc.).

• If the in-channel impairment suddenly changes or goes away, the adaptive equalizer will distort the signal, until new equalizer coefficients for the current channel conditions are derived and the equalizer’s operation updated. This adaptation process is very fast, typically completed in milliseconds.


Adaptive Equalization: Before and After

Graphics courtesy of Sunrise Telecom

Unequalized Equalized


Characterizing Adaptive Equalizers

• Adaptation source

Transmitted training sequence: In conventional zero-forcing or minimum mean square error (MSE) equalizers, a known training sequence is transmitted to the receiver for the purpose of initially adjusting equalizer coefficients.

The signal itself: Equalizers that do not rely upon transmitted training sequences for the initial adjustment of the coefficients are called self-recovering or blind equalizers.

The adaptive equalizer in the downstream receiver of a DOCSIS® cable modem is a blind equalizer (DOCSIS does not specify a training sequence in the downstream signal).


Characterizing Adaptive Equalizers

• Equalization method

Maximum-likelihood sequence detection, linear filter with adjustable coefficients, feedforward, decision-feedback

• Algorithm

Zero-forcing, least-mean-square, tap-leakage, recursive least squares, stochastic gradient


Equalizer Span

• An important parameter in an adaptive equalizer is its span, defined as (number of taps – 1) x tap spacing.*

Tap spacing is the time delay per tap

• The spacing of the upstream pre-equalizer taps in DOCSIS 2.0 is defined as “T-spaced,” or symbol-spaced (the reciprocal of the symbol rate). With a symbol rate of 5.12 Msym/sec, tap spacing is 0.1953125 µsec of time delay per tap.

• With DOCSIS 2.0’s 24 taps, the maximum possible span is (24 – 1) x 0.1953125 µsec = 4.49 µsec.

Another way to calculate this value is (24 – 1)/5.12 = 4.49 µsec

*This example assumes the first tap is the main tap. See speaker notes.


Fractionally Spaced Equalizers

• T-spaced equalizers are the most commonly used

As previously noted, “T-spaced” means the equalizer taps are spaced at the reciprocal of the symbol rate

• A fractionally spaced equalizer (FSE) is based on sampling the incoming signal at least as fast as the Nyquist rate

A ½T-spaced (also written as “T/2-spaced”) equalizer is used in many applications that require a FSE. Other applications may use ¼T-spaced (T/4-spaced), etc.

• FSEs often perform better than T-spaced equalizers in the presence of symbol clock timing errors

FSEs are less sensitive to timing phase

• FSEs are not as common as T-spaced equalizers because of computational complexity and convergence performance


Equalizer Span

• If the channel response contains an echo (micro-reflection) that is further out in delay than the span of the equalizer, the equalizer cannot compensate for that echo.

For example, if there is a significant amount of ISI from a SAW filter’s triple transit, and this ISI is equivalent to an echo at a large delay (beyond the span of the equalizer’s taps), then the equalizer will still do its best on the other impairments. But it won’t be able to cancel the triple transit echoes since they are beyond the limits of the adaptive equalizer’s capabilities—in this case, the equalizer span.


Adaptive Equalizer Block Diagram

Σ

b-2 b-1 b0 b+2

Z-1 Z-1 Z-1

UNEQUALIZEDINPUT

EQUALIZEDOUTPUT

Σ Σ

b+1

Z-1

Σ

Delay element

Multipliers with equalization coefficients

Algorithm fortap gain adjustment

• • • •

•

A T-spaced equalizer means the taps are “spaced” (spacing is the

amount of delay per tap) at the reciprocal of the symbol rate—that is, the reciprocal of the signaling

rate 1/T.

One tap is called the main tap (Z-1 to b0 in this example). The main tap has a gain of 1, and

passes the original signal. Other taps represent either

the “past” or “future” relative to the main tap.

Main tap


Example based on information from Holtzman, Inc.

Micro-reflection Example

Out-of-phase echo with 1 µsec delay and -0.5 amplitude

Am

pli

tud

e

+1.0

+0.5

- 0.5

- 1.0

Time

1.0

- 0.5

1 µsec

Incident signal

Echo • •Null Peak

EchoAmplitude = -0.5

Incident signalAmplitude = +1.0

•

Single Echo:Phasor View

Single Echo:Spectral View

Frequency

Lin

ea

rM

ag

nit

ud

e

Peak

Null•

•

••

••

• ••

••

•

•

••

•

•

0 . 5

1 . 0

1 . 5

0

Dashed arrow is the vector sum of the incident and

echo vectors


30º

30º


Impaired Frequency Response

Here’s a look at the resulting frequency response—magnitude and phase—caused by the micro-reflection:

Am

pli

tud

e

1.0

0.5

Frequency

1.5

Ph

as

e Frequency

MHzs1

1

1

9.54 dB peak-valley!

20log(1.5/0.5) = 9.54

± 30º phase

arcsine(0.5/1) = 30º



Fixing the Impairment

Assuming high SNR (ES/N0), we need an equalizer with the opposite frequency response to cancel the echo:

1.0

0.5

Frequency

1.5

Ph

as

e Frequency

2.0

0.667

1/0.5 = 2.0

1/1.5 = 0.667

The needed magnitude-versus-frequency response also is 9.54 dB peak-valley, but 180º opposite the original magnitude impairment.

The needed phase-versus-frequency response is 180º

opposite the original phase impairment

Am

pli

tud

e

30º

30º

Why is the peak-to-peak linear magnitude of the needed opposite amplitude-versus-frequency response 0.667 to 2.0 rather than the original 0.5 to 1.5? If H(f) is 0.5, then 1/H(f) is 2.0; likewise, if H(f) is 1.5, then 1/H(f) is 0.667. Flat response occurs when H(f) multiplied by

1/H(f) equals 1.00. In this example, 0.5 x 2 = 1.00 and 1.5 x 0.667 = 1.00.


Adaptive Equalization: An Example

Σ

b0 b1 b2 b3

Z-1 Z-1 Z-1IN

OUTΣ Σ

We’ll use the following 4-tap adaptive equalizer to compensate for the impaired frequency response

Main tap


4-Tap Adaptive Equalizer

Multiplier Coefficient

b01.0

b1+0.5

b2+0.25

b3+0.125

Σ

b0 b1 b2 b3

Z-1 Z-1 Z-1IN

OUTΣ Σ

• • •

•

Algorithm fortap gain adjustment

For this example,For this example,assume the algorithmassume the algorithm

derives thesederives thesecoefficients…coefficients…

Am

plit

ud

e

+1.0

+0.5

- 0.5

- 1.0

Time

1.0

- 0.5

• • •

……and each delay elementand each delay elementZZ-1-1 equals 1 equals 1 µµsecsec


The Adaptive Equalizer at Work

Σ

x1.0 b1 b2 b3

Z-1 Z-1 Z-1IN

OUTΣ Σ

• • •

x 1.0 =

Am

plit

ud

e

+1.0

+0.5

- 0.5

- 1.0

Time

1.0

- 0.5

• • • • •

Am

plit

ud

e

+1.0

+0.5

- 0.5

- 1.0

Time

1.0

- 0.5

• • •

Am

plit

ud

e

+1.0

+0.5

- 0.5

- 1.0

Time

1.0

- 0.5

• • •



Σ

x0.5 b2 b3

Z-1 Z-1 Z-1IN

OUTΣ Σ

• • •

x1.0

x 0.5 =

Am

plit

ud

e

+1.0

+0.5

- 0.5

- 1.0

Time

1.0

- 0.5

• • •A

mp

litu

de

+1.0

+0.5

- 0.5

- 1.0

Time

1.0

- 0.5

• • •A

mp

litu

de

+1.0

+0.5

- 0.5

- 1.0

Time

0.5

-0.25

• ••

Am

plit

ud

e

+1.0

+0.5

- 0.5

- 1.0

Time

1.0

- 0.5

• • •

Note the shift or delay of the incident signal and its echo by 1 µsec after they

passed through the first delay element Z-1



Σ

x1.0 b2 b3

Z-1 Z-1 Z-1IN

OUTΣ Σ

• • •

+ =

x0.5

Am

plit

ud

e

+1.0

+0.5

- 0.5

- 1.0

Time

1.0

- 0.5

• • •

Am

plit

ud

e

+1.0

+0.5

- 0.5

- 1.0

Time

1.0

- 0.5

• • •

Am

plit

ud

e

+1.0

+0.5

- 0.5

- 1.0

Time

0.5

-0.25

• ••

Am

plit

ud

e

+1.0

+0.5

- 0.5

- 1.0

Time

-0.25

• ••

1.0



Σ

x1.0 b3

Z-1 Z-1 Z-1IN

OUTΣ Σ

• • •

x0.5 x0.25

x 0.25 =

Am

plit

ud

e

+1.0

+0.5

- 0.5

- 1.0

Time

1.0

- 0.5

• • •

Am

plit

ud

e

+1.0

+0.5

- 0.5

- 1.0

Time

1.0

- 0.5

• • • • •

Am

plit

ud

e

+1.0

+0.5

- 0.5

- 1.0

Time

1.0

- 0.5

• • • • •

Am

plit

ud

e

+1.0

+0.5

- 0.5

- 1.0

Time

-0.125

0.25

• • • •



Σ

x1.0 b3

Z-1 Z-1 Z-1IN

OUTΣ Σ

• • •

x0.5 x0.25

+ =

Am

plit

ud

e

+1.0

+0.5

- 0.5

- 1.0

Time

-0.25

• ••

1.0

Am

plit

ud

e

+1.0

+0.5

- 0.5

- 1.0

Time

-0.125

0.25

• • • •

Am

plit

ud

e

+1.0

+0.5

- 0.5

- 1.0

Time

1.0

- 0.5

• • •

Am

plit

ud

e

+1.0

+0.5

- 0.5

- 1.0

Time

-0.125

• • • •

1.0



Σ

x1.0

Z-1 Z-1 Z-1IN

OUTΣ Σ

• • •

x0.5 x0.25 x0.125

x 0.125 =

Am

plit

ud

e

+1.0

+0.5

- 0.5

- 1.0

Time

1.0

- 0.5

• • • Am

plit

ud

e

+1.0

+0.5

- 0.5

- 1.0

Time

1.0

- 0.5

• • • • • •

Am

plit

ud

e

+1.0

+0.5

- 0.5

- 1.0

Time

1.0

- 0.5

• • • • • •0.125

-0.0625Am

plit

ud

e

+1.0

+0.5

- 0.5

- 1.0

Time• • • • • •


Micro-reflection After Equalization

Σ

x1.0

Z-1 Z-1 Z-1IN

EQUALIZED OUTPUTΣ Σ

• • •

x0.5 x0.25 x0.125

+ =

Original -0.5 amplitude echo was reduced to an amplitude of -0.0625, or an 18 dB improvement. An infinite number of taps is required in a simple feedback-type equalizer like the above to make the echo go to zero. A decision feedback equalizer could further reduce the residual error.

Am

plit

ud

e

+1.0

+0.5

- 0.5

- 1.0

Time

1.0

- 0.5

• • •

0.125

-0.0625Am

plit

ud

e

+1.0

+0.5

- 0.5

- 1.0

Time• • • • • •

Am

plit

ud

e

+1.0

+0.5

- 0.5

- 1.0

Time

-0.125

• • • •

1.0

-0.0625Am

plit

ud

e

+1.0

+0.5

- 0.5

- 1.0

Time• • • • • •

1.0


Micro-reflection After Equalization

Resulting frequency response: magnitude and phase. Ripple is now 1/(4 µsec), or 250 kHz.

Am

pli

tud

e

1.0

0.5

Frequency

1.5

Ph

as

e Frequency

1.0625

0.9375

Now only 1.87 dB p-v

20log(1.0625/0.9375) = 1.87

± 3.58º phase

arcsine(0.0625/1) = 3.58º

30º

30º

3.58º3.58º

kHzs250

4

1


Adaptive Equalization in the Real World

• As mentioned previously, a cable modem uses a blind adaptive equalizer in the device’s downstream QAM receiver. DOCSIS 1.1 and 2.0 cable modems are capable of equalizing—or more accurately, pre-equalizing—the transmitted upstream signal.

DOCSIS 1.1 supports 8-tap upstream pre-equalization, and DOCSIS 2.0 supports 24-tap upstream pre-equalization

• Why pre-equalize in the modem?

The path between each modem and the CMTS is unique

Pre-equalization allows most of the adaptive equalization to be done by the modem before upstream transmission, rather than relying upon the CMTS to do all of the work


Adaptive Equalization in the Real World

• A cable modem has no way of knowing the condition of the channel between its upstream transmitter output and the CMTS’s input. The modem can’t “see” the channel through which the upstream digitally modulated signal is transmitted.

• So how can a cable modem correctly pre-equalize a transmitted upstream signal?


Upstream Adaptive Equalization

b0=1, b1=0.5, b2=0.25, b3=0.125…

The cable modem transmits unequalized ranging bursts to the CMTS. The CMTS’s upstream burst receiver includes an adaptive equalizer that derives equalizer coefficients based on the channel impairment(s) affecting the received signal. Each upstream burst’s preamble is used as a training signal for the CMTS’s equalizer.

The CMTS transmits the derived equalizer coefficients to the modem in a RNG-RSP MAC message.

The cable modem uses the equalizer coefficients in its upstream adaptive equalizer to pre-equalize or pre-distort the transmitted signal, so that when it is received by the CMTS it will, in theory, be unimpaired.

HFC Networ

k

HFC Networ

k

HFC Networ

k

HFC Networ

k

HFC Networ

k

HFC Networ

k


Upstream Adaptive Equalization Example

Upstream 6.4 MHz bandwidth 64-QAM signal

After adaptive equalization:DOCSIS 2.0’s 24-tap adaptive equalization—actually pre-equalization in the modem—was able to compensate for nearly all of the in-channel tilt (with no change in digital channel power). The result: No correctable or uncorrectable FEC errors and the CMTS’s reported upstream MER (SNR) increased to ~36 dB.

Before adaptive equalization:Substantial in-channel tilt caused correctable FEC errors to increment at a rate of about 7000 errored codewords per second (232 bytes per codeword). The CMTS’s reported

upstream MER (SNR) was 23 dB.


Q and A

1 © 2006 cisco systems, inc. all rights reserved. adaptive equalization cisco public adaptive...

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