1. 1 © 2006 Cisco Systems, Inc. All rights reserved. Adaptive Equalization Cisco Public Adaptive Equalization Ron Hranac

2. 2 © 2006 Cisco Systems, Inc. All rights reserved. Adaptive Equalization Cisco Public 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 MHz50 MHz870 MHz 50 MHz870 MHz Equalizer’s frequency response Spectrum’s frequency response after equalization

3. 3 © 2006 Cisco Systems, Inc. All rights reserved. Adaptive Equalization Cisco Public 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

4. 4 © 2006 Cisco Systems, Inc. All rights reserved. Adaptive Equalization Cisco Public What is an Adaptive Equalizer? 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 (E S /N 0 ) 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.

5. 5 © 2006 Cisco Systems, Inc. All rights reserved. Adaptive Equalization Cisco Public What is an Adaptive Equalizer? 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.

6. 6 © 2006 Cisco Systems, Inc. All rights reserved. Adaptive Equalization Cisco Public Adaptive Equalization: Before and After Graphics courtesy of Sunrise Telecom Unequalized Equalized

7. 7 © 2006 Cisco Systems, Inc. All rights reserved. Adaptive Equalization Cisco Public 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).

8. 8 © 2006 Cisco Systems, Inc. All rights reserved. Adaptive Equalization Cisco Public 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

9. 9 © 2006 Cisco Systems, Inc. All rights reserved. Adaptive Equalization Cisco Public 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.

10. 10 © 2006 Cisco Systems, Inc. All rights reserved. Adaptive Equalization Cisco Public 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

11. 11 © 2006 Cisco Systems, Inc. All rights reserved. Adaptive Equalization Cisco Public 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.

12. 12 © 2006 Cisco Systems, Inc. All rights reserved. Adaptive Equalization Cisco Public Adaptive Equalizer Block Diagram Σ b -2 b -1 b0b0 b +2 Z -1 UNEQUALIZED INPUT EQUALIZED OUTPUT Σ Σ b +1 Z -1 Σ Delay element Multipliers with equalization coefficients Algorithm for tap 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 b 0 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

13. 13 © 2006 Cisco Systems, Inc. All rights reserved. Adaptive Equalization Cisco Public Example based on information from Holtzman, Inc. Micro-reflection Example Out-of-phase echo with 1 µsec delay and -0.5 amplitude Amplitude +1.0 +0.5 - 0.5 - 1.0 Time 1.0 - 0.5 1 µsec Incident signal Echo NullPeak Echo Amplitude = -0.5 Incident signal Amplitude = +1.0 Single Echo: Phasor View Single Echo: Spectral View Frequency Linear Magnitude Peak Null 0.5 1.0 1.5 0 Dashed arrow is the vector sum of the incident and echo vectors

14. 14 © 2006 Cisco Systems, Inc. All rights reserved. Adaptive Equalization Cisco Public 30º Example based on information from Holtzman, Inc. Impaired Frequency Response Here’s a look at the resulting frequency response— magnitude and phase—caused by the micro-reflection: Amplitude 1.0 0.5 Frequency 1.5 Phase Frequency 9.54 dB peak-valley! 20log(1.5/0.5) = 9.54 ± 30º phase arcsine(0.5/1) = 30º

15. 15 © 2006 Cisco Systems, Inc. All rights reserved. Adaptive Equalization Cisco Public Example based on information from Holtzman, Inc. Fixing the Impairment Assuming high SNR (E S /N 0 ), we need an equalizer with the opposite frequency response to cancel the echo: 1.0 0.5 Frequency 1.5 Phase 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 Amplitude 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.

16. 16 © 2006 Cisco Systems, Inc. All rights reserved. Adaptive Equalization Cisco Public Adaptive Equalization: An Example Σ b0b0 b1b1 b2b2 b3b3 Z -1 IN OUT Σ Σ We’ll use the following 4-tap adaptive equalizer to compensate for the impaired frequency response Main tap

17. 17 © 2006 Cisco Systems, Inc. All rights reserved. Adaptive Equalization Cisco Public 4-Tap Adaptive Equalizer MultiplierCoefficient b0b0 1.0 b1b1 +0.5 b2b2 +0.25 b3b3 +0.125 Σ b0b0 b1b1 b2b2 b3b3 Z -1 IN OUT Σ Σ Algorithm for tap gain adjustment For this example, assume the algorithm derives these coefficients… Amplitude +1.0 +0.5 - 0.5 - 1.0 Time 1.0 - 0.5 …and each delay element Z -1 equals 1 µsec

18. 18 © 2006 Cisco Systems, Inc. All rights reserved. Adaptive Equalization Cisco Public The Adaptive Equalizer at Work Σ x1.0 b1b1 b2b2 b3b3 Z -1 IN OUT Σ Σ x 1.0 = Amplitude +1.0 +0.5 - 0.5 - 1.0 Time 1.0 - 0.5 Amplitude +1.0 +0.5 - 0.5 - 1.0 Time 1.0 - 0.5 Amplitude +1.0 +0.5 - 0.5 - 1.0 Time 1.0 - 0.5

19. 19 © 2006 Cisco Systems, Inc. All rights reserved. Adaptive Equalization Cisco Public The Adaptive Equalizer at Work Σ x0.5 b2b2 b3b3 Z -1 IN OUT Σ Σ x1.0 x 0.5 = Amplitude +1.0 +0.5 - 0.5 - 1.0 Time 1.0 - 0.5 Amplitude +1.0 +0.5 - 0.5 - 1.0 Time 1.0 - 0.5 Amplitude +1.0 +0.5 - 0.5 - 1.0 Time 0.5 -0.25 Amplitude +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

20. 20 © 2006 Cisco Systems, Inc. All rights reserved. Adaptive Equalization Cisco Public The Adaptive Equalizer at Work Σ x1.0 b2b2 b3b3 Z -1 IN OUT Σ Σ += x0.5 Amplitude +1.0 +0.5 - 0.5 - 1.0 Time 1.0 - 0.5 Amplitude +1.0 +0.5 - 0.5 - 1.0 Time 1.0 - 0.5 Amplitude +1.0 +0.5 - 0.5 - 1.0 Time 0.5 -0.25 Amplitude +1.0 +0.5 - 0.5 - 1.0 Time -0.25 1.0

21. 21 © 2006 Cisco Systems, Inc. All rights reserved. Adaptive Equalization Cisco Public The Adaptive Equalizer at Work Σ x1.0 b3b3 Z -1 IN OUT Σ Σ x0.5x0.25 x 0.25 = Amplitude +1.0 +0.5 - 0.5 - 1.0 Time 1.0 - 0.5 Amplitude +1.0 +0.5 - 0.5 - 1.0 Time 1.0 - 0.5 Amplitude +1.0 +0.5 - 0.5 - 1.0 Time 1.0 - 0.5 Amplitude +1.0 +0.5 - 0.5 - 1.0 Time -0.125 0.25

22. 22 © 2006 Cisco Systems, Inc. All rights reserved. Adaptive Equalization Cisco Public The Adaptive Equalizer at Work Σ x1.0 b3b3 Z -1 IN OUT Σ Σ x0.5x0.25 += Amplitude +1.0 +0.5 - 0.5 - 1.0 Time -0.25 1.0 Amplitude +1.0 +0.5 - 0.5 - 1.0 Time -0.125 0.25 Amplitude +1.0 +0.5 - 0.5 - 1.0 Time 1.0 - 0.5 Amplitude +1.0 +0.5 - 0.5 - 1.0 Time -0.125 1.0

23. 23 © 2006 Cisco Systems, Inc. All rights reserved. Adaptive Equalization Cisco Public The Adaptive Equalizer at Work Σ x1.0 Z -1 IN OUT Σ Σ x0.5x0.25 x0.125 x 0.125 = Amplitude +1.0 +0.5 - 0.5 - 1.0 Time 1.0 - 0.5 Amplitude +1.0 +0.5 - 0.5 - 1.0 Time 1.0 - 0.5 Amplitude +1.0 +0.5 - 0.5 - 1.0 Time 1.0 - 0.5 0.125 -0.0625 Amplitude +1.0 +0.5 - 0.5 - 1.0 Time

24. 24 © 2006 Cisco Systems, Inc. All rights reserved. Adaptive Equalization Cisco Public Micro-reflection After Equalization Σ x1.0 Z -1 IN EQUALIZED OUTPUT Σ Σ x0.5x0.25 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. Amplitude +1.0 +0.5 - 0.5 - 1.0 Time 1.0 - 0.5 0.125 -0.0625 Amplitude +1.0 +0.5 - 0.5 - 1.0 Time Amplitude +1.0 +0.5 - 0.5 - 1.0 Time -0.125 1.0 -0.0625 Amplitude +1.0 +0.5 - 0.5 - 1.0 Time 1.0

25. 25 © 2006 Cisco Systems, Inc. All rights reserved. Adaptive Equalization Cisco Public Micro-reflection After Equalization Resulting frequency response: magnitude and phase. Ripple is now 1/(4 µsec), or 250 kHz. Amplitude 1.0 0.5 Frequency 1.5 Phase 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 3.58º

26. 26 © 2006 Cisco Systems, Inc. All rights reserved. Adaptive Equalization Cisco Public 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

27. 27 © 2006 Cisco Systems, Inc. All rights reserved. Adaptive Equalization Cisco Public 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?

28. 28 © 2006 Cisco Systems, Inc. All rights reserved. Adaptive Equalization Cisco Public Upstream Adaptive Equalization b 0 =1, b 1 =0.5, b 2 =0.25, b 3 =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

29. 29 © 2006 Cisco Systems, Inc. All rights reserved. Adaptive Equalization Cisco Public 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.

30. 30 © 2006 Cisco Systems, Inc. All rights reserved. Adaptive Equalization Cisco Public Q and A

31. 31 © 2006 Cisco Systems, Inc. All rights reserved. Adaptive Equalization Cisco Public